Datasets:

suolyer

/

pile_arxiv

like 3

--- abstract: 'Atomistic tight-binding (TB) simulations are performed to calculate the Stark shift of the hyperfine coupling for a single Arsenic (As) donor in Silicon (Si). The role of the central-cell correction is studied by implementing both the static and the non-static dielectric screenings of the donor potential, and by including the effect of the lattice strain close to the donor site. The dielectric screening of the donor potential tunes the value of the quadratic Stark shift parameter ($\eta_2$) from -1.3 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ for the static dielectric screening to -1.72 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ for the non-static dielectric screening. The effect of lattice strain, implemented by a 3.2% change in the As-Si nearest-neighbour bond length, further shifts the value of $\eta_2$ to -1.87 $\times$ 10$^{-3} \mu$m$^2$/V$^2$, resulting in an excellent agreement of theory with the experimentally measured value of -1.9 $\pm$ 0.2 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. Based on our direct comparison of the calculations with the experiment, we conclude that the previously ignored non-static dielectric screening of the donor potential and the lattice strain significantly influence the donor wave function charge density and thereby leads to a better agreement with the available experimental data sets.' author: - Muhammad Usman - Rajib Rahman - Joe Salfi - Juanita Bocquel - Benoit Voisin - Sven Rogge - Gerhard Klimeck - 'Lloyd L. C. Hollenberg' title: 'Donor hyperfine Stark shift and the role of central-cell corrections in tight-binding theory' --- 0.25cm Introduction ============ Since the Kane proposal for quantum computing using donor spins in silicon [@Kane_Nature_1998], there has been considerable progress towards the realisation of spin-qubit architectures [@Zwanenburg_RMP_2013; @Lloyd_PRB_2006; @Hill_PRB_2005; @Sousa_PRB_2009]. Notable results include single-atom fabricated devices [@Fuechsle_NN_2012; @Watson_NL_2014; @Weber_Science_2012] and control and measurement of individual donor electron and nuclear spins [@Tyryshkin_Nat_Mat_2012; @Saeedi_Science_2013; @Morello_Nature_2010; @Pla_Nature_2013]. However in building a scalable donor-based quantum computer, an important aspect is understanding and controlling the Stark shift of the donor hyperfine levels. Accurate theoretical modelling of the donor hyperfine coupling is a challenging problem. First it requires proper incorporation of the valley-orbit (VO) interaction which has been established as a critical parameter to accurately match the experimentally observed energies of the ground state (A$_1$-symmtery) and the excited states (T$_2$ and E symmetries) [@Wellard_Hollenberg_PRB_2005]. Secondly it is essential to perform the calculation of the ground state donor wave function with high precision through proper implementation of the central-cell effects (short-range potential) and the dielectric screening of long-range Coulomb potential. Earlier studies based on the Kohn and Luttinger’s single-valley effective-mass theory (SV-EMT) [@Kohn_PR_1955] ignored the VO interaction and therefore could not match with the experimental binding energy of the ground state (A$_1$); since then several studies have been performed with incremental improvements in the model. Pantelides and Sah [@Pantelides_Sah_PRB_1974] pointed out that the concept of the central-cell correction is ill defined in SV-EMT and therefore it fails to capture the chemical shift and the splitting of the experimentally observed donor ground state energies which primarily arise from the intervally mixing. Based on this, they presented a multi-valley effective-mass theory (MV-EMT) by explicitly including the central-cell correction along with a non-static dielectric screening of the Coulomb potential representing the donor. Similar EMT based formalisms have been widely applied by various studies later on to investigate the physics of shallow donors [@Saraiva_1; @King_1; @Pica_1]. Overhof and Gerstmann [@Overhof_PRL_2004] applied density-functional theory to successfully calculate the hyperfine ferquency of shallow donors in Si. While their calculations were in excellent agreement with experiment (zero fields), the ab-initio description of the donor wave function was by definition limited to only a few atoms around the donor site. Also they ignored the long-range tail of the Coulomb potential and therefore were unable to match the donor binding energies from their approach. More recently, a much more detailed theoretical calculation was performed by Wellard and Hollenberg [@Wellard_Hollenberg_PRB_2005] based on band-minimum basis (BMB) approach. In their study, by using a core-correcting potential screened by non-static ($k$-dependent) dielectric function, they were able to demostrate excellent agreement with the experimentally measured ground state energy for the P donor in Si (45.5 meV). However the excited state energies remained few meVs off from the experiemental values. Nevertheless, their study clearly highlighted the critical role of the central-cell corrections in theoretical modeling of shallow donors in Si which drastically modifies the charge density of donor wave function and therefore tune the hyperfine coupling and its Stark shift parameters. Atomistic tight-binding (TB) approach, historically used for the deep level impurities [@Vogl_SSP_1981], has been shown to work remarkably well for the shallow level impurities in Si [@Martins_PRB_2004; @Rahman_PRL_2007; @Weber_Science_2012]. The TB method offers several advantages over EMT and DFT based methodologies, including the capability of inherently incorporating the VO intermixings, calculations over very large supercells (containing several million atoms in the simulation domain) and therefore providing much more detailed description of the donor wave funtion, an easy incorporation of externally applied electric field effects in Hamiltonian, and explicit representation of the short-range and the long-range donor potentials, etc. Martin *et al.* [@Martins_PRB_2004] applied second neares-neighbor sp$^3$s$^*$ TB model to study the effects of an applied electric field on P donor wave functions. Later Rahman *et al.* [@Rahman_PRL_2007] applied a much more sophisticated sp$^3$d$^5$s$^3$ TB Hamiltonian to P donors in Si and bench-marked Stark shift of the donor hyperfine coupling against the BMB calculations. The two theoretical models were found to be in remarkable agreement with each other for P donors, and also exhibited very good agreement with the Stark shift measurement data for the Sb donor in Si [@Bradbury_PRL_2006]; however a direct comparison of the hyperfine Stark shift with experiment was not possible due to the unavailability of any experimental data for the P and As donors. The models were also based on minimal central-cell correction, implemented in terms of short-range correction of donor potential at the donor site and a static dielectric screening of the long-range donor potential tail. The previously reported experimental data for the hyperfine coupling of the shallow donors (P, As, Sb etc.) in Si [@Feher_PR_1959], and more recent experimental measurements [@Lo_arxiv_2014] of the hyperfine Stark shift for As donor in Si provide excellent opportunities to directly bench-mark TB theory against the experiment data sets. This work for the first time evaluates the role of central-cell corrections in atomistic TB theory through a direct comparison against the experimental data of the hyperfine interaction for a single As donor in Si. The central-cell corrections in the tight-binding model considered here are implemented by: - Short-range correction of the donor potential: donor potential is truncated at U$_0$ at the donor site. - Dielectric screening of the long-range tail of the donor potential: static vs. non-static dielectric screenings. - Lattice strain around the donor site: changes in the nearest-neighbor bond lengths. We systematically study the critical significance of the central-cell corrections by including the effect of each central-cell component one-by-one and evaluating its role on the hyperfine coupling and its Stark shift parameter. In each case, we first adjust U$_0$ to match the experimentally measured donor binding energies for the ground and excited states within 1 meV accuracy [@Ahmed_Enc_2009]. We then compute charge density at the nuclear site and its character under the influence of an external electrical field. Our calculations demonstrate that the previously ignored central-cell components, non-static dielectric screening of donor potential and lattice strain, produce significant impact on the donor herperfine Stark shift and therefore lead to match the experimental data with an unprecedent accuracy. Such high precision bench-marking of the theory against the experimental data would be useful in accomplishing high precision control over donor wave functions required in quantum computing.  Methodology =========== The atomistic simulations are performed using NanoElectronic MOdeling tool NEMO-3D [@Klimeck_1; @Klimeck_2], which has previously shown to quantitatively match the experimental data sets for a variety of nanostructures and nanomaterials, such as shallow donors in Si  [@Rahman_PRL_2007; @Weber_Science_2012], III-V alloys [@Usman_1; @Usman_2] and quantum dots [@Usman_3; @Usman_4; @Usman_5], SiGe quantum wells [@Neerav_1], etc. The sp$^3$d$^5$s$^*$ tight-binding parameters for Si material are obtained from Boykin *et al*. [@Boykin_PRB_2004], that have been optimised to accurately reproduce the Si bulk band structure. The As donor is represented by a screened Coulomb potential truncated to U($r_0$)=U$_0$ at the donor site, $r_0$. Here, U$_0$ is an adjustable parameter that represents the central-cell correction at the donor site and has been designed to accurately match the ground state binding energy (A$_1$ = 53.8 meV) of the As donor as measured in the experiment. The size of the simulation domain (Si box around the As donor) is chosen as 32 nm $\times$ 65 nm $\times$ 32 nm, consisting of roughly 3.45 million atoms, with closed boundary conditions in all three dimensions. The surface atoms are passivated by our published method [@Lee_PRB_2004] to avoid any spurious states in the energy range of interest. The multi-million atom real-space Hamiltonian is solved by a prallel Lanczos algorithm to calculate donor single-particle energies and wave functions. For the study of the effects of the lattic relaxation, the influence of the changed nearest-neighbor bond lengths on the tight-binding Hamiltonian is computed by a generalization of the Harrison’s scalling law [@Boykin_PRB_2004]. In this formulation, the interatomic interaction energies are taken to vary with the bond length $d$ as $(\dfrac{d_0}{d})^\eta$, where d$_0$ is the unrelaxed Si bond length and $\eta$ is a scalling parameter whose magnitude depends on the type of the interaction being considered and is fitted to obtain hydrostatic deformation potentials. The hyperfine coupling parameter A(0) is directly proportional to the squared magnitude of the ground state wave function at the donor nuclear site, $|\psi(r_0)|^2$ [@Rahman_PRL_2007] and its value is experimentally measured [@Feher_PR_1959] as 1.73 $\times$ 10$^{30}$ m$^{-3}$ for As donor. It is therefore important to theoretically compute the value of $|\psi(r_0)|^2$ at the donor site and compare it with the experimental value. It should be pointed out that in our empirical tight-binding model, the Hamiltonian matrix elements comprising the onsite and nearest-neighbor interactions are optimized numerically to fit the bulk band structure of the host Si material without explicit knowledge of the underlying atomic orbitals. Therefore it is fundamentally not possible to quantitatively determine the value of the hyerperfine coupling A(0) as is possible from the ab-initio type calculations [@Overhof_PRL_2004]. Nevertheless, we apply the methodology published by Lee *et al.* [@Lee_JAP_2005] to estimate the value of $|\psi(r_0)|^2$ from our model, where we have used the value of bulk Si conduction electron at the nuclear site as $\approx$ 9.07 $\times$ 10$^{24}$ cm$^{-3}$ [@Shulman_PR_1956; @Lucy_PRB_2011] and the value of the atomic orbital ratio $\phi_{s^*}(0)/\phi_{s}(0)$ computed to be 0.058 from the assumption of the hydrogen-like atomic orbitals with an effective nuclear charge [@Clementi_JCP_1963]. We believe that this provides a good qualitative comparison of A(0) $ \propto |\psi(r_0)|^2$ from our model with the experimental value, and along with the quantitative match of the donor binding energies (A$_1$, T$_2$, and E) and the Stark shift of hyerfine ($\eta_2$), serve as a benchmark to evaluate the role of the central-cell corrections in the tight-binding theory. We calculate the Stark shift of the hyperfine interaction as follows [@Rahman_PRL_2007]: the potential due to the electrical field is added in the diagonal of the tight-binding Hamiltonian which distorts the donor wave function and pulls it away from the donor site reducing the field dependent hyperfine coupling, A($\overrightarrow{E}$); the hyperfine coupling A($\overrightarrow{E}$) is directly proportional to $|\psi ( \overrightarrow{E}, r_0 )|^2$, where $r_0$ is the location of donor. The change in A($\overrightarrow{E}$) is parametrized as: $$\label{eq:hyperfine_coupling} \Delta A \left( \overrightarrow{E} \right) = A \left( 0 \right) \left( \eta_2 E^2 + \eta_1 E \right)$$ Here $\eta_2$ and $\eta_1$ are the quadratic and linear components of the Stark shift of the hyperfine interaction, respectively. For deeply burried donors (with donor depths typically greater than about 15 nm), the linear component of the Stark shift becomes negligible [@Rahman_PRL_2007]. Therefore we do not provide values of $\eta_1$ in the remainder of this paper which are about two to three orders of magnitude smaller than the values of $\eta_2$. Screening of donor potential by static dielectric constant ========================================================== In the first set of simulations, we apply no central-cell correction (U$_0 \rightarrow - \infty$) and the long-range part of the donor potential is Coulomb potential screened by static dielectric constant ($\epsilon(0)$) as given by Eq. \[eq:Static\_donor\_potential\]: $$\label{eq:Static_donor_potential} U \left( r \right) = \frac{-e^2}{ \epsilon \left( 0 \right) r}$$ where $\epsilon(0)$ = 11.9 is the static dielectric constant of Si and $e$ is the charge on electron. This case is illustrated by schematic of Fig. \[fig:Fig1\] (a). Such setup leads to a six-fold degenerate set of donor states at a binding energy of $\approx$ 29.6 meV, as would be expected from a simple effective-mass type approximation. The donor wave function density at nuclear site is 3.57$\times$10$^{22}$ m$^{-3}$ which is seven orders of magnitude smaller than the experimental value. This clearly highlights the critical role of the central-cell correction at the donor site to accurately capture the splitting of the donor ground state binding energies and the wave function density at the nuclear site as measured in the experiment. ![Electric field response of hyperfine coupling is plotted for bulk As donor for static dielectric screening of donor potential as given by Eq. \[eq:Static\_donor\_potential\]. The data points are computed directly from the TB simulations and the line plots are fittings of Eq. \[eq:hyperfine\_coupling\]. []{data-label="fig:Fig2"}](fig2.png) Next, we setup simulations according to the schematic of Fig. \[fig:Fig1\] (b), where we keep the donor potential U($\overrightarrow{r}$) as a Coulomb potential screened by static dielectric constant for Si as given by Eq. \[eq:Static\_donor\_potential\]. Previous tight binding based theoretical studies for the P donors [@Rahman_PRL_2007; @Martins_PRB_2004] and the As donors [@Lansbergen_Nat_Phys_2008] has also used this type of donor potential. We now include central-cell correction at the donor site as a cut-off potential U$_0$ which is tuned to be 2.6342 eV to accurately reproduce the experimental ground state energy, A$_1$=53.1 meV. Further tuning of the onsite TB $d-$orbital energies [@Ahmed_Enc_2009] allowed to match the experimental excited state energies (T$_2$ and E) as listed in table \[tab:table1\]. By applying this model, we compute the value of $|\psi(r_0)|^2$ at the donor nuclear site as 4.05 $\times$ 10$^{30}$, which comes out to be $\approx$ 2.34 times larger than the experimental value. We also compute the electric field response of the hyperfine coupling for the electric field variation from 0 to 0.5 MV/m as shown in the Fig. \[fig:Fig2\]. The quadratic hyperfine Stark shift parameter $\eta_{2}$ is then calculated from the fitting of the TB data by Eq. \[eq:hyperfine\_coupling\] (details of the calculation methodology have been reported in Ref. ) as -1.32 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ compared to the recent experimental value of -1.9 $\pm$ 0.2 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ for the bulk As donor in Si [@Lo_arxiv_2014]. Table \[tab:table2\] provides an overall summary of results for the static dielectric screening of the donor potential. This shows that even with the static dielectric screening of the donor potential, the central-cell correction part provides a reasonably good description of the donor physics. Since the central-cell effects are implemented through an adjustable parameter U$_0$ at the donor site, we attempt to quantify its effect on the $\eta_2$ by introducing a variation of $\pm$100 meV in its value. Increasing U$_0$ by 100 meV increases the ground state binding energy to 55.6 meV and the value of $\eta_2$ decreases to -1.089 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. On the other hand, decreasing U$_0$ by 100 meV decreases the ground state binding energy to 50.9 meV and the value of $\eta_2$ increases to -1.53 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. This clearly demonstrates that to improve the match with the experimental value of $\eta_2$, the value of U$_0$ should be reduced; however this introduces a large error in the binding energy of the donor ground state which is clearly unacceptable. Therefore we conclude that the current TB model with central-cell parameter U$_0$ and the static dielectric screening of the donor potential provides, at the best, a value of -1.32 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ for the Stark shift of the hyperfine coupling. In the next two sections, we include the effects of non-static dielectric screening of the donor potential and the effect of the nearest-neighbour bond length changes to further evaluate the performance of our TB model. Screening of donor potential by non-static dielectric function ============================================================== With the established tight-binding model as our test system, we start further investigation of the central-cell correction, in particular the screening of the donor potential in the vicinity of the As donor. Previous tight-binding calculations [@Rahman_PRL_2007; @Martins_PRB_2004] for the P donor in Si have been based on the static dielectric screening of the donor potential; however Wellard and Hollenberg [@Wellard_Hollenberg_PRB_2005] have already demonstrated the critical importance of the non-static dielectric screening of the donor potential in their band minimum basis (BMB) calculations. By incorporating a non-static screening of the donor potential given in Ref. , they computed an excellent agreement of the donor ground state binding energy with the experimental value. Furthermore, the effect of the non-static dielectric screening was in particular profound on the spatial distribution of the donor wave function around the donor site. Therefore it is critical to investigate the impact of non-static dielectric screening of the donor potential on the values of $|\psi(r_0)|^2$ and $\eta_2$ computed from the tight-binding model. In this section, we investigate this effect by incorporating various non-static dielectric screenings of the donor potentials as reported in the literature. ![Electric field response of hyperfine coupling is plotted for bulk As donor for various screenings of the donor potential. The data points are computed directly from the TB simulations and the line plots are fittings of Eq. \[eq:hyperfine\_coupling\]. []{data-label="fig:Fig3"}](fig3.png) The screening of the donor potential by a non-static dielectric constant has been a topic of extensive research, and a number of reliable calculations exist for $k$-dependent dielectric function, $\epsilon(k)$, for Si. The most commonly applied dielectric function is obtained by Nara [@Nara_JPSJ_1965]: $$\label{eq:Nonstatic_dielectric} \frac{1}{\epsilon(k)} = \frac{A^2 k^2}{k^2 + \alpha^2} + \frac{\left( 1-A \right) k^2}{k^2 + \beta^2} + \frac{1}{\epsilon \left( 0 \right) } \frac{\gamma^2}{k^2 + \gamma^2}$$ where $A$, $\alpha$, $\beta$, and $\gamma$ are fitting constants and have been numerically fitted by various studies for Si (see table 2 for the fitting values reported by various authors). Based on this $k$-dependent dielectric constant, the new screened donor potential in the real space coordinate system is given by: $$\label{eq:Nonstatic_donor_potential} U \left( r \right) = \frac{-e^2}{ \epsilon \left( 0 \right) r} \left( 1 + A \epsilon \left( 0 \right) \mathrm{e}^{- \alpha r} + \left( 1-A \right) \epsilon \left( 0 \right) \mathrm{e}^{- \beta r} - \mathrm{e}^{- \gamma r} \right)$$ In our next set of simulations, we apply this donor potential and re-adjust the central-cell correction U$_0$ at the donor site to match the ground and excited state binding energies with the experimental values. The new values of U$_0$ and the corresponding values of the binding energies for A$_1$, E, and T$_2$ states are provided in the table \[tab:table3\] for the four non-static dielectric screenings of the donor potential under consideration in this study. After achieving this excellent agreement of the binding energies with the experimental values, we then compute the values of $|\psi(r_0)|^2$ and the electric field response of the hyperfine coupling as plotted in Fig. \[fig:Fig3\] for the various non-static dielectric screening potentials as listed in table \[tab:table2\]. The calculated values of $|\psi(r_0)|^2$ and $\eta_2$ are listed in table \[tab:table3\]. Overall, the non-static dielectric screening of the donor potential works remarkably well in the tight-binding theory and improves the match with the experimental values of $|\psi(r_0)|^2$ and $\eta_2$. For the non-static dielectric screening provided by Nara & Morita, the agreement of the computed $\eta_2$ with the experimental value is within the range of experimental tolerance. We also find a direct relation of $|\psi(r_0)|^2$ with the central-cell correction parameter U$_0$. The smallest value of U$_0$ is for Richard & Vinsome screening which results in the best match of $|\psi(r_0)|^2$ with the experimental value, different only by a factor of 1.5. Effect of lattice relaxation ============================ In the calculations performed above so far, we have assumed the crystal lattice as perfect Si lattice where each atom including the As donor is connected to its four nearest neighbor (NN) atoms by unstrained bond lengths of 0.235 nm. In the past tight-binding studies of the donor hyperfine Stark shift [@Rahman_PRL_2007; @Martins_PRB_2004] the effect of lattice strain has been completely ignored based on the assumption that in the presence of the donor, the NN bond length only negligibly changes. However the recent ab-initio study [@Overhof_PRL_2004] suggested a sizeable increase of 3.2% in the NN bond length for the As donors in Si. Our fully atomistic description of the As donor in Si provides an excellent opportunity to investigate the effect of lattice strain. In our next set of simulations, we increase the bond length of the As donor and its four nearest Si neighbors by 3.2%, thereby increasing it from the unstrained value of 0.235 nm to 0.2425 nm. For this study, we choose the non-static dielectric screening of the donor potential as described by Eq. \[eq:Nonstatic\_donor\_potential\] and the fitting parameters provided by Nara $\&$ Morita as given in the table \[tab:table2\]. This setup is schematically shown in Fig. \[fig:Fig1\] (d). Keeping the central-cell correction fixed at 2.2842 eV, we calculate a significant effect of the NN bond length change on the donor binding energies and the donor wave function confinement at the nuclear site. As evident from the third row of the table \[tab:table4\], the donor ground state binding energy A$_1$ decreases by $\approx$ 10 meV and the value of $|\psi(r_0)|^2$ is decreased by a factor of $\approx$ 2.4 as a result of the lattice strain. Since the binding energies of the donor are adjusted in our model by central-cell correction (by varying U$_0$ and onsite TB energies), we perform further adjustments in U$_0$ by increasing its value to 3.1895 eV to re-establishe the match of the ground state binding energies with the experimental values. Based on this new model, we then recalculate the values of $|\psi(r_0)|^2$ and the Stark shift parameter $\eta_2$, and the corresponding values are provided in the last row of table\[tab:table2\]. The lattice strain only slightly modifies the value of $|\psi(r_0)|^2$ at the donor site, howevere the quadratic Stark shift parameter $\eta_2$ is strongly affected and becomes -1.87 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ which is in remarkable agreement with the exerimental value of -1.9 $\pm$ 0.2 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. Further investigation is needed to establish the connection of the NN bond length change with the value of $|\psi(r_0)|^2$ which would be reported somewhere else. Conclusions =========== In conclusion, this work aims to evaluate and benchmark previously established tight-binding model with the recently measured experimental data of the quadratic Stark shift of the As donor hyperfine interaction. The study is systematically performed to investigate the central-cell correction effects in the tight-binding theory. We include central-cell corrections in terms of donor potential cut-off at the nuclear site, static vs. non-static dielectric screenings of the donor potential, and the effect of the lattice strain by changing the As-Si nearest-neighbor bond lengths. Overall our calculations exhibit that tight-binding theory captures the donor physics remarkably well by reproducing the donor binding energy spectra within 1 meV of the expereimentally measured values. When we include the effects of non-static dielectric screening of the donor potential and lattice strain, the computed value of the quadratic Stark shift parameter ($\eta_2$) is calculated to be -1.87 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ which is in excellent agreement with the experimental value of -1.9 $\pm$ 0.2 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. Such detailed bench-marking of theory against the experimental data would allow us to relaibly investigate the single and two donor electron wave functions, especially those relevant for implementing quantum information processing. ***Acknowledgements:*** This work is funded by the ARC Center of Excellence for Quantum Computation and Communication Technology (CE1100001027), and in part by the U.S. Army Research Office (W911NF-08-1-0527). Computational resources are acknowledged from National Science Foundation (NSF) funded Network for Computational Nanotechnology (NCN) through <http://nanohub.org>. NEMO 3D based open source tools are available at: <https://nanohub.org/groups/nemo_3d_distribution>. [44]{}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 @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [ ()]{} @noop [ ()]{} @noop [ ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [ ()]{} @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{}

--- abstract: 'This paper explores stability of the Einstein universe against linear homogeneous perturbations in the background of $f(\mathcal{G},T)$ gravity. We construct static as well as perturbed field equations and investigate stability regions for the specific forms of generic function $f(\mathcal{G},T)$ corresponding to conserved as well as non-conserved energy-momentum tensor. We use the equation of state parameter to parameterize the stability regions. The graphical analysis shows that the suitable choice of parameters lead to stable regions of the Einstein universe.' author: - | M. Sharif [^1] and Ayesha Ikram [^2]\ Department of Mathematics, University of the Punjab,\ Quaid-e-Azam Campus, Lahore-54590, Pakistan. title: '**Stability Analysis of Einstein Universe in $f(\mathcal{G},T)$ Gravity**' --- [**Keywords:**]{} Stability analysis; Einstein universe; $f(\mathcal{G},T)$ gravity.\ [**PACS:**]{} 04.25.Nx; 04.40.Dg; 04.50.Kd. Introduction ============ The current accelerated expansion of the universe is one of the most astonishing discovery in golden era of cosmology. This has stimulated many researchers to explore the enigmatic nature of dark energy (DE) which is responsible for the phase of cosmic accelerated expansion. Dark energy possesses large negative pressure with repulsive nature but its many salient features are still not known. Modified theories of gravity are considered as the most favorable and optimistic approaches among other proposals to explore the nature of DE. These theories are established by replacing or adding curvature invariants and their corresponding generic functions in the geometric part of general relativity (GR). The Einstein field equations are derived from the first Lovelock scalar dubbed as the Ricci scalar $(R)$ in the Lagrangian density which corresponds to gravity while a particular form of quadratic curvature invariants yields second Lovelock scalar known as Gauss-Bonnet (GB) invariant. This invariant is a linear combination of the form $\mathcal{G}=R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\alpha\beta} R^{\mu\nu\alpha\beta}$, where $R_{\mu\nu}$ and $R_{\mu\nu\alpha\beta}$ represent the Ricci and Riemann tensors, respectively. Gauss-Bonnet invariant is four-dimensional $(4D)$ topological term which has the feature like it is free from spin-2 ghost instabilities [@2]. There are two interesting approaches to discuss the dynamics of $\mathcal{G}$ in $4D$ either by coupling with scalar field or by adding the generic function $f(\mathcal{G})$ in the Einstein-Hilbert action. The first approach naturally appears in the effective low energy action in string theory which effectively discusses the singularity-free cosmological solutions [@3]. Nojiri and Odintsov [@4] introduced second approach as an alternative for DE known as $f(\mathcal{G})$ gravity which elegantly studies the fascinating characteristics of late-time cosmology. Cognola et al. [@a] investigated DE cosmology and found that this theory effectively describes the cosmological structure with a possibility to describe the transition from decelerated to accelerated cosmic phases. De Felice and Tsujikawa [@5] constructed some cosmological viable $f(\mathcal{G})$ models and introduced a procedure to avoid numerical instabilities related with a large mass of the oscillating mode. The same authors [@6] also found that the solar system constraints are consistent for a wide range of cosmological viable model parameters. The captivating issue of cosmic accelerated expansion has successfully been discussed by taking into account modified theories of gravity with matter-curvature coupling. Harko et al. [@7] presented $f(R,T)$ gravity ($T$ is the trace of energy-momentum tensor (EMT)) to study the coupling between geometry and matter. Recently, we introduced another modified theory named as $f(\mathcal{G},T)$ gravity which is a generalization of $f(\mathcal{G})$ gravity [@8]. This modification is based on the coupling of quadratic curvature invariant with matter just as $f(R,T)$ gravity. We studied the non-zero covariant divergence of EMT due to matter-curvature coupling and the massive test particles followed non-geodesic trajectories due to the presence of extra force while the dust particles moved along geodesic lines of geometry. In such matter-curvature coupled theories, cosmic expansion can result from geometric as well as matter component. The stability issue of the Einstein universe (EU) is as old as relativistic cosmology. Einstein tried to find static solution of his field equations to describe isotropic and homogeneous universe. Since the field equations of GR have no static solution, therefore Einstein introduced the term known as cosmological constant $(\Lambda)$ to have static solutions. Einstein universe is described by static FRW universe model with positive curvature filled with perfect fluid in the presence of $\Lambda$. Initially, this model is considered as the most suitable model to discuss static universe but after few years it is found that EU is unstable against small isotropic and homogeneous perturbations [@10]. Harrison [@11] found that the unstable EU for dust distribution becomes oscillatory in the presence of radiations and also observed that stable EU exists against small inhomogeneous perturbations. Gibbons [@12] proved that EU maximizes the entropy against conformal changes if and only if it is stable against speed of sound $(c_{s})$ greater than $\frac{1}{\sqrt{5}}$. Barrow et al. [@13] demonstrated that EU is always neutrally stable in the presence of perfect fluid against small inhomogeneous vector as well as tensor perturbations and also under adiabatic scalar density inhomogeneities until the inequality $5c_{s}^{2}>1$ holds but unstable otherwise. Einstein universe due to its analytical simplicity and fascinating stability properties has always been of great interest to study in the extensions of GR as well as in quantum gravity models. Emergent universe scenario is based on stable EU to resolve the problem of big-bang singularity which is not successful in GR since EU is unstable against homogeneous perturbations [@b]. To find stable static solutions, modified theories have gained much attention to analyze the stability of EU. The stability of EU is studied in braneworld, Einstein-Cartan theory, loop quantum cosmology, non-minimal kinetic coupled gravity etc [@14]. Böhmer et al. [@15] explored its stability using scalar homogeneous perturbations in $f(R)$ gravity and found that stable EU exists for specific forms of $f(R)$ in contrast to GR. Goswami and his collaborators [@16] investigated the existence as well as stability of EU in the background of fourth-order gravity theories. Goheer et al. [@17] studied the existence of EU for power-law $f(R)$ model and found stable solutions. Böhmer and Lobo [@18] discussed the stability of EU in the context of $f(\mathcal{G})$ gravity against scalar homogeneous perturbations and found that stable regions exist for all values of the equation of state parameter $(\omega)$. Böhmer [@c] studied the stability of EU parameterized by the first and second derivatives of scalar potential for linear homogeneous as well as inhomogeneous perturbations in the context of hybrid metric-Palatini gravity and found that a large class of stable solutions. Li et al. [@cc] found stable regions for both open as well as closed universe in modified teleparallel theory against linear homogeneous scalar perturbations. Huang et al. [@d] obtained stable solutions for EU against homogeneous, inhomogeneous scalar, tensor and anisotropic perturbations in Jordan Brans-Dicke theory. The same authors [@e] also found the unstable solutions against homogeneous as well as inhomogeneous scalar perturbations for open universe while stable EU is obtained for a closed universe against homogeneous perturbations in $f(\mathcal{G})$ gravity. Böhmer and his collaborators [@f] analyzed stability regions against both homogeneous and inhomogeneous perturbations in scalar-fluid theories and found stable as well as unstable results against inhomogeneous and homogeneous perturbations, respectively. Darabi et al. [@g] studied the existence and stability of EU in the context of Lyra geometry against scalar, vector as well as tensor perturbations for suitable values of physical parameters. Shabani and Ziaie [@19] analyzed the existence as well as stability of EU in $f(R,T)$ gravity and found stable solutions which were unstable in $f(R)$ gravity. In this paper, we study the stability of EU against scalar homogeneous perturbations in the background of $f(\mathcal{G},T)$ gravity. This analysis is helpful to examine the effects of matter-curvature coupling on the stability of EU. The paper has the following format. In section **2**, we construct the field equations of this theory while section **3** is devoted to analyze the stability under linear homogeneous perturbations around EU for conserved as well as non-conserved EMT. The results are summarized in the last section. Dynamics of $f(\mathcal{G},T)$ Gravity ====================================== The action for $f(\mathcal{G},T)$ gravity is given by [@8] $$\label{1} \mathcal{S}=\int d^{4}x\sqrt{-g}\left[\frac{R+f(\mathcal{G},T)}{2\kappa^2} +\mathcal{L}_{m}\right],$$ where $T=g_{\mu\nu}T^{\mu\nu},~\kappa^2,~g$ and $\mathcal{L}_{m}$ represent coupling constant, determinant of the metric tensor $(g_{\mu\nu})$ and matter Lagrangian density, respectively. The EMT in terms of $\mathcal{L}_{m}$ is defined as [@20] $$\label{2} T_{\mu\nu}=-\frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta g^{\mu\nu}}.$$ If $\mathcal{L}_{m}$ depends on the components of $g_{\mu\nu}$ but does not depend on its derivatives, then Eq.(\[2\]) yields $$\label{3} T_{\mu\nu}=g_{\mu\nu}\mathcal{L}_{m}-2\frac{\partial \mathcal{L}_{m}}{\partial g^{\mu\nu}}.$$ Varying the action (\[1\]) with respect to $g_{\mu\nu}$, we obtain the field equations as follows $$\begin{aligned} \nonumber R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R&=&\kappa^2T_{\mu\nu} -(T_{\mu\nu}+\Theta_{\mu\nu})f_{T}(\mathcal{G},T)+\frac{1}{2}g_{\mu\nu} f(\mathcal{G},T)-(2RR_{\mu\nu}\\\nonumber&-&4R_{\mu}^{\alpha}R_{\alpha\nu} -4R_{\mu\alpha\nu\beta}R^{\alpha\beta}+2R_{\mu}^{\alpha\beta\gamma} R_{\nu\alpha\beta\gamma})f_{\mathcal{G}}(\mathcal{G},T) \\\nonumber&-&(2Rg_{\mu\nu} \Box-4R_{\mu\nu}\Box-2R\nabla_{\mu}\nabla_{\nu} +4R_{\mu}^{\alpha}\nabla_{\nu}\nabla_{\alpha} +4R_{\nu}^{\alpha}\nabla_{\mu}\nabla_{\alpha}\\\label{4}&-&4g_{\mu\nu}R^{\alpha\beta} \nabla_{\alpha}\nabla_{\beta}+4R_{\mu\alpha\nu\beta} \nabla^{\alpha}\nabla^{\beta})f_{\mathcal{G}}(\mathcal{G},T),\end{aligned}$$ where $f_{\mathcal{G}}(\mathcal{G},T)=\partial f(\mathcal{G},T)/\partial\mathcal{G},~f_{T}(\mathcal{G},T)=\partial f(\mathcal{G},T)/\partial T,~\Box=\nabla_{\mu}\nabla^{\mu}$ and $\nabla_{\mu}$ is a covariant derivative whereas $\Theta_{\mu\nu}$ has the following expression $$\label{5} \Theta_{\mu\nu}=g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta g_{\mu\nu}}=-2T_{\mu\nu}+g_{\mu\nu}\mathcal{L}_{m}-2g^{\alpha\beta} \frac{\partial^{2}\mathcal{L}_{m}}{\partial g^{\mu\nu}\partial g^{\alpha\beta}}.$$ The covariant divergence of Eq.(\[4\]) is given by $$\begin{aligned} \nonumber \nabla^{\mu}T_{\mu\nu}&=&\frac{f_{T}(\mathcal{G},T)} {\kappa^2-f_{T}(\mathcal{G},T)}\left[\nabla^{\mu}\Theta_{\mu\nu} -\frac{1}{2}g_{\mu\nu}\nabla^{\mu}T+(T_{\mu\nu}+\Theta_{\mu\nu}) \right.\\\label{6}&\times&\left.\nabla^{\mu}(\ln{f_{T}(\mathcal{G},T)}) \right].\end{aligned}$$ In this theory, the field equations as well as conservation law depend on the contributions from cosmic matter contents, therefore every suitable selection of $\mathcal{L}_{m}$ provides the particular scheme of dynamical equations. The line element for positive curvature FRW universe model is [@15] $$\label{7} ds^2=dt^2-a^2(t)\left(\frac{1}{1-r^2}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)\right),$$ where $a(t)$ is the scale factor. The energy-momentum tensor for perfect fluid is given by $$\label{8} T_{\mu\nu}=(\rho+P)u_{\mu}u_{\nu}-Pg_{\mu\nu},$$ where $\rho,~P$ and $u_{\mu}$ represent the energy density, pressure and four-velocity of the matter distribution, respectively. For perfect fluid as cosmic matter distribution with $\mathcal{L}_{m}=-P$, Eq.(\[5\]) becomes [@7] $$\label{9} \Theta_{\mu\nu}=-2T_{\mu\nu}-Pg_{\mu\nu}.$$ Using Eqs.(\[7\])-(\[9\]) in (\[4\]), we obtain the following set of field equations $$\begin{aligned} \nonumber \frac{3}{a^{2}}(1+\dot{a}^2)&=&\kappa^2\rho+\frac{1}{2}f(\mathcal{G},T) +(\rho+P)f_{T}(\mathcal{G},T)-12\frac{\ddot{a}}{a^3}(1+\dot{a}^2) \\\label{10}&\times&f_{\mathcal{G}}(\mathcal{G},T)+12\frac{\dot{a}}{a^3} (1+\dot{a}^2)\partial_{t}f_{\mathcal{G}}(\mathcal{G},T),\\\nonumber -(1+\dot{a}^2)-2a\ddot{a}&=&\kappa^2a^2P-\frac{1}{2}a^2f(\mathcal{G},T) +12\frac{\ddot{a}}{a}(1+\dot{a}^2)f_{\mathcal{G}}(\mathcal{G},T) \\\label{11}&-&8\dot{a}\ddot{a}\partial_{t}f_{\mathcal{G}}(\mathcal{G},T) -4(1+\dot{a}^2)\partial_{tt}f_{\mathcal{G}}(\mathcal{G},T),\end{aligned}$$ where $$\label{12} \mathcal{G}=\frac{24}{a^{3}}(1+\dot{a}^{2})\ddot{a},\quad T=\rho-3P,$$ and dot represents the time derivative. The conservation equation (\[6\]) for perfect fluid yields $$\begin{aligned} \nonumber \dot{\rho}+3\frac{\dot{a}}{a}(\rho+P)&=&\frac{-1}{\kappa^2 +f_{T}(\mathcal{G},T)}\left[\left(\dot{P}+\frac{1}{2}\dot{T}\right) f_{T}(\mathcal{G},T)+(\rho+P)\right.\\\label{12a}&\times& \left.\partial_{t}f_{T}(\mathcal{G},T)\right].\end{aligned}$$ Stability of Einstein Universe ============================== In this section, we analyze the stability of EU against linear homogeneous perturbations in the background of $f(\mathcal{G},T)$ gravity. For EU, $a(t)=a_{0}=$ constant and consequently, the field equations (\[10\]) and (\[11\]) reduce to $$\begin{aligned} \label{13} \frac{3}{a_{0}^2}&=&\kappa^2\rho_{0}+\frac{1}{2} f(\mathcal{G}_{0},T_{0})+(\rho_{0}+P_{0})f_{T}(\mathcal{G}_{0},T_{0}), \\\label{14}-\frac{1}{a_{0}^2}&=&\kappa^2P_{0}-\frac{1}{2}f(\mathcal{G}_{0},T_{0}),\end{aligned}$$ where $\mathcal{G}_{0}=\mathcal{G}(a_{0})=0,~T_{0}=\rho_{0}-3P_{0},~\rho_{0}$ and $P_{0}$ are the unperturbed energy density and pressure, respectively. To explore the stability regions, we consider linear form of equation of state as $P(t)=\omega\rho(t)$ and define linear perturbations in the scale factor and energy density as follows $$\label{15} a(t)=a_{0}+a_{0}\delta a(t),\quad\rho(t)=\rho_{0}+\rho_{0}\delta\rho(t),$$ where $\delta a(t)$ and $\delta \rho(t)$ represent the perturbed scale factor and energy density, respectively. Applying the Taylor series expansion in two variables upto first order with the assumption that $f(\mathcal{G},T)$ is an analytic function, we have $$\label{16} f(\mathcal{G},T)=f(\mathcal{G}_{0},T_{0})+ f_{\mathcal{G}}(\mathcal{G}_{0},T_{0})\delta\mathcal{G} +f_{T}(\mathcal{G}_{0},T_{0})\delta T,$$ where $\delta\mathcal{G}$ and $\delta T$ have the following expressions $$\label{17} \delta\mathcal{G}=\frac{24}{a_{0}^{2}}\delta\ddot{a},\quad \delta T=T_{0}\delta\rho,$$ where $\delta\ddot{a}=\frac{d^2}{dt^2}(\delta a)$. Using Eqs.(\[13\])-(\[17\]) in (\[10\]) and (\[11\]), we obtain the linearized perturbed field equations as follows $$\begin{aligned} \nonumber &&6\delta a+24\rho_{0}(1+\omega) f_{\mathcal{G}T}(\mathcal{G}_{0},T_{0})\delta\ddot{a}+a_{0}^2\rho_{0}[\kappa^2+(1+\omega) f_{T}(\mathcal{G}_{0},T_{0})\\\label{18}&+&\frac{1}{2}(1-3\omega) f_{T}(\mathcal{G}_{0},T_{0})+\rho_{0}(1+\omega)(1-3\omega) f_{TT}(\mathcal{G}_{0},T_{0})]\delta\rho=0,\\\nonumber &-&\frac{2}{a_{0}^2}\delta a+2\delta\ddot{a}-\frac{96}{a_{0}^4} f_{\mathcal{GG}}(\mathcal{G}_{0},T_{0})\delta a^{(iv)}+\rho_{0} [\kappa^2\omega-\frac{1}{2}(1-3\omega)f_{T}(\mathcal{G}_{0},T_{0})] \delta\rho\\\label{19}&-&4\frac{\rho_{0}}{a_{0}^2}(1-3\omega) f_{\mathcal{G}T}(\mathcal{G}_{0},T_{0})\delta\ddot{\rho}=0.\end{aligned}$$ These equations show that the perturbations in $a(t)$ are related with density perturbations. In the following subsections, we discuss the stability modes for conserved as well as non-conserved EMT. Conserved EMT ------------- In this case, we assume that general conservation law holds in $f(\mathcal{G},T)$ gravity. For this purpose, the right hand side of Eq.(\[12a\]) must be zero which yields $$\label{21} \left(\dot{P}+\frac{1}{2}\dot{T}\right)f_{T}(\mathcal{G},T)+ (\rho+P)\partial_{t}f_{T}(\mathcal{G},T)=0.$$ The conserved matter contents of the universe satisfy the relation given by $$\label{22} \delta\dot{\rho}=-3(1+\omega)\delta\dot{a}.$$ Using this equation in the elimination of $\delta\rho$ from Eqs.(\[18\]) and (\[19\]), we obtain the fourth-order perturbation equation in perturbed $a(t)$ as follows $$\begin{aligned} \nonumber &&\left[6\kappa^2a_{0}\omega-3a_{0}(1-3\omega)f_{T} (\mathcal{G}_{0},T_{0})+2a_{0}\left\{\kappa^2+(1+\omega)f_{T} (\mathcal{G}_{0},T_{0})\right.\right.\\\nonumber&+&\left. \left.\frac{1}{2}(1-3\omega)f_{T}(\mathcal{G}_{0},T_{0})+\rho_{0}(1+\omega)(1-3\omega) f_{TT}(\mathcal{G}_{0},T_{0})\right\}\right]\delta a\\\nonumber&+&\left[24a_{0}\rho_{0}(1+\omega)\left\{ \kappa^2\omega-\frac{1}{2}(1-3\omega)f_{T}(\mathcal{G}_{0},T_{0}) \right\}f_{\mathcal{G}T}(\mathcal{G}_{0},T_{0})\right.\\\nonumber&-&\left. \left\{2+\frac{12\rho_{0}}{a_{0}^2}(1+\omega)(1-3\omega)f_{\mathcal{G}T} (\mathcal{G}_{0},T_{0})\right\}\left\{\kappa^2a_{0}^3+a_{0}^3(1+\omega) f_{T}(\mathcal{G}_{0},T_{0})\right.\right.\\\nonumber&+&\left.\left. \frac{1}{2}a_{0}^3(1-3\omega)f_{T}(\mathcal{G}_{0},T_{0})+a_{0}^3 \rho_{0}(1+\omega)(1-3\omega)f_{TT}(\mathcal{G}_{0},T_{0})\right\}\right] \delta\ddot{a}\\\nonumber&+&\frac{96}{a_{0}^4}\left\{\kappa^2a_{0}^3 +a_{0}^3(1+\omega)f_{T}(\mathcal{G}_{0},T_{0})+\frac{1}{2}a_{0}^3(1-3\omega) f_{T}(\mathcal{G}_{0},T_{0})+a_{0}^3\rho_{0}\right.\\\label{23}&\times& \left.(1+\omega)(1-3\omega)f_{TT}(\mathcal{G}_{0},T_{0})\right\} f_{\mathcal{GG}}(\mathcal{G}_{0},T_{0})\delta a^{(iv)}=0.\end{aligned}$$ Adding Eqs.(\[13\]) and (\[14\]), it follows that $$\label{24} \frac{2}{a_{0}^2}=\rho_{0}(1+\omega)(\kappa^2+f_{T}(\mathcal{G}_{0},T_{0})).$$ Using this expression in Eq.(\[23\]), the resulting perturbation equation yields $$\begin{aligned} \nonumber &&\left[\rho_{0}(1+\omega)\{\kappa^2+f_{T}(\mathcal{G}_{0},T_{0})\}\left\{ \kappa^2(1+3\omega)+(1+\omega)f_{T}(\mathcal{G}_{0},T_{0}) \right.\right.\\\nonumber&-&\left.\left.(1-3\omega) f_{T}(\mathcal{G}_{0},T_{0})+\rho_{0}(1+\omega)(1-3\omega)f_{TT} (\mathcal{G}_{0},T_{0})\right\}\right]\delta a\\\nonumber&+&\left[12\rho_{0}^2(1+\omega)^2\{\kappa^2+f_{T} (\mathcal{G}_{0},T_{0})\}\left\{\kappa^2\omega-\frac{1}{2} (1-3\omega)f_{T}(\mathcal{G}_{0},T_{0})\right\}\right. \\\nonumber&\times&\left.f_{\mathcal{G}T}(\mathcal{G}_{0},T_{0}) -[2+6\rho_{0}^2(1+\omega)^2(1-3\omega)\{\kappa^2+f_{T} (\mathcal{G}_{0},T_{0})\}f_{\mathcal{G}T}(\mathcal{G}_{0},T_{0})] \right.\\\nonumber&\times&\left.\left\{\kappa^2+(1+\omega) f_{T}(\mathcal{G}_{0},T_{0})+\rho_{0}(1+\omega)(1-3\omega) f_{TT}(\mathcal{G}_{0},T_{0})+\frac{1}{2}(1-3\omega)\right.\right. \\\nonumber&\times&\left.\left.f_{T}(\mathcal{G}_{0},T_{0})\right\} \right]\delta\ddot{a}+24\rho_{0}^2(1+\omega)^2\{\kappa^2+f_{T} (\mathcal{G}_{0},T_{0})\}^2\left\{\kappa^2+(1+\omega)\right. \\\nonumber&\times&\left.f_{T}(\mathcal{G}_{0},T_{0}) +\rho_{0}(1+\omega)(1-3\omega)f_{TT}(\mathcal{G}_{0},T_{0})+\frac{1}{2} (1-3\omega)f_{T}(\mathcal{G}_{0},T_{0})\right\}\\\label{25}&\times& f_{\mathcal{GG}}(\mathcal{G}_{0},T_{0})\delta a^{(iv)}=0.\end{aligned}$$ The solution of this equation helps to discuss the stability regions in EU. However, it would be difficult to find stable/unstable solutions due to its complicated nature. We, therefore, consider the particular form of $f(\mathcal{G},T)$ as follows $$\label{26} f(\mathcal{G},T)=f_{1}(\mathcal{G})+f_{2}(T).$$ This choice of model does not involve the direct curvature-matter non-minimal coupling but it can be considered as correction to $f(\mathcal{G})$ gravity. In this case, we have assumed that the EMT is conserved, therefore, we first constrain the above model such that the conservation law holds for it. For this purpose, using the considered form in Eq.(\[21\]), the resulting second order differential equation takes the form $$\nonumber (1-\omega)f_{2}'(T)+2(1+\omega)Tf_{2}''(T)=0,$$ where prime represents derivative with respect to $x~(x=T$ or $\mathcal{G})$. The solution is given by $$\label{27} f_{2}(T)=\frac{c_{1}(1+\omega)}{1+3\omega}T^{\frac{1+3\omega}{2(1+\omega)}}+c_{2},$$ where $c_{i}$’s $(i=1,2)$ are integration constants. This is the unique representation of matter contribution for which conservation law holds with model (\[26\]). The modified GB term $f_{1}(\mathcal{G}_{0})$ acts like an effective $\Lambda$ to the unperturbed field equations. It is worth mentioning here that $f(\mathcal{G})$ gravity is recovered for this choice of $f(\mathcal{G},T)$ model if $f_{2}(T)=0$ [@18]. Inserting the values from Eqs.(\[26\]) and (\[27\]) in (\[25\]), the differential equation takes the form $$\label{28} \Delta_{2}(\Delta_{1}+\Delta_{3})\delta a-2\Delta_{1}\delta \ddot{a}+24\Delta_{1}\Delta_{2}^{2}f_{1}''(\mathcal{G}_{0})\delta a^{(iv)}=0,$$ where $\Delta_{j}$’s $(j=1,2,3)$ are $$\begin{aligned} \nonumber \Delta_{1}&=&\kappa^2+\frac{1}{4}c_{1}(1+5\omega)[\rho_{0}(1-3 \omega)]^{\frac{\omega-1}{2(\omega+1)}}-\frac{1}{4}c_{1}\rho_{0} (1-\omega)(1-3\omega)\\\nonumber&\times&[\rho_{0}(1-3\omega)] ^{\frac{-(3+\omega)}{2(1+\omega)}},\\\nonumber\Delta_{2}&=& \rho_{0}(1+\omega)\left[\kappa^2+\frac{1}{2}\{\rho_{0}(1-3\omega)\} ^{\frac{\omega-1}{2(\omega+1)}}\right],\\\nonumber\Delta_{3}&=& 3\kappa^2\omega-\frac{3}{4}c_{1}(1-3\omega)\{\rho_{0}(1-3\omega)\} ^{\frac{\omega-1}{2(\omega+1)}}.\end{aligned}$$ Equation (\[28\]) provides the following solution $$\nonumber \delta a(t)=d_{1}e^{\Omega_{1}t}+d_{2}e^{-\Omega_{1}t}+d_{3}e^{\Omega_{2}t} +d_{4}e^{-\Omega_{2}t},$$ where $d_{k}$’s $(k=1...4)$ are constants of integration and the parameters $\Omega_{1}$ and $\Omega_{2}$ are frequencies of small perturbations given by $$\label{29} \Omega^{2}_{1,2}=\frac{\Delta_{1}\pm\sqrt{\Delta_{1}^{2}-24 \Delta_{1}\Delta_{2}^{3}(\Delta_{1}+\Delta_{3})f_{1}'' (\mathcal{G}_{0})}}{24\Delta_{1}\Delta_{2}^{2}f_{1}'' (\mathcal{G}_{0})}.$$ In order to avoid the exponential growth of $\delta a(t)$ or collapse, the frequencies are purely complex which lead to the existence of stable EU. Thus, the condition of stability is achieved when $\Omega^{2}_{1,2}<0$. In the limit of GR, $\Omega^{2}_{1}$ diverge while $\Omega^{2}_{2}$ are given by $$\nonumber \Omega^{2}_{2}=\frac{1}{2}\kappa^2\rho_{0}(1+3\omega)(1+\omega),$$ which provide stable region in the range $-1<\omega<-\frac{1}{3}$ [@18]. For simplicity, we introduce a new parameter $\zeta_{1}=24f_{1}''(\mathcal{G}_{0})$ as well as use $\kappa^2=1$ and $\rho_{0}=0.3$ (present day value of density parameter) to discuss the graphical analysis of stable EU [@h2]. Figure **1** shows the stable regions under homogeneous perturbations of EU for $\Omega_{1}^{2}$. It is found that for $c_{1}=1$ in the left plot, the stable EU exists for negative values of $\omega$ while no stable region exists for its positive values. The right panel shows the stable region for $c_{1}=5$ and hence the stability regions decrease as the value of integration constant increases while for negative values of $c_{1}$, no stable regions are found. The regions of stability for frequencies $\Omega_{2}^{2}$ are shown in Figure **2** for both positive as well as negative values of $c_{1}$. The negative values of $\zeta_{1}$ are obtained for $f_{1}''(\mathcal{G}_{0})<0$ which is in agreement with stability condition of $f(\mathcal{G})$ models [@21]. Figure **3** shows the stability regions for both $\Omega_{1}^{2}$ as well as $\Omega_{2}^{2}$ of the whole system. Non-Conserved EMT ----------------- Here, we analyze the stability of $f(\mathcal{G},T)$ model when EMT is not conserved. We consider generic function $f_{1}(\mathcal{G})$ and a linear form of $f_{2}(T)$ in Eq.(\[26\]) as follows $$\label{30} f(\mathcal{G},T)=f_{1}(\mathcal{G})+\kappa^{2}\chi T,$$ where $\chi$ is an arbitrary constant. Substituting in Eq.(\[12a\]), we obtain $$\nonumber \rho=\tilde{\rho}_{0}a^{-3\varphi},\quad\varphi=\frac{2(1+\chi)(1+\omega)} {2+\chi(3-\omega)},$$ where $\tilde{\rho}_{0}$ is an integration constant. The perturbed field equations (\[18\]) and (\[19\]) take the following form $$\begin{aligned} \label{31} &&6\delta a+\kappa^2a_{0}^2\rho_{0}\left[1-\frac{\chi}{2}(\omega-3)\right]\delta\rho=0, \\\label{32}&&2\delta\ddot{a}-\frac{2}{a_{0}^2}\delta a+\kappa^2\left[\omega-\frac{\chi}{2}(1-3\omega)\right]\rho_{0}\delta\rho -\frac{96}{a_{0}^{4}}f_{1}''(\mathcal{G}_{0})\delta a^{(iv)}=0.\end{aligned}$$ The first field equation shows the relationship between the perturbed energy density and scale factor perturbations. Eliminating $\delta\rho$ from Eqs.(\[31\]) and (\[32\]), the resulting differential equation in perturbed $a(t)$ is given by $$\label{33} 2\delta\ddot{a}-\frac{2}{a_{0}^2}\left[1+\frac{3(2\omega-\chi(1-3\omega))} {2-\chi(\omega-3)}\right]\delta a-\frac{96}{a_{0}^4}f_{1}''(\mathcal{G}_{0})\delta a^{(iv)}=0.$$ In this case, the addition of static field equations yields $$\label{34} \frac{2}{a_{0}^2}=\kappa^2\rho_{0}(1+\chi)(1+\omega).$$ Inserting this value of $\frac{2}{a_{0}^2}$ in Eq.(\[33\]), we obtain $$\begin{aligned} \nonumber &&\kappa^2\rho_{0}[\chi(1+\chi)(1-\omega^2)-(1+\chi)^2(1+\omega)(1+3\omega)]\delta a+[2(1+\chi)\\\nonumber&+&\chi(1-\omega)]\delta\ddot{a}-12\kappa^4 \rho_{0}^{2}[2(1+\chi)^{3}(1+\omega)^{2}+\chi(1+\chi)^{2}(1-\omega)(1+\omega)^{2}] \\\label{35}&\times&f_{1}''(\mathcal{G}_{0})\delta a^{(iv)}=0,\end{aligned}$$ whose solution provides the following four frequencies as $$\nonumber \Upsilon^{2}_{1,2}=\frac{-2(1+\chi)-\chi(1-\omega)\pm\sqrt{[2(1+\chi) +\chi(1-\omega)]^{2}-48\kappa^{6}\rho_{0}^{3}\Delta_{4}f_{1}''(\mathcal{G}_{0})}}{24 \kappa^4\rho_{0}^{2}[\chi(1+\chi)^{2}(\omega-1)(1+\omega)^{2} -2(1+\chi)^{3}(1+\omega)^{2}]f_{1}''(\mathcal{G}_{0})},$$ where $$\begin{aligned} \nonumber \Delta_{4}&=&[2(1+\chi)^{3}(1+\omega)^{2}+\chi(1+\chi)^{2}(1-\omega)(1+\omega)^{2}] [(1+\chi)^2(1+\omega)\\\nonumber&\times&(1+3\omega)-\chi(1+\chi)(1-\omega^2)].\end{aligned}$$ When $f_{1}(\mathcal{G}_{0})=0=\chi$, the frequencies $\Upsilon^{2}_{1}$ recover the GR result as obtained in the previous case while frequencies $\Upsilon^{2}_{2}$ diverge. We simplify the expression by introducing a new parameter $\zeta_{2}=-48\kappa^{6}\rho_{0}^{3}f_{1}''(\mathcal{G}_{0})$ which remains positive for $f_{1}''(\mathcal{G}_{0})<0$. Figure **4** shows stable regions against homogeneous perturbations of EU for frequencies $\Upsilon_{1}^{2}$. It is found that when $\chi=1$ (left panel), the stable EU exists for all values of $\omega$ with suitable choice of $\zeta_{2}$ while less stable regions are obtained when $\chi=5$ as shown in the right plot. In the case of non-conserved EMT, the stability regions decrease as the value of model parameter $\chi$ increases while no stable regions are observed for $\chi<0$. The regions of stability in EU for frequencies $\Upsilon^{2}_{2}$ are shown in Figure **5** for considered values of $\chi$ while stability regions for whole system is observed in Figure **6**. Now, we consider the generalized model given by $$\label{a} f(\mathcal{G},T)=f_{1}(\mathcal{G})+\kappa^{2}\chi T^{n},\quad n\neq0.$$ Following the same procedure, we obtain the following fourth-order differential equation in perturbed $a(t)$ as follows $$\begin{aligned} \nonumber &&24\kappa^4\rho_{0}^{2}(1+\omega)^{2}\left[1+n\chi\rho_{0}^{n-1}(1-3\omega) ^{n-1}\right]^{2}f_{1}''(\mathcal{G}_{0})\delta a^{(iv)}-2\delta\ddot{a} \\\nonumber&+&\kappa^2\rho_{0}(1+\omega)\left(1+n\chi\rho_{0}^{n-1}(1-3\omega)^{n-1}\right) \left[1+3\left(\omega-\frac{n}{2}\chi(1-3\omega)^{n}\rho_{0}^{n-1}\right)\right. \\\nonumber&\times&\left.\left(1+n\chi(1-3\omega)^{n-1}\rho_{0}^{n-1}\left[(1+\omega) +\frac{1}{2}(1-3\omega)+(n-1)\right.\right.\right. \\\nonumber&\times&\left.\left.\left.(1+\omega)\right]\right)^{-1}\right]\delta a=0,\end{aligned}$$ whose solution provides the following four frequencies as $$\begin{aligned} \nonumber \Xi^{2}_{1,2}=\frac{1\pm\sqrt{1-24\kappa^6\Delta_{5}f''_{1}(\mathcal{G}_{0})}} {24\left[\kappa^2\rho_{0}(1+\omega)\left(1+n\chi\rho_{0}^{n-1}(1-3\omega)^{n-1} \right)\right]^{2}f_{1}''(\mathcal{G}_{0})},\end{aligned}$$ where $$\begin{aligned} \nonumber \Delta_{5}&=&\rho_{0}^{3}(1+\omega)^{3}\left(1+n\chi\rho_{0}^{n-1}(1-3\omega)^{n-1} \right)^{3}\left[1+3\left(\omega-\frac{n}{2}\chi(1-3\omega)^{n} \right.\right.\\\nonumber&\times&\left.\left.\rho_{0}^{n-1}\right) \left[1+n\chi(1-3\omega)^{n-1}\rho_{0}^{n-1}\left((1+\omega)+\frac{1}{2}(1-3\omega) +(n-1)\right.\right.\right.\\\nonumber&\times&\left.\left.\left. (1+\omega)\right)\right]^{-1}\right].\end{aligned}$$ The graphical analysis of frequencies $\Xi_{2}^{2}$ are shown in Figures **7** and **8** where we have used $\zeta_{3}=-24 \kappa^{6}f_{1}''(\mathcal{G}_{0}),~\kappa^2=1,~\rho_{0}=0.3$ and $\chi=1$. It is found that stable regions are obtained for all the considered values of $n$ while stable EU does not exist for the frequencies $\Xi_{1}^{2}$. In this case, the stability region of whole system is completely described by the frequencies $\Xi_{2}^{2}$. It is interesting to mention here that for $f_{1}(\mathcal{G}_{0})=0=\chi$, the frequencies $\Xi^{2}_{1}$ diverge while GR is recovered for the frequencies $\Xi^{2}_{2}$ as in the previous case. Final Remarks ============= In this paper, we have analyzed the stability issue of EU in the context of $f(\mathcal{G},T)$ gravity which is the extension of $f(\mathcal{G})$ gravity and is based on the ground of matter-curvature coupling. Due to this coupling, the conservation law does not hold as in $f(R,T)$ gravity [@7]. We have considered the isotropic and homogeneous positive curvature FRW line element with perfect fluid as matter content of the universe. The static as well as perturbed field equations are constructed against linear homogeneous perturbations which are parameterized by equation of state parameter. We have formulated the fourth-order perturbed differential equation whose solutions are analyzed for the existence and stability of EU for specific form of $f(\mathcal{G},T)=f_{1}(\mathcal{G})+f_{2}(T)$. For this choice, we have discussed both the models when EMT is conserved as well not conserved and obtained distinct results as compared to $f(\mathcal{G})$ gravity. - We have assumed that EMT is conserved in this gravity and obtained a particular form of $f_{2}(T)$ for which the covariant divergence of EMT becomes zero. We have analyzed the regions of stability around EU and found that stable results are observed for a suitable choice of integration constant $c_{1}$. - Two particular forms of $f_{2}(T)$ are considered for which the covariant divergence of EMT remains non-zero and the value of energy density in terms of scale factor is evaluated. It is found that stable EU exists in this case for both models if the model parameter $\chi$ is chosen appropriately. We conclude that the stable EU universe exists against scalar homogeneous perturbations in the background of $f(\mathcal{G},T)$ for all values of the equation of state parameter if the model parameters are chosen suitably. Einstein universe against vector perturbations (comoving dimensionless vorticity vector) are stable for all equations of state on all scales since any initial vector perturbations remain frozen. The mechanism for stability analysis of EU against tensor perturbations (comoving dimensionless traceless shear tensor) suggests that these fluctuations may not break the stability of EU in the background of $f(\mathcal{G},T)$ gravity [@13]. It would be interesting to investigate complete analysis of tensor as well as inhomogeneous perturbations in this gravity which will be helpful to explore the EU. It is worth mentioning here that our results reduce to $f(\mathcal{G})$ gravity in the absence of matter-curvature coupling [@18]. [30]{} Calcagni, G., Tsujikawa, S. and Sami, M.: Class. Quantum Grav. **22**(2005)3977; De Felice, A., Hindmarsh, M. and Trodden, M.: J. Cosmol. Astropart. Phys. **08**(2006)005. Metsaev, R.R. and Tseytlin, A.A.: Nucl. Phys. B **293**(1987)385; Antoniadis, I., Rizos, J. and Tamvakis, K.: Nucl. Phys. B **415**(1994)497; Kanti, P., Rizos, J. and Tamvakis, K.: Phys. Rev. D **59**(1999)083512; Nojiri, S. and Odintsov, S.D.: Int. J. Geom. Meth. Mod. Phys. **4**(2007)115. Nojiri, S. and Odintsov, S.D.: Phys. Lett. B **631**(2005)1. Cognola, G. et al.: Phys. Rev. D **73**(2006)084007. De Felice, A. and Tsujikawa, S.: Phys. Lett. B **675**(2009)1. De Felice, A. and Tsujikawa, S.: Phys. Rev. D **80**(2009)063516. Harko, T. et al.: Phys. Rev. D **84**(2011)024020. Sharif, M. and Ikram, A.: Eur. Phys. J. C **76**(2016)640. Eddington, A.S.: Mon. Not. R. Astron. Soc. **90**(1930)668. Harrison, E.R.: Rev. Mod. Phys. **39**(1967)862. Gibbons, G.W.: Nucl. Phys. B **292**(1987)784. Barrow, J.D. et al.: Class. Quantum Grav. **20**(2003)L155. Ellis, G.F.R. and Maartens, R.: Class. Quantum Grav. **21**(2004)223; Ellis, G.F.R., Murugan, J. and Tsagas, C.G.: Class. Quantum Grav. **21**(2004)233. Gergely, L.Á. and Maartens, R.: Class. Quantum. Grav. **19**(2002)213; Gruppuso, A. Roessl, E. and Shaposhnikov, M.: J. High Energy Phys. **08**(2004)011; Böhmer, C.G.: Class. Quantum. Grav. **21**(2004)1119; Mulryne, D.J. et al.: Phys. Rev. D **71**(2005)123512; Atazadeh, K. and Darabi, F.: Phys. Lett. B **744**(2015)363; Zhang, K. et al.: Phys. Lett. B **758**(2016)37. Böhmer, C.G., Hollenstein, L. and Lobo, F.S.N.: Phys. Rev. D **76**(2007)084005. Goswami, R., Goheer, N. and Dunsby, P.K.S.: Phys. Rev. D **78**(2008)044011. Goheer, N. Goswami, R. and Dunsby, P.K.S.: Class. Quantum Grav. **26**(2009)105003. Böhmer, C.G. and Lobo, F.S.N.: Phys. Rev. D **79**(2009)067504. Böhmer, C.G., Lobo, F.S.N. and Tamanini, N.: Phys. Rev. D **88**(2013)104019. Li, J.T., Lee, C.C. and Geng, C.Q.: Eur. Phys. J. C **73**(2013)2315. Huang, H., Wu, P. and Yu, H.: Phys. Rev. D **89**(2014)103521. Huang, H., Wu, P. and Yu, H.: Phys. Rev. D **91**(2015)023507. Böhmer, C.G., Tamanini, N. and Wright, M.: Phys. Rev. D **92**(2015)124067. Darabi, F., Heydarzade, Y. and Hajkarim, F.: Can. J. Phys. **93**(2015)1566. Shabani, H. and Ziaie, A.H.: arXiv:1606.07959. Landau, L.D. and Lifshitz, E.M.: *The Classical Theory of Fields* (Pergamon Press, 1971). Ade, P.A.R. et al.: Astron. Astrophys. **594**(2016)A13. Li, B., Barrow, J.D. and Mota, D.F.: Phys. Rev. D **76**(2007)044027. [^1]: msharif.math@pu.edu.pk [^2]: ayeshamaths91@gmail.com

--- abstract: 'We discuss properties of $L^2$-eigenfunctions of Schrödinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce new ones.' address: - | Institute of Mathematics USC RAS, Chernyshevskii str., 112,\ Ufa, 450000, Russia\ & Bashkir State Pedagogical University, October rev. st., 3a,\ Ufa, 450008, Russia,\ E-mail: borisovdi@yandex.ru\ matem.anrb.ru & www.bspu.ru - | Fakultät für Mathematik, Reichenhainer Str. 41,\ Chemnitz, D-09126, Germany,\ www.tu-chemnitz.de/\~mtau - | Fakultät für Mathematik, Reichenhainer Str. 41,\ Chemnitz, D-09126, Germany,\ www.tu-chemnitz.de/stochastik/ author: - 'D. Borisov' - 'M. Tautenhahn' - 'I. Veselić' title: Equidistribution estimates for eigenfunctions and eigenvalue bounds for random operators --- Introduction ============ In this note we present recent results in Harmonic Analysis for solutions of (time-independent) Schr[ö]{}dinger equations and other partial differential equations. They are motivated by interest in techniques relevant for proving localization for random Schr[ö]{}dinger operators. The mentioned Harmonic Analysis results which we present are a quantitative unique continuation principle and an equidistribution property for eigenfunctions, which is scale-uniform. These results, and variants thereof, go under various names, depending on the particular field of mathematics: They are called observability estimate, uncertainty relation, scale-free unique continuation principle, or local positive definiteness. The latter term signifies that a self-adjoint operator is (strictly) positive definite when restricted to a relevant subspace, while it is not so on the whole Hilbert space. For the purpose of motivation we discuss this property in the next section. The term *localization* refers to the phenomenon, that quantum Hamiltonians describing the movement of electrons in certain disordered media exhibit pure point spectrum in appropriately specified energy regions. The corresponding eigenfunctions decay exponentially in space. The (time-dependent) wavepackets describing electrons stay localized essentially in a compact region of space for all times. Nota bene, all mentioned properties hold *almost surely*. This is natural in the context of random operators. An important partial result for deriving localization are Wegner estimates. These are bounds on the expected number of eigenvalues in a bounded energy interval of a random Schr[ö]{}dinger operator restricted to a box. The localization problem has been studied for other classes of random operators beyond those of Schr[ö]{}dinger type. An example are random divergence type operators, see e.g. Refs.  and . This are partial differential operators with randomness in coefficients of higher order terms. In paricular, the second order term is no longer the Laplacian, but a variable coefficient operator. In this context one is again lead to consider the above mentioned questions of Harmonic Analysis for eigenfunctions of differential operators. In this note we present an exposition of recently published results, and an announcement of a quantitative unique continuation principle and an equidistribution estimate for eigenfunctions for a class of elliptic operators with variable coefficients. Motivation: Moving and lifting of eigenvalues {#ss:motivation} --------------------------------------------- Here we discuss some aspects of eigenvalue perturbation theory. It will provide an accessible explanation why one is interested in the results presented in Sections \[sec:schroedinger\] and \[sec:elliptic\] below in the context of random Schr[ö]{}dinger operators and elliptic differential operators, respectively. In fact, to illustrate the main questions it will be for the moment completely sufficient to restrict our attention to the finite dimensional situation, i.e. to perturbation theory for finite symmetric matrices. The focus will be on how (local) positive definiteness of the perturbation relates to lifting of eigenvalues. Let $A$ and $B$ be symmetric $n \times n$ matrices, with $B\geq b>0$ positive definite. The variational min-max principle for eigenvalues shows that for any $k \in \{1,\dots, n\}$ and $t\geq 0$ $$\label{eq:positive_definite_perturbation} \lambda_k(A+tB) \geq \lambda_k(A) + b \, t$$ where $\lambda_k(M)$ denotes the $k$th lowest eigenvalue, counting multiplicities, of a symmetric matrix $M$. Note that the dimension $n$ does not enter in the bound . Without the positive definiteness assumption on $B$ this universal bound will fail, most blatantly if $$A =\begin{pmatrix} A_1 & \ 0 \\ 0 & A_2 \end{pmatrix} \quad \text{and} \quad B =\begin{pmatrix} \operatorname{Id} & \ 0 \\ 0 & -\operatorname{Id} \end{pmatrix} .$$ In this case, all eigenvalue $\lambda_k(A+tB)$ *will move*, even with constant speed w.r.t. the variable $t$, albeit in different directions. If $B$ is singular, some eigenvalues may not move at all. However, for appropriate classes of symmetric matrices $A$, and of positive semidefinite matrices $B$, one may still aim to prove $$\label{eq:positive_semidefinite_perturbation} \forall \, t\geq 0, k \in \{1,\dots, n\} \, \exists \, \kappa >0 \text{ such that } \lambda_k(A+tB) \geq \lambda_k(A) + \kappa t$$ Note however, that $\kappa $ is now not a uniform bound but depends on the class of symmetric matrices from which $A$ is chosen, the class of semidefinite matrices from which $B$ is chosen, the range from which the coupling $t$ is chosen, and the range from which the index $k \in \{1,\dots, n\}$ is chosen. In the case of random operators or matrices one in is interested in the situation where $$\label{eq:multiparameter_family} A(\omega) =A_0+\sum_{j \in Q} \omega_j B_j =\Big(A_0+\sum_{j \in Q, j\neq 0} \omega_j B_j \Big)+ \omega_0 B_0$$ is a multi-parameter pencil. Here $Q$ is some subset of ${\mathbb{Z}}^d$ containing $0$. The real variables $\omega_j$ model random coupling constants determining the strength of the perturbation $B_j$ in each configuration $\omega=(\omega_j)_{j \in Q}$. Now, already suggest to write $A(\omega)$ as $$A(\omega_0^\perp)+tB \quad\text{where} \quad t=\omega_0,\ B=B_0,\ \text{and} \ \omega_0^\perp=(\omega_j)_{j \in Q, j\neq0} .$$ This highlights that if we consider $A(\omega)$ as a function of the single variable $t=\omega_0$, it is clearly a one-parameter family of operators, albeit the “unperturbed part” $A(\omega_0^\perp)$ of $A(\omega)=A(\omega_0^\perp)+tB$ is not a single operator, but varying over the ensemble $(A(\omega_0^\perp))_{\omega_0^\perp}$. To have a useful version of in this situation, the constant $\kappa $ needs to have a uniform lower bound $\inf_{A} \kappa $ where $A=A(\omega_0^\perp)$ varies over all matrices in the ensemble. In what follows we present rigorous results of the type , but where $A$ and $B$ are not finite matrices, but differential and multiplication operators. The relevant operators have all compact resolvent, ensuring that the entire spectrum consists of eigenvalues. Equidistribution property of Schrödinger eigenfunctions {#sec:schroedinger} ======================================================= The following result is taken from Ref. . It is an equidistribution estimate for Schr[ö]{}dinger eigenfunctions, which is uniform w.r.t. the naturally arising length scales, and has strong implications for the spectral theory of random Schrödinger operators. We fix some notation. For $L>0$ we denote by $\Lambda_L = (-L/2 , L/2)^d$ a cube in ${\mathbb{R}}^d$. For $\delta>0$ the open ball centered at $x\in {\mathbb{R}}$ with radius $\delta$ is denoted by $B(x, \delta)$. For a sequence of points $(x_j)_j$ indexed by $j \in {\mathbb{Z}}^d$ we denote the collection of balls $\cup_{j \in {\mathbb{Z}}^d} B(x_j , \delta) $ by $S$ and its intersection with $\Lambda_L$ by $S_L$. We will be dealing with certain subspaces of the standard second order Sobolev space $W^{2,2}({\Lambda}_L)$ on the cube. Let $\Delta$ be the $d$-dimensional Laplacian. Its restriction to the cube ${\Lambda}={\Lambda}_L$ needs boundary conditions to be self-adjoint. The domain of the Dirichlet Laplacian will be denoted by ${\mathcal{D}}(\Delta_{{\Lambda},0})$ and the domain of the Laplacian with periodic boundary conditions by ${\mathcal{D}}(\Delta_{{\Lambda},\mathrm{per}})$. Let $V \colon {\mathbb{R}}^d\to {\mathbb{R}}$ be a bounded measurable function, and $H_L = (-\Delta + V)_{\Lambda_L} $ a Schrödinger operator on the cube $\Lambda_L$ with Dirichlet or periodic boundary conditions. The corresponding domains are still ${\mathcal{D}}(\Delta_{{\Lambda},0})$ and $ {\mathcal{D}}(\Delta_{{\Lambda},\mathrm{per}})$, respectively. Note that we denote a multiplication operator by the same symbol as the corresponding function. The following theorem was proven in Ref. . \[thm:RojasVeselic\] Let $\delta, K_{V} > 0$. Then there exists $C_{\rm sfUC} \in (0,\infty)$ such that for all $L \in 2{\mathbb{N}}+1 $, all measurable $V : {\mathbb{R}}^d \to [-K_{V} , K_{V}]$, all real-valued $\psi \in {\mathcal{D}}(\Delta_{{\Lambda},0}) \cup {\mathcal{D}}(\Delta_{{\Lambda},\mathrm{per}})$ with $(-\Delta + V)\psi = 0$ almost everywhere on $\Lambda_L$, and all sequences $(x_j)_{j \in {\mathbb{Z}}^d} \subset {\mathbb{R}}^d$, such that for all $j \in {\mathbb{Z}}^d$ the ball $B(x_j , \delta) \subset \Lambda_1 + j$, we have $$\label{eq:observability} \int_{S_L} \psi^2 \geq C_{\rm sfUC} \int_{\Lambda_L} \psi^2 .$$ The value of the result is not in the *existence* of the constant $C_{\rm sfUC}$, but in the *quantitative control* of the dependence of $C_{\rm sfUC}$ on parameters entering the model. The very formulation of the theorem states that $C_{\rm sfUC}$ is independent of the position of the balls $B(x_j,\delta)$ within $\Lambda_1 +j$, and independent of the scale $L\in2\mathbb{N} +1$. From the estimates given in Section 2 of Ref.  one infers that $C_{\rm sfUC}$ depends on the potential $V$ only through the norm $\lVert V \rVert_\infty$ (on an exponential scale), and it depends on the small radius $\delta>0$ polynomially, i.e. $C\gtrsim \delta^N$, for some $N\in\mathbb{N}$ which depends on the dimension on $d$ and $\lVert V \rVert_\infty$. The theorem states a property of functions in the kernel of the operator. It is easily applied to eigenfunctions corresponding to other eigenvalues since $$H_L\psi=E\psi \Leftrightarrow (H_L-E)\psi=0 .$$ As a consequence of the energy shift the constant $K_{V}$ has to be replaced with $K_{V-E}$, which may be larger than $K_{V}$. It may always be estimated by $K_{V-E}\leq K_V+|E|$. There is a very natural question supported by earlier results, which was spelled out in Ref. , namely does the following generalisation of Theorem \[thm:RojasVeselic\] hold: Given $\delta >0$, $K\geq0$ and $E\in\mathbb{R}$ there is a constant $C>0$ such that for all measurable $ V\colon \mathbb{R}^d \rightarrow [-K,K] $, all $L \in 2{\mathbb{N}}+1$, and all sequences $(x_j)_{j\in\mathbb{Z}^d} \subset \mathbb{R}^d$ with $B(x_j,\delta) \subset\Lambda_1 +j$ for all $j \in {\mathbb{Z}}^d$ we have $$\label{eq:uncertainty} \chi_{(-\infty,E]} (H_L) \, W_L \, \chi_{(-\infty,E]} (H_L) \geq C~ \chi_{(-\infty,E]} (H_L) ,$$ where $W_L=\chi_{S_L}$ is the indicator function of $S_L$ and $\chi_{I} (H_L)$ denotes the spectral projector of $H_L$ onto the interval $I$. Here $C=C_{\delta, K, E}$ is determined by $\delta, K, E$ alone. Klein obtained a positive answer to the question for sufficiently short subintervals of $(-\infty,E]$. \[thm:Klein-13\] Let $d \in {\mathbb{N}}$, $E\in {\mathbb{R}}$, $\delta\in (0,1/2]$ and $V:{\mathbb{R}}^d \to {\mathbb{R}}$ be measurable and bounded. There is a constant $M_d>0$ such that if we set $$\gamma = \frac{1}{2} \delta^{M_d \bigl(1 + (2\lVert V \rVert_\infty + E)^{2/3}\bigr)} ,$$ then for all energy intervals $I\subset (-\infty, E]$ with length bounded by $2\gamma$, all $L \in 2{\mathbb{N}}+1$, $L\geq 72 \sqrt{d}$ and all sequences $(x_j)_{j\in\mathbb{Z}^d} \subset \mathbb{R}^d$ with $B(x_j,\delta) \subset\Lambda_1 +j$ for all $j \in {\mathbb{Z}}^d$ $$\chi_{I} (H_L) \, W_L \, \chi_{I} (H_L) \geq \gamma^2\chi_{I} (H_L) .$$ This does not answer the above posed question question completely due to the restriction $|I| \leq 2\gamma$. However, the result is sufficient for many questions in spectral theory of random Schrödinger operators. For a history of the questions discussed here and earlier results we refer to Ref. . Random Schr[ö]{}dinger operators {#ss:rSo} -------------------------------- Let ${{\Lambda}_L}$ be a cube of side $L\in2{\mathbb{N}}+1$, $(\Omega, {\mathbb{P}})$ a probability space, $V_0 \colon {{\Lambda}_L}\to {\mathbb{R}}$ a bounded, measurable deterministic potential, $V_\omega \colon {{\Lambda}_L}\to {\mathbb{R}}$ a bounded random potential and $H_{\omega,L}= (-\Delta + V_0+V_\omega)_{{\Lambda}_L}$ a random Schrödinger operator on $L^2({{\Lambda}_L})$ with Dirichlet or periodic boundary conditions. We assume that the random potential is of Delone-Anderson form $$V_\omega(x):= \sum_{j \in{{\mathbb{Z}}^d}} \ \omega_j u_j(x) .$$ The random variables $\omega_j, j\in {{\mathbb{Z}}^d},$ are independent with probability distributions $\mu_j$, such that for some $m>0$ an all $j\in {{\mathbb{Z}}^d}$ we have $\operatorname{\operatorname{supp}}\mu_j \subset [-m, m]$. Fix $0 < \delta_- < \delta_+<\infty$ and $0 < C_- \leq C_+ <\infty$. The sequence of measurable functions $u_j \colon {\mathbb{R}}^d \to {\mathbb{R}}$, $j \in {{\mathbb{Z}}^d}$, is such that $$\begin{aligned} \forall j \in {{\mathbb{Z}}^d}: \quad C_- \chi_{B(z_j,\delta_-)} \leq u_j \leq C_+ \chi_{B(z_j,\delta_+)}, \ \text{and} \ B(z_j,\delta_-) \subset {\Lambda}_1 + j . \end{aligned}$$ Lifting of eigenvalues {#ss:lifting} ---------------------- Let $\lambda_k^L(\omega)$ denote the eigenvalues of $H_{\omega,L}$ enumerated in non-decreasing order and counting multiplicities and $\psi_k=\psi_k^L(\omega)$ the normalised eigenvectors corresponding to $\lambda_k^L(\omega)$. While we suppress the dependence of $\psi_k$ on $L$ and $\omega$ in the notation, it should be kept in mind. Then $$\lambda_k^L(\omega) = {\langle}\psi_k, H_{\omega,L} \psi_k{\rangle}= \int_{{\Lambda}_L} \overline{\psi_k} ( H_{\omega,L} \psi_k ) .$$ Define the vector $ e=(e_j)_{j\in{{\mathbb{Z}}^d}}$ by $e_j=1$ for $j\in{{\mathbb{Z}}^d}$. Consider the monotone shift of $V_\omega$ $$V_{\omega+ {t} \cdot e} = \sum_{j \in{{\mathbb{Z}}^d}} (\omega_j+ {t} ) u_j$$ and set ${Q}={Q}_L= \Lambda_L \cap {{\mathbb{Z}}^d}$. By first order perturbation theory we have $$\frac{\rm d}{{\rm d}{\tau}} \lambda_k^L(\omega+ {\tau} \cdot e) |_{\tau=t} = \langle \psi_k, \sum_{k \in {Q}} u_j \, \psi_k {\rangle}.$$ Note that the right hand side depends on $t$ implicitly through the eigenfunction $\psi_k$. Let us fix some $E_0\in{\mathbb{R}}$ and restrict our attention only to those eigenvalues satisfying $\lambda_n^L(\omega) \leq E_0$. By Theorem \[thm:RojasVeselic\] there exists a constant $C_{\rm sfUC}$ depending on the energy $E_0$, $\delta_-$ and the overall supremum $$\label{eq:Vsupremum} \sup_{|s|\leq m} \ \sup_{|\omega_j|\leq m} \ \sup_{x\in{\mathbb{R}}^d} \big|V_{0}(x) +V_\omega(x) +s \sum_{j\in Q} u_j \big|$$ of the potential, such that $$\sum_{k \in {Q}} {\langle}\psi_k, u_j \, \psi_k {\rangle}\geq C_- \sum_{k \in {Q}}{\langle}\psi_k, \chi_{B(z_k,\delta_-)}\psi_k {\rangle}\geq C_-\cdot C_{\rm sfUC} =: \kappa .$$ Here we used that $\|\psi\|_{L^2 (\Lambda)}=1$. (Note that the quantity $\kappa$ depends a-priori on the model parameters.) Integrating the derivative gives $$\begin{aligned} \nonumber \lambda_k^L(\omega+ {t} \cdot e) &= \lambda_k^L(\omega) + \int_0^{t} \frac{{\mathrm{d}}\lambda_k^L(\omega+ \tau \cdot e) }{{\mathrm{d}}\tau}|_{\tau=s} \, {\mathrm{d}}s \\ & \geq \lambda_k^L(\omega) + \int_0^{t} \kappa \, {\mathrm{d}}s = \lambda_k^L(\omega) + t \kappa . \label{eq:lifting}\end{aligned}$$ This is the lifting estimate for eigenvalues of random (Schrödinger) operators alluded to in §\[ss:motivation\]. It should be compared with there. Indeed, due to the uniform nature of the estimate in Theorem \[thm:RojasVeselic\] we have $$\label{eq:uniform_kappa} \inf_{ L \in 2{\mathbb{N}}+1} \ \inf_{\omega \text{ s.t. } \forall \, k : |\omega_j|\leq m} \ \inf_{ |{t}|\leq m} \ \inf_{n \text{ s.t. } \lambda_n^L(\omega)\leq E_0} \kappa >0 .$$ Thus eigenvalues lifting estimate is almost as uniform as . A parameter, with respect to which the lifting estimate is *not* uniform is the cut-off energy $E_0$. Indeed, if we add in an infimum over $E_0>0$ on the left hand side, it becomes zero, unless $\sum_k\chi_{B(z_k,\delta_-)}\geq 1$ almost everywhere on ${\mathbb{R}}^d$. Wegner estimates ---------------- Here we present a Wegner estimate. Such estimates play an important role in the proof of localization via the multiscale analysis. The latter is an induction argument over increasing length scales. The Wegner bound is used to prove the induction step. Let $ s\colon [0,\infty) \to[0,1]$ be the global modulus of continuity of the family $\{\mu_j\}_{j\in {{\mathbb{Z}}^d}}$, that is, $$\label{definition-s-mu-epsilon} s(\epsilon):= \sup_{j \in {{\mathbb{Z}}^d}} \sup_{a \in {\mathbb{R}}} \, \mu_j\Big(\Big[a-\frac{\epsilon}{2},a+\frac{\epsilon}{2}\Big]\Big)$$ The main result of Ref.  on the model described in the last paragraph is a Wegner estimate which is valid for all compact energy intervals. \[t:Wegner\] Let $H_{\omega,L}$ be a random Schrödinger operator as in §\[ss:rSo\]. Then for each $E_0\in {\mathbb{R}}$ there exists a constant $C_W$, such that for all $E\le E_0$, $\epsilon \le 1/3$, and all $L\in 2{\mathbb{N}}+1$ we have $$\label{eq:WE} {\mathbb{E}}\{{{\mathop{\mathrm{Tr} \,}}}[ \chi_{[E-\epsilon,E+\epsilon]}(H_{\omega, L}) ]\} \le C_W \ s(\epsilon) \, \lvert \ln \, \epsilon \rvert^d \ \lvert \Lambda_L \rvert .$$ The Wegner constant $C_W$ depends only on $E_0$, $\|V_0\|_\infty$, $m$, $C_-$, $C_+$, $\delta_-$, and $\delta_+$. Klein[@Klein-13] obtains an improvement over this result based on his above quoted Theorem \[thm:Klein-13\]. There are many earlier, related Wegner estimates. For an overview we refer to Ref. . Comparison of local $L^2$-norms ------------------------------- An important step in the proof of Theorem \[thm:RojasVeselic\] is the following result which compares $L^2$-norms of the restrictions of a PDE-solution to two distinct subsets. In our applications the solution will be an eigenfunction of the Schrödinger operator. Various estimates of this type have been given in Refs. , and . We quote here the version from the last mentioned paper. \[thm:quantitative-UCP\] Let $K, R, \beta\in [0, \infty), \delta \in (0,1]$. There exists a constant $C_{\rm qUC}=C_{\rm qUC}(d,K, R,\delta, \beta) >0$ such that, for any $G\subset {\mathbb{R}}^d$ open, any $\Theta\subset G$ measurable, satisfying the geometric conditions $$\operatorname{diam} \Theta + \operatorname{dist} (0 , \Theta) \leq 2R \leq 2 \operatorname{dist} (0 , \Theta), \quad \delta < 4R, \quad B(0, 14R ) \subset G,$$ and any measurable $V\colon G \to [-K,K]$ and real-valued $\psi\in W^{2,2}(G)$ satisfying the differential inequality $$\label{eq:subsolution} \lvert \Delta \psi \rvert \leq \lvert V\psi \rvert \quad \text{a.e.on } G \quad \text{ as well as } \quad \int_{G} \lvert \psi \rvert^2 \leq \beta \int_{\Theta} \lvert \psi \rvert^2 ,$$ we have $$\label{eq:aim} \int_{B(0,\delta)} \lvert \psi \rvert^2 \geq C_{\rm qUC} \int_{\Theta} \lvert \psi\rvert^2 .$$ plot\[smooth cycle\] coordinates[(1,1) (1,6) (4,6) (8,7.5) (10,7) (9,2) ]{}; plot\[smooth cycle\] coordinates[(1,1) (1,6) (4,6) (8,7.5) (10,7) (9,2) ]{}; (5.5,3.5) circle (0.5cm); (5.5,3.5) circle (0.5cm); (5.5,3.5) circle (1pt); (4.4,3.7) node [$B(0,\delta)$]{}; (7,4) rectangle (8,5); (7,4) rectangle (8,5); (8.3,4.5) node [$\Theta$]{}; (5.51,3.51)–(7,4.1); (5.5,3.48)–(5.8,1.095); (6.35,4.1) node [$R$]{}; (5.3,2) node [$14R$]{}; (4,6.3) node [$G$]{}; Equidistribution property eigenfunctions of second order elliptic operators {#sec:elliptic} =========================================================================== Notation -------- Let ${\mathcal{L}}$ be the second order partial differential operator $${\mathcal{L}}u = -\sum_{i,j=1}^d \partial_i \left( a^{ij} \partial_j u \right)$$ acting on functions $u$ on ${\mathbb{R}}^d$. Here $\partial_i$ denotes the $i$th weak derivative. Moreover, we introduce the following assumption on the coefficient functions $a^{ij}$. \[ass:elliptic+\] Let $r,{\vartheta}_1 , {\vartheta}_2 > 0$. The operator ${\mathcal{L}}$ satisfies $A(r,{\vartheta}_1 , {\vartheta}_2)$, if and only if $a^{ij} = a^{ji}$ for all $i,j \in \{1,\ldots , d\}$ and for almost all $x,y \in B(0,r)$ and all $\xi \in {\mathbb{R}}^d$ we have $$\label{eq:elliptic} {\vartheta}_1^{-1} \lvert \xi \rvert^2 \leq \sum_{i,j=1}^d a^{ij} (x) \xi_i \xi_j \leq {\vartheta}_1 \lvert \xi \rvert^2 \quad\text{and}\quad \sum_{i,j=1}^d \lvert a^{ij} (x) - a^{ij} (y) \rvert \leq {\vartheta}_2 \lvert x-y \rvert .$$ A quantitative unique continuation principle -------------------------------------------- We first present an extension of the quantitative continuation principle, formulated for Schrödinger operators in Theorem \[thm:quantitative-UCP\], to elliptic operators with variable coefficients. \[thm:qUC-elliptic\] Let $R\in (0,\infty)$, $K_V, \beta \in [0,\infty)$ and $\delta \in (0, 4 R]$. There is an $\epsilon> 0$, such that if $ A(14R, 1+\epsilon, \epsilon)$ holds then there is a constant $C_{\rm qUC} > 0$, such that for any open $G\subset {\mathbb{R}}^d$ containing the origin and $\Theta \subset G$ measurable satisfying $$\operatorname{diam} \Theta + \operatorname{\operatorname{dist}}(0 , \Theta) \leq 2R \leq 2 \operatorname{\operatorname{dist}}(0 , \Theta) \quad \text{and} \quad B(0,14R) \subset G,$$ any measurable $V : G \to [-K_V , K_V]$ and real-valued $\psi \in W^{2,2} (G)$ satisfying the differential inequality $$\label{eq:psi} \lvert {\mathcal{L}}\psi \rvert \leq \lvert V\psi \rvert \quad \text{a.e.\ on $G$} \quad \text{as well as} \quad \frac{\lVert \psi \rVert_G^2}{\lVert \psi \rVert_\Theta^2} \leq \beta ,$$ we have $$\lVert \psi \rVert_{B(x,\delta)}^2 \geq C_{\rm qUC} \lVert \psi \rVert_{\Theta}^2 .$$ Scale-free unique continuation principle ---------------------------------------- We move on to discuss the equidistribution property or scale-free unique continuation principle for eigenfunctions. The aim is to formulate an analog of Theorem \[thm:RojasVeselic\] for variable coefficient elliptic operators. As presented below, for the moment we have solved only the situation where the second order term is sufficiently close to the Laplacian. As before, we denote by ${\Lambda}_L$ a box of side $L\in {\mathbb{N}}$. By $V$ we indicate a bounded measurable potential on ${\mathbb{R}}^d$ taking values in $[-K_V,K_V]$, where $K_V$ is a positive constant. We restrict the operator ${\mathcal{L}}$ on ${\Lambda}_L(0)$ and add either periodic or Dirichlet boundary conditions. In the former case we denote such an operator by ${\mathcal{L}}_{L,0}$, and its domain ${\mathcal{D}}({\mathcal{L}}_{L,0})$ is the subspace of $W^{2,2}({\Lambda}_L)$ consisting of functions vanishing on $\partial {\Lambda}_L$. The notation for the operator with periodic boundary condition is ${\mathcal{L}}_{L,\mathrm{per}}$ and its domains ${\mathcal{D}}({\mathcal{L}}_{L,\mathrm{per}})$ consists of the functions in $W^{2,2}({\Lambda}_L)$ satisfying periodic boundary conditions. \[ass:periodicCoefficients\] For each pair $i,j$ the function $a^{ij}\colon {\mathbb{R}}^d \to {\mathbb{R}}$ is ${\mathbb{Z}}^d$-periodic. Assume that in the case of operator ${\mathcal{L}}_{L,0}$ its coefficients $a^{ij}$, $i\not= j$ vanish on the sides of box ${\Lambda}_L$, while the coefficients $a^{ii}$ satisfy periodic boundary conditions on the sides of box ${\Lambda}_L$. In the case of operator ${\mathcal{L}}_{L,\mathrm{per}}$ suppose that all its coefficients satisfy periodic boundary conditions on the sides of box ${\Lambda}_L$. \[thm:equidistribution-elliptic\] Fix $K_V\in [0,+\infty)$, $\delta\in(0,1]$. Assume $A(\sqrt{d},1+\epsilon,\epsilon)$ with $\epsilon>0$ as in Theorem \[thm:qUC-elliptic\] . Assume \[ass:periodicCoefficients\]. Then there exists a constant $C_{sfUC}>0$ such that for any $L\in 2{\mathbb{N}}+1$, any sequence $$\label{d1.1} Z:=\{z_k\}_{k\in{\mathbb{Z}}^d} \ \text{ in }\ {\mathbb{R}}^d \quad \text{such that} \ B(z_k,\delta)\subset {\Lambda}_1(k) \text{ for each } k\in{\mathbb{Z}}^d,$$ any measurable $V: {\Lambda}_L\mapsto [-K_V,K_V]$ and any real-valued $\psi\in{\mathcal{D}}({\mathcal{L}}_{L,0})$, respectively $\psi\in {\mathcal{D}}({\mathcal{L}}_{L,\mathrm{per}})$ satisfying $$\label{d1.2} |{\mathcal{L}}\psi|\leqslant |V\psi|\quad \text{a.e.}\quad {\Lambda}_L$$ we have $$\label{d1.3} \int\limits_{S_L} |\psi(x)|^2 dx=\sum\limits_{k\in Q_L} \|\psi\|_{L_2(B(z_k,\delta))}^2\geqslant C_{sfUC} \|\psi\|_{L_2({\Lambda}_L)}^2,$$ where $S_L:=S\cap{\Lambda}_L=\cup_{k\in Q_L} B(z_k,\delta)$, $Q_L={\Lambda}_L\cap {\mathbb{Z}}^d$, and $S:=\cup_{k\in {\mathbb{Z}}^d} B(z_k,\delta)$. As a *Corollary* we obtain immediately an eigenvalue lifting estimate analogous to , where $\kappa$ is again uniform w.r.t. many parameters, as spelled out in subsection \[ss:lifting\] explicitly. The proof of Theorem \[thm:equidistribution-elliptic\] is based on the strategy implemented in Ref. . First one uses the conditions on the coefficients $a^{ij}$ described in Assumption \[ass:periodicCoefficients\] to extend $\psi$ as well as the differential expression ${\mathcal{L}}$ to the whole of ${\mathbb{R}}^d$ while keeping the $W^{2,2}$-regularity and the differential inequality originally satisfied by $\psi$. Then one uses the comparison Theorem \[thm:qUC-elliptic\] for local $L^2$-norms. Note that now the condition concerning the minimal distance to the boundary of $G$ plays no role, since $\psi$ has been extended to the whole of ${\mathbb{R}}^d$. From this point the combinatorial and geometric arguments of Ref  take over. In fact, one can prove a abstract meta-theorem: Once the comparison of local $L^2$-norms of $\psi$ holds up to the boundary, an equidistribution property for $\psi$ follows. Interestingly, such an argument no longer uses the fact that $\psi$ is a solution of an differential equation or inequality. Acknowledgments {#acknowledgments .unnumbered} =============== D.B. was partially supported by RFBR, the grant of the President of Russia for young scientists - doctors of science (MD-183.2014.1), and the fellowship of Dynasty foundation for young mathematicians. M.T. and I.V. have been partially supported by the DAAD and the Croatian Ministry of Science, Education and Sports through the PPP-grant “Scale-uniform controllability of partial differential equations”. M.T. and I.V. have been partially supported by the DFG. [1]{} A. Figotin and A. Klein, [*Commun. Math. Phys.*]{} [**180**]{}, 265 (1996). P. Stollmann, [*Isr. J. Math.*]{} [**107**]{}, 125 (1998). C. Rojas-Molina and I. Veseli[ć]{}, [*Commun. Math. Phys.*]{} [**320**]{}, 245 (2013). A. Klein, [*Commun. Math. Phys.*]{} [**323**]{}, 1229 (2013). F. Germinet and A. Klein, [*J. Eur. Math. Soc.*]{} [**15**]{}, 53 (2013). J. Bourgain and A. Klein, [*Invent. Math.*]{} [**194**]{}, 41 (2013). D. I. Borisov, M. Tautenhahn and I. Veseli[ć]{}, Equidistribution properties of eigenfunctions of divergence form operators, in preparation.

--- abstract: 'In this paper, we introduce Adaptive Cluster Lasso(ACL) method for variable selection in high dimensional sparse regression models with strongly correlated variables. To handle correlated variables, the concept of clustering or grouping variables and then pursuing model fitting is widely accepted. When the dimension is very high, finding an appropriate group structure is as difficult as the original problem. The ACL is a three-stage procedure where, at the first stage, we use the Lasso(or its adaptive or thresholded version) to do initial selection, then we also include those variables which are not selected by the Lasso but are strongly correlated with the variables selected by the Lasso. At the second stage we cluster the variables based on the reduced set of predictors and in the third stage we perform sparse estimation such as Lasso on cluster representatives or the group Lasso based on the structures generated by clustering procedure. We show that our procedure is consistent and efficient in finding true underlying population group structure(under assumption of irrepresentable and beta-min conditions). We also study the group selection consistency of our method and we support the theory using simulated and pseudo-real dataset examples.' address: Indian Statistical Institute author: - Niharika Gauraha - 'and Swapan K. Parui' bibliography: - 'niharika\_arXiv.bib' title: 'Efficient Clustering of Correlated Variables and Variable Selection in High-Dimensional Linear Models' --- High-Dimensional Data Analysis ,Correlated Variable Selection ,Adaptive Cluster Lasso ,Adaptive Cluster Representative Lasso ,Adaptive Cluster Group Lasso Introduction ============ We consider the usual linear regression model $$\begin{aligned} \label{eq:lr} \textbf{Y} &= \textbf{X} \beta^{0}+\epsilon,\end{aligned}$$ with response vector $Y_{n \times 1}$, design matrix $X_{ n \times p}$, true underlying coefficient vector $\beta^0_{p \times 1}$ and error vector $\epsilon_{n\times 1}$. When the number of predictors (p) is much larger than the number of observations (n), $p >> n$, the ordinary least squares estimator is not unique and may over fit the data. The parameter vector $\beta^{0}$ can only be estimated based on given very few observations under assumption of sparsity in $\beta^{0}$. To infer the true active set $S_0 = \{j; \beta^{0}_{j} \neq 0\}$, the Lasso([@Tibshirani]), its variants and other regularized regression methods are mostly used for sparse estimation and variable selection. However, variable selection in situations involving high empirical correlation remains one of the most important issues. This problem is encountered in many applications such as in microarray analysis, a group of genes which participate in the same biological pathway tend to have highly correlated expression levels(see [@Segal]) and it is often required to consider all of them if they are related to the underlying biological process. It has been proven that the design matrix must satisfy the following conditions for the Lasso to perform exact variable selection: irrepresentable(IR) condition([@Zhao]) and beta-min condition([@Buhlmann1]). Having highly correlated variables implies that the design matrix violates the IR condition. To deal with variable selection with correlated variables, mainly two approaches has been proposed: simultaneous clustering and model fitting and clustering followed by the sparse estimation (i.e. group lasso). The former approach imposes restrictive conditions on the design matrix. However, the time complexity for clustering of variables severely limits the dimension of data sets that can be processed by the later approach. Moreover, group selection in models with a larger number of groups is more difficult(see [@Wei]). To overcome these limitations we propose a three stage procedure, Adaptive Cluster Lasso method. Basically, we try to reduce the dimensions first using the Lasso(or its adaptive or thresholded version) before clustering of variables. At a high level our method works as follows. At the first stage, the Lasso is used to do initial selection of variables then we also select those variables which are not selected by the Lasso but they are strongly correlated with the variables selected by the Lasso. If the design matrix satisfies the beta-min condition, then after the first stage, the selected set of variables contains the true active set and the dimensionality of the problem is reduced by a huge amount. At the second stage, we perform clustering of variables on the reduced model, so that strongly correlated variables are grouped together in disjoint clusters. In the third stage, we do group-wise sparse estimation based on the structures generated by clustering procedure. The second and third stages of ACL together is the same as Cluster Group Lasso(CGL) or Cluster Representative Lasso(CRL) with ordinary hierarchical correlation based clustering, defined in [@Buhlmann2]. Hence, ACL method is an extension of the clustering lasso methods proposed in [@Buhlmann2]. Mainly, there are two lines of thought, the one is to find an appropriate and efficient clustering of correlated variables and the other line of thought is to avoid the false negatives. With these thoughts in mind, we develop a computationally efficient variable selection procedure $\hat{S}_{ACL}$, which identifies the appropriate correlated group structures and selects all variables from a group of correlated variables where at least one of them is active. Assuming Group Irrepresentable(GIR) and group beta-min condition on the design matrix, we prove that the $\hat{S}_{ACL}$ selects the true model, with much less computational complexity. We show that the dimensionality reduction and subsequent clustering and CGL(or CRL) improves over the plain clustering and CGL(or CRL). We illustrate the proposed method and compare it with the methods proposed in [@Buhlmann2] by extensive simulation studies and we also apply it to a pseudo-real dataset. The rest of this paper is organized as follows. In section 2, we provide notations, assumptions, review of relevant work and we discuss our contribution. In section 3, we review mathematical theory of the Lasso and group Lasso. In section 4, we describe the proposed algorithm which mostly selects more adequate models in terms of model interpretation and prediction performance. In section 5, we study theoretical properties of the proposed method. We also show that the variable selection is consistent for high dimensional sparse problems. In section 6, we provide numerical results based on simulated and pseudo real dataset. Section 7 contains the computational details and we shall provide conclusion in section 8. Background and Notations ======================== In this section, we state notations and assumptions, we define required concepts and we also provide review of the relevant work. Notations and Assumptions ------------------------- The following notations,assumptions and definitions are applied throughout this paper. We consider the usual linear regression set up with univariate response variable $Y \in R$ and p-dimensional predictors $X_i \in R^p$: $$\begin{aligned} \label{eq:lr2} Y_i& = \sum_{j=1}^{p} X_{i}^{(j)}\beta^{0}_{j} + \epsilon_i \quad i=1,...,n \; \;j=1,...,p\end{aligned}$$ or, in matrix notation (as in Equation \[eq:lr\]) $$\begin{aligned} \textbf{Y} &= \textbf{X} \beta^{0}+\epsilon\end{aligned}$$ where $\beta^{0} \in R^p$ are unknown true regression coefficients to be estimated, and the components of the noise vector $\epsilon \in R^n$ are i.i.d. $N(0, \sigma^2)$. The columns of the design matrix X are denoted by $X^{j}$. We assume that the design matrix **X** is fixed, the data is centred and the predictors are standardized, so that $\sum_{i=1}^{n} Y_i = 0$, $\sum_{i=1}^{n} X^{j}_{i} = 0$ and $\frac{1}{n} X^{j'}X^{j} = 1$ for all $j=1,...,p$.\ The $L_1$-norm is defined as: $$\begin{aligned} \label{eq:l1} \|\beta\|_1 = \textstyle \sum_{j=1}^p |\beta_j|\end{aligned}$$ $L_2$-norm squared is defined as: $$\begin{aligned} \label{eq:l2} \|\beta\|^{2}_2 = \textstyle \sum_{j=1}^p \beta^{2}_{j}\end{aligned}$$ The $L_\infty-$ norm is defined as: $$\begin{aligned} \label{eq:linf} \|\beta\|_{\infty} = \textstyle max_{1 \leq i \leq n |} |\beta_j|\end{aligned}$$ The true active set $S_0$ denotes the support of the subset selection solution($S_0 = supp(\beta_0)$) and defined as $$\begin{aligned} \label{eq:s0} S_0 &= \{j; \beta^{0}_{j} \neq 0\}\end{aligned}$$ The sign function is defined as: $$\begin{aligned} sign(x) = \left\lbrace \begin{array}{ll} -1 & \text{ if } x < 0 \\ 0 & \text{ if } x = 0\\ 1 & \text{ if } x > 0 \end{array} \right.\end{aligned}$$ The (scaled)Gram matrix(covariance matrix) is defined as $$\hat{\Sigma}= \frac{X'X}{n}$$ The $\beta_S$ has zeroes outside the set S,as $$\beta_S = \{ \beta_j I(j \in S) \}$$ and $\beta = \beta_S + \beta_{S^c}$.\ For the given $S \subset \{1,2,...,p \}$, the covariance matrix can be partitioned as: $$\begin{aligned} \label{sigmaPart} \Sigma = \left[ \begin{array}{cc} \Sigma_{11} = \Sigma(S) & \Sigma_{12}(S)\\ \Sigma_{21}(S)\quad \quad &\Sigma_{22} = \Sigma(S^c) \end{array} \right]\end{aligned}$$ Minimum eigenvalue of a matrix A is denotes as $\Lambda_{min}(A)$. Clustering of Variables {#sub:clust} ----------------------- We use correlation based, bottom-up agglomerative hierarchical clustering methods to cluster predictors, which forms groups of variables based on correlations between them. For further details on grouping of variables and determining the number of clusters, we refer to [@Buhlmann2]. The Lasso and the Group Lasso ----------------------------- The Least Absolute Shrinkage and Selection Operator (Lasso) was introduced by Tibshirani [@Tibshirani]. It is a penalized least squares method that imposes an L1-penalty on the regression coefficients, which does both shrinkage and automatic variable selection simultaneously due to the nature of the L1-penalty. We denote $\hat{\beta}$, as a Lasso estimated parameter vector. Assume $\lambda$ is the regularization parameter, then then Lasso estimator is computed as: $$\begin{aligned} \hat{\beta} \in \mathop{arg min}_{\beta \in \mathbb{R}^p} \{\frac{1}{n} \| \textbf{y} - \textbf{X} \beta \|_{2}^{2}+ \lambda \|\beta\|_1 \}\end{aligned}$$ and the estimated active set is denoted as $\hat{S}$ and defined as $$\begin{aligned} \label{eq:s1} \hat{S} = \{j; \hat{\beta}_{j} \neq 0\}\end{aligned}$$ The Lasso error vector is defined as $$\begin{aligned} \Delta = \hat{\beta} - \beta^0\end{aligned}$$ One of the major disadvantages of the the lasso is that, the Lasso tends to select single or a few variables, from a group of highly correlated variables. When the distinct groups or clusters among the variables are known a priory and it is desirable to select or drop the whole group instead of single variables. Then the Group Lasso (see [@Yuan]) or its variants are used, that imposes an $L_2$-penalty on the coefficients within each group to achieve such group sparsity. Here we define some more notations and state assumptions for the group Lasso. We may interchangeably use $\beta^{0}$ and $\beta$ for the true regression coefficient vector, the later one is without the superscript. Let us assume that the parameter vector $\beta$ is structured into groups, $G = \{ G_1, . . . , G_q \}$, where $q < n $, denotes the number of groups. The partition $G$ basically builds a partition of the index set $\{ 1,...,p\}$ with $ \cup_{r=1}^{q} G_r = \{1,... , p\}$ and $G_r \cap G_l = \emptyset, \quad r\neq l$. The parameter vector $\beta$, then has the structure $\beta = \{ \beta_{G_1}, ..., \beta_{G_q} \}$ where $\beta_{G_j} = \{\beta_r: r \in G_j \}$. The columns of the each group is represented by $X^{G_j}$. $$X = (X^{(1)}, ..., X^{(p)}) = (X^{(G_1)}, ... , X^{(G_q)})$$ The response vector Y can also be written as $$Y = \sum_{j=1}^{q} X^{(G_j)} \beta_{G_j} + \epsilon$$ where $X^{(G_j)} \beta_{G_j} = \sum_{k=1}^{m_k} X^{(G_j)}_k (\beta_{G_j})_k $. The loss function of the group Lasso is same as the loss function of the Lasso $\frac{1}{n}\|Y-X\beta \|_{2}^{2}$. The group Lasso penalty is defined as $$\|\beta\|_{2,1} = \textstyle \sum_{j=1}^q \| X^{G_j} \beta_{G_j} \|_2 \sqrt{\frac{m_j}{n}}$$ where $m_j = |G_j|$ is the group size. Since the penalty is invariant under parametrizations within-group. Therefore, without loss of generality, we can assume $\Sigma_{rr} = I$, the $m_r \times m_r$ identity matrix. Hence the group Lasso penalty can be written as $$\|\beta\|_{2,1} = \sum_{j=1}^q \sqrt{m_j} \|\beta_{G_j} \|_2$$ The Group Lasso estimator(with known q groups) is defined as $$\begin{aligned} \label{eq:grpLasso} \hat{\beta}_{grp}\in \mathop{argmin}_{\beta} \{ \frac{1}{n}\|Y-X\beta \|_{2}^{2} + \lambda \|\beta\|_{2,1} \}\end{aligned}$$ The group Lasso has the following properties: - The group Lasso behaves like the lasso at the group level, depending on the value of the regularization parameter $\lambda$, the whole group of variables may drop out of the model. - For singleton groups (when the group sizes are all one), it reduces exactly to the lasso. - The group Lasso penalty is invariant under orthonormal transformation within the groups. - The group Lasso estimator has similar oracle inequalities as the standard Lasso for prediction accuracy and estimation error. It has group wise variable selection property(We discuss mathematical theory in the section \[secLassoTheory\]). Let W denote the actives group set , $W \subset \{ 1,...,q \}$, with cardinality $w = |W|$. Throughout the article, the following assumption are made for the group Lasso: - The size of the each group is less than the number of observations. $$m_{max} < n$$. - The number of active groups, w, is less than the number of observations (sparsity assumption). ### Cluster Group Lasso When the group structure is not known then clusters $G_1, . . . , G_q$ are generated from the design matrix X( using correlation based method etc.). Then the group Lasso is applied to the resulting clusters. We denote the clusters selected by the group Lasso as $\hat{S}_{clust}$, and is defined as $$\begin{aligned} \hat{S}_{clust} = \{r: \text{ cluster }G_r \text{ is selected, } r = 1,...,q\}\end{aligned}$$ The union of the selected clusters gives the selected set of variables. $$\begin{aligned} \hat{S}_{CGL} = \cup_{r \in \hat{S}_{clust}} G_r\end{aligned}$$ ### Cluster Representative Lasso Similar to the CGL, the cluster representative Lasso, first identifies groups among the variables and then applies the lasso for cluster representatives (see [@Buhlmann2]). When sign of the regression coefficients within a group is the same then taking group representatives is advantageous, whereas when near cancellation among $\beta^{0}_{j}(j \in G_r)$ takes place then CGL is preferred.\ We define representative for each cluster as $$\begin{aligned} \bar{X}^{(r)} = \frac{1}{|G_r|} \sum_{j \in G_r} X^{(j)}, \quad r = 1,...,q.\end{aligned}$$ The design matrix of cluster representatives is denoted as $\bar{X}_{n \times q}$. Then optimization problem for CRL is defined using response $Y$ and the design matrix of cluster representatives $\bar{X}$ as: $$\begin{aligned} \hat{\beta}_{CRL} \in \arg\!\min_{\beta} ( \|{ \textbf{y} - \bar{\textbf{X}} \beta} \|^{2}_{2}+ \lambda_{CRL} \|\beta \|_1 )\end{aligned}$$ The selected clusters are then denoted as: $$\begin{aligned} \hat{S}_{clust,CRL} = \{r; \hat{\beta}_{CRL,r} \neq 0, r = 1, . . . , q \}\end{aligned}$$ and the selected variables are obtained as the union of the selected clusters as: $$\begin{aligned} \hat{S}_{CRL} = \cup_{r \in \hat{S}_{clust,CRL}} G_r\end{aligned}$$ Review of Relevant work and our Contribution -------------------------------------------- Here, we provide a brief review of relevant work in this area, and we also show that how our proposal differs or extends the previous studies. The Lasso can not do variable selection in the situations where predictors are highly correlated. As mentioned before, to handle correlated covariates in variable selection methods, two algorithmic approaches have been developed in the past: clustering of variables and model fitting either simultaneously or at two different stages. Examples of the methods that do clustering and model fitting simultaneously are Elastic Net([@Hui]), Fused LASSO([@Fused]), octagonal shrinkage and clustering algorithm for regression(OSCAR, [@oscar]) and Mnet([@Mnet] ) etc. The Elastic Net uses a combination of the $L_1$ and $L_2$ penalties, OSCAR uses a combination of $L_1$ norm and and $L_{\infty}$ norm and Mnet uses a combination of $L_2$ and Minimum Concave Penalty(MCP). We note that these methods use only combination of penalties, they do not use any specific information on the correlation pattern among the predictors and hence they do not reveal any group structure in the data. Now, we discuss a few methods that perform clustering and model fitting at different stages, i.e. Cluster Group Lasso(CGL, [@Buhlmann2]), Cluster representative Lasso(CRL,[@Buhlmann2]), Stability Feature Selection using Cluster Representative LASSO (SCRL, [@Niharika]) and sparse Laplacian shrinkage estimator(SLS, [@SLS]). CRL, CGL and SCRL use correlation based and canonical correlation methods to perform hierarchical clustering. SLS also considers the correlation patterns among predictors but requires that highly correlated variables should have similar predictive effects.The main disadvantage of this approach is mainly due to clustering in the presence of unstructured data or noise features. It is difficult to determine the exact group structures or the exact number of groups in high-dimensions and in the presence of noise features. Moreover, the CPU time taken by clustering algorithms is unacceptable when the number of predictors are huge. To address these problems, we propose to reduce the dimensionality before performing clustering which makes our proposal different from previous work. Basically, our work can be viewed as an extension of the two stage procedure, Cluster Lasso Methods with correlation based clustering, proposed in [@Buhlmann2]. The extension is that we add a dimensionality reduction stage prior to performing the clustering, which leads to clustering of variables more accurately and efficiently and thus consistent group variable selection. In particular, we consider Adaptive Clustering Group Lasso(ACGL), where the Lasso is used as preprocessing step at the first stage, correlation based clustering at second stage and the group Lasso in the third stage(defined in section 4). We also consider the CGL method with ordinary hierarchical clustering, denoted by CGLcor, see [@Buhlmann2]. We compare ACGL and CGLcor in terms of predictive performance, variable selection and CPU time expended, in section 5. Our extensive simulation studies show that ACGL outperforms the CGLcor. Mathematical Theory of the Lasso and the Group Lasso {#secLassoTheory} ==================================================== In this section, we review the results required for proving consistent variable selection( and group variable selection) in high dimensional linear models. For more details on the mathematical theory for the lasso and group lasso, we refer to: [@Sara2], [@Zhao], [@Wei] [@Buhlmann2] and [@Buhlmann1]. The Lasso compatibility condition holds for a fixed set $S \subset \{1,...,p\}$ with cardinality $s = |S|$, a constant $\phi_{comp}(S) > 0$ and if for all $ \| \Delta_{S^c}\|_1 \leq 3 \| \Delta_{S}\|_1 \neq 0$ the following holds $$\begin{aligned} \| \Delta_{S}\|_{1}^{2} \leq \left\lbrace \frac{s\frac{1}{n} \| X \Delta \|_{2}^{2} }{ \phi^{2}(S)} \right\rbrace\end{aligned}$$ where $\phi_{comp}(S) $ is called the compatibility constant. The constant 3 is due to the condition $\lambda \geq 2\lambda_0$, which is required to overrule the stochastic process part,(see [@Buhlmann1] for details). Without loss of generality we can assume $S = \{1,...,s \}$ and partition the covariance matrix in block-wise form as given in equation \[sigmaPart\]. Assuming $\Sigma^{-1}_{11}$ is invertible, the various form of Irrepresentable(IR) Conditions are defined as follows. The strong irrepresentable condition is said to be met for the set S, with cardinality $s = |S|$, if the following holds: $$\begin{aligned} \label{IR1} \|\Sigma_{12}(S) \Sigma^{-1}(S) \tau_S \|_{\infty} < 1, \quad \forall \tau_S \in \mathbb{R}^s \; such \; that \; \| \tau_S \|_{\infty} \leq 1\end{aligned}$$ The weak irrepresentable condition holds for a fixed $\tau_S \in \mathbb{R}^s$ if $$\begin{aligned} \label{IR2} \|\Sigma_{12}(S) \Sigma^{-1}(S) \tau_S \|_{\infty} \leq 1\end{aligned}$$ For some $0<\theta <1$, the $\theta$ uniform irrepresentable condition holds if $$\begin{aligned} \label{IR3} \mathop{max}_{\| \tau_S \|_{\infty} \leq 1} \|\Sigma_{12}(S) \Sigma^{-1}(S) \tau_S \|_{\infty} \leq \theta\end{aligned}$$ Sufficient conditions(eigenvalue and mutual incoherence) on design matrix to hold IR are discussed in [@Zhao] and [@Hastie]. the Beta Min Condition is met for the regression coefficient $\beta^0$, if $\min |\beta^0| \geq \frac{4 \lambda s_0}{\phi^{2}(S)}$ Under the following assumptions the Lasso selects the true active set $S_0$ with high probability: - Irrepresentable Condition holds for $S_0$. - beta-min condition holds for $\beta^0$. The following inequality shows the bounds for prediction error and estimation error of the Lasso estimator.( for derivation and proof we refer to [@Buhlmann1]). $$\begin{aligned} \frac{1}{n}\| X \Delta \|_{2}^{2} + \lambda \|\Delta \|_1 & \leq \frac{4\lambda^2 s}{ \phi^{2}_{comp}} \end{aligned}$$ Our error analysis for the group Lasso is based on the pure active group and pure noise group assumptions, that is,\ $(A5):$ all variables are active variables within an active group and no variables are active in a noise group.\ We define the group Lasso error as $\Delta_{G_r} = \beta_{G_r} - \beta^{0}_{G_r} $, and also assume the following.\ $(A6)$: We assume that clustering process identifies the group structure correctly. The group Lasso compatibility condition holds for a fixed set $W \subset \{1,...,q\}$ with cardinality $w = |W|$, a constant $\phi_{grp}(W) > 0$ and if for all $ \sum_{r \in W^c}\| \Delta_{G_r} \|_{2} \leq 3 \sum_{r \in W}\| \Delta_{G_r} \|_{2} \neq 0$ the following holds $$\begin{aligned} (\sum_{r \in W}\| \Delta_{G_r} \|_{2})^2 \leq \left\lbrace \frac{ w\frac{1}{n} \| X \Delta \|_{2}^{2} }{\phi_{grp}^{2}(W)} \right\rbrace\end{aligned}$$ where $\phi_{grp}(W) $ is called the group Lasso compatibility constant. The Lasso compatibility condition implies the group Lasso compatibility condition, it is explained by the following Lemma(See Lemma 8.2 of the book [@Buhlmann1], for the proof). Let $W \subset \{ 1,...,q \}$ be a group index set, say, $W = \{ 1, ..., w\}$ Consider the full index set corresponding to W: $$S = \{ (1,1), ..., (1,m1),..., (w,1),...,(w,m_w) \} = \{ 1,...,s \}$$, where (i,j) denotes jth member of ith group and $s = \sum^{w}_{j=1} m_j$. If compatible condition holds for S with compatiblility constant $\phi(S)$ then the compatibility condition holds for the $\phi_{grp}(W)$, and $\phi_{grp}(W) \geq \phi(S)$ The group IR condition is met for the set W with a constant $0< \theta < 1$, if for all $\tau \in \mathbb{R}^s$ with $\|\tau \|_{2, \infty} = \mathop{max}_{1\leq r \leq q} \| \tau_{G_r} \|_2 \leq 1 $, the following holds $$\begin{aligned} \frac{1}{m_r} \|(\Sigma_{21} \Sigma^{-1}_{11} K \tau)_{G_r} \| \quad\forall r \not\in W,\end{aligned}$$ where $K = diag(m_1 I_{m_1} , ..., m_w I_{m_w})$ We note that the GIR definition reduces to the Lasso IR condition for singleton groups(see [@Basu]). The group beta-min Condition is met for $\beta^0$ , if $\|\beta^{G_r}\|_{\infty} \geq \frac{D \lambda \sqrt{m_r}}{n} \quad \forall r \in W $, where $D>0$ is a constant which depends on $\sigma, \phi_{grp}$ and other constants used in cone constraints and GIR conditions. We note that, only one component of the $\beta^{G_r}, \forall r \in W$ has to be sufficiently large, because we aim to select groups as a whole, and not individual variables. For its exact form, we refer to [@Florentina]. Under the following assumptions the group Lasso selects the true active groups $W$ with high probability: - GIR Condition holds for $W$. - Group beta-min condition holds for $\beta^{G_r}, \forall r \in W$. Next, we discuss sufficient condition for the GIR to hold. We denote $\Sigma_{r,l} = X^{G_r^{'}} X^{G_l}/n$, $r,l \in \{ 1,...,q\}$. We partition the covariance matrix group wise. (here we assume that each $\Sigma_{r,r}$ is non-singular, or we may use the pseudo inverse) $$\begin{aligned} R_W = \left[ \begin{array}{cccc} I & \Sigma^{-1/2}_{11}\Sigma_{12}\Sigma^{-1/2}_{22} & ... & \Sigma^{-1/2}_{11}\Sigma_{1w}\Sigma^{-1/2}_{ww} \\ \Sigma^{-1/2}_{22}\Sigma_{21}\Sigma^{-1/2}_{11} & I & ...& \Sigma^{-1/2}_{22}\Sigma_{2w}\Sigma^{-1/2}_{ww} \\ \vdots & \vdots & \ddots & \vdots\\ \Sigma^{-1/2}_{ww}\Sigma_{w1}\Sigma^{-1/2}_{11} & \Sigma^{-1/2}_{ww}\Sigma_{w2}\Sigma^{-1/2}_{22} &... & I \end{array} \right]\end{aligned}$$ We note that diagonal elements are $I_{m_r \times m_r}$ identity matrix due to parameterization invariance properties. Now Suppose that $R_W$, has smallest eigenvalue $\Lambda_{min}(R_{W}) > 0$ and that canonical correlations between groups are small enough that intern implies the incoherence assumptions. Therefore, under the eigenvalue and incoherence condition of $R_W$, the group irrepresentable condition holds(see [@Buhlmann2]). Now we prove that the Lasso IR condition implies the group Lasso IR(GIR) condition. Let $W \subset \{ 1,...,q \}$ be a group index set, say, $W = \{ 1, ..., w\}$ Consider the full index set corresponding to W: $$S = \{ (1,1), ..., (1,m1),..., (w,1),...,(w,m_w) \} = \{ 1,...,s \}$$ where $s = \sum^{w}_{j=1} m_j$. If the Lasso IR condition holds for the set S then the group Lasso IR condition holds for the set W. Proof is trivial, the IR condition on the set S implies that $ \Sigma_{11}$ is invertible , $\Lambda_{min}(\Sigma_{11})>0$, and correlation between variables in S and between variables in $S$ and $S^c$ are small enough. That implies small enough canonical correlations within the groups in active groups W, and between the groups in $W$ and $W^c$. The small enough canonical correlations between groups ensure the incoherence assumptions and therefore the GIR condition holds. The following inequality shows the similar bounds for prediction error and estimation error of the group Lasso estimator([@Buhlmann2]). $$\begin{aligned} \frac{1}{n}\| X \Delta \|_{2}^{2} + \lambda \sum_{r =1}^{q}\| \Delta_{G_r} \|_{2} & \leq \frac{24\lambda^2 \sum_{r \in W} m_r}{ \phi^{2}_{grp}(W)} \end{aligned}$$ The Adaptive Cluster Lasso Methods ================================== It is known that the Lasso tends to select one or few variables from the group of highly correlated variables, even though many or all of them belong to the active set. We aim to avoid false negatives and solve clustering problem efficiently and more accurately. To solve clustering problem efficiently, we propose a preprocessing step to reduce the dimensionality using the Lasso methods before clustering of variables. To avoid the false negative, we use the concept of clustering the correlated variables and then selecting or dropping the whole group instead of single variables same as the CGLcor(CRLcor) proposed in [@Buhlmann2]. The proposed procedure, ACL is a 3-stage procedure, where we can choose to use different methods at different stages depending on the nature of the problem. The different stages of the ACL procedure is explained as follows. 1. Dimensionality Reduction\ For selecting initial set of variables, we use the Lasso(or its adaptive or thresholded version). Since the Lasso tends to select one or a few variables from the group of strongly correlated variables, therefore we use the Lasso to select the group representative predictors. After we have selected the initial set of variables(group representative members), we get the rest of the group members by simple correlated screening. In section 3, we have shown that for highly correlated structures, the variables set selected by this approach always contains the true active set under assumption of GIR and GBM on the design matrix. Let the variables set selected by Lasso is given by $$\begin{aligned} \hat{S}_{Lasso} &= \{j; \hat{\beta}_{Lasso,j}(\lambda_1) \neq 0\} .\end{aligned}$$ Then we select correlated variables as $$\begin{aligned} \hat{S}_{corr} &= \{k; \quad k \in \{1,...,p \}\setminus \hat{S}_{Lasso}, j \in \hat{S}_{Lasso} \text{ and }corr(X_j, X_k) > \rho\},\end{aligned}$$ where $\lambda_1$ is the tuning parameter used by Lasso and $\rho > 0.7$ denotes the strong correlation between two variables. Then the selected set of variable are given by $$\begin{aligned} \hat{S}_{1} &= \hat{S}_{Lasso} \cup \hat{S}_{corr}\end{aligned}$$ 2. Clustering of Variables\ After first stage there may be huge amount of reduction in the dimensionality, we denote the reduced design matrix as **$X_{red} = \{X_j; j \in \hat{S}_{1}\}$**. On the reduced set of predictors, we apply correlation based clustering methods to group strongly correlated variables into disjoint groups. We denote the inferred clusters as $G_1, . . . , G_q $. 3. Supervised Selection of Clusters\ From the reduced design matrix **$X_{red}$**, and inferred clusters $G_1, . . . , G_{q}$ as described in previous stages, we select the variables in a group-wise fashion which involves selecting or dropping the group as a whole. Various methods have been proposed to achieve grouping effect in case of highly correlated variables, i. e. the group Lasso([@Yuan]), Group Square-Root Lasso([@Florentina]), Adaptive group Lasso([@Wei]) and Lasso on cluster representatives etc. Suppose, the selected set of groups are denoted by $$\hat{S}_{G} = \{r: \text{ group } G_r \text{ is selected}\}$$ The final selected set of variables is then the union of the selected groups. $$\hat{S}_{ACL} = \cup_{r} r \in \hat{S}_{G}$$ **Input:** dataset $(Y,X)$\ **Output:** $\hat{S}$:= set of selected variables\ **Steps:** Perform Lasso on data $(Y,X)$, Denote $\hat{S}_{Lasso}$ as variable set selected\ $S_1$ := $\hat{S}_{Lasso}$\ Let $X_{red} = X^{S_1}$ be the reduced design matrix\ Perform Clustering of variables on data $X_{red}$,\ Denote clusters as $G_1, ..., G_q$ and partition variable set as\ $\hat{S_{G_1}}$, ..., $\hat{S_{G_q}}$\ Perform group Lasso on $(Y,X_{red})$ with group information $G_1, ..., G_q$, denote the selected set of groups as\ $\hat{S}_{cluster} = \{r; \text{ cluster } G_r \text{is selected, } r = 1, . . . , q\}.$\ The union of the selected groups is then the selected set of variables\ $\hat{S}_{ACL} = \cup_{r} r \in \hat{S}_{cluster} $\ **return** $\hat{S}$ Complexity Analysis of the ACL Method ------------------------------------- In this section, we compute time complexity of the ACL method at different stages. 1. First stage\ The time complexity of the first stage consists of the time complexity of the Lasso plus time required for variable screening. suppose $\hat{s} = |\hat{S}_{Lasso}|$, denotes the number of variables selected by Lasso, then time taken by variable screening is $\hat{s}*(p-\hat{s})$, which is $O(p\hat{s})$.\ 2. Second stage\ Computationally, The second step can be completely avoided. Clustering can be done while screening correlated variables at the first stage itself. However, We opted to state the clustering method separately for transparency and for deriving its theoretical properties, in particular comparing it with other methods where clustering is performed at different stages i.e. CGLcor. But while implementing the algorithm, we can efficiently combine variable screening and clustering. So no extra computational cost is added at this stage.\ 3. Third stage\ The same computational complexity as for the group Lasso, which depends on the number of groups and size of each group. Hence the overall time complexity of the proposed method is dominated by the time required for the Lasso and the group Lasso, see [@Julien] for complexity analysis of the Lasso. Theoretical Properties of the ACL Procedure =========================================== In this section, we study the theoretical properties of the ACL methods, and we show that nothing is lost by using ACL methods instead of Clustering Lasso methods with correlation based clustering as proposed in [@Buhlmann2]. Particularly, under the GIR and group beta-min condition, the ACGL method has the same accuracy as the CGLcor, in terms of estimation, prediction and variable selection. The gain is in terms of computations since the ACGL performs clustering on the reduced set of predictors. We introduce the following theorems which are needed for proving variable selection consistency for the proposed algorithm. Suppose that, the uniform-$\theta$ IR condition is met for the true active set $S := S_0$, which in turn implies that with large probability, the Lasso does not make false positive selection of variables. Then for any $S_1 \subset S $, the uniform-$\theta_1( \leq \theta)$ IR condition holds for the set $S_1$. **Proof** We invoke the result given in the book [@Buhlmann1], Corollary 7.2. Since the uniform-$\theta$ IR condition holds for the set S, then the following inequality also holds. $$\frac{\sqrt{s} \mathop{max}_{j \not\in S} \sqrt{\sum_{k \in S} \sigma^{2}_{jk}}} {\Lambda^{2}_{min}(\Sigma_{11}(S)) } \leq \theta$$ where $\sigma_{jk}$ denotes the $(jk)^{th}$ entry of $\Sigma$.\ Now we delete some variables from S, and denote the reduced subset as $S_1$ and corresponding partition of variance-covariance matrix is $\Sigma_{11}(S_1)$ and $\Sigma_{21}(S_1)$. Since $S1 \subset S$, the following inequality holds because any symmetric minor of $\Sigma_{11}(S)$ will have min-eigenvalues at least as big as $\Lambda_{min}(\Sigma_{11}(S))$. $$\Lambda_{min}(\Sigma_{11}(S_1)) \geq \Lambda_{min}(\Sigma_{11}(S))$$ Therefore $$\theta_1 = \frac{\sqrt{s_1} \; max_{j \not\in S_1} \sqrt{\sum_{k \in S_1} \sigma^{2}_{jk}}} {\Lambda^{2}_{min}(\Sigma_{11}(S_1)) } \leq \frac{\sqrt{s} \; max_{j \not\in S} \sqrt{\sum_{k \in S} \sigma^{2}_{jk}}} {\Lambda^{2}_{min}(\Sigma_{11}(S)) } \leq \theta$$ Hence the IR condition holds for the set $S_1 \subset S$. The similar result holds for the group Lasso which is given in the following lemma. Suppose that, the uniform-$\theta$ IR condition is met for the set of true active groups $W \subset \{1,...,q \}$, It implies that with large probability, the Lasso does not make false positive selection of groups. Then the uniform-$\theta_1( \leq \theta)$ GIR condition holds for the following cases: - for any $W_1 \subset W $, when the number of groups are reduced. - when $|W_1| = |W| $ but $\{G_r, r \in W_1\} \subset \{ G_r, r \in W\} $, group sizes are reduced for some groups. - when group size as well as number of groups are reduced, $W_1 \subset W $ and $\{G_r: r \in W_1\} \subset \{ G_r: r \in W \} $. Proof is trivial. Suppose that IR condition holds for the active set $S$. If w disjoint groups are formed within S such that $ S= \sum_{j \in W} m_j$, then IR for the Lasso imply IR condition for the group Lasso. Since the Reduced-set IR will hold for $S_1 \subset S$, where the reduced set can be interpreted as change in group structure in terms of reduced number of groups and/or reduced size of groups, as $$\begin{aligned} S_1 &= \sum_{r \in W_1} m_r, \quad W_1 \subset W % \\ %S_1 &= \sum_{r \in W} m_{r}, \quad |m_{r_1}| \leq |m_r| \\ %S_1 &= \sum_{r \in W_1} m_{r_1}, \quad W_1 \subset W , \quad |m_{r_1}| \leq |m_r| \end{aligned}$$ Therefore Reduced-group IR will hold for the set $S_1$. Case Studies ------------ In this section, we illustrate the variable selection consistency of the proposed method using a couple of scenarios under assumption of GIR, group beta-min and no noise case. ### Orthonormal Case The case $ \Sigma \approx I $ corresponds to uncorrelated variables and hence IR condition holds for any $S \subset \{ 1,...,p\}$ and we also assume that beta-min condition holds for a fixed S. We claim that ACGL selects the true active set $\hat{S}_{ACGL} = S_0$ with high probability.\ Proof:\ First, we perform the Lasso operation on the pair (Y,X), to get the initial set of variables, say $\hat{S}_1$. Then with high probability $\hat{S_1} = S_0$ under assumption of IR and beta-min condition. Since variables are uncorrelated no additional variable will be pulled in when we do correlation screening. At the second stage, we perform clustering on the reduced set of predictors. Clustering process will report the singleton groups due to independence structure, and finally at the third step CGL will select the true active set $S_0$ again, due to reduced-set IR condition. We proved that ACGL consistently selects the true active set for the orthonormal case. It is obvious that, there is no advantage of using AGCL or the plain CGL/CRL methods over the standard Lasso for this case. But ACL outperforms over CGL/CRL in terms of computations, since AGCL considers the reduced set of predictors for clustering and the group lasso is called for reduced number of groups, whereas plain CGL/CRL considers all p variables for clustering which requires huge computations when the dimension is ultra high. ### Block Diagonal Case The case $ \Sigma \approx diag(\mathcal{T}_1, \mathcal{T}_2, ..., \mathcal{T}_q) $ corresponds to uncorrelated groups and hence group IR condition holds for any $W \subset \{ 1,...,q\}$ and we also assume that group beta-min condition holds. Each $\mathcal{T}_i$ is a ${m_i \times m_i}$ matrix with elements as $$(\mathcal{T}_i){j,k} = \left\lbrace \begin{array}{cc} 1, & j=k\\ \rho, & else \end{array} \right.$$ where $\rho > 0.7$, since variables are highly correlated within each group. Without loss of generality we can assume that all the variables are ordered in a way such that all active groups come first. $W \subset \{1,...,q\} $ then $W = \{1,...,w \}. $ and we also assume pure active or pure noise group. Let $$\begin{aligned} S_0 &= \{ (1,1),..., (1,m_1), ..., (w,1),..., (w,m_w) \} = \{1,...,s_0 \}, %W^{c} &= \{1,...,p \}/W, \text{ be the noise group }\end{aligned}$$ be the true active set. The Lasso tends to select(depending on the amount of regularization) one variable from each active block. Without loss of generality we assume that the Lasso selects (j,1) variable from each $j \in W$, and the selected variable set is $\hat{S}_{Lasso} = \{ (1,1),(2,1),...,(w,1) \}$, Now we add all variables from $\{1,...,p \}/S_{Lasso}$ which are strongly correlated with atleast of the the variable from $S_{Lasso}$. Therefore we get $$\begin{aligned} \hat{S}_1 &= \{1,...,s \}\\ \implies & \hat{S}_1 = S_0 \end{aligned}$$ Hence, after the first stage of dimensionality reduction, the selected set of variables contains the true active variables. Assuming that the clustering procedure correctly identifies the true underlying group structure, then the group Lasso at the third stage correctly selects all the w groups, due to the sub-group IR condition for the group Lasso. Hence, the proposed method consistently selects the true active set under the assumption of GIR and group beta-min condition for the block diagonal case as well. One may argue that, there is no need for the second and the third stage. Specifically, when the Lasso selects one variable per active group then correlation screening will bring in those correlated variables which were not selected by the Lasso. We refer to the discussion in [@Junzhou], on using the group Lasso over Lasso. Numerical Results ================= In this section, we consider three simulation settings and a pseudo real data example in order to empirically compare the performance of the proposed method with the other existing methods. Since the comparison between the Lasso, CGL and CRL have already been studied in the paper [@Buhlmann2], here we only report the results for ACGL and CGLcor methods. Simulation Study ---------------- Three examples are considered in this simulation. In each example, data is simulated from the linear model in (equation 1) with fixed design **X**. All the three examples are the same as simulation examples used in the paper [@Buhlmann2]. For each example, our simulated dataset consisted of a training and an independent validation set and 50 such datasets were simulated for each example. The models were fitted on the training data for each 50 datasets and the model with the lowest test set Mean Squared Error(MSPE) was selected as the final model. For model interpretation, we consider true positive rate as a measure of performance. We also measure the CPU time expended by each methods. The MSE and the true positive rate are defined as follows. $$\begin{aligned} MSE &= \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2\\ TPR &= f(|\hat{S}|) = \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}\end{aligned}$$ ### Block Diagonal Model We generate covariates from $N_p(0,\Sigma_1)$, where $\Sigma_1$ consists of 100 block matrices $\mathcal{T}$, and $\mathcal{T} $ is a ${10 \times 10}$ matrix, defined as $$\mathcal{T}_{j,k} = \left\lbrace \begin{array}{cc} 1, & j=k\\ .9, & else \end{array} \right.$$ For the regression coefficient $\beta^0$ the following four configurations are considered:\ (E1.1) $S_0 = \{1, 2, . . . , 20\}$ and for any $j \in S_0$ we sample $\beta^0_{j}$ from the set $\{.1,.2,.3, . . . , 2\}$ without replacement (a new for each simulation run). This set up has all the active variables in the first two blocks of highly correlated variables.\ (E1.2) $S_0 = \{1,2,11,12 . . . , 91,92\}$ and for any $j \in S_0$ we sample $\beta^0_{j}$ from the set $\{.1,.2,.3, . . . , 2\}$ without replacement (a new for each simulation run). This set up has all the active variables in the first and the second variables of the first ten blocks.\ (E1.3) The $\beta^0$ has the same configuration as in (E1.1) but we change the sign of randomly chosen half of the active parameters (a new for each simulation run).\ (E1.4) The $\beta^0$ has the same configuration as in (E1.2) but we change the sign of randomly chosen half of the active parameters (a new for each simulation run).\ Simulation results are reported in table \[table:E11\](MSE and standard deviation), figure \[fig:E1\](TPR) and table \[table:E13\](CPU time). $\sigma$ Method E1.1 E1.2 E1.3 E1.4 ---------- -------- ---------------- ---------------- ---------------- ---------------- 3 ACGL 12.46 (1.76) 21.95 (2.98) 9.308 (2.40) 20.90 (4.81) CGLcor 14.97 (2.40) 37.05 (5.21) 13.34 (2.06) 24.31 (6.50) 12 ACGL 188.23 (23.32) 149.97 (28.98) 129.29 (22.93) 165.91 (22.36) CGLcor 206.19 (29.97) 186.61 (25.69) 160.31 (23.04) 168.26 (24.70) : MSE(sd) for Example block diagonal model[]{data-label="table:E11"} ![Plot of $ \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}$ vs. $|\hat{S}|$ for block diagonal model. ACGL(green solid line) and CGLcor(blue dashed-dotted line)[]{data-label="fig:E1"}](figure1) $\sigma$ Method E2.1 E2.2 E2.2 E2.4 ---------- -------- ------ ------ ------ ------ 3 ACGL 4.25 5.04 2.09 6.46 CGLcor 2510 2510 2510 2510 12 ACGL 3.91 5.86 1.08 3.33 CGLcor 2510 2510 2510 2510 : CPU times(in seconds) for block diagonal model[]{data-label="table:E13"} From table \[table:E11\], we see that the ACGL method has lower prediction error than the CGLcor for all four configurations and figure \[fig:E1\] shows that the ACGL has higher TPR. From table \[table:E13\] we see that ACGL is much efficient, the CPU time required for ACGL for all four configurations are much less than as compared to the CPU time expended by CGLcor. Please note that CPU time for CGLcor is approximately the same for all configurations. ### Single Block Design We generate covariates from $N_p(0,\Sigma_2)$, where $\Sigma_2$ consisted of a single group of strongly correlated variables of size 30, it is defined as $$\Sigma_{2;j,k} = \left\lbrace \begin{array}{cc} 1, & j=k\\ 0.9& i, j \in \{1, . . . , 30\} \text{ and } i != j, \\ 0& otherwise \end{array} \right.$$ The remaining 970 variables are uncorrelated. For the regression coefficient $\beta^0$ we consider the following four configurations:\ (E2.1) $S_0 = \{1, 2, . . . , 15\} \cup \{31, 32, . . . , 35\}$ and for any $j \in S_0 $ we sample $\beta^0_{j}$ from the set $\{.1,.2,.3, . . . , 2\}$ without replacement (new for each simulation run). The correlated block contains 15, the most of the active predictors and the remaining five active predictors are uncorrelated.\ (E2.2) $S_0 = \{1, 2, . . . , 5\} \cup \{31, 32, . . . , 45\}$ and for any $j \in S_0 $ we sample $\beta^0_{j}$ from the set $\{.1,.2,.3, . . . , 2\}$ without replacement (new for each simulation run). Here the correlated block contains only 5 active predictors, and the remaining 15 predictors are uncorrelated.\ (E2.3) The $\beta^0$ has the same configuration as in (E2.1) but we change the sign of randomly chosen half of the active parameters (new for each simulation run).\ (E2.4) The $\beta^0$ has the same configuration as in (E2.2) but we change the sign of randomly chosen half of the active parameters (new for each simulation run). Simulation results are reported in table \[table:E21\], table \[table:E23\] and figure \[fig:E2\]. $\sigma$ Method E2.1 E2.2 E2.3 E2.4 ---------- -------- ---------------- ---------------- ---------------- ---------------- 3 ACGL 11.40 (4.2) 29.94 (5.34) 15.01 3.28) 27.03 (3.9) CGLcor 247.52 (28.74) 54.73 (10.59) 21.37 (9.51) 31.58 (14.17) 12 ACGL 146.17(23.46) 192.64 (12.81) 127.91 (22.02) 159.62 (26.40) CGLcor 384.78 (48.26) 191.26 (25.55) 159.40 (23.88) 174.49 (25.40) : MSE(sd) for single block model[]{data-label="table:E21"} ![Plot of $ \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}$ vs. $|\hat{S}|$ for single block model. ACGL(green solid line) and CGLcor(blue dashed-dotted line)[]{data-label="fig:E2"}](figure2) $\sigma$ Method E2.1 E2.2 E2.3 E2.4 ---------- -------- ------ ------ ------ ------ 3 ACGL 7.58 5.52 2.63 5.63 CGLcor 2463 2463 2463 2463 12 ACGL 3.17 4.92 2.37 2.47 CGLcor 2463 2463 2463 2463 : CPU times(in seconds) for single block model[]{data-label="table:E23"} Table \[table:E21\] shows that the ACGL method has lower predictive performance than the CGLcor for all four configurations and figure \[fig:E2\] shows that in terms of variable selection, ACGL is clearly better than CGLcor. From table \[table:E21\], we see that CPU time required(in seconds) for ACGL for all four configurations are much less than as compared to the CGLcor. The CPU time for CGLcor is approximately the same for all configurations. ### Duo Block Model We generate covariates from $N_p(0,\Sigma_3)$, where $\Sigma_3$ consists of 500 block matrices $\mathcal{T}, $ and $\mathcal{T} $ is a ${2 \times 2}$ matrix, defined as $$\mathcal{T}_{j,k} = \left\lbrace \begin{array}{cc} 1, & j=k\\ .9, & else \end{array} \right.$$ For the regression coefficient $\beta_0$ we consider $S_0 = \{1, 2, . . . , 20\}$ and for any $j \in S_0$ $$\beta^0_{j} = \left\lbrace \begin{array}{cc} 2, & j \in \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\},\\ \frac{\frac{1}{3} \sqrt{\frac{\log p}{n}} \sigma} {1.9}, & j \in \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \end{array} \right.$$ In this setup, the $\beta^0$ is the same for all 50 simulation runs. The Lasso would not select the variables from $\{2, 4, 6, . . . , 20\}$, since they do not satisfy the beta-min condition but it would select the other from $\{1, 3, 5, . . . , 19\}$. The Table \[table:E31\], Table \[table:E32\] and Figure \[fig:E3\] show the simulation results for the duo block model. $\sigma$ Method MSE(sd) ---------- -------- ---------------- 3 ACGL 20.82 (5.94) CGLcor 32.00 (6.50) 12 ACGL 179.11 (19.35) CGLcor 193.97 (27.05) : MSE(sd) for duo block model[]{data-label="table:E31"} ![Plot of $ \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}$ vs. $|\hat{S}|$ for duo block model. ACGL(green solid line) and CGLcor(blue dashed-dotted line)[]{data-label="fig:E3"}](figure3) $\sigma$ Method CPU time ---------- -------- ---------- 3 ACGL 7.69 CGLcor $>$ 2500 12 ACGL 2.94 CGLcor $>$ 2500 : CPU time(in seconds) for duo block model[]{data-label="table:E32"} The results show that the ACGL performs better in terms of predictive performance, variable selection and CPU time required. (We stopped recording CPU time for CGLcor after 2500sec, here clustering of a thousand of variables and then the group lasso for $\approx$500 clusters make the process very slow. Pseudo Real Data ---------------- We consider here a real dataset, riboflavin$(n = 71, p = 4088)$ data for the design matrix X with synthetic regression coefficients $\beta^0$ and simulated Gaussian errors $N_n(0, \sigma^2 I)$. See [@HDview] for the details on riboflavin dataset. We select the first thousand covariates which have largest empirical variances. We fix the size of the active set to $s_0 =10$. For the true active set, we randomly select a variable, say variable k(a new in each simulation), and then we select other nine variables which have highest absolute correlation to the variable k, and for each $j \in S_0$ we set $\beta_j = 1$. This configuration is exactly the same as pseudo real example used in [@Buhlmann2]. The performance measures are reported in table \[table:Ribo1\], figure \[fig:Ribo\] and table \[table:Ribo2\], based on 50 independent simulation runs. Here we compare CRLcor([@Buhlmann2]) with Adaptive Cluster Representative Lasso(ACRL) where the Lasso is used as preprocessing step at the first stage, correlation based clustering at second step and the Lasso for cluster representatives in the third stage. The Group Lasso is not appropriate for this setup, since k is chosen arbitrarily and the group size may exceed the number of observations. $\sigma$ Method MSPE(std) ---------- -------- --------------- 3 ACRL 2.36 (0.52) CRLcor 39.02 (25.15) 15 ACRL 25.02 (4.03) CRLcor 50.40 (27.68) : MSE(sd) for Riboflavin dataset[]{data-label="table:Ribo1"} \[h!\] ![Plot of $ \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}$ vs. $|\hat{S}|$ for Riboflavin dataset. ACRL(green solid line) and CRLcor(magenta dashed-dotted line)[]{data-label="fig:Ribo"}](figure4) $\sigma$ Method E3 ---------- -------- ------ 3 ACGL 1.47 CRLcor 2347 12 ACGL 1.48 CRLcor 2471 : CPU times(in seconds) for Riboflavin dataset[]{data-label="table:Ribo2"} The table \[table:Ribo1\], figure \[fig:Ribo\] and table \[table:Ribo2\] show that ACRL performs better than CRLcor in terms of prediction, variable selection and CPU time consumption. Computational Details ===================== Statistical analysis was performed in R 3.2.2. We used, the package “glmnet" for penalized regression methods(the Lasso and adaptive Lasso), the package “gglasso" to perform group Lasso and the package “ClustOfVar" for clustering of variables. Conclusion and Future Work ========================== In this article, we proposed a three stage procedure for variable selection for high-dimensional linear model with strongly correlated variables. Our procedure is an extension of the algorithms proposed in [@Buhlmann2]. A technical extension compared with [@Buhlmann2] is that we propose to reduce the dimension at the first stage using Lasso(or its adaptive or thresholded version) prior to clustering at the second stage and then supervised selection of clusters in the third stage. We proved that the variables selected by our algorithm contains the true active set consistently under GIR and group beta-min conditions. Our simulation studies show that reducing dimension improves the speed and accuracy of the clustering process and then considering correlation structure improves variable selection and predictive performance. Since the theoretical results we developed for our algorithms are not restricted to the squared error loss, it can be extended to the generalized linear models, i.e, the preprocessing step of dimensionality reduction can be added to the group Lasso for the logistic regression([@Meier]), and this is our future work. References {#references .unnumbered} ==========

NIKHEF 2015-026\ DAMTP-2015-45 1.5cm [**Quantum corrections in Higgs inflation: the Standard Model case**]{} [ **Damien P. George$^{1,2}$[^1], Sander Mooij$^{3}$and Marieke Postma$^4$**]{} [*$^1$ -.1truecm Department of Applied Mathematics and Theoretical Physics,\ Centre for Mathematical Sciences, University of Cambridge,\ Wilberforce Road, Cambridge CB3 0WA, United Kingdom* ]{} [*$^2$ -.1truecm Cavendish Laboratory, University of Cambridge,\ JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom* ]{} [*$^3$ -.1truecm FCFM, Universidad de Chile,\ Av. Blanco Encalada 2008,\ 837.0415 Santiago, Chile* ]{} [*$^4$ -.1truecm Nikhef,\ Science Park 105,\ 1098 XG Amsterdam, The Netherlands* ]{} 0.5cm [**ABSTRACT**]{}\ We compute the one-loop renormalization group equations for Standard Model Higgs inflation. The calculation is done in the Einstein frame, using a covariant formalism for the multi-field system. All counterterms, and thus the betafunctions, can be extracted from the radiative corrections to the two-point functions; the calculation of higher n-point functions then serves as a consistency check of the approach. We find that the theory is renormalizable in the effective field theory sense in the small, mid and large field regime. In the large field regime our results differ slightly from those found in the literature, due to a different treatment of the Goldstone bosons. Introduction ============ In Higgs inflation the Higgs field of the Standard Model (SM) is coupled non-minimally to gravity [@Salopek:1988qh; @bezrukov1; @bezrukov2; @bezrukov_loop; @bezrukov3; @bezrukov4]. Apart from this single non-minimal coupling, no new physics is needed to describe inflation and the subsequent period of reheating, and the theory seems to be extremely predictive. However, this presupposes the parameters of the theory at high and low scale are related by renormalization group (RG) flow. The betafunctions in the low scale regime are the usual SM ones; in this paper we calculate the RG flow in both the mid field and the high scale (inflationary) regime. The idea of using the Higgs as the inflaton is an attractive one, not least because it allows one to connect collider observables with measurements of the early universe. Since its inception the model itself has come under a lot of scrutiny and criticism. First, unitarity is lost at high energies and the perturbative theory can only be trusted for energies below the unitarity cutoff [@burgess1; @barbon; @burgess2; @Hertzberg1; @cliffnew]. Although it is uncertain how to interpret this result as the cutoff is field dependent (according to [@bezrukov4; @linde_higgs; @moss; @He1], all relevant physical scales are always below the unitarity bound), it is clear that any new physics living at this scale may affect the inflationary predictions [@cliffnew; @Hertzberg2]. Second, given the currently measured central values for the top and Higgs mass, the Higgs potential becomes unstable around $10^{11}$ GeV, which would be disastrous for Higgs inflation. However, the verdict is not yet out, as it only takes $2-3\sigma$ deviations to push the instability bound all the way to the Planck scale [@disc1; @disc2; @branchina; @branchina2; @archil; @alexss] (in the very recent note [@kniehl] absolute stability of the electroweak vacuum is reported to be excluded by only 1.3 $\sigma$). Even though these claims are still debated, and SM Higgs inflation may still be alive, it is worth noting that constraints may be avoided in modified set-ups with an extended Higgs sector. Our results apply for large non-minimal coupling, but apart from that they are equally applicable to the various implementations of Higgs inflation. The renormalization group equations (RGEs) in Higgs inflation have been derived by several groups [@bezrukov_loop; @bezrukov3; @wilczek; @barvinsky; @barvinsky2; @barvinsky3], but they differ in details. The main source of disagreement comes from the choice of frame, and the treatment of the Higgs sector (Does the Higgs decouple from all fields? And the Goldstone bosons?). In previous work [@damien; @volpe] we have shown that the Jordan and Einstein frame describe exactly the same physics, and that any difference stems from an erroneous comparison of quantities defined in different frames. In this work we will work in the Einstein frame. Although dimensional analysis indicates that some of the Goldstone boson (GB)-loop corrections are large, and seem to spoil renormalization, gauge symmetry kicks in leading to cancellations of these large contributions. We find Higgs inflation is renormalizable in the effective field theory (EFT) sense [^2], and for energies below the unitarity cutoff. The small-field regime of Higgs inflation is where $\phi_0 \ll {m_{\rm p}}/\xi$, with $\phi_0$ the value of the background Higgs field, $\xi$ the non-minimal Higgs-gravity coupling which is of order $10^4$ (well below experimental bounds [@Calmet; @He2]), and ${m_{\rm p}}$ the Planck mass. In this regime the theory is effectively like the SM and therefore renormalizable in the EFT sense. In the large field regime ($\phi_0 \gg {m_{\rm p}}/\sqrt{\xi}$, corresponding to inflation) the potential has an approximate shift symmetry, which restricts the form of the loop corrections. As a result, all one-loop corrections can be absorbed in the parameters of the classical theory, and the EFT is renormalizable. Somewhat surprisingly, we find the same in the mid-field regime (${m_{\rm p}}/\xi < \phi_0< {m_{\rm p}}/\sqrt{\xi}$), even though it is far away from both an IR fixed point and the region in which the shift symmetry applies. In [@damien] we have studied the renormalization of the non-minimally coupled Higgs field in isolation, without any gauge or fermion fields, and our findings were in line with the literature. In this work we want to extend this previous analysis to the full SM. At first glance this does not seem to be problematic. Due to the non-minimal coupling to gravity, the coupling of the radial Higgs to both gauge field and fermions is suppressed in the large field regime. One can simply neglect all diagrams with these couplings. For example, loop diagrams with a fermion or gauge boson loop always dominate over the corresponding diagram with a Higgs loop. Effectively the Higgs decouples from the theory. However, the situation for the Goldstone bosons (GBs) is more complex: their coupling to the gauge fields is also suppressed, while the GB-fermion coupling is not. Upon going to unitary gauge, this corresponds to a coupling of the fermion to the longitudinal polarization of the gauge fields, and both the transverse and longitudinal polarizations couple with the usual SM strength to the fermions. All calculations are performed in the Einstein frame. For a discussion of the equivalence of Einstein and Jordan frame, see [@damien; @volpe]. One of the main complications in the calculation is that after transforming to the Einstein frame one ends up with non-canonical kinetic terms for the Higgs and Goldstone field. Due to the nonzero curvature of the field space, it is impossible to make a field transformation that brings the kinetic terms to their canonical form. Our approach here is to expand the action around a large classical background value for the inflaton field, and use the formalism of [@jinnouk; @seery; @Kaiser2] so that this background expansion can be done maintaining covariance in the field space metric. In our calculation we have neglected the time-dependence of the background field, as well as FLRW corrections and the backreaction from gravity; we argue that these corrections are at most subleading. (The inclusion of gravity corrections to the Higgs part of the theory has been addressed, in a covariant way, in [@moss].) Moreover, we are neglecting higher order kinetic terms by evaluating the field metric on the background. It would be an interesting but equally challenging task to develop a framework that can get around this latter limitation. Our main results are the SM Higgs inflation RGEs in the three regimes, where we included only the top-Yukawa coupling $y_t$. We find that Higgs/GB self-interactions and Higgs-fermion-interactions (but not GB-fermion) can be neglected in the mid and large field regime; Higgs/GB-gauge interactions decouple in the large field regime. This gives the following betafunctions: $$\begin{aligned} (4\pi)^2 \beta_\lambda &= 24 \lambda^2\fac + A +(4\pi)^2 \cdot 4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= -\frac{\fac}{4} (3g_1^2+9 g_2^2) + 3y_t^2 \nn \\ (4\pi)^2\beta_{g_3} &= -7 g_3^3,\nn\\ (4\pi)^2\beta_{g_2} &= - \frac{(20 - \fac)}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{(40 + \fac )}{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= {\left[}\frac32 \fac y_t^3 -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t{\right]}+(4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi\big|_{\rm mid, large} &= (4\pi)^2 \cdot2 \gamma_\phi \xi\end{aligned}$$ with $A= (3/8)(2g_2^4 + (g_2^2+g_1^2)^2) -6y_t^4$ and = { [ll]{} 1, & [small]{} ,\ 0, &[mid]{} , . . \[fac\] These betafunctions break down at the boundary of the regimes, where the EFT expansion in a small parameter is no longer valid; this gives additional threshold corrections which we have not calculated. All sign conventions used in this paper follow the QFT textbook by Srednicki [@srednicki], except for the sign of the Yukawa interaction terms, which is opposite to Srednicki’s. Higgs inflation {#s:hi} =============== In this section we give a brief overview of Higgs inflation and set our notation. Lagrangian ---------- The Jordan frame Lagrangian[^3] is (using $-+++$ metric signature) Ł\^J = $$-\frac12 {m_{\rm p}}^2 {\left(}1+ \frac{2\xi \Phi^\dagger \Phi}{{m_{\rm p}}^2} {\right)}R[g^J] + \L^J_{\rm SM}$$, with $$\begin{aligned} \L^J_{\rm SM} &= -\frac14 (f^a_{\mu\nu})^2 -\frac14 (F^a_{\mu\nu})^2 -\frac14 B_{\mu\nu}^2 -(D_\mu \Phi)^\dagger (D^\mu \Phi) - \lambda(\Phi^\dagger \Phi - v^2/2)^2 \nn\\ &\hspace{0.45cm}+ \bar Q_L (i\slashed{D}) Q_L +\bar u_R (i\slashed{D}) u_R +\bar d_R (i\slashed{D}) d_R - (y_d \bar Q_L \cdot \Phi d_R + y_u \bar u_R (i\sigma^2) \Phi^\dagger Q_L +{\rm h.c} ), \label{L_SM}\end{aligned}$$ where $Q_L =(u \; d)^\top_L$. Further $f^a_{{\mu\nu}},F^a_{{\mu\nu}},B_{{\mu\nu}}$ are the SU(3), SU(2) and U(1) field strengths respectively. The Higgs field is SU(2) complex doublet, which we parameterize = 1 $ \begin{array}{c} {\varphi}^+ \\ \phi_0+{\varphi}+ i \theta_3 \end{array} $ , \[Hfields\] with $\phi_0$ the classical background, ${\varphi}$ the Higgs field and ${\varphi}^+ = \theta_1+i\theta_2, \theta_3$ the GBs. The covariant derivative acts on the Higgs field and fermions as $$\begin{aligned} D_\mu \Phi &= {\left(}\partial_\mu -i g_2 A^a_\mu \tau^a - i Y_\phi g_1 B_\mu{\right)}\Phi ,\nn \\ D_\mu Q_L &=(\partial_\mu - i g_3 f^a_\mu t^a- i g_2 A^a_\mu \tau^a - i Y_Q g_1 B_\mu)Q_L, \nn \\ D_\mu u_R &=(\partial_\mu - i g_3 f^a_\mu t^a - i Y_u g_1 B_\mu)u_R, \label{DH}\end{aligned}$$ with $\tau^a = \sigma/2$ for the spinor representation. The hypercharges are $Y_\phi =1/2$, $Y_Q = 1/6$ and $Y_u=2/3$. At leading order, the only fermion that matters to find the running of the SM couplings is the top quark. We reach the Einstein frame after a conformal transformation: $ g^E_{\mu\nu} = \Omega^2 g^J_{\mu\nu}$, with \^2 = $1+ \frac{2\xi \Phi^\dagger \Phi}{{m_{\rm p}}^2}$. \[Omega\] The Einstein frame Lagrangian becomes Ł\^E =$$-\frac12 {m_{\rm p}}^2 R[g^E] + \L_{\rm mat}^E$$. We neglect the expansion of the universe, and take a Minkowski metric. The gauge kinetic terms are conformally invariant. The fermionic kinetic terms can be made canonical via a rescaling $\psi^E = \psi/\Omega^{3/2}$; the net effect is then a rescaling of the Yukawa interaction. All non-trivial effects of the non-minimal coupling are in the Higgs sector. $$\begin{aligned} \L^E_{\rm mat} &= -\frac14 (f^a_{\mu\nu})^2 -\frac14 (F^a_{\mu\nu})^2 -\frac14 B_{\mu\nu}^2 -\frac{1}{\Omega^2}(D_\mu \Phi)^\dagger (D^\mu \Phi) - \frac{3 \xi^2}{{m_{\rm p}}^2\Omega^4} \partial_\mu (\Phi^\dagger \Phi) \partial^\mu (\Phi^\dagger \Phi) \nn \\ &\hspace{0.45cm}+ \bar Q^E_L (i\slashed{D}) Q^E_L +\bar u^E_R (i\slashed{D}) u_R^E +\bar d^E_R (i\slashed{D}) d^E_R \nn\\ &\hspace{.43cm} - \frac{\lambda}{\Omega^4}(\Phi^\dagger \Phi - v^2/2)^2 - (\frac{y_d}{\Omega} \bar Q^E_L \cdot \Phi d^E_R + \frac{y_u}{\Omega} \bar u^E_R (i\sigma^2) \Phi^\dagger Q^E_L +{\rm h.c.} )\end{aligned}$$ The Higgs kinetic term is non-minimal. Let $\phi^i = \{\phi_R=\phi_0 + \varphi,\theta_i\}$ run over the Higgs field and Goldstone bosons. Then the metric in field space in component form is Ł\^E\_[mat]{} - 12 \_[ij]{} \_i \_j = -12 $$\frac{\delta_{ij}}{ \Omega^2} + \frac{ 6 \xi^2}{{m_{\rm p}}^2\Omega^4} \chi_i \chi_j$$ \_i \_j. \[non\_can\] The curvature on field space $R[\gamma_{ij}] \neq 0$ (this point was made, and further generalized, in [@Kaiser1], and the kinetic terms cannot be diagonalized. At most one can diagonalize the quadratic kinetic terms at one specific point in field space. Consider the electroweak sector. For the gauge bosons the kinetic terms remain canonical in the Einstein frame. As far as the quadratic action is concerned the action for the massive gauge bosons and Goldstone bosons is simply three times the action of a U(1) theory. To see this explicitly, consider the Higgs kinetic terms $$\begin{aligned} \L_{\rm higgs} &\supset -\frac{1}{\Omega^2}(D_\mu \Phi)^\dagger (D^\mu \Phi) \nn \\ &=-\frac{1}{2\Omega^2}{\left[}\partial_\mu {\varphi}\partial^\mu {\varphi}+ \sum_{a=1}^3 (\partial_\mu \theta_a \partial^\mu \theta_a -2g_a A^a_\mu (\phi \partial^\mu \theta_a -\theta_a \partial^\mu \phi) + g_a^2 \phi^2A^a_\mu A_a^\mu) +...{\right]}. $$ The gauge boson mass eigenstates are $\{A_1,A_2, Z,A_\gamma\}$ with Z= (g\_2 A\_3-g\_1 B),A\_= (g\_1 A\_3+g\_2B) , and couplings g\_a = 12 {g\_2,g\_2,,0}. \[ga\] This corresponds to three massive and one massless field. Note that we took the mass eigenstates as real gauge fields, and used the real and imaginary parts of $W_+$, rather than the complex states $W_\pm$. From now on we will work in the Einstein frame. For convenience we drop the superscript $E$, and work in Planck units ${m_{\rm p}}=1$. Three regimes {#s:regimes} ------------- Higgs inflation is non-renormalizable as the field space metric and potential are non-polynomial. But this does not exclude that the theory is renormalizable in the EFT sense over a limited field space. Our demands are that in a given field regime the theory can be expanded in a small parameter $\delta$, and that all loop corrections can be absorbed in counterterms order by order. Truncating the theory at some finite order in $\delta$ gives a renormalizable EFT with a finite number of counterterms. #### Small field regime The small field regime corresponds to $\delta_s \equiv \xi \phi_0 \ll 1$. To leading order in the expansion parameter $\delta_s$, the Lagrangian reduces to the SM Lagrangian. #### Mid field regime The mid field regime corresponds to $1/\xi < \phi_0 < 1/\sqrt{\xi}$. In this regime we rescale $\xi \to \delta_m^{-2}\xi$ and $\phi_0 \to \delta_m^{3/2} \phi_0$, such that both $\xi \phi_0^2 \propto \delta_m$ and $1/(\xi \phi_0)^2 \propto \delta_m$, and use $\delta_m$ as our expansion parameter. (We should admit that formally this expansion can only be trusted in the middle of this regime.) #### Large field regime Inflation takes place for field values $\delta_l \equiv 1/(\xi \phi_0^2) \ll 1$. The expansion in $\delta$ is equivalent to an expansion in slow-roll parameters, since $\eta=\mathcal{O}\left(\delta\right)$ and ${\epsilon}=\mathcal{O}\left(\delta^2\right)$. Covariant formalism and counterterms {#s:cov} ==================================== We want to investigate how the loop corrections and counterterms change in the small, mid and large field regime. For simplicity, we first focus on a U(1) Abelian Higgs model coupled to a left- and right-handed fermion. The generalization to the full SM Higgs inflation is postponed till sec. \[s:abelian\_beta\]. This way the effects of the non-minimal coupling can be studied in a simple set-up, without all intricacies of the chiral SM. Another advantage of the U(1) model is that gauge invariance and the Ward identities assure that many counterterms are independent of the gauge choice, which makes it easier to check the calculation. Lagrangian in covariant fields ------------------------------ This subsection reviews the covariant formalism introduced in [@jinnouk] and further worked out in [@seery; @Kaiser2]. Given the curvature of field space, it is very convenient to adopt an approach that maintains the covariance of the equations. For a U(1) theory with a complex Higgs field and a left- and right-handed Weyl fermion the Einstein frame matter Lagrangian is $$\begin{aligned} \L &= -\frac14 F_{{\mu\nu}}F^{{\mu\nu}}-\frac12 \gamma_{ab} \partial_\mu \phi^a \partial^\mu \phi^b +i\bar \psi \slashed{\partial}\psi- V(\phi^a) - \bar \psi F(\phi^a) \psi \nn\\ & \hspace{.43cm} -g A (G^\theta \partial \phi-G^\phi \partial \theta) -\frac12 g^2 A^2 G + {\left(}g q_L \bar \psi \slashed{A} P_L \psi +g q_R \bar \psi \slashed{A} P_R \psi{\right)}, \label{LU1}\end{aligned}$$ with $$\begin{aligned} V(\phi^a) &= \frac{\lambda}{4} \frac{|\phi_0+\varphi+i\theta |^4}{\Omega^4},\qquad F(\phi^a) = \frac{y}{\sqrt{2}}\frac{\phi_0+ \varphi+ i\gamma^5\theta}{\Omega}, \nn\\ G^\phi &= \frac{\phi}{\Omega^2},\qquad G^\theta = \frac{\theta}{\Omega^2},\qquad G = \frac{(\phi^2 + \theta^2)}{\Omega^2}. \label{VF}\end{aligned}$$ Now expand the Lagrangian around the background $\phi^a = (\phi_0(t) +\varphi(x,t), \theta(x,t))$. The fluctuation fields $\delta\phi^a=(\varphi,\theta)$ are not in the tangent space at $\phi_0^a$, and therefore do not transform as a tensor. We are led to introduce the covariant fluctuation $Q^a = (h,\chi)$, which is related to $\delta\phi^a$ via \^a = Q\^a -1[2!]{} \^a\_[bc]{} Q\^b Q\^c + 1[3!]{}$\Gamma_{be}^a \Gamma^e_{cd} -\Gamma^a_{bc,d}$ Q\^b Q\^c Q\^d + ... \[Qdef\] This is the notation we will use throughout this paper: $({\varphi},\theta)$ are the fluctuations of the original Jordan frame field (with $\phi_0$ the classical background field), and $(h,\chi)$ are the covariant fields. Further we define the covariant time derivative D\_t = \_a. Note that in the limit $\dot \phi_0 =0$ this reduces to the usual derivative $D_t = \partial_t$. Now we can expand the action in covariant fluctuations. We neglect FLRW corrections and the backreaction from gravity, as well as the time-dependence of the background field $\phi_0$; we come back to this in Sec. \[s:checks\]. The result for the interaction Lagrangian is $$\begin{aligned} \L_{\rm int} &= -\ (V + V_{;a} Q^a +\frac1{2!} V_{;ab} Q^a Q^b +...) -\bar\psi \ (F + F_{;a} Q^a +\frac1{2!} F_{;ab} Q^a Q^b +...) \psi \nn\\ & \hspace{.43cm} - g A \partial h ( G^\theta_{;a} Q^a +\frac1{2!} G^\theta_{;ab} Q^a Q^b +...) + g A \partial \chi ( G^\phi_{;a} Q^a +\frac1{2!} G^\phi_{;ab} Q^a Q^b +...) \nn\\ & \hspace{.43cm} -\frac12 g^2 A^2 ( G_{;a} Q^a +\frac1{2!} G_{;ab} Q^a Q^b +...) + {\left(}g q_L \bar \psi \slashed{A} P_L \psi +g q_R \bar \psi \slashed{A} P_R \psi{\right)}. \label{L_cov}\end{aligned}$$ All coefficients are evaluated on the background. The subscript with a semi-colon denotes the covariant derivative. We just found the Lagrangian for the covariant fields by Taylor expanding using covariant derivatives. An equivalent way of deriving the same Lagrangian is solving the relation $\phi^i(Q^j)$ [(\[Qdef\])]{} explicitly, substituting in the Lagrangian [(\[LU1\])]{}, [^4] and then Taylor expand in the fields $Q^i$ (using partial derivatives). This point of view will be useful when defining the counterterms in the next section. Here we just give the explicit form of [(\[Qdef\])]{} relating the original Langragian fields $({\varphi},\theta)$ to the covariant fields $(h,\chi)$: $$\begin{aligned} {\varphi}&= {\left(}h + \frac{ (h^2-\chi^2)}{2\phi_0} + ..{\right)}- \frac1\xi{\left(}\frac{h^2}{\phi_0^3} + \frac{h^3}{3\phi_0^4} + .... {\right)}+ \frac{1}{\xi^2} {\left(}\frac{h^2+\chi^2}{12\phi_0^3} + .... {\right)}.\nn \\ \theta & = {\left(}\chi + \frac{h \chi}{\phi_0} +..{\right)}- \frac1\xi{\left(}\frac{h \chi}{\phi_0^3} + \frac{4h^2 \chi}{3\phi_0^4} + .... {\right)}+ \frac{1}{\xi^2} {\left(}\frac{h \chi}{\phi_0^5} + \frac{h^2 \chi}{12\phi_0^4} + .... {\right)}. \label{Q_expl}\end{aligned}$$ We checked that substituting this in the Lagrangian and expanding, we indeed retrieve [(\[L\_cov\])]{}. Gauge fixing ------------ We have to add a gauge fixing and ghost Lagrangian, which can also be expanded in covariant fields. We fix the gauge via Ł\^E\_[GF]{} =-1[2\_G]{} $\partial^\mu A_\mu - g G^\phi(\phi_0) \xi_G \theta $\^2 . \[L\_GF\] This removes the quadratic $A\partial \theta$ couplings from the Lagrangian. In the small field regime $\Omega_0 \equiv \Omega(\phi_0) =1$ and we retrieve the standard $R_\xi$-gauge. We choose to write the gauge fixing term in terms of the Jordan frame fields (as opposed to the covariant fields) as these have a well defined gauge transformation. We work in Landau gauge ${{\xi_G}}= 0$. Then the ghost field decouples Ł\_[FP]{}\^E |\_[[[\_G]{}]{}=0]{} = -\_|c \^c. Feynman rules ------------- Now we can derive the Feynman rules from the above action. First we define the effective couplings $$\begin{aligned} \L_{\rm int}& = - \lambda_{m h n\chi} h^m \chi^n - y_{m h n \chi} h^m \chi^n \bar \psi (i\gamma^5)^\alpha \psi - (g_{ A \partial h mh n \chi } \partial h - g_{ A \partial \chi mh n\chi} \partial \chi ) A h^m \chi^n \nn \\ & \hspace{.43cm} -g_{2A mh n\chi} A^2 h^m \chi^n +g_L \bar \psi \slashed{A} P_L \psi + g_R \bar \psi \slashed{A} P_R \psi \label{vertices1}\end{aligned}$$ with $\alpha =1$ if the number $n= $ odd, and $\alpha=0$ otherwise (signs are absorbed in the couplings). All interactions are defined with a minus sign (the only exception is for one of the derivative interactions and the fermion-gauge interaction), and without numerical factors. This means that for a vertex with $m$ $h$-fields and $n$ $\chi$-fields and with or without fermion/gauge lines we have, respectively: $$\begin{aligned} V^{(m hn\chi)}&= (-i) m! n! \lambda_{m hn\chi},\nn\\ V^{(m hn\chi2\psi)}&= (-i) m! n! y_{m\phi n\chi}(i\gamma^5)^\alpha,\nn \\ V^{(m hn\chi2A)}&= (-i) 2! m! n! g_{2Am hn\chi}. \label{vertices2}\end{aligned}$$ For the derivative interaction we get V\^[(A h m h n)]{} = -i g\_[(A h m h n)]{} (-i k\^),V\^[(A m h n)]{} = i g\_[(A m h n)]{} (-i k\^), with $k$ the momentum running through the vertex. The fermion, scalar and gauge propagators are given by: $$\begin{aligned} -i D_\psi(k) &= \frac{-i(-\slashed{k}+m_\psi)}{k^2+m_\psi^2-i{\epsilon}}, \nn \\ -i D_{Q^a}(k) &= \gamma^{aa} \frac{-i}{k^2 + (m^2)_a^a-i{\epsilon}}, \nn \\ -i D_{\mu\nu}(k) &\stackrel{\xi_G = 0}{ =}-i \frac{g_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}}{k^2 +m_A^2 -i{\epsilon}}, \label{propagators}\end{aligned}$$ with masses $m_\psi = F(\phi_0)$, $m_A = G(\phi_0)$, and $(m^2)^a_b = \gamma^{ac} V_{;cb}(\phi_0)$. The scalar mass is diagonal, which we used in the scalar propagator. The propagators are standard except for the metric factor in the scalar propagator. The ghost field decouples. The above expansion of the Lagrangian, and corresponding Feynman rules are equally valid in all field regimes (although in the small field regime the notation is overkill). The explicit expressions for the vertices are given in Appendix \[s:vertices\]. The Higgs/GB mass and self-interactions are suppressed in the mid and large field regime, and we can neglect Higgs/GB loops in the quantum corrections. Likewise, the interaction with the gauge field is suppressed in the large field regime. The Higgs-fermion coupling is small in the mid and large field regime, but the GB-fermion coupling is not: it has the standard SM strength. This does not come as a surprise. The gauge-fermion interactions are unaffected by the non-minimal coupling. And since the GB is eaten to become the longitudinal polarization of the gauge boson, it should have the same interaction strength as the transverse polarizations. A priori it is not clear what the effects are on the betafunctions. The explicit computation is done in the next section. Counterterms ------------ In this section we introduce counterterms. It proves convenient, or maybe even necessary, to define the wave functions $Z_i$ in terms of the original Jordan frame fields, rather than the covariant fields. The usual U(1) symmetry relations between the various $Z_i$-factors then apply. To define the counterterms we start with the Einstein frame Lagrangian in terms of the [*Jordan frame*]{} fields [(\[LU1\])]{} and rescale the bare fields (with label “b") to physical fields (without label) via $$\begin{aligned} \phi_b &= \sqrt{ Z_\phi} \phi, & \theta_b &= \sqrt{ Z_\theta} \theta, & \psi_b &= \sqrt{Z_\psi} \psi,& A^\mu_b& = \sqrt{Z_A} A^\mu, \nn \\ \lambda_b &= Z_\lambda \lambda, & \xi_b &= Z_\xi \xi, & y_b &= Z_y y, & g_b &= Z_g g, \label{Zelement}\end{aligned}$$ with $\phi = \phi_0 + {\varphi}$. In Landau gauge $\xi_G=0$ the wavefunctions are $Z_{\varphi}=Z_{\phi_0} =Z_\theta$. We further define Z\_i = 1+\_i. We can then split the Lagrangian $\L =\L_{\rm renormalized} + \L_{\rm ct}$ with the counterterms proportional to $\delta_i$. The total Lagrangian is $$\begin{aligned} \L &= -\frac12 (Z_{g_{ab}} g_{ab}) Z_\phi \partial \phi^a \partial \phi^b + Z_\psi \bar \psi i\slashed{\partial}\psi -\frac{1}{4}Z_A F_{\mu\nu} F^{\mu\nu} \nn \\ & - \frac{Z_\lambda Z_\phi^2 \lambda ( \phi^2 + \theta^2)^2}{ 4(1+ Z_\xi Z_\phi \xi \phi^2)^2} - \frac{Z_\psi \sqrt{Z_\phi} Z_y y}{\sqrt{2} (1+ Z_\xi Z_\phi \xi \phi_R^2)} \bar \psi (\phi + i\theta \gamma^5) \psi \nn\\ & -\frac1{ (1+ Z_\xi Z_\phi \xi \phi_R^2)} {\left(}\sqrt{Z_A} Z_\phi Z_g g A (\phi \partial \theta -\theta \partial \phi) + Z_A Z_\phi Z_g^2 \frac12 g^2 A^2 (\phi^2+\theta^2) {\right)}\nn \\ &+Z_\psi Z_g Z_A^{1/2} {\left(}g q_L \bar \psi \slashed{A} P_L \psi +g q_R \bar \psi \slashed{A} P_R \psi{\right)}. \label{Lstart}\end{aligned}$$ This can be simplified using the U(1) Ward identity Z\_g = Z\_A\^[-1/2]{}. \[ward\] Moreover, as we will derive in the next section, we have in the mid and large field regime (in the small field regime, $\xi$ drops out of the Lagrangian at leading order, and no counterterm can be determined at this order) Z\_= 1[Z\_]{}, &gt; . \[Z\_xi\] This means $\Omega^2 = 1 + \xi \phi^2$ does not run (in the large field regime it means $\delta = 1/(\xi \phi_0^2) \sim {{\rm e}}^{-h_0}$ does not run). We can now use $\phi^i(Q^a)$ in [(\[Q\_expl\])]{} to rewrite the Lagrangian in terms of the covariant fields. It is then clear that all interactions arising from expanding the potential have the same renormalization factor, namely $Z_\lambda Z_\phi^2$. Similar statements can be made for the gauge and Yukawa interactions. We can also define “composite” wavefunctions for the interactions of the covariant fields via $$\begin{aligned} \L \supset&= - Z_{\lambda; mh n\chi} \lambda_{mh n\chi} h^m \chi^n - Z_{y; m h n \chi} y_{m h n \chi} h^m \chi^n \bar \psi (i\gamma^5)^\alpha \psi \nn \\ & - Z_{2A; m h n \chi} g_{2A m h n \chi} A^2 h^m \bar \chi^n - Z_{A \partial Q^a; m h n \chi} g_{A \partial Q^a m h n \chi} A \partial Q^a h^m \bar \chi^n . \label{Lstart2}\end{aligned}$$ It is straightforward to rewrite the composite wavefunctions in terms of elementary ones [(\[Zelement\])]{}, by comparing terms in the two actions above (\[Lstart\],\[Lstart2\]), when both written out in covariant fields. As an explicit example, consider the potential in the large field regime; using [(\[Lstart\])]{} it can be expanded as V = + $ - \frac{1}{\phi_0^4 \xi^3} h^2 - \frac{1}{3\phi_0^6 \xi^3} h^4 + \frac{1}{12 \phi_0^6 \xi^5} \chi^2 - \frac{1}{108 \phi_0^{10} \xi^7} \chi^4 + ...$ \[Zcompo\] from which we read off Z\_[V\_0]{} = ,Z\_[2h]{} = Z\_[4h]{} = Z\_[2]{} = Z\_[4]{} = . \[Z\_lambda\] Ignoring $\dot \phi_0$ corrections the Higgs kinetic terms are Ł-12 Z\_[2h]{} g\_[h h]{} (h)\^2 = -12 Z\_[2h]{} (h)\^2 = -12 (h)\^2. \[Z\_kin1\] The leading counterterm vanishes. The kinetic term for the GBs is Ł-12 Z\_[2]{} g\_ ()\^2 = -12 Z\_[2]{} ()\^2 = -12 Z\_h ()\^2. \[Z\_kin2\] And thus Z\_[2h]{} -1 =Ø(), Z\_[2]{} = Z\_. To derive these results we have used $Z_{h} = Z_{\chi} =Z_\phi$, i.e. the same counterterms for the Jordan frame and covariant fields. As we will discuss below, although this approximation is valid for the kinetic terms, it is not in general. ### On the wavefunctions of the covariant fields {#s:cov_wave} Instead of defining the wavefunctions of the Jordan frame fields, we could have tried to work with those of the covariant fields. Rather than beginning from [(\[Zelement\])]{}, we would then introduce $Z_{\phi_0}$, $Z_h$ and $Z_\chi$. Consider the large field regime. The potential is expanded in covariant fields as in [(\[Zcompo\])]{}. We can read off the counterterms from the parametric dependence of the various terms. For the quadratic Higgs and GB interactions we find $Z_h^{-1} Z_\xi^{-3}$ and $Z_h^{-2} Z_\xi^{-5}$ respectively, which should be equal by gauge invariance. This excludes setting $Z_{h} = Z_{\chi} =Z_\phi$ in the potential, as it gives inconsistent results.[^5] As we will now discuss, this approach breaks down for the GB interactions. We can understand where this stems from. To derive the $h^2$ and $h^4$ interactions at leading order, it is enough to only keep the first term in the expansion [(\[Q\_expl\])]{}. Then taking the relevant terms in [(\[Q\_expl\])]{}, and setting $Q_b^i = \sqrt{Z_Q} Q^i$ we get $$\begin{aligned} \sqrt{Z_\phi} {\varphi}&= \sqrt{Z_{h}} {\left(}h + \frac{ (h^2-\chi^2)}{2\phi_0} + ..{\right)}, \\ \sqrt{Z_\phi} \theta & =\sqrt{Z_{\chi}} {\left(}\chi + \frac{h \chi}{\phi_0} +..{\right)}.\end{aligned}$$ Thus for $h$ interactions we can take $Z_{h} = Z_{\chi} = Z_\phi$. Similarly, for the kinetic terms, only the first order expansion is needed, which is what we used above. However, to derive the GB interactions, the leading and subleading terms cancel, and to get the correct interaction one needs to expand $\phi^i(Q^j)$ to sufficient high order in $\delta$. In particular, one needs also to take into account the last term in the expansion in [(\[Q\_expl\])]{}. But then $Z_{h} = Z_{\chi} = Z_\phi =1/Z_\xi$ is no longer a consistent solution; it would give the inconsistent relation = (Q + Q\^2 + ...) + Z\_\^[3/2]{} ( Q\^2+ Q\^3 +...). Thus for GB scattering one cannot really define $Z_{\chi}$, $Z_{h}$ in terms of the elementary wavefunctions [(\[Zelement\])]{}. Fortunately, this is also not necessary, because we can simply use [(\[Zcompo\])]{}. Approximations used {#s:checks} ------------------- Before diving into the calculation we first list here the approximations made. 1. [We have dropped the time-dependence of the background field: $\dot \phi_0 \to 0$]{} 2. [We have neglected FLRW corrections and the backreaction of gravity]{} 3. [We have evaluated the field metric on the classical background.]{} 1\. In [@mp] we calculated the effective action in the SM regime, taking into account the rolling of the classical background field $\dot \phi_0$. Generalizing standard techniques to calculate the effective action to the time-dependent situation, we found the radiative corrections to both the classical potential and the kinetic terms. This allowed us to extract both the $\delta_\lambda$ and $\delta_\phi$ counterterms from the effective action. We retrieved the standard results. The time-dependence does not affect the form of the counterterms. Had we done the calculation in a time-independent way, by neglecting $\dot \phi_0$, we would have found the same $\delta_\lambda$ counterterm. In the large field regime the time-dependence enters also the kinetic terms, which are non-minimal, and it may not be obvious that we can neglect these effects. However, the large field regime is the inflationary regime, and all time-dependent corrections are slow roll suppressed. Working at leading order in the expansion parameter, as we do, they can be neglected. 2\. In [@damien; @GMP] we calculated the effective action in the SM regime, in a FLRW background. We showed that when working in the [*Einstein frame*]{}, the backreaction from gravity can be neglected. The reason is that the corrections are of the order of the slow roll parameter ${\epsilon}\sim \delta^2$, which are small compared to $\eta \sim \delta$ and thus can be neglected at leading order. Doing the calculation in a FLRW background will give order $\mathcal{O}(H^2)$ corrections to the scalar masses, to the Higgs and GB mass in our case. However, these masses only appear in diagrams with a Higgs and GB in the loop, which thus also involve suppressed GB/Higgs couplings. There is however one diagram that becomes of leading order in the $\delta$-expansion, which is the last term of [(\[Pi\_fermion\])]{}, giving the GB loop correction to the fermion propagator. Nevertheless, this diagram is still suppressed by $1/\xi$. Hence, to be really sure that FLRW corrections will not affect our results we have to work in the large $\xi \gg 1$ limit. 3\. The kinetic terms for the GB/Higgs field are of the form Ł- 12 \_[ij]{} Q\^i Q\^j = - 12 \_[ij]{}(\_0) Q\^i Q\^j + ... where we have expanded the field space metric around the background. The first term is quadratic and determines the structure of the propagators. The higher order terms, denoted by the ellipses above, can then be treated as additional interaction terms. It is hard to systematically take into account the effects of higher order interactions, and we have neglected them in our calculations in the next section. Unfortunately, it seems that for at least one diagram this is not a good approximation, as we discuss in section \[s:painful\]. One-loop corrections {#s:loop} ==================== To derive the one-loop betafunctions for the gauge, Yukawa and Higgs interactions, $g,y,\lambda$, it is enough to calculate the corrections to the gauge, fermion and scalar propagator, which is what we will do in this section. We will also compute corrections to 3 and 4-point interactions. These will serve as consistency checks on the result, which provide further checks on the validity of our approximations (discussed in the previous section). Another consistency check is the comparison with the Coleman-Weinberg effective action [@CW]. No field independent counterterms can be defined for the whole regime, but it may be possible to define renormalizable EFTs in the three different regimes. Then the hope is that the threshold corrections in patching them together are small. To find the result in a given regime, we plug in the explicit form of the couplings expanded in the expansion parameter valid in this regime. The expansion parameters were defined in subsection \[s:regimes\]; the explicit form of the couplings can be found in Appendix \[s:vertices\]. Coleman-Weinberg effective action --------------------------------- The Coleman-Weinberg calculation for a dynamical background field has been performed in [@mp]. From this we can extract $Z_{V_0}, Z_{2h}$, which should be consistent with the loop corrections to the Higgs/GB propagator and self-scattering. The effective action gets contributions from the bosonic, mixed and fermionic loops respectively, and is for ${{\xi_G}}=0$ $$\begin{aligned} \Gamma_{\rm CW} &= \frac{1}{32\pi^2{\epsilon}} {\left[}m_h^4 +m_\theta^4 +3 m_A^4 -4 m_f^4 + \frac{3}{2} m_{A\theta}^4 + 4 m_f \ddot m_f {\right]},\end{aligned}$$ with $m_{A\theta}^2 = -2 g \dot{\phi}_0/\Omega_0^2$. Adding classical and all one-loop contributions gives $$\begin{aligned} \Gamma & = \frac12 \gamma_{hh} \dot{\phi}_0^2{\left[}-Z_{2h} + \frac{1}{8\pi^2{\epsilon}} {\left(}\frac{3 g^2}{1+ \xi \phi_0^2(1+6\xi)} - \frac{y^2}{\Omega_0^2 (1+ \xi \phi_0^2(1+6\xi))} {\right)}{\right]}\nn \\ &\hspace{0.45cm} + \frac{\lambda \phi_0^4}{4\Omega_0^4} {\left[}-Z_{V_0} + \frac{1}{8\pi^2{\epsilon}} {\left(}\lambda s(\phi_0) +3 \frac{g^4}{\lambda} - \frac{y^4}{\lambda}{\right)}{\right]}, \label{Gamma}\end{aligned}$$ with s(\_0) = +. It is clear that no field-independent counterterms can be defined over the whole regime. Expanding the corrections in the respective regimes we find (still applying the notation $Z_i=1+\delta_i$) \_[V\_0]{}= 1[8\^2]{}$$10 \fac\lambda + 3\frac{g^4}{\lambda} - \frac{y^4}{\lambda}$$, \[dV0\] where we used notation [(\[fac\])]{}. Note that $\delta_{V_0} = \delta_\lambda + 2\delta_\phi$ in the small and mid field regime, but $\delta_{V_0} = \delta_\lambda -2\delta_\xi$ for large field. As we will see, consistency with Higgs/GB n-point functions requires $\delta_\phi = -\delta_\xi$, as in [(\[Z\_xi\])]{}, and thus $\delta_{V_0}$ constrains the same elementary counterterms in the whole regime. Furthermore, we find \_[[2h]{}]{} = 1[8\^2]{}$$\fac 3g^2 - \fac y^2 +\O(\delta)$$. \[d2h\] In the large field regime $\delta_{{2h}} = \O(\delta)$ is a consistency check, but does not put any constraints on the elementary counterterms [(\[Zelement\])]{}. In the mid field regime we find $\delta_{2h} =2(\delta_\phi+\delta_\xi) = \O(\delta)$, and thus to lowest order [(\[Z\_xi\])]{} is satisfied. In the small field regime $\delta_{{2h}} = \delta_\phi$, and we find an answer consistent with [(\[Zphi\])]{} below. Higgs/GB interactions --------------------- We start with the corrections to the Higgs propagator. Compared to the standard small-field calculation, the fermion loop is different because of the presence of new fermion-Higgs/GB couplings. The gauge loop proceeds as in the small field regime, with the only exception that the diagram with derivative interactions cancels (at first order in the expansion parameter) as in the large field regime $g_{A \partial h \chi} =- g_{A\partial \chi h}$ have opposite sign, instead of being equal. The result for the counterterm, fermion and gauge, mixed gauge-GB and Higgs/GB loops is $$\begin{aligned} \Pi^{h} &= -\delta_{2h} \gamma_{hh} k^2 -\delta_{\lambda_{2h}} 2\lambda_{2h} + \frac1{8\pi^2{\epsilon}} \bigg[ - 12 y_h^2 m_\psi^2 - 8 y_{2h} m_\psi^3 -2y_h^2 k^2 \nn \\ &\hspace{0.45cm} +6 g_{2A 2h} m_A^2 +6 g_{2A h}^2 +3 k^2 \gamma^{\chi\chi}{\left(}\frac{g_{A \partial h \chi}+ g_{A\partial \chi h}}{2} {\right)}^2 \nn \\ &\hspace{0.45cm} +12 \gamma^{hh}\lambda_{4h} m_h^2 +2 \gamma^{\chi\chi} \lambda_{2h2\chi} m_\chi^2 + 18 (\gamma^{hh})^2\lambda_{3h}^2 + 2 (\gamma^{\chi\chi})^2 \lambda_{h2\chi}^2 \bigg] . \label{Pi_h}\end{aligned}$$ The $\gamma^{aa}$ factors stem from the Higgs and GB propagators. Further, we used $m_h^2 = \gamma^{hh} (2\lambda_{2h})$ and similar for the GB mass. This yields \_[\_[2h]{}]{} = 1[8\^2]{}$$10 \fac \lambda + 3\frac{g^4}{\lambda} - \frac{y^4}{\lambda}$$, while for the kinetic term we retrieve [(\[d2h\])]{}. Comparing with the CW result we find that $\delta_{\lambda_{2h}} =\delta_{V_0}$. In the large field regime this gives the equality $\delta_\lambda -\delta_\phi -3\delta_\xi =\delta_\lambda -2 \delta_\xi$, from which we get \_= -\_, \[delta\_xi\] which assures that $\Omega$ does not run. This is the same as derived in the mid field regime from $\delta_{2h} = \O(\delta)$. In the small field regime $\xi$ drops out of the Lagrangian at leading order, and no relation for $\delta_\xi$ can be derived at this order. The correction to the GB propagator is $$\begin{aligned} \Pi^{\chi} &= - \delta_{2\chi} \gamma_{\chi\chi} k^2 -\delta_{\lambda_{2\chi}} 2\lambda_{2\chi} + \frac1{8\pi^2{\epsilon}} \bigg[ - 4 y_\chi^2 m_\psi^2 - 8 y_{2\chi} m_\psi^3 -2y_\chi^2 k^2 \nn \\ &\hspace{0.45cm} +6 g_{2A 2\chi} m_A^2 +3 k^2 \gamma^{hh}{\left(}\frac{g_{A \partial h \chi}+ g_{A\partial \chi h}}{2} {\right)}^2 \nn \\ &\hspace{0.45cm} +12 \gamma^{\chi\chi}\lambda_{4\chi} m_\chi^2 +2 \gamma^{hh} \lambda_{2h2\chi} m_h^2 + 4 \gamma^{hh}\gamma^{\chi\chi} \lambda_{h2\chi}^2 \bigg] . \label{Pi_chi}\end{aligned}$$ We find $\delta_{\lambda_{2h}} =\delta_{\lambda_{2\chi}}$ in all three regimes, as required by gauge invariance. Further we have \_[2]{} = \_= 1[8\^2]{}$$3\fac g^2 -y^2$$. \[Zphi\] It is interesting to note that in the large field regime the counterterm $\delta_{\lambda 2\chi} = \O(\delta^3)$ whereas the individual fermion and gauge loop diagrams in [(\[Pi\_chi\])]{} are $\O(\delta^2)$. Renormalizability thus requires the two fermion diagrams to cancel at leading order, to end up with an $\O(\delta^3)$ loop correction. This is indeed what happens. This intricate cancellation is even more pronounced when we consider corrections to higher $n$-point GB scattering. For example, the structure of the fermion contribution to the four-point GB vertex is $$\begin{aligned} V^{(4\chi)} &= -\delta_{\lambda 4\chi} 4! \lambda_{4\chi} \nn \\ &\hspace{0.45cm} + \frac{ 4!} {8\pi^2{\epsilon}} \Bigg[ 36 \lambda_{4\chi}^2(\gamma^{\chi\chi})^2 + \lambda_{2h2\chi}^2(\ghh)^2 -m_h^2 \lambda_{2h4\chi} \ghh -15 m_\chi^2 \lambda_{6\chi}\gamma^{\chi\chi} + 8\lambda_{h4\chi} \lambda_{h2\chi} \ghh \gamma^{\chi\chi} \nn \\ & \hspace{1 cm} - 4 y_{4\chi} m_\psi^3 - 4y_{3\chi} y_\chi (m_\psi^2+\frac12 k^2)- 6 y_{2\chi}^2 (m_\psi^2+\frac16 k^2) -4 y_{2\chi} y_\chi^2 m_\psi - y_\chi^4 \nn \\ & \hspace{1 cm} + 3(g_{2A2\chi}^2 + g_{2A4\chi} m_A^2) \Bigg] . \label{4theta}\end{aligned}$$ Now the counterterm on the first line is $\delta_{\lambda 4\chi} = \O(\delta^5)$. The GB and Higgs loop diagrams on the second line above give $\O(\delta^6)$ corrections and can be neglected. All individual fermion loop diagrams — the terms on the third line — and all individual gauge loop diagrams — the terms on the fourth line — are $\O(\delta^3)$, much larger than the counterterm. Thus both the leading and subleading contributions need to cancel when adding the diagrams. This is indeed what happens and we find $\delta_{\lambda_{2h}} = \delta_{\lambda_{4\chi}}$ as required by gauge invariance. This intricate cancellation, and the need to go to sub-sub-leading order in the $\delta$-expansion, is the reason we cannot easily define the wavefunctions for the covariant fields, as discussed in section [(\[s:cov\_wave\])]{}. Note, however, that the $k^2$-term in [(\[4theta\])]{} above does not cancel, and gives a correction that cannot be absorbed. [^6] For this we have to add a new dimension-6 counterterm which is a four-point $\chi$-interaction with two derivatives; very schematically Ł \^2 ()\^2, with a cutoff \~(y\_[3]{}y\_+12 y\_[2]{}\^2)\^[-1/2]{} \_[unitarity]{} \[CU1\] that is equal to or larger than the unitarity cutoff \_[unitarity]{} \~ {1[ ]{} , \_0 , } \[unitarity\] in the small, mid and large field regime regime. The EFT breaks down for energy scales beyond the unitarity cutoff. The new counterterms needed to absorb divergencies enter at even higher scales. As such they do not put further constraints on the domain of validity of the EFT. Yukawa interactions ------------------- We first calculate the corrections to the fermion propagator. The fermion-gauge coupling is standard over the whole field range, and the gauge loop gives the same result in all three regimes. This is not the case for the Higgs and GB loop, as the former is suppressed in the mid and large field regime. The fermion two-point function is $$\begin{aligned} \Pi^{(2\psi)} &= -\delta_\psi \slashed{k} - \delta_{m_\psi} m_\psi + \frac1{8\pi^2{\epsilon}} \bigg[ -3 m_\psi g^2 q_L q_R \nn \\ &\hspace{0.45cm} +y_h^2 \gamma^{h h} (m_\psi -\frac12 \slashed{k}) -y_\chi^2 \gamma^{\chi\chi} (m_\psi +\frac12 \slashed{k}) +y_{2h}\gamma^{h h} m_h^2 +y_{2\chi}\gamma^{\chi\chi} m_\chi^2 \bigg], \label{Pi_fermion}\end{aligned}$$ where we used $g_{A\bar \psi_{L,R} \psi_{L,R}} = g q_{L,R}$ in all three regimes. There is a minus sign difference between the two $\slashed{k}$ terms, which originates from the $(i\gamma^5)$ in vertices with an odd number of GBs. This gives for the counterterms $$\begin{aligned} \delta_\psi &= -\frac1{8\pi^2{\epsilon}} {\left[}\frac{y^2}{4}(\fac+1) {\right]}, \nn \\ \delta_{m_\psi}&=\delta_y +\delta_\psi+ \frac12 \delta_\phi =\frac1{8\pi^2{\epsilon}}{\left(}-3 g^2 q_L q_R + \frac12(\fac-1) y^2{\right)}. \label{Z_2psi}\end{aligned}$$ The GB and Higgs contribution add in $\delta_\psi$ and cancel in $\delta_{m_\psi}$ in the small field regime; in the mid and large field regime only the GB contribution survives. The Yukawa interactions, both Higgs-fermion and GB-fermion, should give consistent results. Indeed we find $$\begin{aligned} V^{\Psi \bar{\Psi}h}_{\rm tot} &=- \frac{1}{8\pi^2{\epsilon}}\Bigl( -y_h^3\ghh +y_h y_\chi^2\gamma^{\chi \chi} -3m_h^2 y_{3h}\ghh -m_\chi^2 y_{h2\chi}\gamma^{\chi \chi} -2y_h y_{2h}\ghh \left(\slashed{k}+2m_\Psi\right) \nn\\ & \quad -\frac{y_\chi y_{h\chi}\gamma^{\chi \chi}}{2} \left(\slashed{k}-2m_\Psi\right) -6 y_{2h} \lambda_{3h} (\ghh)^2 -2 y_{2\chi} \lambda_{h2\chi}(\gamma^{\chi \chi})^2 + 3 q_L q_R g_{\bar\Psi A\Psi}^2 y_h\Bigr) -\delta_{y_h}y_h,\end{aligned}$$ and $$\begin{aligned} V^{\Psi \bar{\Psi}\chi}_{\rm tot} &=- \frac{i \gamma^5}{8\pi^2{\epsilon}}\Bigg( y_h^2 y_\chi \ghh - y_\chi^3\gamma^{\chi \chi} -m_h^2 y_{2h\chi}\ghh -3 m_\chi^2 y_{3\chi}\gamma^{\chi \chi} -\frac{y_h y_{h\chi}\ghh}{2} \left(\slashed{k}+2m_\Psi\right) \nn\\ & \quad -2y_\chi y_{2\chi}\gamma^{\chi \chi} \left(\slashed{k}+2m_\Psi\right) - 2y_{h\chi} \lambda_{h2\chi}\ghh\gamma^{\chi \chi} +3q_L q_R g_{\bar\Psi A\Psi}^2 y_\chi \Biggr) - i\gamma_5\delta_{y_\chi} y_\chi. \end{aligned}$$ This indeed gives $\delta_{y_h} = \delta_{y_\chi} = \delta _{m_\psi}$, with the latter given in [(\[Z\_2psi\])]{}. However, once again there is a small glitch as the $\slashed{k}$ terms in both expressions do not cancel. New non-renormalizable counterterms need to be added, schematically of the form Ł |(h + i\^5 ) with cutoff \~(\^y\_y\_[2]{})\^[-1]{} \_[unitarity]{}. \[CU2\] Since the cutoff exceeds the unitarity cutoff, these terms do no affect the range of validity of the EFT in the three regimes. Gauge interactions ------------------ We begin with the gauge boson propagator. $$\begin{aligned} \Pi^A_{{\mu\nu}}&= -\delta_A( k^2 g_{\mu\nu} - k_\mu k_\nu) -\delta_{m_A} m_A^2 g_{{\mu\nu}}\nn \\ &\hspace{0.45cm} + \frac1{8\pi^2 {\epsilon}}\Bigg[ {\left(}3 \gamma^{hh} g_{h2A}^2 + 2 \gamma^{hh} g_{2h2A} m_h^2 + 2\gamma^{\chi\chi} g_{2\chi 2A} m_\chi^2{\right)}g^{{\mu\nu}}\nn \\ &\hspace{0.45cm}+ \gamma^{hh} \gamma^{\chi \chi} {\left[}-\frac14 (g_{A\partial h \chi}+g_{A\partial \chi h})^2 {\left(}\frac{k^2}{3} +m_h^2 +m_\theta^2{\right)}g^{{\mu\nu}}+ (g_{A\partial h \chi}^2-g_{A\partial h \chi}g_{A\partial \chi h}+g_{A\partial \chi h}^2)\frac13k^\mu k^\nu{\right]}\nn \\ &\hspace{0.45cm} -\frac23 (k^2 g^{{\mu\nu}}-k^\mu k^\nu){\left(}g_L^2 +g_R^2{\right)}-2 (g_L -g_R)^2 m_\psi^2 \Bigg]\end{aligned}$$ It should be remembered that we normalized $q_\phi =1$. The counterterms are \_A = -1[8\^2 ]{} g\^2$ \fac \frac13 +\frac23 q_L^2 +\frac23 q_R^2$. Using the Ward identity $2\delta_g= -\delta_A $ [(\[ward\])]{}, it follows that $\delta_{m_A} = 2\delta_g + \delta_\phi + \delta_A = \delta_\phi$. Reading off $\delta_{m_A} $ from the above expression, and comparing with our earlier result [(\[Zphi\])]{} for $\delta_\phi$, we indeed find agreement. In the large field regime there is also a derivative interaction at leading order that is not transversal and cannot be absorbed in $\delta_A$. We find a term \^A 1[8\^2 ]{} k\^k\^. \[transversal\] This term can be neglected only for $\xi \gg 1$. This is the only place where this extra condition is needed. The transverse term breaks the Ward identities in the Landau gauge, and should not be there. It arises as a consequence of our approximations, discussed in more detail in section \[s:painful\]. We are not too worried about this term, as it is absent in the large $\xi$ limit. But moreover, it is also a gauge dependent term. We could have chosen a gauge fixing Ł\^E\_[GF]{} =-1[2\_G]{} $\partial^\mu A_\mu - g \frac{\phi_0}{\Omega_0} \xi_G \chi $\^2 defined in terms of the covariant fields rather than the Jordan frame fields [(\[L\_GF\])]{}. In Landau gauge, this gauge fixing gives the same results for all other diagrams, but now also the transversal part [(\[transversal\])]{} vanishes. As a consistency test we also calculated the $2A2h$ interaction, which gives $$\begin{aligned} V^{2A2h} &=\frac{g^{\mu\nu}}{8\pi^2{\epsilon}} \Bigl( 48 \lambda_{4h} g_{2A2h}q_\Phi^2(\ghh)^2 +8 \lambda_{2h2\chi} g_{2A2\chi} (\gamma^{\chi\chi})^2 - (g_{A\partial h \chi } +g_{A\partial \chi h })^2\ghh (\gamma^{\chi\chi})^2 \lambda_{2h2\chi} \nn\\ & \qquad - 6 (g_{A\partial h \chi } +g_{A\partial \chi h })^2(\ghh)^2 \gamma^{\chi\chi} \lambda_{4h} \ +24 g_{2A2h}^2 \ghh - 4 (q_L-q_R)^2y_h^2 g_{\bar \Psi A \Psi}^2 \nn\\ & \qquad -8 (q_L-q_R)^2 y_{2h}g_{\bar \Psi A\Psi}^2 m_\Psi \Bigr) -4 \delta_{g_{2A2h}} g_{2A2h} g^{\mu\nu},\end{aligned}$$ yielding $\delta_{m_A} = \delta_{g_{2A2h}}$, as it should. Gauge-fermion vertex {#s:painful} -------------------- There is one interaction that does not give a consistent result, which is the fermion-gauge coupling. To calculate it the important terms in the Lagrangian are Ł= - \_(\^) \^(\^) - + \_[i=L,R]{}i|\_i \_i - . \[oerlag\] There are three one-loop diagrams: 1) a GB loop with the photon attached to the fermion line, 2) a Higgs loop with the photon attached to the fermion line, and 3) a mixed Higgs-GB loop with the photon attached via a derivative interaction to $\Phi$. The result is $$\begin{aligned} V_{\rm loop}^{ \bar \Psi A_\mu \Psi} &= -\frac{\gamma^\mu}{16\pi^2 {\epsilon}} \Bigg[ g_{\bar \Psi A \Psi} (y_\chi^2 \gamma^{\chi\chi} +y_h^2 \ghh) \left(q_L P_R+q_R P_L\right) \nn \\ & \qquad + \left(g_{A\partial h \chi }+g_{A\partial \chi h }\right) y_h y_\chi \ghh \gamma^{\chi\chi} \left(q_\Phi P_L - q_\Phi P_R\right) \Bigg], \nn \\ V_{\rm CT}^{ \bar \Psi A_\mu \Psi} &= -\delta_{ \bar \Psi A_\mu \Psi}g_{\bar \Psi A)\mu \Psi} \gamma^\mu(q_L P_L + q_R P_R) .\end{aligned}$$ In the small field regime this reduces to V\_[loop]{}\^[ |A\_]{} =-(q\_L P\_L + q\_R P\_R ), where we used gauge invariance: $ q_\Phi - q_L + q_R =0$. This expression can be absorbed in the counterterm, which is proportional to $\left(q_L P_L + q_R P_R \right)$ as well. In the large field regime, however, the diagrams with a Higgs loop are suppressed and we get V\_[loop]{}\^[ |A\_]{} = . We cannot combine the two parts, and will not get something proportional to $\left(q_L P_L + q_R P_R \right)$. The same problem arises in the mid-field regime. This result in the large field regime breaks gauge invariance explicitly. How did it arise? When we repeat the calculation without the first term in [(\[oerlag\])]{}, the Higgs and GB field still have the same propagator. As a result all three diagrams contribute and the result adds up to something gauge invariant. However, when we include the first term, things go wrong as the Higgs propagator is now suppressed compared to the GB propagator. Note however, that the first term is explicitly gauge invariant. It is our approximation that breaks the gauge invariance, when we evaluate the metric on the background $\gamma_{ij} (\phi,\theta) = \gamma_{ij} (\phi_0)$. In particular, for the first term we set Ł- \_(\^) \^(\^) = - (\_)\^2+ ... where the ellipses denote neglected higher order derivative interactions (to be precise: higher n-point interactions with two derivatives). We listed this as the third approximation in subsection \[s:checks\]. These higher order terms need to be included to obtain a gauge invariant result. Unfortunately, it does not seem straightforward to do so. We would like to postpone the setup of a framework able to handle higher order derivative terms to future work, leaving a loose thread to our current calculation. However, since the calculation of the two-point function involves lower order vertices, we expect it to be less prone to our approximation. RGE equations {#s:beta} ============= First we give the betafunctions for the Abelian-Higgs model with a non-minimal coupling, then in subsection \[s:SM\_beta\] we generalize to full SM Higgs inflation. Abelian Higgs model {#s:abelian_beta} ------------------- First we list all the counterterms, found in the previous section: $$\begin{aligned} \delta_\phi =-\delta_\xi& =\frac1{8\pi^2{\epsilon}} {\left(}3g^2 \fac - y^2{\right)},\nn \\ \delta_{\lambda_{2h}} = \delta_{\lambda_{4h}}=\delta_{\lambda_{2\chi}} = \delta_{\lambda_{4\chi}}&=\frac1{8\pi^2{\epsilon}}{\left(}10 \lambda \fac +3 \frac{g^4}{\lambda} -\frac{y^4}{\lambda} {\right)},\nn\\ \delta_\psi &= -\frac{1}{8\pi^2{\epsilon}}\left(\frac{y^2}{4} (\fac+1) \right),\nn\\ \delta_{m_\psi} = \delta_{y_{h\bar \psi \psi}} =\delta_{y_{\chi\bar \psi \psi}} & =\frac1{8\pi^2{\epsilon}}{\left(}-3 g^2 q_L q_R+\frac12 y^2 (\fac-1) {\right)},\nn\\ \delta_A &= -\frac{1}{8\pi^2{\epsilon}}g^2 (\frac{1}{3}q_\phi^2 \fac +\frac{2}{3}q_L^2+\frac{2}{3}q_R^2). \nn\\\end{aligned}$$ The counterterms for the couplings in the Lagrangian are then given by $\delta_\lambda=\delta_{\lambda_{4h}} -2 \delta_\phi$, $\delta_y = \delta_{m_\psi} -1/2 \delta_\phi -\delta_\psi$, and the Ward identiy $\delta_g =-1/2 \delta_A$ respectively. From this the beta-functions can be found via $\beta_\lambda = \lambda ({\epsilon}\delta_\lambda)$, and similarly for the Yukawa, gauge and non-minimal Higgs-gravity coupling. This gives $$\begin{aligned} \beta_\lambda & = \frac1{8\pi^2}{\left(}10 \fac \lambda^2 +3 g^4 -y^4 -6 \fac g^2 \lambda +2 y^2\lambda {\right)},\nn \\ \beta_y & = \frac1{8\pi^2}{\left(}-3 q_L q_R g^2 y-\frac32 \fac (q_L-q_R)^2g^2y +\frac14(1+3\fac) y^3{\right)}\nn \\ \beta_g&=\frac{1}{8\pi^2} g^3 {\left(}\fac\frac{1}{6}q_\phi^2 +\frac{1}{3}q_L^2+\frac{1}{3}q_R^2{\right)},\nn \\ \beta_\xi \big|_{\rm mid, large} &=-\frac1{8\pi^2} (\fac 3 g^2-y^2) \xi. \label{betaU1}\end{aligned}$$ We have not derived $\beta_\xi$ in the small field regime. At leading order all $\xi$ dependence drops out of the Lagrangian in the SM regime. Since our computation relies on the approximations listed in section \[s:checks\], which fail beyond the leading order, we have to leave the computation of $\beta_\xi$ in the small field regime open. SM Higgs inflation {#s:SM_beta} ------------------ Our results for a U(1) theory can be extended to the full Standard Model (SM) Higgs inflation. Working in background field gauge [^7], the symmetries of the classical effective action are similar to those of the U(1) theory. The main difference is that now there are 3 GBs, the top quark has three colors, and one needs to sum over the strong, weak and hypercharge interactions. #### Higgs coupling First we extend the U(1) results in the small field regime to the full SM beta-functions, which can be found for example in [@sher]. The SM betafunction for the Higgs self-coupling is a straightforward generalization of the U(1) result: $$\begin{aligned} \beta_\lambda &= \frac{1}{8\pi^2} {\left[}(9+n_\theta) \lambda^2 +3 \sum_a g_a^4 - n_c y_t^4 - 2\lambda ( 3\sum_a g_a^2 -n_c y_t^2) {\right]}\nn \\ &= \frac{1}{8\pi^2} {\left[}12 \lambda^2 + \frac{3}{16} {\left(}2 g_2^4 + (g_2^2+g_1^2)^2{\right)}-3 y_t^4 - 2\lambda (\frac94 g_2^2 +\frac34 g_1^2 -3y_t^2) {\right]},\end{aligned}$$ with $n_\theta =3$ the number of GBs, $n_c =3$ the number of colors and the $g_a$ are given in [(\[ga\])]{}. We have only included the running of the top Yukawa. Generalizing from the U(1) model, it follows that the $\lambda^2$ and the $\lambda g_{1,2}^2 $ terms are suppressed in the mid and large field regime. This gives \_ = $$24 \lambda^2\fac + \frac{3}{8} {\left(}2 g_2^4 + (g_2^2+g_1^2)^2{\right)}-6 y_t^4 - \lambda (9 g_2^2 +3{g_1}^2)\fac +12 y_t^2 \lambda)$$. #### Gauge coupling For an $SU(N)$ group the betafunction is (g) |\_[SU(N)]{}= - $ 22N - 2 n_f - n_H$, with $n_f$ the number of Weyl fermions and $n_s$ the number of complex Higgs fields, both in the fundamental representation. For an Abelian group there is no contribution from the gauge field, and the formula becomes (g) |\_[U(N)]{}= $4 \sum q_f^2 +\sum q_s^2$, with $q_f,q_s$ the charges of the Weyl fermion and real scalars respectively. This reproduces our result in the small field regime. In the mid and large field the Higgs and GB contributions to the running are absent; this does not affect QCD, but for the EW sector we get $$\begin{aligned} \beta_{g_3} &= -\frac{7}{(4\pi)^2} g_3^3,\nn\\ \beta_{g_2} &= -\frac{1}{(4\pi)^2} \frac{(20 - \fac)}{6}g_2^3, \nn\\ \beta_{g_1} &= \frac{1}{(4\pi)^2} \frac{(40 + \fac)}{6}g_1^3. \nn\end{aligned}$$ #### Yukawa coupling The running of the top Yukawa follows from the counterterm $\delta_{y} = \delta_{y_{h\bar \psi \psi}}-1/2(\delta_\phi +\delta_{t_L} +\delta_{t_R})$. Explicit expressions can e.g. be found in [@zhou; @tubitak]. For the top quark the SM counterterms are: $$\begin{aligned} \delta_{t_L} &=-\frac{1}{8\pi^2{\epsilon}} {\left(}\frac{\fac}{4} y_t^2 + \frac14 y_t^2 + \frac12 y_b^2{\right)}, \nn \\ \delta_{t_R} &=-\frac{1}{8\pi^2{\epsilon}} {\left(}\frac{\fac}{4} y_t^2 + \frac14 y_t^2 + \frac12 y_t^2{\right)}.\end{aligned}$$ The $y^2$ contributions stem from loops with $h$, $\chi$ and $\phi^+ = \chi_1-i\chi_2$ respectively. For $t_L$ the $\phi^+$ loop can only be made with a bottom quark in the loop, which gives a contribution to $y_b^2$ — which we neglect in the following. In our U(1) toy model we only had the first two contributions from the $h$ and $\chi$-loop; indeed this matches the counterterm we found before. In the mid and large field regime the Higgs loop is suppressed, but not the GB loop. Likewise, we expect the $\phi^+$-loop to contribute in the large field regime, as these are GBs, with the same structure of interactions as $\chi$. The Higgs counterterm is \_=- $$n_c y_t^2 -3 \fac\sum_a g_a^2$$ =- $$3 y_t^2 -\frac34 \fac ( 3 g_2^2 +g_1^2)$$, which generalizes our previous results. Here $n_c$ is the number of colors. The gauge contribution is suppressed in the large field regime. The vertex correction is \_[y\_[h|]{}]{}= - $$-\frac12y_t^2 (\fac-1) +3 Y_{tL} Y_{tR} g_1^2+ 3 C_2(R_t)g_3^2$$, \[delta\_PhiQ\] with $C_2(R_t) =4/3$ for the fundamental in SU(3), and $Y_{tL}=1/6,\, Y_{tR}=2/3$. For the U(1) model we found that the $y^2$-correction from the $h$ and $\chi$-loop cancels in the small field regime. However, in the large field regime the Higgs contribution is negligible, and there is a net contribution from the GB $\chi$. The $\phi^+$ loop gives a contribution $\propto y_b^2$ and can be neglected. The gauge terms stem from top-top-gauge loops, and since the fermion-gauge couplings are standard in the large field regime, these are unaffected. This gives for the betafunction \_[y\_t]{} = 1[(4)\^2]{} $$\frac32 (2+\fac) y_t^3 -{\left(}\frac{8+9\fac }{12} g_1^2 +\frac{9\fac}{4} g_2^2 +8g_3^2 {\right)}y_t$$. In the small field regime $\fac =1$ and we get the standard result. #### Non-minimal coupling Further, we found in the large and mid field regime that $\delta_\phi =-\delta_\xi$. This gives the betafunction for the non-minimal coupling \_|\_[mid, large]{} = $$6 y^2 -\frac32 \fac ( 3 g_2^2 +g_1^2)$$. ### End results $$\begin{aligned} (4\pi)^2 \beta_\lambda &= 24 \lambda^2\fac + A +(4\pi)^2 \cdot 4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= -\frac{\fac}{4} (3g_1^2+9 g_2^2) + 3y_t^2 \nn \\ (4\pi)^2\beta_{g_3} &= -7 g_3^3,\nn\\ (4\pi)^2\beta_{g_2} &= - \frac{(20 - \fac)}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{(40 + \fac )}{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= \frac32 \fac y_t^3 -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t+(4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi\big|_{\rm mid, large} &= (4\pi)^2 \cdot2 \gamma_\phi \xi \label{final}\end{aligned}$$ with $A= (3/8)(2g_2^4 + (g_2^2+g_1^2)^2) -6y_t^4$. Discussion {#s:concl} ========== Comparison with literature. --------------------------- In recent years several groups have presented renormalization group equations for SM Higgs inflation. The disagreement between these results has been a major motivation to write this paper which follows, in our opinion, the most systematic approach so far. In this section we compare our findings to some encountered in the recent literature. Let us first quickly compare this work to our own previous work [@damien]. There we have studied the renormalization of just a (complex) non-minimally coupled scalar, leaving the inclusion of fermions and gauge fields and the generalization to full SM Higgs inflation to this work. Our findings here generalize those in [@damien]. Note in particular that in that work, we concluded that we could not say anything about the RG flow in the mid-field regime, as the corrections were an order of $\delta$ smaller than the counterterms. However, the fermionic and some of the gauge corrections that we have found now are of the same order as the counterterms. That is why we can now present expressions for the running couplings in the mid field regime (barring threshold corrections) without getting in contradiction with our previous work. Now for the comparison to other authors. In general, it seems that all other approaches follow some predefined treatment for Higgs and Goldstone bosons. In some cases only the Higgs contributions are kept, in other cases only the GBs, in yet other cases only loop contributions are excluded, etcetera. Our result does not respect any of these guidelines. For example, we find that the GB contribution to the effective potential is suppressed, while the GB loops do contribute to the Yukawa corrections. To see exactly which field contributes to which loop correction all correction diagrams need to be properly computed. (Although the differences are never very dramatic.) We first compare to reference [@bezrukov3], which states that in large field the action is just the action of the chiral SM with $v = {m_{\rm p}}/\sqrt{\xi}$, and the Higgs (but not the GBs) decouples. The RGEs quoted for the large field regime are $$\begin{aligned} (4\pi)^2 \beta_\lambda &= A + (4\pi)^2 \cdot4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= -\frac{1}{4} (3g_1^2+6 g_2^2) + 3y_t^2 \nn \\ (4\pi)^2\beta_{g_2} &= - \frac{(20 - 1/2)}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{(40 + 1/2)}{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t +(4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi &= (4\pi)^2 \cdot 2 \gamma_\phi \xi.\end{aligned}$$ Here a gauge contribution in $\gamma_\phi$ has been included, which explains the difference in $\beta_\lambda$, $\beta_\xi$ and $\beta_{y_t}$ with our work. In the betafunctions for the gauge coupling only the real Higgs field is excluded, whereas we also exclude the GB. The chiral SM is non-renormalizable, and new operators have to be included. At one-loop level there is a correction to the Z-boson mass, which depends on the running coefficients of two of these new operators. There is no such thing in our set-up, which is renormalizable in the EFT sense. We furthermore note that in this work the running of $\xi$ is computed with an approach very different from ours, via the running of the SM vev $v$. However, apart from our disagreement over the gauge contribution to $\gamma_\phi$, we find the same answer. Reference [@wilczek] states that in the inflationary regime quantum loops involving the Higgs field are heavily suppressed. The proposed prescription (originally introduced in [@Salopek:1988qh]) is to assign one factor of $s(\phi)$ for every off-shell Higgs that runs in a quantum loop, with s() = 1[\^2]{} \_ = . Although the metric factor in $s$ above is for the real Higgs field, judging from the RGEs presented the prescription has been applied to the complex field (nothing is said explicitly about Higgs and GBs). In large field, the quoted RGEs reduce to: $$\begin{aligned} (4\pi)^2 \beta_\lambda &= A + (4\pi)^2 \cdot4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= -\frac{1}{4} (3g_1^2+9 g_2^2) + 3y_t^2 \nn \\ (4\pi)^2\beta_{g_2} &= - \frac{20}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{40 }{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= -3 y_t^3 -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t + (4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi &= 2 ((4\pi)^2 \cdot\gamma_\phi + 6\lambda) ( \xi+1/6).\end{aligned}$$ There is a gauge contribution to $\gamma_\phi$, which explains the difference in $\beta_\lambda$, $\beta_\xi$ and partly $\beta_{y_t}$ with our result. In $\beta_{g_i}$ the full Higgs doublet is taken out in the large field regime, in agreement with our result. $\beta_\xi$ is found by taking gravity as a classical background, following the pioneering work in [@Odintsov1; @Odintsov2; @Odintsov3]. We think that this is not a good approximation in the Jordan frame. This same s-factor formalism was followed in References [@rose; @kyle], with the modification that now only the loops of the real Higgs field are excluded, and not those of the GBs. However our final answers agree with neither of the results obtained there. Reference [@barvinsky3] writes that Goldstone modes, in contrast to the Higgs particle, do not have mixing with gravitons in the kinetic term. Therefore, their contribution is not suppressed by the $s$-factor. We disagree with this. The GBs cannot be treated as usual, as in polar coordinates $\rho^2 (\partial \theta)^2$ the radial field is not the canonical one. In Cartesian coordinates, all fields are equally coupled to the Ricci tensor via $\xi R \sum (\phi^i)^2$. The quoted results are $$\begin{aligned} (4\pi)^2 \beta_\lambda &= 6 \lambda^2 + A - (4\pi)^2 \cdot 4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= \frac{1}{4} (3g_1^2+9 g_2^2) - 3y_t^2 \nn \\ (4\pi)^2\beta_{g_2} &= - \frac{(20 - 1/2)}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{(40 + 1/2)}{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= {\left[}y_t^3 -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t{\right]}-(4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi &= 6\xi \lambda -(4\pi)^2 \cdot2\gamma_\phi \xi.\end{aligned}$$ A gauge contribution to $\gamma_\phi$ is included, which partly explains the difference in $\beta_\lambda$, $\beta_\xi$ and $\beta_{y_t}$ with our results. For $\beta_{y_t}$ the contribution of one GB $y_t^2$-term has been excluded instead of 3GB $y_t^2$-terms. In $\beta_{g_i}$ only the GB is taken out in the large field regime, in disagreement with our result. This should be the identical to the chiral model of [@bezrukov3], as the Higgs field is decoupled in the large field limit. However the RGEs are still different. Lastly, [@bezrukov3; @wilczek] use two different normalization conditions, one with a field independent cutoff in the Jordan frame, or with a field dependent cutoff in the Jordan frame. However, two frames give identical physics. It is often quoted that a field independent cutoff in the Jordan frame corresponds to a field dependent cutoff in the Einstein frame, and vice versa. However, dimensionful quantities by themselves have no invariant meaning, their values depend on the unit system. If we express the cutoff in Planck units (the Planck mass is frame dependent), a constant cutoff in the one frame is equivalent to a constant cutoff in the other frame. The conformal rescaling only rescales [*all*]{} length scales, which does not change the physics. See also our discussion in [@damien]. On-shell equivalence between the frames has also been established in [@christian]. The question about a field dependent or independent cutoff is a frame invariant question when expressed in Planck units. The choice of cutoff has thus nothing to do with a choice of frame. One can still debate whether the results depend on the (field dependent) choice of cutoff in the Einstein frame. A priori, this is not expected; the cutoff is only introduced to regularize the divergent integrals, but is at the end taken to infinity. Requiring the counterterms to be field independent, the different field dependent and independent cutoffs lead to different normalization conditions. In practice one can only relate physical measurements at different energy scales. The translation between the observable and the coupling defined in the normalization condition will be different in each case. The end result is that when comparing physical observables at different scales, the cutoff dependence drops out. Conclusions ----------- We have calculated the one-loop corrections to Higgs inflation in the small, mid and large field regime. We have done the calculations for the Abelian Higgs model; the results can then rather straightforwardly be generalized to full Standard Model Higgs inflation. We have found that in all three regimes the model is renormalizable in the effective field theory sense. The RGEs for SM Higgs inflation we found are given in [(\[final\])]{}. The results for the mid field regime are new. The running of the non-minimal coupling can be derived in the mid and large field regime, and follows from the consistency of the radiative corrections to the potential and to the two-point functions. In the small field regime all dependence on the non-minimal coupling drops out of the equations at leading order in the small field expansion, and nothing can be said about its running. The computation of the radiative corrections was done in the Einstein frame, in the Landau gauge, using a covariant formalism for the multi-field system. The one-loop corrections to the propagators are sufficient to determine the full set of counterterms, and thus the betafunctions. As extra checks, we have calculated many higher n-point functions as well. Especially the results for four-point scattering of the Goldstone bosons are impressive in this regard: in the large (and mid) field regime both the leading and subleading divergencies exactly cancel, yielding a consistent counterterm. However, we have stumbled on some potential problematic outcomes as well. First of all, to cancel all divergencies, new non-renormalizable counterterms need to be added, see (\[CU1\], \[CU2\]). However, the cutoff implied by these new counterterms always exceeds the unitarity cutoff (\[unitarity\]). Therefore they do not put further constraints on the validity of the EFTs in the various regimes. Second, in (\[transversal\]) we have seen that the gauge boson propagator picks up a transverse part, that should be absent by the Ward identities. This term vanishes in the large $\xi$ limit. Moreover, it is gauge dependent, and we believe it should vanish in a full calculation. Thirdly, the one-loop gauge-fermion vertex gives a gauge symmetry breaking result as well. We can trace it back to the explicit symmetry breaking in our approximation of the non-minimal kinetic terms. In a full calculation the symmetry should be restored, yielding a result consistent with the gauge propagator corrections that we have found, but we leave this for further work. In conclusion, we have computed the full set of RGE equations for Standard Model Higgs inflation. The Higgs-fermion part has withstood an impressive set of consistency checks. When including the gauge symmetry our result obtained from propagator corrections has failed one consistency test, which we think can be ascribed to the intrinsic limitations in our approach (neglecting higher order kinetic terms by evaluating the field metric on the background). It would be an interesting but equally challenging task to develop a framework that can get around these limitations. Acknowledgments {#acknowledgments .unnumbered} =============== DG is funded by a Herchel Smith fellowship. SM is supported by the Fondecyt 2015 Postdoctoral Grant 3150126 and by the “Anillo” project ACT1122, funded by the “Programa de Investigación Asociativa." MP is funded by the Netherlands Foundation for Fundamental Research of Matter (FOM) and the Netherlands Organisation for Scientific Research (NWO). We thank Mikhail Shaposhnikov and Sergey Sibiryakov for illuminating discussions. Couplings {#s:vertices} ========= We list the couplings for the Abelian U(1) model. The explicit values in the small, mid, and large field regime are found expanding in $\delta_i$: $$\begin{aligned} & \delta_s = \xi \phi_0, & {\rm small} \nn \\ &\xi \to \delta_m^{-2}\xi,\;\phi_0 \to \delta_m^{3/2} \phi_0, & {\rm mid} \nn \\ & \delta_l = 1/(\xi \phi_0^2) . & {\rm large}\end{aligned}$$ Note that contrary to the small and large field regime, the $\delta_m$ parameter is just a rescaling parameter. In the small field we express the results in $\phi_0$ rather than $\delta_s$, as this form is more familiar. Below we give the leading expression for the metric and for the relevant couplings; the three values between the braces correspond to the small, mid and large field regime. The metric is: $$\begin{aligned} \gamma_{hh} & =\left\{1,~\frac{6 \phi_0^2 \xi ^2}{\delta _m}, ~(6 \xi +1) \delta _l\right\} \nn \\ \gamma_{\chi\chi} & =\left\{1,~1,~\delta _l\right\}.\end{aligned}$$ The Higgs and GB self-interactions are $$\begin{aligned} \lambda_{2h} &= \frac{1}{2!} V_{;\phi\phi} \Big|_{\rm bg} =\lambda \left\{ \frac32 \phi_0^2, ~ \phi_0^2 \delta_m^3, ~ -\frac1\xi \delta_l^2 \right\} \nn\\ \lambda_{2\chi} &= \frac{1}{2!} V_{;\theta\theta} \Big|_{\rm bg} =\lambda \left\{ \frac12 \phi_0^2, ~ \frac{1}{12\xi^2} \delta_m^4, ~ \frac{ \delta_l^3}{2\xi(1+6\xi)} \right\} \nn\\ \lambda_{3h} &= \frac{1}{3!} V_{;\phi\phi\phi} \Big|_{\rm bg} =\lambda \left\{ \phi_0,~ {\left(}\frac1{18\phi_0 \xi^2} -2\phi_0^3 \xi{\right)}\delta_m^{5/2}, ~ \frac{2\delta_l^{5/2}}{3\sqrt{\xi}} \right \} \nn\\ \lambda_{h2\chi} &= \frac{1}{3!} \left(V_{;\phi\theta\theta} + V_{;\theta \phi \theta} + V_{;\theta \theta \phi} \right) \Big|_{\rm bg} =\lambda\left\{\phi_0, ~ \frac{\delta_m^{5/2}}{18 \phi_0 \xi^2},~ -\frac{4 \delta_l^{7/2}}{3\sqrt{\xi}(1+6\xi)} \right\} \nn\\ \lambda_{4h} &= \frac{1}{4!}V_{;\phi\phi\phi\phi} \Big|_{\rm bg} = \lambda\left\{ \frac14,~ -\frac{\delta_m}{18\phi_0^2 \xi^2}, ~ ~- \frac{\delta_l^3}{3} \right\} \nn\\ \lambda_{2h2\chi} &= \frac{1}{4!} \left( V_{;\phi\phi\theta\theta} + {\rm 5 perms} \right) \Big|_{\rm bg} =\lambda \left\{ \frac12, ~ -\frac{\delta_m}{18\phi_0^2\xi^2},~ \frac{11 \delta_l^4}{6(1+6\xi)} \right\} \nn\\ \lambda_{4\chi} &= \frac{1}{4!}V_{;\theta\theta\theta\theta} \Big|_{\rm bg} =\lambda \left\{\frac14,~ \frac{\delta_m^2}{432 \phi_0^4 \xi^4},~ -\frac{\delta_l^5}{3(1+6\xi)^2} \right\} \nn\\ \lambda_{5h} &= {\rm etc.}\end{aligned}$$ The Yukawa interactions are $$\begin{aligned} m_\psi & =F^\phi \Big|_{\rm bg} =\frac{y}{\sqrt{2}} \left\{\phi_0 ,~\phi_0 \delta _m^{3/2},~\frac{1}{\sqrt{\xi }}\right\} \nn \\ y_h & =F^\phi_{;\phi} \Big|_{\rm bg} =\frac{y}{\sqrt{2}} \left\{1,~1,~\delta_l^{3/2}\right\} \nn\\ y_{2h} &= \frac1{2!}F^\phi_{;\phi\phi} \Big|_{\rm bg} =\frac{y}{\sqrt{2}} \left\{-(3 \xi^2 +\xi) \phi_0,~ -\frac{1}{2\phi_0 \delta _m^{3/2}}, -\sqrt{\xi } \delta _l^2 \right\} \nn \\ y_{3h} & =\frac{1}{3!}F^\phi_{;\phi\phi\phi} \Big|_{\rm bg} = \frac{y}{\sqrt{2}} \left\{-\frac13 (3 \xi^2 +\xi),~-\frac{1}{2 \phi_0^2 \delta _m^{3}},~\frac23 \xi \delta_l^{5/2} \right\} \nn \\ y_{4h} &= {\rm etc.} \nn \\ y_\chi &=F^\theta_{;\theta} \Big|_{\rm bg} = \frac{y}{\sqrt{2}}\left\{1,~1,~ \sqrt{\delta_l} \right\} \nn \\ y_{2\chi} &=\frac1{2!} F^\phi_{;\theta\theta} \Big|_{\rm bg} =\frac{y}{\sqrt{2}}\left\{-(3 \xi^2 +\xi) \phi_0 ,~-\frac{1}{2 \phi_0 \delta _m^{3/2}},~-\frac12\sqrt{\xi } \delta _l \right\} \nn \\ y_{3\chi} &=\frac{1}{3!}F^\theta_{;\theta\theta\theta} \Big|_{\rm bg} = \frac{y}{\sqrt{2}} \left\{-\frac13 (3 \xi^2 +\xi),~-\frac{1}{6 \phi_0^2 \delta _m^{3}},~-\frac16 \xi \delta_l^{3/2} \right\} \nn \\ y_{4\chi} &=\frac1{4!}F^\phi_{;\theta\theta\theta\theta} \Big|_{\rm bg} = \frac{y}{\sqrt{2}} \left\{{\left(}\frac{\xi^2}{3} +3 \xi^3 +\frac{15\xi^4}{2}{\right)}\phi_0,~\frac{1}{24 \phi_0^3 \delta _m^{9/2}},~\frac1{24} \xi^{3/2} \delta_l^{2} \right\}\end{aligned}$$ with F\^= , F\^= , as follows from [(\[VF\])]{}. Finally, the gauge interactions are $$\begin{aligned} m_A^2 & =g^2 G \Big|_{\rm bg} = g^2\left\{\ \phi_0^2, ~\phi_0^2 \delta_m^3, ~ \frac1\xi \right\} \nn \\ g_{2Ah} & =g^2 G_{;\phi} \Big|_{\rm bg} = g^2\left\{\ \phi_0, ~\phi_0 \delta_m^{3/2}, ~ \frac{\delta_l^{3/2}}{\sqrt{\xi}} \right\} \nn \\ g_{2A2h} & =\frac1{2!} g^2 G_{;\phi\phi} \Big|_{\rm bg} = g^2\left\{\ \frac12, ~{\left(}\frac1{12\phi_0^2 \xi^2} -\phi_0^2\xi{\right)}\delta_m, ~ -\delta_l^{2} \right\} \nn\\ g_{2A2\chi} & =\frac1{2!} g^2 G_{;\theta\theta} \Big|_{\rm bg} = g^2\left\{\ \frac12, ~\frac{\delta_m}{12\phi_0^2 \xi^2}, ~ \frac{\delta_l^{3}}{2(1+6\xi) }\right\} \nn\\ g_{A \chi \partial h } &= g G^\phi_{;\theta} \Big|_{\rm bg} =g\left\{1,~1,~\delta _l \right\} \nn \\ g_{Ah \partial \chi } &= g G^\theta_{;\phi} \Big|_{\rm bg} =g\left\{1,~1,~-\delta _l\right\} \nn\\ g_{A \bar \psi_{L,R} \psi_{L,R}} & = g q_{L,R} \end{aligned}$$ with $G,G^\phi,G^\theta$ defined in [(\[VF\])]{}. To get the two derivative couplings, we have simply set $\partial \phi=\partial h$ and $\partial \theta=\partial\chi$. For higher derivative couplings (to be precise: with still one derivative but more fields, such as $g_{A 2 h \partial \chi }$) we need to go beyond the first terms in these expansions, by using [(\[Q\_expl\])]{}. [9]{} D. S. Salopek, J. R. Bond, J. M. Bardeen, Phys. Rev.  [**D40** ]{} (1989) 1753. F. L. Bezrukov, M. Shaposhnikov, Phys. Lett.  [**B659** ]{} (2008) 703-706. \[arXiv:0710.3755 \[hep-th\]\]. F. Bezrukov, D. Gorbunov, M. Shaposhnikov, JCAP [**0906** ]{} (2009) 029. \[arXiv:0812.3622 \[hep-ph\]\]. F. L. Bezrukov, A. Magnin and M. Shaposhnikov, Phys. Lett. B [**675**]{} (2009) 88 \[arXiv:0812.4950 \[hep-ph\]\]. F. Bezrukov, M. Shaposhnikov, JHEP [**0907** ]{} (2009) 089. \[arXiv:0904.1537 \[hep-ph\]\]. F. Bezrukov, A. Magnin, M. Shaposhnikov and S. Sibiryakov, JHEP [**1101**]{} (2011) 016 \[arXiv:1008.5157 \[hep-ph\]\]. C. P. Burgess, H. M. Lee and M. Trott, JHEP [**0909**]{} (2009) 103 \[arXiv:0902.4465 \[hep-ph\]\]. J. L. F. Barbon and J. R. Espinosa, Phys. Rev. D [**79**]{} (2009) 081302 \[arXiv:0903.0355 \[hep-ph\]\]. C. P. Burgess, H. M. Lee and M. Trott, JHEP [**1007**]{} (2010) 007 \[arXiv:1002.2730 \[hep-ph\]\]. M. P. Hertzberg, JHEP [**1011**]{}, 023 (2010) \[arXiv:1002.2995 \[hep-ph\]\]. C. P. Burgess, S. P. Patil and M. Trott, JHEP [**1406**]{}, 010 (2014) \[arXiv:1402.1476 \[hep-ph\]\]. S. Ferrara, R. Kallosh, A. Linde, A. Marrani and A. Van Proeyen, Phys. Rev. D [**83**]{} (2011) 025008 \[arXiv:1008.2942 \[hep-th\]\]. I. G. Moss, arXiv:1409.2108 \[hep-th\]. J. Ren, Z. Z. Xianyu and H. J. He, JCAP [**1406**]{}, 032 (2014) \[arXiv:1404.4627 \[gr-qc\]\]. M. P. Hertzberg, JCAP [**1208**]{}, 008 (2012) \[arXiv:1110.5650 \[hep-ph\]\]. F. Bezrukov, M. Y. .Kalmykov, B. A. Kniehl and M. Shaposhnikov, JHEP [**1210**]{}, 140 (2012) \[arXiv:1205.2893 \[hep-ph\]\]. G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori and A. Strumia, JHEP [**1208**]{}, 098 (2012) \[arXiv:1205.6497 \[hep-ph\]\]. V. Branchina and E. Messina, Phys. Rev. Lett.  [**111**]{}, 241801 (2013) \[arXiv:1307.5193 \[hep-ph\]\]. V. Branchina and E. Messina, arXiv:1507.08812 \[hep-ph\]. A. Kobakhidze and A. Spencer-Smith, arXiv:1404.4709 \[hep-ph\]. A. Spencer-Smith, arXiv:1405.1975 \[hep-ph\]. A. V. Bednyakov, B. A. Kniehl, A. F. Pikelner and O. L. Veretin, arXiv:1507.08833 \[hep-ph\]. A. De Simone, M. P. Hertzberg and F. Wilczek, Phys. Lett.  B [**678**]{} (2009) 1 \[arXiv:0812.4946 \[hep-ph\]\]. A. O. Barvinsky, A. Y. .Kamenshchik and A. A. Starobinsky, JCAP [**0811**]{} (2008) 021 \[arXiv:0809.2104 \[hep-ph\]\]. A. O. Barvinsky, A. Y. .Kamenshchik, C. Kiefer, A. A. Starobinsky and C. Steinwachs, JCAP [**0912**]{}, 003 (2009) \[arXiv:0904.1698 \[hep-ph\]\]. A. O. Barvinsky, A. Y. Kamenshchik, C. Kiefer, A. A. Starobinsky and C. F. Steinwachs, arXiv:0910.1041 \[hep-ph\]. D. P. George, S. Mooij and M. Postma, JCAP [**1402**]{}, 024 (2014) \[arXiv:1310.2157 \[hep-th\]\]. M. Postma and M. Volponi, Phys. Rev. D [**90**]{} (2014) 10, 103516 \[arXiv:1407.6874 \[astro-ph.CO\]\]. M. Atkins and X. Calmet, Phys. Rev. Lett.  [**110**]{}, no. 5, 051301 (2013) \[arXiv:1211.0281 \[hep-ph\]\]. Z. Z. Xianyu, J. Ren and H. J. He, Phys. Rev. D [**88**]{}, 096013 (2013) \[arXiv:1305.0251 \[hep-ph\]\]. J. -O. Gong and T. Tanaka, JCAP [**1103**]{}, 015 (2011) \[Erratum-ibid.  [**1202**]{}, E01 (2012)\] \[arXiv:1101.4809 \[astro-ph.CO\]\]. J. Elliston, D. Seery and R. Tavakol, JCAP [**1211**]{}, 060 (2012) \[arXiv:1208.6011 \[astro-ph.CO\]\]. D. I. Kaiser, E. A. Mazenc and E. I. Sfakianakis, Phys. Rev. D [**87**]{}, 064004 (2013) \[arXiv:1210.7487 \[astro-ph.CO\]\]. A. Salvio and A. Mazumdar, arXiv:1506.07520 \[hep-ph\]. M. Srednicki, “Quantum field theory,” Cambridge, UK: Univ. Pr. (2007) 641 p D. I. Kaiser, Phys. Rev. D [**81**]{}, 084044 (2010) \[arXiv:1003.1159 \[gr-qc\]\]. S. Mooij and M. Postma, JCAP [**1109**]{}, 006 (2011) \[arXiv:1104.4897 \[hep-ph\]\]. D. P. George, S. Mooij and M. Postma, JCAP [**1211**]{} (2012) 043 \[arXiv:1207.6963 \[hep-th\]\]. S. R. Coleman, E. J. Weinberg, Phys. Rev.  [**D7** ]{} (1973) 1888-1910. M. Sher, Phys. Rept.  [**179**]{} (1989) 273. B. -R. Zhou, Commun. Theor. Phys.  [**26**]{} (1996) 97. M. Ozer, Turk. J. Phys.  [**22**]{} (1998) 351. I. L. Buchbinder and S. D. Odintsov, Yad. Fiz.  [**42**]{}, 1268 (1985) \[Class. Quant. Grav.  [**2**]{}, 721 (1985)\]. I. L. Buchbinder, S. D. Odintsov and I. L. Shapiro, “Effective action in quantum gravity,” Bristol, UK: IOP (1992) 413 p E. Elizalde and S. D. Odintsov, Phys. Lett. B [**321**]{}, 199 (1994) \[hep-th/9311087\]. R. N. Lerner and J. McDonald, Phys. Rev. D [**83**]{}, 123522 (2011) \[arXiv:1104.2468 \[hep-ph\]\]. K. Allison, JHEP [**1402**]{}, 040 (2014) \[arXiv:1306.6931 \[hep-ph\]\]. A. Y. Kamenshchik and C. F. Steinwachs, Phys. Rev. D [**91**]{}, no. 8, 084033 (2015) \[arXiv:1408.5769 \[gr-qc\]\]. [^1]: [dpg39@cam.ac.uk]{}, , [^2]: Higgs inflation is non-renormalizable as the field space metric and potential are non-polynomial. But this does not exclude that the theory is renormalizable in the EFT sense (as is the case in the IR). Our demands are that in the large and mid field regime the theory can be expanded in a small parameter $\delta$, and that all loop corrections can be absorbed in counterterms order by order. Truncating the theory at some finite order in $\delta$ gives a renormalizable EFT with a finite number of counterterms. [^3]: For considerations about initial conditions for Higgs inflation, and the possible inclusion of a $R^2$-term, see [@anupam] [^4]: Equivalently, and probably more easily, the expansion $\phi^i(Q^i)$ can be found by $\phi^i = \phi^i_{:a} Q^a + \frac1{2!} \phi^i_{:ab} Q^a Q^b+ ...$. [^5]: One could leave the counterterms $Z_{h}$, $Z_{\chi}$, $Z_\phi$ unrelated a priori, and determine them by the requirement to absorb all (1-loop) divergencies. We tried this approach, and it fails. There is no consistent choice of counterterms that renders the theory finite. [^6]: Here $k$ is the momentum flowing in the loop at one of the vertices; the propagator structure is $\int {{\rm d}}^4 l D_\psi(l) D_\psi(l+k)$. [^7]: In practice, it is not so easy to calculate diagrams in the background field gauge, as it is unclear how to expand the Lagrangian in covariant fields with a shifted metric.

--- abstract: 'This work deals with partial MDS (PMDS) codes, a special class of locally repairable codes, used for distributed storage system. We first show that a known construction of these codes, using Gabidulin codes, can be extended to use any maximum rank distance code. Then we define a standard form for the generator matrices of PMDS codes and use this form to give an algebraic description of PMDS generator matrices. This implies that over a sufficiently large finite field a randomly chosen generator matrix in PMDS standard form generates a PMDS code with high probability. This also provides sufficient conditions on the field size for the existence of PMDS codes.' author: - 'Alessandro Neri and Anna-Lena Horlemann-Trautmann' bibliography: - 'PMDS\_stuff.bib' title: Random Construction of Partial MDS Codes --- Introduction ============ In a distributed storage system a file $x\in {{\mathbb{F}}}_q^k$ (where ${{\mathbb{F}}}_q$ denotes the finite field of cardinality $q$), is encoded and stored as some codeword $c\in {{\mathbb{F}}}_q^n$, over several storage nodes. Each of these nodes is assumed, for simplicity, to store exactly one coordinate of $c$. In case some of the nodes fail, we want to be able to recover the lost information using as little effort as possible. The *locality* of a code plays an important role in this context, and it denotes the number of nodes one has to contact for repairing a lost node. We call the set of nodes one has to contact if a given node fails, the locality group of that node. For this work we assume that the locality groups are distinct. *Partial MDS codes* are *maximally recoverable* codes in this setting, i.e., any erasure pattern that is information theoretically correctable is correctable with such a code. It is known that maximally recoverable codes in general, and PMDS codes in particular, exist for any locality configuration if the field size is large enough [@ch07]. Furthermore, some constructions of PMDS codes are known, e.g. [@bl13; @bl14; @bl16; @ch15; @go14]. In this paper we describe a random construction of PMDS codes, by prescribing a generator matrix of the respective code in a specific form, which we will call *PMDS standard form*. If we fill the non-prescribed coordinates of this generator matrix with random values, by high probability, the resulting code is PMDS, if the underlying field size is large enough. We derive a lower bound on this probability (depending on the field size). This gives rise to a lower bound on the necessary field size for PMDS codes to exist. With some final adjustments on the random construction we get a lower bound that improves the one of [@ch07]. Preliminaries ============= PMDS codes ---------- Consider a distributed storage system with $m$ disjoint locality groups, where the $i$-th group is of size $n_i$ ($i=1,\dots,m$) and can correct any $r_i$ erasures. First we set the locality for the code to be $\ell \in \mathbb N$. We can divide the coordinates of the code into blocks of length $n_1, \dots, n_m$, where $n_i=\ell+r_i$, such that each block represents a locality group. We denote an MDS code of length $n$ and dimension $k$ by $[n,k]$-MDS code. We use the definition of PMDS codes given in [@ht17], which generalizes the definition of [@bl13]. Let $\ell,m, r_1,\dots,r_m \in \mathbb N$. Define $n:=\sum_{i=1}^m (r_i+\ell)$ and let $C\subseteq {{\mathbb{F}}}_q^n$ be a linear code of dimension $k<n$ with generator matrix $$\label{eq:genmatrix} G= \left( B_1 \mid \dots \mid B_m \right) \in {{\mathbb{F}}}_q^{k\times n}$$ such that $B_i\in {{\mathbb{F}}}_q^{k\times ( r_i + \ell)}$. Then $C$ is a $[n,k, \ell ; r_1,\dots,r_m]$-*partial-MDS (PMDS) code* (with locality $\ell$) if - for $i\in \{1,\dots,m\}$ the row space of $B_i$ is a $[r_i+\ell, \ell ]$-MDS code, and - for any $r_i$ erasures in the $i$-th block ($i=1,\dots, m$), the remaining code (after puncturing the coordinates of the erasures) is a $[m\ell, k]$-MDS code. The erasure correction capability of PMDS codes is as follows: \[lem:PMDScap\][@ht17 Lemma 3] A $[n,k, \ell ; r_1,\dots,r_m]$-PMDS code can correct any $r_i$ erasures in the $i$-th block (simultaneously) plus $s:= m\ell -k$ additional erasures anywhere in the code. We can see that the definition of PMDS codes given makes sense only for $k\geq \ell$. In case of equality, or in the case that $m=1$ there exist only trivial PMDS codes, i.e. the only PMDS codes are MDS codes. It was shown in [@ht17] that a code is a $[n,k, \ell; r_1,\dots,r_m]$-PMDS code if and only if it is maximally recoverable (for the respective locality group configuration). The same results had previously been shown in [@go14 Lemma 4] for the case $r_1=r_2=\dots = r_m$. Now we give a summary on known results about PMDS codes. [@ch07]\[prop4\] Maximally recoverable (MR) codes of length $n$ and dimension $k$ exist for any locality configuration over any finite field of size $q> \binom{n-1}{k-1}$. MR codes are PMDS codes for disjoint locality blocks. Therefore, Proposition \[prop4\] implies that PMDS codes exist for any set of parameters when the field size is large enough. A construction of PMDS codes based on rank-metric and MDS codes was given in [@ca17], when $r_1=r_2=\dots=r_m$. This gives the following existence result: [@ca17] $[n,k,\ell; r,\dots,r]$-PMDS codes with $m$ locality blocks of the same length exist over a finite field of size $q^{n-mr}$. Furthermore, some specific constructions of PMDS codes, for particular values of $s$ or of the $r_i$, are given in [@bl13; @bl14; @bl16; @go14]. In particular, a general construction for PMDS codes with $s=1$ was given in [@ht17]. This construction is based on the concatenation of several MDS codes as building blocks. [@ht17 Corollary 14] 1. For any integers $m\geq 2 $ and $\ell,r_1,\dots,r_m \geq 1$ there exists a $[n,k=m\ell-1, \ell; r_1,\dots,r_m]$-PMDS code over any field ${{\mathbb{F}}}_q$ with $q\geq \max_i\{r_i\}+\ell$. 2. If there exists $h\in \mathbb N$ such that $\ell\in \{3,2^h-1\}$ and $\max_i \{r_i\} +\ell = 2^h+1$, then there exists a $[n,k=m\ell-1, \ell; r_1,\dots,r_m]$-PMDS code over ${{\mathbb{F}}}_q$ with $q=2^h= \max_i\{r_i\}+\ell -1$. In [@ht17] the authors also show that this construction is basically the only one possible, i.e., every PMDS with $s=1$ is of this form, giving thus a characterization for this set of parameters. However, for $s\geq 2$ there is no characterization yet for PMDS codes. Zarisky topology over finite fields ----------------------------------- Let ${{\mathbb{F}}}$ be a field, and ${{\mathbb{F}}}[x_1,\ldots,x_N]$ be the polynomial ring over ${{\mathbb{F}}}$. Denote by $\bar{{{\mathbb{F}}}}$ the algebraic closure of ${{\mathbb{F}}}$. For a subset $S\subseteq {{\mathbb{F}}}[x_1,\ldots,x_N]$ we define the *algebraic set* $$V(S): = \{{\boldsymbol}\alpha \in \bar{{\mathbb{F}}}_q^r \mid f({\boldsymbol}\alpha) = 0, \forall f \in S\} .$$ The *Zariski topology* on $\bar{{{\mathbb{F}}}}^N$ is defined as the topology whose closed sets are the algebraic sets, while the complements of the Zariski-closed sets are the *Zariski-open sets* [@ha13b Ch. I, Sec. 1]. A subset $A\subset\bar{{{\mathbb{F}}}}^N$ is called a *generic set* if $A$ contains a non-empty Zariski-open set. In classical geometry one studies the Zariski topology over the complex numbers. In this framework, a generic set inside $\mathbb C^N$ is dense and its complement is contained in an algebraic set of dimension at most $N-1$. If one wants to consider generic sets restricted to a finite field ${{\mathbb{F}}}_q$, the situation is slightly different. Here, for a subset $T\subseteq{{\mathbb{F}}}_q^N$ one can always find a set of polynomials $S\subseteq {{\mathbb{F}}}_q[x_1,\dots,x_N]$ such that $$T=\{{\boldsymbol}\alpha \in {{\mathbb{F}}}_q^N \mid f({\boldsymbol}\alpha) = 0, \forall f \in S\}.$$ and therefore the Zariski topology restricted to ${{\mathbb{F}}}_q^N$ is the discrete topology. This means that it is not useful to extend the notion of generic sets to finite fields since it would not give any information. However, given a set of polynomials $S\subseteq{{\mathbb{F}}}_q[x_1,\ldots,x_N]$, we can define the set of *${{\mathbb{F}}}_q$-rational points* as $$V(S;{{\mathbb{F}}}_{q}): = \{{\boldsymbol}\alpha \in {{\mathbb{F}}}_{q}^N \mid f({\boldsymbol}\alpha) = 0, \forall f \in S\} .$$ In this setting the Schwartz-Zippel Lemma implies an analog result to the one of generic sets, as explained in the following. [@le98p Lemma 1.1]\[lem:SZ\] Let $f\in {{\mathbb{F}}}_q[x_1,\dots,x_N]$ be a non-zero polynomial of total degree $d \geq 0$. Let $T\subseteq \bar{{{\mathbb{F}}}}$ be a finite set and let $\alpha_1, \dots, \alpha_N$ be selected at random independently and uniformly from $T$. Then $$\Pr\big(f(\alpha_1,\ldots,\alpha_N)=0\big)\leq\frac{d}{|T|}.$$ As a consequence of this result we have that, in case the size of $S$ and the total degrees of the polynomials in $S$ do not depend on the finite field, the proportion between the cardinality of $ V(S;{{\mathbb{F}}}_{q})$ and the cardinality of the whole space ${{\mathbb{F}}}_q^N$ goes to $0$ as $q$ grows. Vice versa, for growing $q$ the probability that a random point is in the complement of $ V(S;{{\mathbb{F}}}_{q})$ tends to $1$. This result will be crucial in Section \[sec:topprob\] for our random construction of PMDS codes. Rank-metric codes ----------------- We now give some known facts about rank-metric codes. Recall that ${\mathbb{F}_{q^N}}$ is isomorphic to ${\mathbb{F}_q}^N$ as an ${\mathbb{F}_q}$- vector space. From this it easily follows that ${\mathbb{F}_{q^N}}^n\cong {\mathbb{F}_q}^{N\times n}$. Then we can give the following definition. The *rank distance* $d_R$ on ${{\mathbb{F}}}_q^{N\times n}$ is defined by $$d_R(U,V):= {\mathrm{rank}}(U-V) , \quad U,V \in {{\mathbb{F}}}_q^{N\times n}.$$ Analogously, if $\boldsymbol u,\boldsymbol v \in {\mathbb{F}_{q^N}}^n$, then $d_R(\boldsymbol u,\boldsymbol v)$ is the rank of the difference of the respective matrix representations in ${{\mathbb{F}}}_q^{N\times n}$. Observe that the definition of rank distance in the case of vectors in ${\mathbb{F}_{q^N}}^n$ does not depend on the choice of the basis. Moreover it can be shown that the function $d_R: {\mathbb{F}_{q^N}}^n \times {\mathbb{F}_{q^N}}^n \rightarrow {{\mathbb{R}}}_{\geq 0}$ is a metric. An *${\mathbb{F}_{q^N}}$-linear rank-metric code* ${\mathcal{C}}$ of length $n$ and dimension $k$ is a $k$-dimensional subspace of ${\mathbb{F}_{q^N}}^n$ equipped with the rank distance. The *minimum distance* of ${\mathcal{C}}$ is defined as $$d_R({\mathcal{C}}):=\min\left\{d_R({\boldsymbol}u,{\boldsymbol}v) \mid {\boldsymbol}u,{\boldsymbol}v\in {\mathcal{C}}, {\boldsymbol}u\neq {\boldsymbol}v\right\}.$$ \[th:SB\][@ro91 Theorem 1] Let ${\mathcal{C}}\subseteq {\mathbb{F}_{q^N}}^n$ be an ${\mathbb{F}_{q^N}}$-linear rank-metric code of dimension $k$. Then $$d_R({\mathcal{C}})\leq n-k+1.$$ Codes attaining the Singleton-like bound are called *Maximum Rank Distance (MRD) Codes*. A necessary and sufficient condition for the existence of MRD codes is that $n\leq N$. In this framework, a characterization for ${\mathbb{F}_{q^N}}$-linear MRD codes in terms of their generator matrices was given in [@ho16 Corollary 2.12], which in turn is based on a result given in [@ga85]. For this we define the set $${\mathcal E}_q(k,n):=\left\{E \in {{\mathbb{F}}}_q^{k\times n} \mid {\mathrm{rank}}(E)=k\right\}.$$ \[prop:MRDCrit\] Let $G\in {{\mathbb{F}}}_{q^m}^{k\times n}$ be a generator matrix of a rank-metric code $\mathcal{C}\subseteq {{\mathbb{F}}}_{q^m}^n$. Then $\mathcal{C}$ is an MRD code if and only if $${\mathrm{rank}}(GE^T) =k$$ for all $E\in {\mathcal E}_q(k,n)$. General construction using rank metric codes ============================================ In this section we generalize the construction given in [@ca17]. In that work the authors use Gabidulin codes in order to build $[n,k, \ell, r,\ldots,r]$-PMDS codes. We will show that this construction also works for different $r_i$, and that Gabidulin codes can be replaced by any linear MRD codes. Fix $n, k,\ell, r_1,\ldots,r_m$, and let $\widetilde{G} \in {\mathbb{F}_{q^N}}^{k\times m\ell}$ be the generator matrix of a MRD code. For the existence of an MRD code we need $N\geq m\ell$. Moreover, for every $i=1,\ldots, m$, we consider a $[\ell+r_i,\ell]$-MDS code over ${\mathbb{F}_q}$ with generator matrix $M_i$, and define $$\label{eq:MDSblock} M:=\left(\begin{array}{cccc} M_1 & 0 & \ldots & 0 \\ 0 & M_2 &\ldots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & \ldots & 0 & M_m \end{array}\right) \in {\mathbb{F}_q}^{m\ell \times n}.$$ We can now formulate our PMDS construction. \[thm16\] Let $\widetilde{G} \in {\mathbb{F}_{q^N}}^{k\times m\ell}$ be the generator matrix of a MRD code and let $M$ be the matrix defined in (\[eq:MDSblock\]). Then the matrix $\widetilde{G}M$ is a generator matrix for a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code over ${\mathbb{F}_{q^N}}$. Let $G:=\widetilde{G}M$ and let $S\in \mathcal T_{k,\ell}(G)$ be the submatrix obtained by selecting columns $h_1,\ldots,h_{k_j}$ from the $j$th block for $j=1,\ldots,m$, where $k_i\leq \ell$ and $k_1+\ldots+k_m=k$. $S$ is equal to $\widetilde{G}\widetilde{M}$, where $$\widetilde{M}=\left(\begin{array}{cccc} N_1 & 0 & \ldots & 0 \\ 0 & N_2 &\ldots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & \ldots & 0 & N_m \end{array}\right),$$ and $N_j$ is the $\ell \times k_j$ submatrix of $M_j$ obtained by the respective selected columns. Since $M_i$ generates an $[\ell+r_i, \ell]$-MDS code, any $\ell$ columns of $M_i$ are linearly independent. Thus, ${\mathrm{rank}}(N_i)=k_i$ and ${\mathrm{rank}}(\widetilde{M})=k_1+\ldots+k_m=k$. By Proposition \[prop:MRDCrit\] we have that $\det(\widetilde{G}\widetilde{M})\neq 0$, and we conclude the proof using Proposition \[prop:PMDS\]. \[cor:exPMDS\] Let $m\geq 2 $ and $\ell,r_1,\dots,r_m \geq 1$, $k\geq \ell$ be positive integers. Then, for every prime $p$ and every positive integer $L\geq n_0m\ell$ there exists a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code over ${{\mathbb{F}}}_{p^L}$, where $$n_0=\min\{j \in {{\mathbb{N}}}\mid p^j \geq \ell + r_i-1, \mbox{ for } i=1,\ldots,m\}.$$ A MRD code in ${\mathbb{F}_{q^N}}^{ m\ell}$ exists if $N\geq m\ell$. Suitable MDS codes over ${{\mathbb{F}}}_q$ for the matrix in exist if $q\geq \max\{\ell + r_i - 1\}$. The statement follows from Theorem \[thm16\] with $q=p^{n_0}$ and $L=n_0N$. Algebraic description of PMDS codes =================================== We will now define a standard form for generator matrices of PMDS codes. This standard form is the main tool for the random construction of PMDS codes. \[thm:stform\] Let $m\geq 2 $ and $s, \ell,r_1,\dots,r_m \geq 1$ and let ${\mathcal{C}}$ be a $[n,k=m\ell-s, \ell; r_1,\dots,r_m]$-PMDS code over a field ${{\mathbb{F}}}_q$. Then ${\mathcal{C}}$ has a generator matrix of the form $$\label{eq:stform} {G}= \left({B}_1 \mid \dots \mid {B}_m \right),$$ where - ${B}_i=({C}_i \mid {D}_i)$, ${C}_i \in {{\mathbb{F}}}_q^{k \times \ell}$ and ${D}_i \in {{\mathbb{F}}}_q^{k\times r_i}$ for $i=1,\ldots, m$, and - the submatrix ${G}_{C}=\left({C}_1 \mid \dots \mid {C}_m \right)$ is of the form $${G}_C=\left[ I_{k} \mid A \right],$$ with $A$ being superregular. Let $\widetilde{G}$ be a generator matrix for ${\mathcal{C}}$ of the form (\[eq:genmatrix\]), i.e. $$\widetilde{G}= \left(\widetilde{B}_1 \mid \dots \mid \widetilde{B}_m \right).$$ Puncturing every block $\widetilde{B}_i$ in the last $r_i$ columns, we get that the submatrix $\widetilde{G}_C$ is the generator matrix of a $[m\ell,k]$-MDS code. Operating on the rows of such a submatrix we can transform it to a matrix $G_C=\left[ I_{k} \mid A \right]$, with $A$ superregular. I.e., there exists an invertible matrix $P \in \mathrm{GL}_k({\mathbb{F}_q})$ such that $P\widetilde{G}_C=\left[ I_{k} \mid A \right]$, and therefore the matrix $G:=P\widetilde{G}$ is a generator matrix of ${\mathcal{C}}$ of the required form. We now consider the entries $a_{w,z}$ of $A$ as variables $x_{w,z}$ for $w=1,\ldots,k$ amd $z=1,\ldots,s$. We know that the column space of ${D}_i$ is inside the column space of ${C}_i$, by the parameters of the block MDS codes. This means that every column in ${{D}_i}$ is a linear combination of the columns of ${{C}_i}$. If we denote by ${{D}_i}^{(j)}$ the $j$th column of ${{D}_i}$, then $$\label{eq:columns} {{D}_i}^{(j)}=\sum_{t=1}^{\ell}y_{t,i,j}{{C}_i}^{(t)}$$ for some $y_{t,i,j}$, which we also consider variable. This way we can consider a $k\times n$ generator matrix as a matrix in ${{\mathbb{F}}}_q[x_{w,z}, y_{t,i,j}]^{k\times n}$ (where ${{\mathbb{F}}}_q[x_{w,z}, y_{t,i,j}]$ denotes the polynomial ring in all $x_{w,z}, y_{t,i,j}$). Let $R=\sum_{i=1}^m r_i$. We denote $\boldsymbol{\alpha} := (\alpha_{w,z})_{w,z}\in {{\mathbb{F}}}_q^{sk}$ and $\boldsymbol{\beta} := (\beta_{t,i,j})_{t,i,j} \in {{\mathbb{F}}}_q^{\ell R}$. If we replace the variables $x_{w,z}, y_{t,i,j}$ described above in a matrix in PMDS standard form by the values $\alpha_{w,z}, \beta_{t,i,j}$, we denote the corresponding generator matrix by $$G(\boldsymbol{\alpha},\boldsymbol{\beta}).$$ Analogously we will denote the variable form by $ G(\boldsymbol{x},\boldsymbol{y})$. However, a general matrix of this form is not necessarily a generator matrix of a PMDS code for any values $\boldsymbol{\alpha},\boldsymbol{\beta}$. The following proposition shows what needs to be fulfilled to generate a PMDS code: \[prop:PMDS\] A matrix $G\in {{\mathbb{F}}}_q^{k\times n}$ generates a $[n,k=m\ell-s, \ell; r_1,\dots,r_m]$-PMDS code if and only if, every submatrix in the set $$\mathcal T_{k,\ell}(G):=\left\{S\in {{\mathbb{F}}}^{k \times k} \mid \begin{array}{l}S \mbox{ is a submatrix of } G \mbox{ with }\\ \mbox{ at most } \ell \mbox{ columns per block } B_i \end{array}\right\}$$ has non-zero determinant. This follows from the definition of PMDS, cf. also [@ht17]. The above results give an algebraic description of the generator matrix of a $[n,k=m\ell-s, \ell; r_1,\dots,r_m]$-PMDS code over ${{\mathbb{F}}}_q$, as follows. If we consider the variable form of a generator matrix $G$ as above, and the polynomial $$\label{eq:poly} p(\boldsymbol{x},\boldsymbol{y}):= \mathrm{lcm} \{ \det S \mid S\in\mathcal T_{k,\ell}(G)\} \in {{\mathbb{F}}}_q[x_{w,z}, y_{t,i,j}],$$ then, we have that $G(\boldsymbol{\alpha},\boldsymbol{\beta})$ generates a $[n,k=m\ell-s, \ell; r_1,\dots,r_m]$-PMDS code over ${{\mathbb{F}}}_q$ if and only if $p(\boldsymbol{\alpha},\boldsymbol{\beta})$ is non-zero. Topological and probability results {#sec:topprob} =================================== In this section we first deal with the algebraic description of the generator matrix of a PMDS code in the algebraic closure of the finite field where we want our code to be built. After that, we analyze the probability that a code whose generator matrix is of the form $G(\boldsymbol{\alpha},\boldsymbol{\beta})$ is PMDS. Moreover, we also study the existence of PMDS codes for given parameters $n,k,\ell, s, r_1,\ldots,r_m$ and $R=\sum_{i=1}^m r_i$, giving sufficient conditions on the field size. Although for $s=1$ this problem was completely solved in [@ht17], for $s\geq 2$ this is still an open problem. We denote the set of valid entries for PMDS generator matrices over the algebraic closure of the finite field ${\mathbb{F}_q}$ by $$\small{\mathcal A_{\mathrm{PMDS}}:=\left\{(\boldsymbol{\alpha},\boldsymbol{\beta}) \in \bar{{{\mathbb{F}}}}_q^{sk}\times \bar{{{\mathbb{F}}}}_q^{\ell R} \mid \mathrm{row space}(G(\boldsymbol{\alpha},\boldsymbol{\beta})) \mbox{ is PMDS}\right\},}$$ Then the following result holds. \[th:gen\] $\mathcal A_{\mathrm{PMDS}}$ is a generic set. By Proposition \[prop:PMDS\] we have that $$\mathcal A_{\mathrm{PMDS}}:=\left\{(\boldsymbol{\alpha},\boldsymbol{\beta}) \in \bar{{{\mathbb{F}}}}_q^{sk}\times \bar{{{\mathbb{F}}}}_q^{\ell R} \mid p(\boldsymbol{\alpha},\boldsymbol{\beta})\neq 0 \right\},$$ and therefore $\mathcal A_{\mathrm{PMDS}}$ is a Zariski open set. Concerning the non-emptiness, let $q=p^{t_0}$. From Corollary \[cor:exPMDS\] there exists a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code ${\mathcal{C}}$ over ${{\mathbb{F}}}_{p^L}$, for some $L$ multiple of $t_0$. By Theorem \[thm:stform\], ${\mathcal{C}}$ has a generator matrix of the form $G(\boldsymbol{\alpha},\boldsymbol{\beta})$, therefore $(\boldsymbol{\alpha},\boldsymbol{\beta})\in \mathcal A_{\mathrm{PMDS}}$. This means that over the algebraic closure, by probability $1$, for randomly chosen $\boldsymbol{\alpha},\boldsymbol{\beta}$ the matrix $G(\boldsymbol{\alpha},\boldsymbol{\beta})$ generates a PMDS code. For underlying *finite* fields, this implies that for growing field size this probability will tend to $1$. We now derive a probability formula depending on the field size. We can easily observe that the entries of $G(\boldsymbol{x},\boldsymbol{y})$ are polynomials of total degree $0,1$ or $2$. In particular, if $t:=\lceil\frac{s}{\ell}\rceil$ and we write $G(\boldsymbol{x},\boldsymbol{y})$ as in , then the entries of the blocks ${{D}_i}$ are polynomials of degree at most $1$ for $i=1,\ldots, m-t$, and of degree at most $2$ for the last $t$ blocks. To estimate the degree of $p(\boldsymbol{x},\boldsymbol{y}) $ we need the following lemma. \[lem:Vid\] $$\sum_{j=0}^r (r-j)\binom{m}{j}\binom{n}{r-j}=n\binom{m+n-1}{r-1}.$$ \[lem:deg\] The total degree of the polynomial $p(\boldsymbol{x},\boldsymbol{y})$, defined as in (\[eq:poly\]), satisfies the inequality $$\deg p(\boldsymbol{x},\boldsymbol{y}) \leq 2(n-k)\binom{n-1}{k-1}.$$ It holds that $$\mathcal T_{k,\ell}(G) \subset \mathcal M_k(G):=\{ S \in {\mathbb{F}_q}^{k\times k} \mid S \mbox{ is a submatrix of } G\},$$ hence the polynomial $p(\boldsymbol{x},\boldsymbol{y})$ divides the polynomial $$q(\boldsymbol{x},\boldsymbol{y}):=\mathrm{lcm}\{\det S \mid S\in \mathcal M_k(G)\}.$$ Observe that the entries of the first $k$ columns of the submatrix ${G}_C$ have degree 0. Let $t:=\lceil\frac{s}{\ell}\rceil$. Then the entries of the columns corresponding to the blocks ${{D}_i}$ for $i=1,\ldots, m-t$ have degree at most $1$, as well as the last $m\ell -k$ columns of ${G}_C$. Finally, the columns of the blocks ${{D}_i}$ for $i=m-t+1,\ldots, m$, have degree at most $2$. In particular, all the entries of the blocks ${{D}_i}$ and the last $m\ell -k$ columns of ${G}_C$ have degree at most $2$. Therefore, $$\begin{aligned} \deg q(\boldsymbol{x},\boldsymbol{y}) &\leq \sum_{S \in \mathcal M_k(G)} \deg \det S \\ &\leq \sum_{j_0=0}^k2(k-j_0) \binom{k}{j_0}\binom{n-k}{k-j_0}\\ &=2(n-k)\binom{n-1}{k-1}\end{aligned}$$ where the last equality follows from Lemma \[lem:Vid\]. Since $\deg p(\boldsymbol{x},\boldsymbol{y}) \leq \deg q(\boldsymbol{x},\boldsymbol{y})$ we conclude the proof. We can now formulate a lower bound for the probability that a randomly chosen generator matrix in PMDS standard form generates a PMDS code over a finite field ${{\mathbb{F}}}_q$: \[thm:Prob\] Let the entries of $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ be uniformly and independently chosen at random in ${\mathbb{F}_q}$. Then $$\mathrm{Pr}\{\mathrm{row space}(G(\boldsymbol{\alpha},\boldsymbol{\beta})) \mbox{ is PMDS }\}\geq 1- \frac{2(n-k)\binom{n-1}{k-1}}{q}.$$ We have $$\begin{aligned} &\mathrm{Pr}\{\mathrm{row space}(G(\boldsymbol{\alpha},\boldsymbol{\beta})) \mbox{ is PMDS }\} \\ =&\mathrm{Pr}\{(\boldsymbol{\alpha},\boldsymbol{\beta}) \notin V(p(\boldsymbol{x},\boldsymbol{y});{\mathbb{F}_q})\} \\ =&1-\mathrm{Pr}\{p(\boldsymbol{\alpha},\boldsymbol{\beta})=0\} \\ \geq& 1-\frac{\deg p(\boldsymbol{x},\boldsymbol{y})}{q}\geq 1 - \frac{2(n-k)\binom{n-1}{k-1}}{q},\end{aligned}$$ where the last two inequalities follow from Lemmas \[lem:SZ\] and \[lem:deg\], respectively. From this we can deduce an existence result for PMDS codes over finite fields of a given minimal size. If $q>2(n-k)\binom{n-1}{k-1}$ then there exists a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code over the finite field ${\mathbb{F}_q}$. One notices that this is not an improvement over the known existence result from Proposition \[prop4\]. However, we can improve the above result, considering a step-by-step construction. We will again consider a generator matrix in PMDS standard form as in . We start with an $[m\ell,k]$-MDS code over a finite field ${\mathbb{F}_q}$ and write its generator matrix as $(C_1 \mid \dots \mid C_m)$. For this purpose it is sufficient that $q\geq m\ell -1$. Then we construct the first column $D_1^{(1)}$ of the block $D_1$ as in (\[eq:columns\]). Every entry will be a degree $1$ polynomial in the variables $y_{t,1,1}$ for $t=1,\ldots,\ell$. Imposing that every $k\times k$ minor of $$G':=\left( {C}_1\mid {D}_1^{(1)} \mid {C}_2\mid \ldots \mid {C}_m \right)$$ is non-zero, we get the condition $p'(y_{t,1,1})\neq 0$, where $$p'(y_{t,1,1})=\mathrm{lcm}\{ \det S \mid S \in \mathcal T_{k,\ell}(G')\}.$$ Using Lemma \[lem:SZ\], we obtain $$\mathrm{Pr}\{p'(\beta_{t,1,1})= 0\}\leq \frac{\deg p'}{q}.$$ In this situation $\deg p' \leq \binom{m\ell}{k-1}$, therefore for $q> \binom{m\ell}{k-1}$ we have that there exists at least one evaluation of $p'$ that is non-zero and such that $G'(\beta_{t,1,1})$ generates a $[n,k,\ell, 1,0,\ldots,0]$-PMDS code. Repeating this construction step by step, we get a $[n,k,\ell,r_1,\ldots,r_{m-1},r_{m}-1]$-PMDS code. From that code we build the last column ${D}_m^{(r_m)}$ of the block ${D}_m$ again as in (\[eq:columns\]): $${D}_m^{(r_m)}=\sum_{t=1}^{\ell}y_{t,m,r_m}{C}_m^{(t)}.$$ In the end we get the matrix $$\label{eq:G} {G}(y_{t,m,r_m})=({C}_1\mid {D}_1 \mid \ldots \mid {C}_m \mid {D}_m ),$$ where the matrix ${G}(y_{t,m,r_m})$ without the last column generates a $[n,k,\ell,r_1,\ldots,r_{m-1},r_{m}-1]$-PMDS code, and the entries of the last column are polynomials of total degree at most $1$ in the variables $y_{t,m,r_m}$, for $t=1,\ldots,\ell$. Let $m, n, k, n_1,\ldots,n_m, f_1,\ldots,f_m$ be positive integers such that $n=\sum_i n_i$. Let $N_0:=0$, $N_i:=\sum_{j=1}^i n_j$ and $J_i=\{N_{i-1}+1,\ldots,N_i\}$ for $i=1,\ldots,m-1$. We define the set $$\mathcal M(k;n_1\ldots,n_m;f_1,\ldots,f_m) =\> \left\{I \subset \{1,\ldots,n\}\mid |I|=k, |I\cap J_i|\leq f_i\right\}$$ and $M(k;n_1\ldots,n_m;f_1,\ldots,f_m)$ as its cardinality. Let ${G}(y_{t,m,r_m})$ be as in (\[eq:G\]). The total degree of the polynomial $$\tilde{p}(y_{t,m,r_m}):=\mathrm{lcm}\{\det S \mid S\in \mathcal T_{k,\ell}({G}(y_{t,m,r_m}))\}$$ is less or equal to $$M(k-1;\ell+r_1\ldots,\ell+r_{m-1},\ell+r_m-1;\ell,\ldots,\ell,\ell-1) =: M^*.$$ The polynomial $\tilde{p}(y_{t,m,r_m})$ has degree less or equal to $\sum \deg \det S$. By assumption all the determinants $\det S$ for $S$ not containing the last column are non zero elements in ${\mathbb{F}_q}$. The only polynomials with degree $1$ are the determinants of $k\times k$ submatrices involving the last column, and they are exactly $M^*$ many. \[cor25\] If $q>M^*$ then there exists a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code over the finite field ${\mathbb{F}_q}$. To our knowledge there is no closed formula for $M^*$. However, it is easy to see that $M^* \leq \binom{n-1}{k-1}$ and that the inequality is strict if any of the conditions $|I\cap J_i|\leq f_i$ is non-empty. Hence, Corollary \[cor25\] improves upon Proposition \[prop4\]. Conclusion ========== We gave a generalization of a known PMDS code construction based on rank-metric codes. Furthermore, we investigated a random construction of PMDS codes by prescribing a PMDS standard form. We derived a lower bound on the probability that a randomly filled matrix in PMDS standard form generates a PMDS code. This probability implies a lower bound on the field size needed for such codes to exist. In the end we gave a step-by-step construction of such a generator matrix to improve this lower bound on the necessary field size.

--- abstract: | We present effective pre-training strategies for neural machine translation (NMT) using parallel corpora involving a pivot language, i.e., source-pivot and pivot-target, leading to a significant improvement in source$\rightarrow$target translation. We propose three methods to increase the relation among source, pivot, and target languages in the pre-training: 1) step-wise training of a single model for different language pairs, 2) additional adapter component to smoothly connect pre-trained encoder and decoder, and 3) cross-lingual encoder training via autoencoding of the pivot language. Our methods greatly outperform multilingual models up to +2.6% <span style="font-variant:small-caps;">Bleu</span> in WMT 2019 French$\rightarrow$German and German$\rightarrow$Czech tasks. We show that our improvements are valid also in zero-shot/zero-resource scenarios.\ author: - | Yunsu Kim$^{1\hspace{-0.1em}}\Thanks{\hspace{0.5em}Equal contribution.}$ Petre Petrov$^{1,2*}$ Pavel Petrushkov$^{2}$ Shahram Khadivi$^{2}$ Hermann Ney$^{1}$\ $^{1}$RWTH Aachen University, Aachen, Germany\ [{surname}@cs.rwth-aachen.de]{}\ $^{2}$eBay, Inc., Aachen, Germany\ [{petrpetrov,ppetrushkov,skhadivi}@ebay.com]{}\ bibliography: - 'references.bib' title: | Pivot-based Transfer Learning for Neural Machine Translation\ between Non-English Languages --- Introduction ============ Machine translation (MT) research is biased towards language pairs including English due to the ease of collecting parallel corpora. Translation between non-English languages, e.g., French$\rightarrow$German, is usually done with pivoting through English, i.e., translating French (*source*) input to English (*pivot*) first with a French$\rightarrow$English model which is later translated to German (*target*) with a English$\rightarrow$German model [@de2006catalan; @utiyama2007comparison; @wu2007pivot]. However, pivoting requires doubled decoding time and the translation errors are propagated or expanded via the two-step process. Therefore, it is more beneficial to build a single source$\rightarrow$target model directly for both efficiency and adequacy. Since non-English language pairs often have little or no parallel text, common choices to avoid pivoting in NMT are generating pivot-based synthetic data [@bertoldi2008phrase; @chen2017teacher] or training multilingual systems [@firat2016zero; @johnson2017google]. In this work, we present novel transfer learning techniques to effectively train a single, direct NMT model for a non-English language pair. We pre-train NMT models for source$\rightarrow$pivot and pivot$\rightarrow$target, which are transferred to a source$\rightarrow$target model. To optimize the usage of given source-pivot and pivot-target parallel data for the source$\rightarrow$target direction, we devise the following techniques to smooth the discrepancy between the pre-trained and final models: - Step-wise pre-training with careful parameter freezing. - Additional adapter component to familiarize the pre-trained decoder with the outputs of the pre-trained encoder. - Cross-lingual encoder pre-training with autoencoding of the pivot language. Our methods are evaluated in two non-English language pairs of WMT 2019 news translation tasks: high-resource (French$\rightarrow$German) and low-resource (German$\rightarrow$Czech). We show that NMT models pre-trained with our methods are highly effective in various data conditions, when fine-tuned for source$\rightarrow$target with: - Real parallel corpus - Pivot-based synthetic parallel corpus (*zero-resource*) - None (*zero-shot*) For each data condition, we consistently outperform strong baselines, e.g., multilingual, pivoting, or teacher-student, showing the universal effectiveness of our transfer learning schemes. The rest of the paper is organized as follows. We first review important previous works on pivot-based MT in Section \[sec:related\]. Our three pre-training techniques are presented in Section \[sec:methods\]. Section \[sec:results\] shows main results of our methods with a detailed description of the experimental setups. Section \[sec:analysis\] studies variants of our methods and reports the results without source-target parallel resources or with large synthetic parallel data. Section 6 draws conclusion of this work with future research directions. Related Work {#sec:related} ============ In this section, we first review existing approaches to leverage a pivot language in low-resource/zero-resource MT. They can be divided into three categories: 1. \[sec:pivoting\] **Pivot translation (pivoting).** The most naive approach is reusing (already trained) source$\rightarrow$pivot and pivot$\rightarrow$target models directly, decoding twice via the pivot language [@kauers2002interlingua; @de2006catalan]. One can keep $N$-best hypotheses in the pivot language to reduce the prediction bias [@utiyama2007comparison] and improve the final translation by system combination [@costa2011enhancing], which however increases the translation time even more. In multilingual NMT, modify the second translation step (pivot$\rightarrow$target) to use source and pivot language sentences together as the input. 2. \[sec:pivot-synth\] **Pivot-based synthetic parallel data.** We may translate the pivot side of given pivot-target parallel data using a pivot$\rightarrow$source model [@bertoldi2008phrase], or the other way around translating source-pivot data using a pivot$\rightarrow$target model [@de2006catalan]. For NMT, the former is extended by to compute the expectation over synthetic source sentences. The latter is also called teacher-student approach [@chen2017teacher], where the pivot$\rightarrow$target model (teacher) produces target hypotheses for training the source$\rightarrow$target model (student). 3. **Pivot-based model training.** In phrase-based MT, there have been many efforts to combine phrase/word level features of source-pivot and pivot-target into a source$\rightarrow$target system [@utiyama2007comparison; @wu2007pivot; @bakhshaei2010farsi; @zahabi2013using; @zhu2014improving; @miura2015improving]. In NMT, jointly train for three translation directions of source-pivot-target by sharing network components, where use the expectation-maximization algorithm with the target sentence as a latent variable. deploy intermediate recurrent layers which are common for multiple encoders and decoders, while share all components of a single multilingual model. Both methods train the model for language pairs involving English but enable zero-shot translation for unseen non-English language pairs. For this, encode the target language as an additional embedding and filter out non-target tokens in the output. combine the multilingual training with synthetic data generation to improve the zero-shot performance iteratively, where applies the NMT prediction score and a language model score to each synthetic example as gradient weights. Our work is based on transfer learning [@zoph2016transfer] and belongs to the third category: model training. On the contrary to the multilingual joint training, we suggest two distinct steps: pre-training (with source-pivot and pivot-target data) and fine-tuning (with source-target data). With our proposed methods, we prevent the model from losing its capacity to other languages while utilizing the information from related language pairs well, as shown in the experiments (Section \[sec:results\]). Our pivot adapter (Section \[sec:adapter\]) shares the same motivation with the interlingua component of , but is much compact, independent of variable input length, and easy to train offline. The adapter training algorithm is adopted from bilingual word embedding mapping [@xing2015normalized]. Our cross-lingual encoder (Section \[sec:cross-enc\]) is inspired by cross-lingual sentence embedding algorithms using NMT [@schwenk2017learning; @schwenk2018filtering]. Transfer learning was first introduced to NMT by , where only the source language is switched before/after the transfer. and use shared subword vocabularies to work with more languages and help target language switches. propose additional techniques to enable NMT transfer even without shared vocabularies. To the best of our knowledge, we are the first to propose transfer learning strategies specialized in utilizing a pivot language, transferring a source encoder and a target decoder at the same time. Also, for the first time, we present successful zero-shot translation results only with pivot-based NMT pre-training. Pivot-based Transfer Learning {#sec:methods} ============================= ![Plain transfer learning.[]{data-label="fig:plain"}](plain-transfer.pdf){width="\linewidth"} Our methods are based on a simple transfer learning principle for NMT, adjusted to a usual data condition for non-English language pairs: lots of source-pivot and pivot-target parallel data, little (low-resource) or no (zero-resource) source-target parallel data. Here are the core steps of the plain transfer (Figure \[fig:plain\]): 1. Pre-train a source$\rightarrow$pivot model with a source-pivot parallel corpus and a pivot$\rightarrow$target model with a pivot-target parallel corpus. 2. Initialize the source$\rightarrow$target model with the source encoder from the pre-trained source$\rightarrow$pivot model and the target decoder from the pre-trained pivot$\rightarrow$target model. 3. Continue the training with a source-target parallel corpus. If we skip the last step (for zero-resource cases) and perform the source$\rightarrow$target translation directly, it corresponds to zero-shot translation. Thanks to the pivot language, we can pre-train a source encoder and a target decoder without changing the model architecture or training objective for NMT. On the contrary to other NMT transfer scenarios [@zoph2016transfer; @nguyen2017transfer; @kocmi2018trivial], this principle has no language mismatch between transferor and transferee on each source/target side. Experimental results (Section \[sec:results\]) also show its competitiveness despite its simplicity. Nonetheless, the main caveat of this basic pre-training is that the source encoder is trained to be used by an English decoder, while the target decoder is trained to use the outputs of an English encoder — not of a source encoder. In the following, we propose three techniques to mitigate the inconsistency of source$\rightarrow$pivot and pivot$\rightarrow$target pre-training stages. Note that these techniques are not exclusive and some of them can complement others for a better performance of the final model. Step-wise Pre-training {#sec:step-wise} ---------------------- ![Step-wise pre-training.[]{data-label="fig:step-wise"}](incremental-pre-training.pdf){width="\linewidth"} A simple remedy to make the pre-trained encoder and decoder refer to each other is to train a single NMT model for source$\rightarrow$pivot and pivot$\rightarrow$target in consecutive steps (Figure \[fig:step-wise\]): 1. Train a source$\rightarrow$pivot model with a source-pivot parallel corpus. 2. Continue the training with a pivot-target parallel corpus, while freezing the encoder parameters of 1. In the second step, a target decoder is trained to use the outputs of the pre-trained source encoder as its input. Freezing the pre-trained encoder ensures that, even after the second step, the encoder is still modeling the source language although we train the NMT model for pivot$\rightarrow$target. Without the freezing, the encoder completely adapts to the pivot language input and is likely to forget source language sentences. We build a joint vocabulary of the source and pivot languages so that the encoder effectively represents both languages. The frozen encoder is pre-trained for the source language in the first step, but also able to encode a pivot language sentence in a similar representation space. It is more effective for linguistically similar languages where many tokens are common for both languages in the joint vocabulary. Pivot Adapter {#sec:adapter} ------------- ![Pivot adapter.[]{data-label="fig:adapter"}](pivot-adapter.pdf){width="\linewidth"} Instead of the step-wise pre-training, we can also postprocess the network to enhance the connection between the source encoder and the target decoder which are pre-trained individually. Our idea is that, after the pre-training steps, we adapt the source encoder outputs to the pivot encoder outputs to which the target decoder is more familiar (Figure \[fig:adapter\]). We learn a linear mapping between the two representation spaces with a small source-pivot parallel corpus: 1. Encode the source sentences with the source encoder of the pre-trained source$\rightarrow$pivot model. 2. Encode the pivot sentences with the pivot encoder of the pre-trained pivot$\rightarrow$target model. 3. Apply a pooling to each sentence of 1 and 2, extracting representation vectors for each sentence pair: ($\mathbf{s}$, $\mathbf{p}$). 4. Train a mapping $\mathbf{M}\in\mathbb{R}^{d \times d}$ to minimize the distance between the pooled representations $\mathbf{s}\in\mathbb{R}^{d \times 1}$ and $\mathbf{p}\in\mathbb{R}^{d \times 1}$, where the source representation is first fed to the mapping: $$\begin{aligned} \hat{\mathbf{M}} = \operatorname*{argmin}_{\mathbf{M}} \sum_{\mathbf{s},\mathbf{p}} \|\mathbf{M}\mathbf{s}-\mathbf{p}\|^2 \label{eq:min-vec} \end{aligned}$$ where $d$ is the hidden layer size of the encoders. Introducing matrix notations $\mathbf{S}\in\mathbb{R}^{d \times n}$ and $\mathbf{P}\in\mathbb{R}^{d \times n}$, which concatenate the pooled representations of all $n$ sentences for each side in the source-pivot corpus, we rewrite Equation \[eq:min-vec\] as: $$\begin{aligned} \mathbf{\hat{M}} = \operatorname*{argmin}_{\mathbf{M}} \|\mathbf{M}\mathbf{S}-\mathbf{P}\|^2 \label{eq:min-mtx}\end{aligned}$$ which can be easily computed by the singular value decomposition (SVD) for a closed-form solution, if we put an orthogonality constraint on $\mathbf{M}$ [@xing2015normalized]. The resulting optimization is also called Procrustes problem. The learned mapping is multiplied to encoder outputs of all positions in the final source$\rightarrow$target tuning step. With this mapping, the source encoder emits sentence representations that lie in a similar space of the pivot encoder. Since the target decoder is pre-trained for pivot$\rightarrow$target and accustomed to receive the pivot encoder outputs, it should process the mapped encoder outputs better than the original source encoder outputs. Cross-lingual Encoder {#sec:cross-enc} --------------------- ![Cross-lingual encoder.[]{data-label="fig:cross-lingual"}](cross-lingual-encoder.pdf){width="\linewidth"} As a third technique, we modify the source$\rightarrow$pivot pre-training procedure to force the encoder to have cross-linguality over source and pivot languages; modeling source and pivot sentences in the same mathematical space. We achieve this by an additional autoencoding objective from a pivot sentence to the same pivot sentence (Figure \[fig:cross-lingual\]). The encoder is fed with sentences of both source and pivot languages, which are processed by a shared decoder that outputs only the pivot language. In this way, the encoder is learned to produce representations in a shared space regardless of the input language, since they are used in the same decoder. This cross-lingual space facilitates smoother learning of the final source$\rightarrow$target model, because the decoder is pre-trained to translate the pivot language. The same input/output in autoencoding encourages, however, merely copying the input; it is said to be not proper for learning complex structure of the data domain [@vincent2008extracting]. Denoising autoencoder addresses this by corrupting the input sentences by artificial noises [@hill2016learning]. Learning to reconstruct clean sentences, it encodes linguistic structures of natural language sentences, e.g., word order, better than copying. Here are the noise types we use [@edunov2018understanding]: - Drop tokens randomly with a probability $p_\mathrm{del}$ - Replace tokens with a `<BLANK>` token randomly with a probability $p_\mathrm{rep}$ - Permute the token positions randomly so that the difference between an original index and its new index is less than or equal to $d_\mathrm{per}$ We set $p_\mathrm{del}=0.1$, $p_\mathrm{rep}=0.1$, and $d_\mathrm{per}=3$ in our experiments. The key idea of all three methods is to build a closer connection between the pre-trained encoder and decoder via a pivot language. The difference is in when we do this job: Cross-lingual encoder (Section \[sec:cross-enc\]) changes the encoder pre-training stage (source$\rightarrow$pivot), while step-wise pre-training (Section \[sec:step-wise\]) modifies decoder pre-training stage (pivot$\rightarrow$target). Pivot adapter (Section \[sec:adapter\]) is applied after all pre-training steps. Main Results {#sec:results} ============ We evaluate the proposed transfer learning techniques in two non-English language pairs of WMT 2019 news translation tasks[^1]: French$\rightarrow$German and German$\rightarrow$Czech. **Data** We used the News Commentary v14 parallel corpus and newstest2008-2010 test sets as the source-target training data for both tasks. The newstest sets were oversampled four times. The German$\rightarrow$Czech task was originally limited to unsupervised learning (using only monolingual corpora) in WMT 2019, but we relaxed this constraint by the available parallel data. We used newstest2011 as a validation set and newstest2012/newstest2013 as the test sets. Both language pairs have much abundant parallel data in source-pivot and pivot-target with English as the pivot language. Detailed corpus statistics are given in Table \[tab:corpus-stat\]. **Preprocessing** We used the Moses[^2] tokenizer and applied true-casing on all corpora. For all transfer learning setups, we learned byte pair encoding (BPE) [@sennrich2016neural] for each language individually with 32k merge operations, except for cross-lingual encoder training with joint BPE only over source and pivot languages. This is for modularity of pre-trained models: for example, a French$\rightarrow$English model trained with joint French/English/German BPE could be transferred smoothly to a French$\rightarrow$German model, but would not be optimal for a transfer to e.g., a French$\rightarrow$Korean model. Once we pre-train an NMT model with separate BPE vocabularies, we can reuse it for various final language pairs without wasting unused portion of subword vocabularies (e.g., German-specific tokens in building a French$\rightarrow$Korean model). On the contrary, baselines used joint BPE over all languages with also 32k merges. ----------- ------- ----------- ---------- Words Usage Data Sentences (Source) fr-en 35M 950M en-de 9.1M 170M Fine-tune fr-de 270k 6.9M de-en 9.1M 181M en-cs 49M 658M Fine-tune de-cs 230k 5.1M ----------- ------- ----------- ---------- : Parallel training data statistics.[]{data-label="tab:corpus-stat"} **Model and Training** The 6-layer base Transformer architecture [@vaswani2017attention] was used for all of our experiments. Batch size was set to 4,096 tokens. Each checkpoint amounts to 10k updates for pre-training and 20k updates for fine-tuning. Each model was optimized with Adam [@kingma2014adam] with an initial learning rate of 0.0001, which was multiplied by 0.7 whenever perplexity on the validation set was not improved for three checkpoints. When it was not improved for eight checkpoints, we stopped the training. The NMT model training and transfer were done with the <span style="font-variant:small-caps;">OpenNMT</span> toolkit [@klein-etal-2017-opennmt]. Pivot adapter was trained using the <span style="font-variant:small-caps;">Muse</span> toolkit [@conneau2018word], which was originally developed for bilingual word embeddings but we adjusted for matching sentence representations. ---------------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Direct source$\rightarrow$target 14.8 75.1 16.0 75.1 11.1 81.1 12.8 77.7 Multilingual many-to-many 18.7 71.9 19.5 72.6 14.9 76.6 16.5 73.2 Multilingual many-to-one 18.3 71.7 19.2 71.5 13.1 79.6 14.6 75.8 Plain transfer 17.5 72.3 18.7 71.8 15.4 75.4 18.0 70.9 + Pivot adapter 18.0 71.9 19.1 71.1 15.9 75.0 18.7 70.3 + Cross-lingual encoder 17.4 72.1 18.9 71.8 15.0 75.9 17.6 71.4 + Pivot adapter 17.8 72.3 19.1 71.5 15.6 75.3 18.1 70.8 Step-wise pre-training 18.6 70.7 19.9 70.4 15.6 75.0 18.1 70.9 + Cross-lingual encoder **19.5** **69.8** **20.7** **69.4** **16.2** **74.6** **19.1** **69.9** ---------------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- **Baselines** We thoroughly compare our approaches to the following baselines: 1. *Direct source$\rightarrow$target*: A standard NMT model trained on given source$\rightarrow$target parallel data. 2. *Multilingual*: A single, shared NMT model for multiple translation directions [@johnson2017google]. - *Many-to-many*: Trained for all possible directions among source, target, and pivot languages. - *Many-to-one*: Trained for only the directions *to* target language, i.e., source$\rightarrow$target and pivot$\rightarrow$target, which tends to work better than many-to-many systems [@aharoni2019massively]. In Table \[tab:main\], we report principal results after fine-tuning the pre-trained models using source-target parallel data. As for baselines, multilingual models are better than a direct NMT model. The many-to-many models surpass the many-to-one models; since both tasks are in a low-resource setup, the model gains a lot from related language pairs even if the target languages do not match. Plain transfer of pre-trained encoder/decoder without additional techniques (Figure \[fig:plain\]) shows a nice improvement over the direct baseline: up to +2.7% <span style="font-variant:small-caps;">Bleu</span> for French$\rightarrow$German and +5.2% <span style="font-variant:small-caps;">Bleu</span> for German$\rightarrow$Czech. Pivot adapter provides an additional boost of maximum +0.7% <span style="font-variant:small-caps;">Bleu</span> or -0.7% <span style="font-variant:small-caps;">Ter</span>. Cross-lingual encoder pre-training is proved to be not effective in the plain transfer setup. It shows no improvements over plain transfer in French$\rightarrow$German, and 0.4% <span style="font-variant:small-caps;">Bleu</span> worse performance in German$\rightarrow$Czech. We conjecture that the cross-lingual encoder needs a lot more data to be fine-tuned for another decoder, where the encoder capacity is basically divided into two languages at the beginning of the fine-tuning. On the other hand, the pivot adapter directly improves the connection to an individually pre-trained decoder, which works nicely with small fine-tuning data. Pivot adapter gives an additional improvement on top of the cross-lingual encoder; up to +0.4% <span style="font-variant:small-caps;">Bleu</span> in French$\rightarrow$German and +0.6% <span style="font-variant:small-caps;">Bleu</span> in German$\rightarrow$Czech. In this case, we extract source and pivot sentence representations from the same shared encoder for training the adapter. Step-wise pre-training gives a big improvement up to +1.2% <span style="font-variant:small-caps;">Bleu</span> or -1.6% <span style="font-variant:small-caps;">Ter</span> against plain transfer in French$\rightarrow$German. It shows the best performance in both tasks when combined with the cross-lingual encoder: up to +1.2% <span style="font-variant:small-caps;">Bleu</span> in French$\rightarrow$German and +2.6% <span style="font-variant:small-caps;">Bleu</span> in German$\rightarrow$Czech, compared to the multilingual baseline. Step-wise pre-training prevents the cross-lingual encoder from degeneration, since the pivot$\rightarrow$target pre-training (Step 2 in Section \[sec:step-wise\]) also learns the encoder-decoder connection with a large amount of data — in addition to the source$\rightarrow$target tuning step afterwards. Note that the pivot adapter, which inserts an extra layer between the encoder and decoder, is not appropriate after the step-wise pre-training; the decoder is already trained to correlate well with the pre-trained encoder. We experimented with the pivot adapter on top of step-wise pre-trained models — with or without cross-lingual encoder — but obtained detrimental results. Compared to pivot translation (Table \[tab:zero\]), our best results are also clearly better in French $\rightarrow$German and comparable in German$\rightarrow$Czech. Analysis {#sec:analysis} ======== In this section, we conduct ablation studies on the variants of our methods and see how they perform in different data conditions. Pivot Adapter {#pivot-adapter} ------------- ------------------ ---------- ---------- Adapter Training None 18.2 70.7 Max-pooled 18.4 70.5 Average-pooled **18.7** **70.3** Plain transfer 18.0 70.9 ------------------ ---------- ---------- : Pivot adapter variations (German$\rightarrow$Czech). All results are tuned with source-target parallel data.[]{data-label="tab:adapter"} Firstly, we compare variants of the pivot adapter (Section \[sec:adapter\]) in Table \[tab:adapter\]. The row “None” shows that a randomly initialized linear layer already guides the pre-trained encoder/decoder to harmonize with each other. Of course, when we train the adapter to map source encoder outputs to pivot encoder outputs, the performance gets better. For compressing encoder outputs over positions, average-pooling is better than max-pooling. We observed the same trend in the other test set and in French$\rightarrow$German. We also tested nonlinear pivot adapter, e.g., a 2-layer feedforward network with ReLU activations, but the performance was not better than just a linear adapter. Cross-lingual Encoder {#cross-lingual-encoder} --------------------- ------------ ------- ---------- ---------- Trained on Input Clean 15.7 77.7 Noisy 17.5 73.6 Clean 15.9 77.3 Noisy **18.0** **72.7** ------------ ------- ---------- ---------- : Cross-lingual encoder variations (French$\rightarrow$ German). All results are in the zero-shot setting with step-wise pre-training.[]{data-label="tab:cross-enc"} Table \[tab:cross-enc\] verifies that the noisy input in autoencoding is indeed beneficial to our cross-lingual encoder. It improves the final translation performance by maximum +2.1% <span style="font-variant:small-caps;">Bleu</span>, compared to using the copying autoencoding objective. As the training data for autoencoding, we also compare between purely monolingual data and the pivot side of the source-pivot parallel data. By the latter, one can expect a stronger signal for a joint encoder representation space, since two different inputs (in source/pivot languages) are used to produce the exactly same output sentence (in pivot language). The results also tell that there are slight but consistent improvements by using the pivot part of the parallel data. Again, we performed these comparisons in the other test set and German$\rightarrow$Czech, observing the same tendency in results. --------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Multilingual many-to-many 14.1 79.1 14.6 79.1 5.9 - 6.3 99.8 Pivot translation 16.6 72.4 17.9 72.5 16.4 74.5 **19.5** **70.1** Teacher-student 18.7 70.3 20.7 69.5 16.0 75.0 18.5 70.9 Plain transfer 0.1 - 0.2 - 0.1 - 0.1 - Step-wise pre-training 11.0 81.6 11.5 82.5 6.0 92.1 6.5 87.8 + Cross-lingual encoder 17.3 72.1 18.0 72.7 14.1 76.8 16.5 73.5 + Teacher-student **19.3** **69.7** **20.9** **69.3** **16.5** **74.6** 19.1 70.2 --------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Zero-resource/Zero-shot Scenarios --------------------------------- If we do not have an access to any source-target parallel data (*zero-resource*), non-English language pairs have two options for still building a working NMT system, given source-English and target-English parallel data: - *Zero-shot*: Perform source$\rightarrow$target translation using models which have not seen any source-target parallel sentences, e.g., multilingual models or pivoting (Section \[sec:related\].\[sec:pivoting\]). - *Pivot-based synthetic data*: Generate synthetic source-target parallel data using source$\leftrightarrow$English and target$\leftrightarrow$English models (Section \[sec:related\].\[sec:pivot-synth\]). Use this data to train a model for source$\rightarrow$target. Table \[tab:zero\] shows how our pre-trained models perform in zero-resource scenarios with the two options. Note that, unlike Table \[tab:main\], the multilingual baselines exclude source$\rightarrow$target and target$\rightarrow$source directions. First of all, plain transfer, where the encoder and the decoder are pre-trained separately, is poor in zero-shot scenarios. It simply fails to connect different representation spaces of the pre-trained encoder and decoder. In our experiments, neither pivot adapter nor cross-lingual encoder could enhance the zero-shot translation of plain transfer. Step-wise pre-training solves this problem by changing the decoder pre-training to familiarize itself with representations from an already pre-trained encoder. It achieves zero-shot performance of 11.5% <span style="font-variant:small-caps;">Bleu</span> in French$\rightarrow$German and 6.5% <span style="font-variant:small-caps;">Bleu</span> in German$\rightarrow$Czech (newstest2013), while showing comparable or better fine-tuned performance against plain transfer (see also Table \[tab:main\]). With the pre-trained cross-lingual encoder, the zero-shot performance of step-wise pre-training is superior to that of pivot translation in French$\rightarrow$German with only a single model. It is worse than pivot translation in German$\rightarrow$Czech. We think that the data size of pivot-target is critical in pivot translation; relatively huge data for English$\rightarrow$Czech make the pivot translation stronger. Note again that, nevertheless, pivoting (second row) is very poor in efficiency since it performs decoding twice with the individual models. For the second option (pivot-based synthetic data), we compare our methods against the sentence-level beam search version of the teacher-student framework [@chen2017teacher], with which we generated 10M synthetic parallel sentence pairs. We also tried other variants of , e.g., $N$-best hypotheses with weights, but there were no consistent improvements. Due to enormous bilingual signals, the model trained with the teacher-student synthetic data outperforms pivot translation. If tuned with the same synthetic data, our pre-trained model performs even better (last row), achieving the best zero-resource results on three of the four test sets. We also evaluate our best German$\rightarrow$Czech zero-resource model on newstest2019 and compare it with the participants of the WMT 2019 unsupervised news translation task. Ours yield 17.2% <span style="font-variant:small-caps;">Bleu</span>, which is much better than the best single unsupervised system of the winner of the task (15.5%) [@marie-etal-2019-nicts]. We argue that, if one has enough source-English and English-target parallel data for a non-English language pair, it is more encouraged to adopt pivot-based transfer learning than unsupervised MT — even if there is no source-target parallel data. In this case, unsupervised MT unnecessarily restricts the data condition to using only monolingual data and its high computational cost does not pay off; simple pivot-based pre-training steps are more efficient and effective. ---------------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Direct source$\rightarrow$target 20.1 69.8 22.3 68.7 11.1 81.1 12.8 77.7 + Synthetic data 21.1 68.2 22.6 68.1 15.7 76.5 18.5 72.0 Plain transfer 21.8 67.6 23.1 67.5 17.6 73.2 20.3 68.7 + Pivot adapter 21.8 67.6 23.1 67.6 **17.6** **73.0** **20.9** **68.3** + Cross-lingual encoder 21.9 67.7 23.4 67.4 17.5 73.5 20.3 68.7 + Pivot adapter **22.1** **67.5** 23.3 67.5 17.5 73.2 20.6 68.5 Step-wise pre-training 21.8 67.8 23.0 67.8 17.3 73.6 20.0 69.2 + Cross-lingual encoder 21.9 67.6 **23.4** **67.4** 17.5 73.1 20.5 68.6 ---------------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Large-scale Results ------------------- We also study the effect of pivot-based transfer learning in more data-rich scenarios: 1) with large synthetic source-target data (German$\rightarrow$Czech), and 2) with larger real source-target data in combination with the synthetic data (French$\rightarrow$German). We generated synthetic parallel data using pivot-based back-translation [@bertoldi2008phrase]: 5M sentence pairs for German$\rightarrow$Czech and 9.1M sentence pairs for French$\rightarrow$German. For the second scenario, we also prepared 2.3M more lines of French$\rightarrow$German real parallel data from Europarl v7 and Common Crawl corpora. Table \[tab:large\] shows our transfer learning results fine-tuned with a combination of given parallel data and generated synthetic parallel data. The real source-target parallel data are oversampled to make the ratio of real and synthetic data to be 1:2. As expected, the direct source$\rightarrow$target model can be improved considerably by training with large synthetic data. Plain pivot-based transfer outperforms the synthetic data baseline by up to +1.9% <span style="font-variant:small-caps;">Bleu</span> or -3.3% <span style="font-variant:small-caps;">Ter</span>. However, the pivot adapter or cross-lingual encoder gives marginal or inconsistent improvements over the plain transfer. We suppose that the entire model can be tuned sufficiently well without additional adapter layers or a well-curated training process, once we have a large source-target parallel corpus for fine-tuning. Conclusion ========== In this paper, we propose three effective techniques for transfer learning using pivot-based parallel data. The principle is to pre-train NMT models with source-pivot and pivot-target parallel data and transfer the source encoder and the target decoder. To resolve the input/output discrepancy of the pre-trained encoder and decoder, we 1) consecutively pre-train the model for source$\rightarrow$pivot and pivot$\rightarrow$target, 2) append an additional layer after the source encoder which adapts the encoder output to the pivot language space, or 3) train a cross-lingual encoder over source and pivot languages. Our methods are suitable for most of the non-English language pairs with lots of parallel data involving English. Experiments in WMT 2019 French$\rightarrow$German and German$\rightarrow$Czech tasks show that our methods significantly improve the final source$\rightarrow$target translation performance, outperforming multilingual models by up to +2.6% <span style="font-variant:small-caps;">Bleu</span>. The methods are applicable also to zero-resource language pairs, showing a strong performance in the zero-shot setting or with pivot-based synthetic data. We claim that our methods expand the advances in NMT to many more non-English language pairs that are not yet studied well. Future work will be zero-shot translation without step-wise pre-training, i.e., combining individually pre-trained encoders and decoders freely for a fast development of NMT systems for a new non-English language pair. Acknowledgments {#acknowledgments .unnumbered} =============== {width="25.00000%"} {width="20.00000%"} This work has received funding from the European Research Council (ERC) (under the European Union’s Horizon 2020 research and innovation programme, grant agreement No 694537, project “SEQCLAS”) and eBay Inc. The work reflects only the authors’ views and none of the funding agencies is responsible for any use that may be made of the information it contains. [^1]: [^2]:

--- abstract: | [\ ]{} Heavy neutrinos with additional interactions have recently been proposed as an explanation to the MiniBooNE excess. These scenarios often rely on marginally boosted particles to explain the excess angular spectrum, thus predicting large rates at higher-energy neutrino-electron scattering experiments. We place new constraints on this class of models based on neutrino-electron scattering sideband measurements performed at MINER$\nu$A and CHARM-II. A simultaneous explanation of the angular and energy distributions of the MiniBooNE excess in terms of heavy neutrinos with light mediators is severely constrained by our analysis. In general, high-energy neutrino-electron scattering experiments provide strong constraints on explanations of the MiniBooNE observation involving light mediators. author: - 'Carlos A. Argüelles' - Matheus Hostert - 'Yu-Dai Tsai' bibliography: - 'main.bib' title: | Testing New Physics Explanations of MiniBooNE Anomaly\ at Neutrino Scattering Experiments --- **Introduction –** Non-zero neutrino masses have been established in the last twenty years by measurements of neutrino flavor conversion in natural and human-made sources, including long- and short-baseline experiments. The overwhelming majority of data supports the three-neutrino framework. Within this framework, we have measured the mixing angles that parametrize the relationship between mass and flavor eigenstates to few-percent-level precision [@Esteban:2018azc]. The remaining unknowns are the absolute scale of neutrino masses and their origin, the CP-violating phase, and the mass ordering of the neutrinos. Nevertheless, anomalies in short-baseline accelerator and reactor experiments [@Athanassopoulos:1996jb; @Aguilar:2001ty; @AguilarArevalo:2007it; @Aguilar-Arevalo:2018gpe] challenge this framework and are yet to receive satisfactory explanations. Minimal extensions of the three-neutrino framework to explain the anomalies introduce the so-called sterile neutrino states, which do not participate in Standard Model (SM) interactions in order to agree with measurements of the Z-boson invisible decay width [@ALEPH:2010aa]. Unfortunately, these minimal scenarios are disfavoured as they fail to explain all data [@Collin:2016aqd; @Capozzi:2016vac; @Dentler:2018sju; @Diaz:2019fwt]. This has led the community to explore non-minimal scenarios. Along this direction, it is interesting to study well-motivated neutrino-mass models that can also explain the short-baseline anomalies and are testable in the laboratory. In this work, we will investigate a class of neutrino-mass-related models that have been proposed as an explanation of the anomalous observation of $\nu_e$-like events in MiniBooNE [@Aguilar-Arevalo:2018gpe]. MiniBooNE is a mineral oil Cherenkov detector located in the Booster Neutrino Beam (BNB), at Fermilab [@AguilarArevalo:2008yp; @AguilarArevalo:2008qa]. Using data collected between 2002 and 2017, the experiment has observed an excess of $\nu_e$-like events that is currently in tension with the standard three-neutrino prediction and is beyond statistical doubt at the $4.7 \sigma$ level [@Aguilar-Arevalo:2018gpe]. While it is possible that the excess is fully or partially due to systematic uncertainties or SM backgrounds (see, *e.g.*, [@AguilarArevalo:2008rc; @Aguilar-Arevalo:2012fmn; @Hill:2010zy]), many Beyond the Standard Model (BSM) explanations have been put forth. These new physics (NP) scenarios typically require the existence of new particles, which can: participate in short-baseline oscillations [@Murayama:2000hm; @Strumia:2002fw; @Barenboim:2002ah; @GonzalezGarcia:2003jq; @Barger:2003xm; @Sorel:2003hf; @Barenboim:2004wu; @Zurek:2004vd; @Kaplan:2004dq; @Pas:2005rb; @deGouvea:2006qd; @Schwetz:2007cd; @Farzan:2008zv; @Hollenberg:2009ws; @Nelson:2010hz; @Akhmedov:2010vy; @Diaz:2010ft; @Bai:2015ztj; @Giunti:2015mwa; @Papoulias:2016edm; @Moss:2017pur; @Carena:2017qhd], change the neutrino propagation in matter [@Liao:2016reh; @Liao:2018mbg; @Asaadi:2017bhx; @Doring:2018cob], or be produced in the beam or in the detector and its surroundings [@Gninenko:2009ks; @Gninenko:2010pr; @Dib:2011jh; @McKeen:2010rx; @Masip:2012ke; @Masip:2011qb; @Gninenko:2012rw; @Magill:2018jla]. These models either increase the conversion of muon- to electron-neutrinos or produce electron-neutrino-like signatures in the detector, where in the latter category one typically exploits the fact that the LSND and MiniBooNE are Cherenkov detectors that cannot distinguish between electrons and photons. Although many MiniBooNE explanations lack a connection to other open problems in particle physics, recent models [@Bertuzzo:2018ftf; @Bertuzzo:2018itn; @Ballett:2018ynz; @Ballett:2019cqp; @Ballett:2019pyw] are motivated by neutrino-mass generation via hidden interactions in the heavy-neutrino sector. In particular, a common prediction of these models is the upscattering of a light neutrino into a heavy neutrino, usually with masses in the tens to hundreds of MeV, which subsequently decays into a pair of electrons. To reproduce the MiniBooNE excess angular distribution either the heavy neutrino must have moderate boost factors and the pair of electrons produced need to be collimated [@Bertuzzo:2018itn], or the heavy neutrino two-body decays must be forbidden [@Ballett:2018ynz]. In this article, we introduce new techniques to probe models that rely on the ambiguity between photons and electrons to explain the MiniBooNE observation, using the dark neutrino model from [@Bertuzzo:2018itn; @Bertuzzo:2018ftf] as a benchmark scenario. Our analysis extends to all models with new marginally boosted particles produced in coherent-like neutrino interactions, as they predict large number of events at higher energies [@Gninenko:2009ks; @Gninenko:2010pr; @Dib:2011jh; @McKeen:2010rx; @Masip:2012ke; @Masip:2011qb; @Gninenko:2012rw; @Magill:2018jla]. Thus, our analysis uses high-energy neutrino-electron scattering measurements [@Auerbach:2001wg; @Deniz:2009mu; @Bellini:2011rx; @Park:2013dax; @Valencia:2019mkf; @Park:2015eqa; @Valencia-Rodriguez:2016vkf; @DeWinter:1989zg; @Geiregat:1992zv; @Vilain:1994qy]. This process is currently used to normalize the neutrino fluxes, due to its well-understood cross section, and has been a fertile ground for light NP searches [@Pospelov:2017kep; @Lindner:2018kjo; @Magill:2018tbb]. Here, however, we expand the capability of these measurements to probe BSM-produced photon-like signatures, by developing a new analysis using previously neglected sideband data. Our technique is complementary to recent searches for coherent single-photon topologies [@Abe:2019cer]. Since the upscattering process has a threshold of tens to hundreds of MeV, we focus on two high-energy neutrino experiments:  [@Park:2013dax; @Valencia:2019mkf; @Park:2015eqa; @Valencia-Rodriguez:2016vkf], a scintillator detector in the Neutrinos at the Main Injector (NuMI) beamline at Fermilab, and CHARM-II [@DeWinter:1989zg; @Geiregat:1992zv; @Vilain:1994qy], a segmented calorimeter detector at CERN along the Super Proton Synchrotron (SPS) beamline. These experiments are complementary in the range of neutrino energies they cover and have different background composition. In all cases a relevant sideband measurement exists, allowing us to take advantage of the excellent particle reconstruction capabilities of and the precise measurements at CHARM-II to constrain NP. ![[*Upscattering cross section compared to the quasi-elastic.*]{} The quasi-elastic cross section on Carbon ($6p^+$) is shown as a function of the neutrino energy (solid black line). The coherent (solid blue) and incoherent (dashed blue) scattering NP cross sections are also shown for the benchmark point of [@Bertuzzo:2018itn]. In the background, we show the BNB flux of $\nu_\mu$ at MiniBooNE (light gray), and the NuMI beam neutrino flux at MINER$\nu$A for the LE (light golden) and ME (light blue) runs in neutrino mode.\[fig:cross\_section\]](cross_sections.pdf){width="49.00000%"} **Model –** We consider a minimal realisation of dark neutrino models [@Bertuzzo:2018ftf; @Bertuzzo:2018itn; @Ballett:2018ynz; @Ballett:2019cqp; @Ballett:2019pyw] that can explain MiniBooNE. This comprises of one Dirac heavy neutrino[^1], $\nu_4$, with its associated flavor state, $\nu_D$. The dark neutrino $\nu_D$ is charged under a new local U$(1)^\prime$ gauge group, which is part of the particle content and gauge structure needed for mass generation. The dark sector is connected to the SM in two ways: kinetic mixing between the new gauge boson and hypercharge, and neutrino mass mixing. We start by specifying the kinetic part of the NP Lagrangian $$\mathscr{L}_{\rm kin} \supset \;\; \frac{1}{4} \hat{Z}^{\prime}_{\mu \nu} \hat{Z}^{\prime \mu \nu} + \frac{\sin{\chi}}{2} \hat{Z}^{\prime}_{\mu \nu} \hat{B}^{\mu \nu} + \frac{m_{\hat{Z}^\prime}^2}{2} \hat{Z}^{\prime \mu} \hat{Z}^\prime_{\mu},$$ where $\hat{Z}^{\prime \mu}$ stands for the new gauge boson field, $\hat{Z}^{\prime \mu\nu}$ its field strength tensor, and $\hat{B}^{\mu \nu}$ the hypercharge field strength tensor. After usual field redefinitions [@Chun:2010ve], we arrive at the physical states of the theory. Working at leading order in $\chi$ and assuming $m_{Z^\prime}^2/m_{Z}^2$ to be small, we can specify the relevant interaction Lagrangian as $$\mathscr{L}_{\rm int} \supset \;\;g_D \overline{\nu}_D \gamma_\mu \nu_D Z^{\prime \mu} + e \varepsilon Z'^{\mu}J^{\rm EM}_{\mu},$$ where $J^{\rm EM}_{\mu}$ is the SM electromagnetic current, $g_D$ is the U$(1)^\prime$ gauge coupling assumed to be $\mathcal{O}(1)$, and $\varepsilon \equiv c_{\rm w} \chi$, with $c_{\rm w}$ being the cosine of the weak angle. Additional terms would be present at higher orders in $\chi$ and mass mixing with the SM $Z$ is also possible, though severely constrained. After electroweak symmetry breaking, $\nu_D$ is a superposition of neutrino mass states. The flavor and mass eigenstates are related via $$\nu_\alpha = \sum^{4}_{i=1} U_{\alpha i}\nu_{i}, \quad (\alpha=e,\mu,\tau,D),$$ where $U$ is a $4\times4$ unitary matrix. It is expected that $|U_{\alpha 4}|$ is small for $\alpha = e, \mu, \tau$, but $|U_{D4}|$ can be of $\mathcal{O}(1)$ [@Parke:2015goa; @Collin:2016aqd]. ![*New physics prediction at ME and CHARM-II.* Neutrino-electron scattering data in $dE/dx$ at (top) and in $E\theta^2$ at CHARM-II (bottom). Error bars are too small to be seen. For both experiments, we show the $\nu-e$ signal and total background prediction quoted (after tuning at MINER$\nu$A), as well as the NP prediction (divided by 10 at CHARM-II). The cuts in our analysis our shown as vertical lines. \[fig:NP\_events\]](both_cartoon.pdf){width="50.00000%"} **MiniBooNE signature and region of interest–** The heavy neutrino is produced from an active flavour state upscattering on a nuclear target $A$, $\nu_\alpha A \to \nu_4 A$. The upscattering cross section is proportional to $\alpha_D \alpha_\textsc{qed}\varepsilon^2 |U_{\alpha 4}|^2$, dominated by $|U_{\mu 4}|$ since all current accelerator neutrino beams are composed mainly of muon neutrinos. This production can happen off the whole nucleus in a coherent way or off individual nucleons. For $m_{Z^\prime} \lesssim 100$ MeV, the production will be mainly coherent, but for heavier masses, such as the ones considered in [@Ballett:2018ynz], incoherent upscattering dominates. In Fig. \[fig:cross\_section\], we show the NP cross section at the benchmark point of [@Bertuzzo:2018itn] and compare it with the quasi-elastic cross section. By superimposing the cross section on the neutrino fluxes of and MiniBooNE, we make it explicit that the larger energies at and CHARM-II are ideal to produce $\nu_4$. Once produced, $\nu_4$ predominantly decays into a neutrino and a dielectron pair, $\nu_4 \to \nu_\alpha e^+ e^-$, either via an on-shell [@Bertuzzo:2018itn] or off-shell [@Ballett:2018ynz] $Z^\prime$ depending on the choice of $m_4$ and $m_{Z^\prime}$. In this work, we restrict our discussion to the $m_4 > m_{Z^\prime}$ case, where the upscattering is mainly coherent and is followed by a chain of prompt two body decays $\nu_4 \to \nu_\alpha (Z^\prime \to e^+ e^-)$. The on-shell $Z^\prime$ is required to decay into an overlapping $e^+e^-$ pair, setting a lower bound on its mass of a few MeV. Experimentally, however, $m_{Z^\prime} > 10$ MeV for $e \epsilon \sim 10^{-4}$ to avoid beam dump constraints [@Bauer:2018onh]. Increasing $m_{Z^\prime}$ increases the ratio of incoherent to coherent events and makes the electron pair less overlapping. Even though we focus on overlapping $e^+e^-$ pairs, we note that a significant fraction of events would appear as well-separated showers or as a pair of showers with large energy asymmetry, similarly to neutral current (NC) $\pi^0$ events. The asymmetric events also contribute to the MiniBooNE excess and offer a different target for searches in $\nu-e$ scattering data. A fit to the neutrino energy spectrum at MiniBooNE was performed in [@Bertuzzo:2018itn] and is reproduced in [Fig. \[fig:final\_plot\]]{}. We have performed our own fit to the MiniBooNE energy spectrum using the data release from [@Aguilar-Arevalo:2018gpe], and our results agree with [@Bertuzzo:2018itn], when we simulate the signal at MiniBooNE and the analysis cuts in the same way. This fit leads to preferred values of $m_4$ close to 100 MeV and $|U_{\mu 4 }| \sim 10^{-4}$. Unfortunately, this energy-only fit neglects the distribution of the excess events as a function of their angle $\theta$ with respect to the beam. This is important, as the total observed excess contains only $\approx 50\%$ of the events in the most forward bin ($0.8 < \cos{\theta} < 1.0$), with a statistical uncorrelated uncertainty of 5% on this quantity. As was recently pointed out in [@Jordan:2018qiy], few NP scenarios can reproduce the angular distribution of the MiniBooNE excess. Among these are models where new unstable particles are produced in inelastic collisions in the detector, such as the present case. Here, large $\theta$ can be achieved by tweaking the mass of the heavy neutrino; the signal becomes less forward as $\nu_4$ becomes heavier. To show this, we use our dedicated Monte Carlo (MC) simulation to asses the values of $m_4$ preferred by MiniBooNE data [^2]. For $m_{Z^\prime} = 30$ MeV and $m_4 = 100$, $200$, and $400$ MeV, we find that 98%, 87%, and 70% of the NP events would lie in the most forward bin, respectively. We find the predicted angular distribution to be more forward than [@Bertuzzo:2018itn] due to an improved MiniBooNE simulation; see Supplementary Material for details. This simulation discrepancy is understood and only strengthens our conclusions. Thus the relevant region for the MiniBooNE angular distribution is $m_4 \gtrsim 400$ MeV for $m_{Z^\prime} = 30$ MeV. [**Our analysis –**]{} Neutrino-electron scattering measurements predicate their cuts in the following core ideas: no hadronic activity near the interaction vertex, small opening angle from the beam, $E_e \theta^2 \lesssim 2 m_e$, and the requirement that the measured energy deposition, $dE/dx$, be consistent with that of a single electron. For the NP events, when the coherent process dominates and the mass of the $Z^\prime$ is small, the first two conditions are often satisfied. However, the requirement of a single-electron-like energy deposition removes a significant fraction of the new-physics induced events. This presents a challenge, as the NP events are mostly overlapping electron pairs and will potentially be removed by the $dE/dx$ cut. In order to circumvent this problem, we perform our analysis not at the final-cut level, but at an intermediate one. This is done differently for CHARM-II and MINER$\nu$A: the CHARM-II experiment provides data as a function of $E_e \theta^2$ without the $dE/dx$ cut, and provides data as a function of the measured $dE/dx$ after analysis cuts on $E_e \theta^2$. We have developed our own MC simulation for candidate electron pair events in MiniBooNE, MINER$\nu$A and CHARM-II; see the Supplementary Material for more details on detector resolutions, precise signal definition, and resulting distributions. We only consider the coherent part of the cross section to avoid hadronic-activity cuts, which is conservative. We also select only events with small energy asymmetries and small opening electron angles. When required, we assume the mean $dE/dx$ in plastic scintillator to follow the same shape as the NC $\pi^0$ prediction. Our prediction for new physics events for the BP point is show in Fig. \[fig:NP\_events\] on top of the ME and CHARM-II data and MC prediction. The CHARM-II analysis is mostly based on Fig. 1 of [@Vilain:1994qy]. This sample is shown as a function of $E\theta^2$ and does not have any cuts on $dE/dx$. It contains all events with shower energies between $3$ and $24$ GeV, and our final cut on $E\theta^2$ is fixed at $28$ MeV. For , the event selection is identical for the low-energy (LE) and medium-energy (ME) analyses [@Park:2015eqa; @Valencia:2019mkf]. The minimum shower energy required is $0.8$ GeV in order to remove the $\pi^0$ background and have reliable angular and energy reconstruction. Events are kept only when they meet the following angular separation criterion: $E_e \theta^2 < 3.2\times 10^{-3}~{\rm ~GeV ~rad^2}$. A final cut is applied, ensuring $dE/dx < 4.5~{\rm MeV} / 1.7~{\rm cm}$. The analyses use the data outside the previous $dE/dx$ cut to constrain backgrounds. This sideband is defined by all events with $E_e\theta^2 > 5 \times 10^{-3} {\rm ~GeV ~rad^2}$ and $dE/dx < 20~{\rm MeV}/1.7~{\rm cm}$. Using this sideband measurement, the collaboration tunes their backgrounds by ($0.76$, $0.64$, $1.0$) for ($\nu_e$CCQE, $\nu_\mu$NC, $\nu_\mu$CCQE) processes in the LE mode. Our LE analysis uses the data shown in Fig. 3 of [@Park:2015eqa] where all the cuts are applied except for the final $dE/dx$ cut. In our final event selection, we require that the sum of the energy deposited be more than $4.5$ MeV$/ 1.7$ cm, compatible with an $e^+e^-$ pair and yielding an efficiency of $90\%$. The recent ME data contains an excess in the region of large $dE/dx$ [@Valencia:2019mkf], where the NP events would lie. However, this excess is attributed to NC $\pi^0$ events, and grows with the shower energy undershooting the rate require to explain the MiniBooNE anomaly. With normalization factors as large as 1.7, the collaboration tunes primarily the NC $\pi^0$ prediction in an energy dependent way. After tuning, the total NC $\pi^0$ sample corresponds to $20\%$ of the total number of events before the $dE/dx$ cut. To place our limits, we perform a rate-only analysis by means of a Pearson’s $\chi^2$ as test statistic; detailed definition is given in the Supplementary Material. We incorporate uncertainties in background size and flux normalization as nuisance parameters with Gaussian constraint terms. For the neutrino-electron scattering and BSM signal, we allow the normalization to scale proportionally to the same flux uncertainty parameter. The background term also scales with the flux-uncertainty parameter but has an additional nuisance parameter to account for its unknown size. We obtain our constraint as a function of heavy neutrino mass $m_4$, and mixing $|U_{\mu 4}|$ assuming a $\chi^2$ with two degrees of freedom [@Tanabashi:2018oca]. In our nominal LE (ME) analysis, we allow for 10% uncertainty on the flux [@Aliaga:2016oaz], and 30% (40%) uncertainty on the background motivated by the amount of tuning performed on the original backgrounds. Note that the nominal background predictions in the LE (ME) analysis overpredicts (underpredicts) the data before tuning, and that tuning parameters are measured at the 3% (5%) level [@Park:2013dax; @Valencia:2019mkf]. We also perform a background-ignorant analysis in which we assume 100% uncertainty for the background normalization, which changes our conclusions by only less than a factor of two. This emphasizes the robustness of our bound, since the NP typically overshoots the low number of events in the sideband. For the benchmark point of [@Bertuzzo:2018itn], we predict a total signal of 232 (4240) events for LE (ME). For CHARM-II, the NP signal lies mostly in a region with small $E\theta^2$. Thus, we constrain backgrounds using the data from $28 < E\theta^2 < 60$ MeV rad$^2$. This sideband measurement constrains the normalization of the backgrounds in the signal region at the level of $3\%$. The extrapolation of the shape of the background to the signal region introduces the largest uncertainty in our analysis. For this reason, we raise the uncertainty of the background normalization from $3\%$ to a conservative $10 \%$ when setting the limits. Flux uncertainties are assumed to be $4.7\%$ and $5.2\%$ for neutrino and antineutrino mode [@Allaby:1987bb], respectively, and are applicable to the new-physics signal, $\nu-e$ scattering prediction, and backgrounds. Uncertainties in the $\nu-e$ scattering cross sections are expected to be sub-dominant and are neglected in the analysis [@deGouvea:2006hfo]. For CHARM-II, the NP also yields too many events in the signal region, namely $\approx 2.2\times10^{5}$ events for the benchmark point of [@Bertuzzo:2018itn] in antineutrino mode. If we lower $|U_{\mu4}| = 10^{-4}$ and $m_4 = 100$ MeV, CHARM-II would still have $\approx 3 \times 10^3$ new physics events. ![[*New constraints on dark neutrinos as a MiniBooNE explanation.*]{} The fit to the MiniBooNE energy distribution from [@Bertuzzo:2018itn] is shown as closed yellow (orange) region for one (three) sigma C.L., together with the benchmark point (${\bf\odot}$). Our constraints are shown at 90% C.L. for LE in blue (solid – 30% background normalization uncertainty, dashed – conservative 100% case), for ME in cyan (solid – 40% background normalization uncertainty, dashed – conservative 100% case), and for CHARM-II in red (solid – 3% background normalization from the sideband constraint, dashed – conservative 10% case). Vertical lines show the percentage of excess events at MiniBooNE that lie in the most forward angular bin. Exclusion from heavy neutrino searches is shown as a hatched background. Other relevant assumed parameters are shown above the plot; changing them does not alter our conclusion.\[fig:final\_plot\]](bounds.pdf){width="3.38in"} [**Results and conclusions –**]{} The resulting limits on dark neutrinos using neutrino-electron scattering experiments are shown in the $|U_{\mu 4}|$ vs $m_4$ plane at 90% confidence level (CL) in [Fig. \[fig:final\_plot\]]{}. The MiniBooNE fit from [@Bertuzzo:2018itn] is shown, together with vertical lines indicating the percentage of events at MiniBooNE that populate the most forward angular bin. We have chosen the same values of $\varepsilon$, $\alpha_D$, and $m_{Z^\prime}$ as used in [@Bertuzzo:2018itn], and shown their benchmark point ($m_4 = 420$ MeV and $|U_{\mu 4}|^2 = 9 \times 10^{-7}$) as a dotted circle. For these parameters, we can conclude that a good angular distribution at MiniBooNE is in large tension with neutrino-electron scattering data. We note that the MiniBooNE event rate scales identically to our signal rate in all the couplings, and the dependence on $m_{Z^\prime}$ is subleading due to the typical momentum transfer to the nucleus, provided $m_{Z^\prime} \lesssim 100$ MeV . This implies that changing the values of these parameters does not modify the overall conclusions of our work. In addition, for this realization of the model, larger $m_{Z^\prime}$ implies larger values of $m_4$, increasing their impact on neutrino-electron scattering data. Our and CHARM-II results are mutually reinforcing given that they impose similar constraints for $m_4 \lesssim 200 $ MeV. For larger masses, the kinematics of the signal becomes less forward and the production thresholds start being important. This explains the upturns visible in our bounds, where we observe it first in and later in CHARM-II as we increase $m_4$, since CHARM-II has higher beam energy. We emphasize that our analysis is general, and can be adapted to other models. In fact, any MiniBooNE explanation with heavy new particles faces severe constraints from high-energy neutrino-electron scattering data if the signal is free from hadronic activity. This is realised, for instance, in scenarios with heavy neutrinos with dipole interactions [@Gninenko:2009ks; @Gninenko:2010pr; @Dib:2011jh; @McKeen:2010rx; @Masip:2012ke; @Masip:2011qb; @Gninenko:2012rw; @Magill:2018jla]. Our bounds can also be adapted to other scenarios with dark neutrinos and heavy mediators [@Ballett:2018ynz; @Ballett:2019pyw]. For those, however, we do not expect our bounds to constrain the region of parameter space where the MiniBooNE explanation is viable, since most of the signal at MiniBooNE contains hadronic activity which would be visible at and CHARM-II. In the near future, our new analysis strategy could be used in the up-coming ME results on antineutrino-electron scattering. The NP cross section, being the same for neutrino and antineutrinos, is thus more prominent on top of backgrounds. This class of analyses will also greatly benefit from improved calculations and measurements of coherent $\pi^0$ production and single-photon emitting processes. This is particularly important given the excess seen in the ME analysis. A new result can also be obtained by neutrino-electron scattering measurements at NO$\nu$A, which will sample a different kinematic regime as its off-axis beam peaks at lower energies and expects fewer NC $\pi^0$ events per ton. Beyond neutrino-electron scattering, the BSM signatures we consider could be lurking in current measurements of $\pi^0$ production, *e.g.*, at MINOS [@Adamson:2016hyz] and MINER$\nu$A [@Wolcott:2016hws] [^3], and in analyses like the single photon search performed by T2K [@Abe:2019cer]. Thus, if dark neutrinos are indeed present in current data, our technique will be crucial to confirm it. To summarize, a variety of measurements are underway to further lay siege to this explanation of the MiniBooNE observation and, simultaneously, start probing testable neutrino mass generation models, as well as other similar NP signatures. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Janet Conrad, Kareem Farrag, Alberto Gago, Gordan Krnjaic, Trung Le, Pedro Machado, Kevin Mcfarland, and Jorge Morfin for useful discussions, and Jean DeMerit for carefully proofreading our work. The authors would like to thank Fermilab for the hospitality at the initial stages of this project. Also, the authors would like to thank Fermilab Theory Group and the CERN Theory Neutrino Platform for organizing the conference “Physics Opportunities in the Near DUNE Detector Hall,” which was essential to the completion of this work. CAA would especially like to thank Fermilab Center for Neutrino Physics summer visitor program for funding his visit. MH’s work was supported by Conselho Nacional de Ciência e Tecnologia (CNPq). CAA is supported by U.S. National Science Foundation (NSF) grant No. PHY-1801996. This document was prepared by YDT using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359. **Supplementary Material** Our analysis discussed in the main text is now described in more detail and all assumptions used in our simulations are summarized. We start by discussing our statistical method, and then discuss our Monte Carlo (MC) simulation, stating our signal definitions more precisely. Later, we show a few kinematical distributions from our dedicated MC simulation, including the angular distributions at MiniBooNE used to obtain the vertical lines in [Fig. \[fig:final\_plot\]]{}. In order to furher aid reproducibility of our result, we also make our Monte Carlo events for some parameter choices available on <span style="font-variant:small-caps;">g</span>it<span style="font-variant:small-caps;">h</span>ub [^4]. Statistical Analysis ==================== Our statistical analysis uses the Pearson-$\chi^2$ as a test statistic, where the expected number of events is scaled by nuisance parameters to incorporate systematic uncertainties. Our test statistic reads $$\begin{aligned} \chi^2(\vec\theta, \alpha, \beta) = \frac{ (N_{\rm data} - N_\mathrm{pred}(\vec\theta, \alpha, \beta) )^2 }{ N_\mathrm{pred}(\vec\theta, \alpha, \beta)} + \left(\frac{\alpha}{\sigma_\alpha}\right)^2 + \left(\frac{\beta}{\sigma_\beta}\right)^2,\end{aligned}$$ with the number of predicted events given by $$\begin{aligned} N_\mathrm{pred}(\vec\theta, \alpha, \beta) = (1+\alpha + \beta) \mu_{\rm MC}^{\rm BKG} + (1+\alpha) \mu_{\rm MC}^{\nu-e} + (1 + \alpha) \mu_{\rm BSM}(\vec\theta),\end{aligned}$$ where $\vec\theta$ are the model parameters, while $\alpha$ and $\beta$ are nuisance parameters that incorporate uncertainties from the overall rate and the background rate, respectively. Here, $N_{\rm data}$ stands for the total rate observed in the experiment, $\mu_{\rm MC}^{\rm BKG}$ the quoted background rates, $\mu_{\rm MC}^{\nu-e}$ the quoted $\nu-e$ events, and $\mu_{\rm BSM}(\vec\theta)$ the predicted number of BSM events calculated using our MC. We discuss the choice of these systematic uncertainties, namely the choice of $\sigma_\alpha$ and $\sigma_\beta$, when describing the simulation of each experiment below. To obtain our results we use the test statistic profiled over the nuisance parameters, namely $\chi^2(\vec\theta) = \min_{(\alpha, \beta)} \left(\chi^2(\vec\theta, \alpha, \beta) \right)$, and use the test-statistic thresholds given in [@pdg]. Simulation Details ================== Experiment Detector Resolution Overlapping Analysis Cuts ------------- ----------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------- ----------------------------------------------------------------------------------------- MiniBooNE $\sigma_E/E = 12\%/\sqrt{E_e/{\rm GeV}}$ $\sigma_\theta = 4{^\circ}$ $E_{+} > 30$ MeV $E_{-} > 30$ MeV $\Delta \theta_{\pm} < 13^\circ$ N/A MINER$\nu$A $\sigma_E/E = 5.9\%/\sqrt{E_e/{\rm GeV}} + 3.4\%$ $\sigma_\theta = 1{^\circ}$ $E_{+} > 30$ MeV $E_{-} > 30$ MeV $\Delta \theta_{\pm} < 8^\circ$ $E_{\rm vis} > 0.8$ GeV $E_{\rm vis} \theta^2 < 3.2$ MeV $Q^2_{\rm rec} < 0.02$ GeV$^2$ CHARM-II $\sigma_E/E = 9\%/\sqrt{E/{\rm GeV}} + 11\%$ $\sigma_\theta/{\rm mrad} = \frac{27 (E/{\rm GeV})^2 +14}{\sqrt{E/{\rm GeV}}} + 1$ $E_{+} > 30$ MeV $E_{-} > 30$ MeV $\Delta \theta_{\pm} < 4^\circ$ $3$ GeV $<E_{\rm vis} < 24$ GeV $E_{\rm vis} \theta^2 < 28$ MeV We generate events distributed according to the upscattering cross section for the process $\nu_\mu A \to \nu_4 A$, where $A$ is a nuclear target. We only discuss upscattering on nuclei, as the number of incoherent scattering on protons is much smaller for the relevant $Z^\prime$ masses; see [Fig. \[fig:cross\_section\]]{}. We then implement the chain of two-body decays: $\nu_4 \to \nu_\mu Z^\prime$ followed by $Z^\prime \to e^+ e^-$. To go from our MC true quantities to the predicted experimental observables, we perform three procedures. First, we smear the energy and angles of the $e^+$ and $e^-$ originating from the decay of the $Z^\prime$ according to detector-dependent Gaussian resolutions. Next, we select all events with that satisfy the $e^+e^-$ overlapping condition given in [Table \[tab:parameters\]]{}. Namely, if the condition is satisfied they are assumed to be reconstructed as a single electromagnetic (EM) shower. This guarantees that the events behave like a photon shower inside the detector [^5]. Finally, for and CHARM-II, these samples are subject to analysis-dependent kinematic cuts to determine if they contribute to the $\nu-e$ scattering sample. Detector resolutions, requirements for the dielectron pair to be overlapping, and analysis-dependent cuts are summarized in [Table \[tab:parameters\]]{}. We now list the experimental parameters used in our simulations for each individual detector. #### CHARM-II The CHARM-II experiment is simulated using the CERN West Area Neutrino Facility (WANF) wide band beam  [@Vilain:1998uw]. The total number of protons-on-target (POT) is $2.5 \times 10^{19}$ for the $\nu$ and $\overline{\nu}$ runs combined. We assume glass to be the main detector material, (SiO$_2$), such that we can treat neutrino scattering off an average target with $\langle Z\rangle=11$ and $\langle A \rangle = 20.7$ [@DeWinter:1989zg; @Vilain:1993sf]. The fiducial volume in our analysis is confined to a transverse area of $320$cm$^2$, which corresponds to a fiducial mass of $547$t, and the detection efficiency is taken to be $76\%$; efficiency for $\pi^0$ sample is quoted at $82\%$ [@Vilain:1992wx]. We reproduce the total number of $\nu-e$ scattering events with $3$ GeV $< E_{\rm vis} <24$ GeV, namely $2677+2752$, to within a few percent level when setting the number of POTs in $\nu$ mode to be $1.69$ of that in the $\overline{\nu}$ mode [@Geiregat:1991md]. We assume a flux uncertainty of $\sigma_\alpha = 4.7\%$ for neutrino, and $\sigma_\alpha = 5.2\%$ for antineutrino beam [@Vilain:1992wx]. The background uncertainty is constrained to be $\sigma_\beta = 3\%$ using the data with $E_{\rm vis} \theta^2 > 28$ MeV, where the number of new physics events is negligible. #### MINER$\nu$A For our MINER$\nu$A simulation, we use the low-energy (LE) and medium-energy (ME) NuMI neutrino fluxes [@AliagaSoplin:2016shs]. The total number of POT is $3.43\times 10^{20}$ for LE data, and $11.6\times10^{20}$ for ME data. The detector is assumed to be made of CH, with a fiducial mass of $6.10$ tons and detection efficiencies of $73\%$ [@Parke:2015goa; @Valencia:2019mkf]. We assume a flux uncertainty of $\sigma_\alpha = 10\%$ for both the LE and ME modes [@Aliaga:2016oaz]. Due to the tuning performed in the sideband of interest, the uncertainties on the background rate are much larger. For the LE, we take $\sigma_\beta = 30\%$, while for the ME data $\sigma_\beta = 50\%$. Although tuning is significant for the coherent $\pi^0$ production sample, the overall rate of backgrounds in the sideband with large $dE/dx$ does not vary by more than $20\%$ ($40\%$) in the LE (ME) tuning. #### MiniBooNE To simulate MiniBooNE, we use the Booster Neutrino Beam (BNB) fluxes from [@AguilarArevalo:2008yp]. Here, we only discuss the neutrino run, although the predictions for the antineutrino run are very similar. We assume a total of $12.84 \times 10^{20}$ POT in neutrino mode. The fiducial mass of the detector is taken as $450$t of CH$_2$. In order to apply detector efficiencies, we compute the reconstructed neutrino energy under the assumption of CCQE scattering $$\begin{aligned} E_\nu^{CCQE} = \frac{E_{\rm vis} m_p}{m_p - E_{\rm vis} (1 - \cos{\theta}) },\end{aligned}$$ where $E_{\rm vis} = E_{e^+} + E_{e^-}$ is the total visible energy after smearing. Under this assumption, we can apply the efficiencies provided by the MiniBooNE collaboration [@Aguilar-Arevalo:2012fmn]. Using our MC we can reproduce well the distributions obtained using the MiniBooNE Monte Carlo data release provided for oscillation analyses. Kinematic Distributions ======================= As an important check of our calculation and of the explanation of the MiniBooNE excess within the model of interest, we plot the MiniBooNE neutrino data from 2018 [@Aguilar-Arevalo:2018gpe] against our MC prediction in Suppl. Fig. \[fig:MB\_distributions\]. We do this for three different new physics parameter choices. We set $m_{Z^\prime} = 30$ MeV, $\alpha \epsilon^2 = 2\times10^{-10}$ and $\alpha_D = 1/4$ for all points, but vary $|U_{\mu 4}|^2$ and $m_4$ so that the final number of excess events predicted by the model at MiniBooNE equals 334. {width="49.00000%"} {width="49.00000%"} To verify that the new physics signal is important in neutrino-electron studies, we also plot kinematical distributions for the benchmark point (BP) discussed in the main text for different detectors. This corresponds to $m_{Z^\prime} = 30$ MeV, $\alpha \epsilon^2 = 2\times10^{-10}$, $\alpha_D = 1/4$, $|U_{\mu 4}|^2 = 9\times10^{-7}$ and $m_4 = 420$ MeV. The interesting variables are the energy asymmetry of the dielectron pair $$|E_{\rm asym}| = \frac{|E_+ - E_-|}{E_+ + E_-},$$ as well as the separation angle $\Delta \theta_{e^+e^-}$ between the two electrons. These variables are plotted in Suppl. Fig. \[fig:other\_distributions\] at MC truth level, before any smearing or selection takes place. We also plot the total reconstructed energy $E_{\rm vis} = E_{e^+} + E_{e^-}$ and the quantity $E_{\rm vis} \theta^2$, where $\theta$ stands for the angle formed by the reconstructed EM shower and the neutrino beam. The visible energy, $E_{\rm vis}$, and angle, $\theta$, are computed after smearing, but before the selection into overlapping pairs takes place. {width="49.00000%"} {width="49.00000%"}\ {width="49.00000%"} {width="49.00000%"} [^1]: Models with the decay of Majorana particles will lead to greater tension with the angular distribution at MiniBooNE due to their isotropic nature [@Formaggio:1998zn; @Balantekin:2018ukw]. [^2]: Since the released MiniBooNE data do not provide the correlation between angle and energy, and their associated systematics, an energy-angle fit is not possible. [^3]: This $\nu_e$CCQE measurement by observes a significant excess of single photon-like showers attributed to diffractive $\pi^0$ events. These are abundant in similar realizations of this NP model [@Ballett:2018ynz]. [^4]: [[github.com/mhostert/DarkNews]{}](https://github.com/mhostert/DarkNews). [^5]: For MiniBooNE, we also include events that are highly asymmetric in energy, *i.e.*, $E_{\pm} > 30$ MeV and $E_{\mp} < 30$ MeV, where the most energetic shower defines the angle with respect to the beam.

--- abstract: 'We study the elastic response of bilayer membranes with fixed projected area to both stretching and shape deformations. A surface tension is associated to each of these deformations. By using model amphiphilic membranes and computer simulations, we are able to observe both the types of deformation, and thus, both the surface tensions, related to each type of deformation, are measured for the same system. These surface tensions are found to assume different values in the same bilayer membrane: in particular they vanish for different values of the projected area. We introduce a simple theory which relates the two quantities and successfully apply it to the data obtained with computer simulations.' author: - Alberto Imparato title: Surface tension in bilayer membranes with fixed projected area --- Bilayer membranes are composed of amphiphilic molecules whose hydrophobic part is strongly insoluble in water. Such molecules tends to form interfaces with the solvent in order to reduce the interfacial energy [@Isr]: a bilayer is thus formed by two adjacent sheets of amphiphilic molecules separating two aqueous phases. The interest in such structures resides in the fact that they play an outstanding role in the organization of biological cells: bilayer membranes of lipids form the basic structure of a cell’s plasma membrane and of internal membranes, which surround the organelles in eukaryotic cells, such as the nucleus, mitochondria and the Golgi apparatus [@ABL; @L1]. In a fluid membrane fluctuating freely in a solvent, all internal degrees of freedom, related, e.g, to the hydrocarbon chain conformation, and to to the local molecular density, relax on a fast time scale. The membrane is then characterized by a zero shear modulus, i.e. any shear deformation induces a flow of the amphiphilic molecules within the membrane. Such a system is then subject to only two types of elastic deformations: bending deformations and stretching deformations. The first are deformations normal to the membrane plane which change the membrane shapes, while the latter are in-plane deformations which change the local area per molecule projected onto the membrane midplane. The surface of a fluctuating membrane can thus be viewed as an interface with the solvent, and we will call [*fluctuation*]{} surface tension the surface tension of the bilayer membrane associated to shape fluctuations. The surface tension of an interface saturated by surfactant molecules is expected to vanish (see, e.g., [@Saf]), and thus the fluctuation surface tension of a fluid membrane fluctuating freely in a solvent is usually assumed to be zero. However in the case of a constrained bilayer membrane, such as those fixed on a frame, the geometrical constrain can lead to a non vanishing surface tension. Non vanishing fluctuation surface tension have been observed, e.g., in [@MM]. On a macroscopic length scale, a membrane fixed on a frame can be treated as an elastic sheet: if one of the sides of the frame is movable and one tries to change the frame area by pulling or pushing on this side, an elastic force will appear which will tend to restore the preferred value of the frame area. The parameter describing such elastic behaviour is the [*mechanical*]{} surface tension and can assume non negative values too [@GL1; @LE2]. Intuitively, one would expect that the two surface tensions associated with the two different membrane deformations, namely shape deformation and lateral stretching, are different quantities, see ref. [@DL] and discussion therein. However, in experiments, they are usually measured by observing the one or the other type of deformation, and so one cannot compare them experimentally for the same system. On the contrary, the two surface tensions can be measured using computer simulations of model membranes, and this has been done in two previous works [@FP0; @Ste]. But in one case, the two surface tensions have been found to be equal within the statistical errors [@FP0], while in the other case they have been found to be proportional with respect to each other [@Ste]. The aim of this paper is thus the systematical study and comparison between the fluctuation surface tension and the mechanical surface tension, measured by observing both the types of deformation in the same bilayer membrane. In the present paper we consider bilayer membrane with fixed projected area. The projected area correspond to the area of the frame in the picture drawn above. By using model bilayer membrane and computer simulations, we measure both the surface tension associated to shape deformation and that associated to stretching deformations. We show, for the first time, that in the same system these two quantities assume different values, and in particular they do not vanish for the same value of the projected area. The model used here for the computer simulations, have been largely used in previous works and has been quite successful in describing many dynamic and elastic properties of real bilayer membranes [@GL1; @GL2; @alb1; @alb2]. The paper is organized as follows: in section \[model\], we describe the model bilayer, together with the computer simulation techniques which we use in the present work. In section \[mis\_S\] we discuss the theory of the mechanical surface tension and the method adopted to measure it. We then show the results for such quantity obtained by computer simulations. In section \[mis\_s\], after reviewing the classical elasticity theory for the shape fluctuations of bilayer membranes, we show the results for the shape fluctuation surface tension, obtained by computer simulations. In section \[compare\_sec\], the two quantities are compared, and a simple theory which connects them is introduced and discussed. We discuss our results and conclude in section \[concl\]. In the following, we consider a bilayer membrane made up of $N$ amphiphilic molecules, at temperature $T$, spanning a square frame of area $A_p=L^2$, its effective area being $A$. Model bilayers and computer simulations {#model} ======================================= Coarse grained model of amphiphilic bilayers -------------------------------------------- We adopt the same coarse grained model introduced and used in refs.[@GL1; @GL2; @alb1; @alb2], which turned out to be an effective model to study several properties of bilayer membranes such as surface tension, bending rigidity and diffusion characteristics. The amphiphilic molecules are represented as linear chains of beads, a single bead representing the molecule hydrophilic head (H), or several $\mathrm{CH}_2/ \mathrm{CH}_3$ groups of the amphiphile hydrocarbon chain (C) see fig. \[mol\]. The water molecule is also represented as a single bead (W). The hydrophobic interaction of C with W and H particles is modelled by soft core potential $$U{_{SC}}({r_{ij}})=4 {\epsilon_{\alpha \beta}}{\left({\frac{ \ell'}{{r_{ij}}}}\right)}^9 \, , \label{usc}$$ while attractive interactions W-W, W-H, H-H, C-C, are modelled by a Lennard-Jones potential $$U{_{LJ}}({r_{ij}})=4 {\epsilon_{\alpha \beta}}{\left[{ {\left({\frac{\ell}{{r_{ij}}}}\right)}^{12}-{\left({\frac{\ell}{{r_{ij}}}}\right)}^{6}}\right]} \, , \label{ulj}$$ where $\alpha/\beta\, \epsilon \, \{ \text W, \text H, \text C \} $. Adjacent beads along a single model molecule interact via the harmonic potential $$U_2(r_{i,i+1})=k_2{\left({r_{i,i+1}-\ell}\right)}^2\, , \label{u2}$$ where $i$ and $i+1$ indicate two successive particles along the chain. The effects of hydrocarbon chain stiffness is modelled by letting all the particles within a single amphiphile molecule interact via the three-body bending potential $$U_3({\mathbf{r}}_{i-1,i},{\mathbf{r}}_{i,i+1})=k_3{\left({1-\frac{{\mathbf{r}}_{i-1,i}\cdot {\mathbf{r}}_{i,i+1}}{r_{i-1,i}\, r_{i,i+1}}}\right)}=k_3(1-\cos \phi_i) \, , \label{u3}$$ where ${\mathbf{r}}_{i,i+1}={\mathbf{r}}_{i+1}-{\mathbf{r}}_i$, see fig. \[mol\]. Simulation method and model parameters -------------------------------------- In the present work we combine both Monte-Carlo (MC) and Molecular Dynamics (MD) simulation methods. Our simulations are carried out for a cuboidal boxes of constant volume and periodic boundary conditions. The MC algorithm is used to make the system relax towards configurations of minimal potential energy, the output of $500\times N$ MC steps are used as starting configurations for the MD part of the simulations. For the parameters characterizing the above potentials we choose the values used in refs. [@GL1; @GL2; @alb1; @alb2]. For the interaction ranges of the LJ potentials we take $\ell=1/3$ nm which is of the same order as the LJ lengths of interactions for pairs of ${\text{CH}_2}$ groups, ${\text{CH}_3}$ groups or water molecules, as discussed in [@EB]. The characteristic length of the repulsive SC potential (\[u3\]), is taken to be $\ell'=1.05 \ell$ as in [@GL1]: with this choice the hard-core repulsion of the SC potential is of the same order of the repulsive part of the LJ potential. We take $\epsilon=2/N_{A_{V}}$ kJ ($N_{A_{V}}$ is the Avogadro number): this value is bigger than the LJ energies for pairs of ${\text{CH}_2}$ and/or water molecules reported in [@EB], but it takes into account that in our model one C particle corresponds to three or four ${\text{CH}_2}$ groups [@GL1]. The strength of the harmonic bond potential (\[u2\]) and of the three-body bending potentials (\[u3\]), are taken to be $k_2=5000 \epsilon/\ell^2$ and $k_3=2 \epsilon$ [@GL1]. Since we run MD simulations, two additional parameters are involved in the present model: the masses of the different beads, which enter the equations of motions, and the time step $\delta t$, used to discretize such equations. For simplicity’s sake, all the beads are taken to have the same mass $m=0.036/N_{A_{V}}$ kg, which is approximately the mean value of the mass of a water molecule and the mass of four ${\text{CH}_2}$ groups. Using this value of $m$ one obtains the characteristic time scale $\tau=\sqrt{m \ell^2 /\epsilon}=1.4$ ps. In the MD simulations, the integration time step $\delta t$ is taken to be $\delta t=\tau/2000=0.7$ fs. The [*leap-frog*]{} algorithm at constant temperature [@AT] with $k_B T=1.35 \epsilon$ (corresponding to $T=325$ K) is used for the integration of the equations of motions. In the following, if not differently specified, all the quantities will be expressed in units of the tree basic parameters $\ell$, $\epsilon$ and $m$. In the present paper we consider bilayers with three different values of the number of amphiphilic molecules $N=512$, $N=768$ and $N=1152$. For each value of $N$, the simulations are run with different values of the projected area $A_p$, which corresponds to the area of the simulation box side parallel to the bilayer. In changing the projected area, we keep constant the number of water particles $N_W$, as well as the overall simulation box volume $V$, i.e., we keep the overall system density constant. The values of $N$, $N_W$ and $V$ used in the simulations described in the following are reported in table \[tab1\]. [|c|c|c|]{} $N$& $N_W$& $V(\ell^3)$\ 512 & 3200 & 8640\ 768 & 4800& 12960\ 1152 & 7200 & 19440\ Stress tensor and mechanical surface tension {#mis_S} ============================================ On a macroscopic length scale, a membrane fixed on a frame can be viewed as an elastic sheet: the system will be characterized by some equilibrium value of the frame area where the force exerted on the frame sizes vanishes, and small changes in the frame area cause restoring forces to appear. If $F$ is the free energy of a framed bilayer, the surface tension $\Sigma$ conjugated to its projected area $A_p$ is defined as $$\Sigma={\frac{\partial F}{\partial A_p}}. \label{defSigma}$$ In the present work, the bilayer surface tension is measured using a method first introduced by Schofield and Henderson [@SH], and then extended by Goetz and Lipowsky [@GL1]. A fluctuating membranes can be considered isotropic and homogeneous along any direction parallel to the membrane plane, if the amplitude of the fluctuations is not too high. Thus, the system stress tensor will be a function of the coordinate $z$ perpendicular to the bilayer plane. The surface tension is related to the system stress tensor via $$\Sigma=\int {{\mathrm d}}z {\left[{\Sigma_T(z) -\Sigma_N(z)}\right]}, \label{defS}$$ where $\Sigma_T(z)$ and $\Sigma_N(z)$ are the components of the stress tensor perpendicular and normal to the bilayer surface, respectively. The integral can be extended to infinity because the stress tensor in isotropic in the bulk water and so $\Sigma_T(z)=\Sigma_N(z)$. By noting that the stress tensor is defined as the negative of the pressure tensor, one sees that eq. (\[defS\]) is equivalent to the original expression of the surface tension of a planar liquid-vapor interface, as formulated by Kirkwood and Buff [@KB]. The macroscopic stress tensor $\mathbf \Sigma$ can be expressed in terms of the microscopic stress $\mathbf s$ which depends on the momenta and on the positions of the system particles within a small volume. The microscopic and the macroscopic stress tensor are related via $${\mathbf \Sigma}={\left\langle \mathbf s \right\rangle}, \label{medias}$$ where the brackets denote ensemble average. The microscopic stress tensor ${\mathbf s}$ consists of two contributions one arising from the kinetic energy and the other from the interaction energy of the system particles within a small volume ${\mathbf s}={\mathbf s}_K+{\mathbf s}_{in}$. The kinetic contribution ${\mathbf s}_K$ to the macroscopic stress tensor, as given by eq. (\[medias\]), can be neglected, since it is isotropic, and must vanish on average. Thus, one is left with the interaction part of the microscopic stress tensor ${\mathbf s}_{in}$ only, and eq. (\[medias\]) becomes $${\mathbf \Sigma}={\left\langle \mathbf s \right\rangle}_{in}, \label{mediasin}$$ Furthermore a planar bilayer membrane in solvent can be considered homogeneous along any direction parallel to the the bilayer plane, and so is the stress tensor. The system can thus be divided in thin slices parallel to the bilayer plane, and the stress tensor can be averaged over each of these slices. Thus the microscopic stress tensor is given by [@GL1; @SH; @LE2] $${\mathbf s}_{in}(z)=f(z_i,z_j,z) \frac{1}{\Delta V}\sum_{i>j} {\mathbf F}_{ij}\otimes {\mathbf r}_{ij}, \label{defsz}$$ where the sum is extended to all the system particle pairs, $\Delta V$ is the volume of the slice, the function $f(z_i,z_j,z)$ determines the actual contribution of the pair $i,j$ to the stress tensor in the slice of coordinate $z$, and the symbol $\otimes$ denotes the tensorial product. If both the particles are inside the current slice we take $f(z_i,z_j,z)=1$ . Let $\Delta z$ be the thickness of each slice, if both the particles are external to the current slice and on opposite sides, we take $f(z_i,z_j,z)=\Delta z/|z_i-z_j|$, where, using periodic boundary conditions, the shortest distance $|z_i-z_j|$ between them is considered. If just one of the two particles lies within the current slice, we take $f(z_i,z_j,z)=\Delta z/(2|z_i-z_j|)$. In all the other cases, the contribution of the particles $i,j$ to the stress tensor associated to the slice of coordinate $z$ vanishes, and thus we take $f(z_i,z_j,z)=0$. Inspection of eq. (\[defsz\]) suggests that its rhs can be viewed as a local density of the macroscopic virial tensor ${\mathbf W}=\sum_{i>j} {\mathbf F}_{ij}\otimes {\mathbf r}_{ij}$, and thus the function $f(z_i,z_j,z)$ determines the fraction of virial tensor density to be associated to the slice of coordinate $z$. Measurements of the microscopic stress tensor --------------------------------------------- Using the model amphiphilic bilayer described in section \[model\], the microscopic stress tensor $\mathbf s_{in}(z)$, as given by eq. (\[defsz\]), was measured for the three values of the number of amphiphilic molecules $N$ here considered, $N=512$, $N=768$, and $N=1152$. For each value of $N$, $\mathbf s_{in}(z)$ was measured for different values of the bilayer projected area $A_p$ keeping constant the total volume $V$, see table \[tab1\]. The simulation boxes have been divided into 140 $z$-slices, which corresponds to $\Delta z\sim 0.1\ell$, for the simulation box dimensions here used. The off-diagonal elements of the system stress tensor are found to vanish as expected, as well as the average of its kinetic part (data not shown). For each value of $A_p$, the mechanical surface tension $\Sigma$ is obtained using eq. (\[defS\]), by performing a discrete integration. For each value of $N$ and $A_p$ we run 5 simulations of $2\times 10^5$ MD steps, and a run of $6\times 10^6$ MC steps was interposed after each run of $2\times 10^5$ MD steps. The MC steps are inserted to allow the system to reach regions of its phase space not easily accessible with a single MD trajectory. It is worth noting that the code for the stress tensor measuring is extremely time-consuming: a single run of $2\times 10^5$ MD steps, for a system with $N=1152$ amphiphiles, needs about one month of machine time, on a computer equipped with a 1 GHz processor. The relative fluctuations of the surface tension with respect to its average value are very large, as found in other work [@GL1]. Following ref. [@GL1], in order to reduce the effect of short time fluctuations, the total simulation time of each run is divided in blocks of 5000 MD steps, and the surface tension is subaveraged over each of these blocks. This average value over the 5000 MD steps is then used as the sample value for the current block. The surface tension obtained from simulations is plotted in fig. \[compare\_S\], as a function of the projected area per amphiphile $a_p=A_p/(N/2)$, for the three values of $N$ here considered. According to the classical elasticity theory, the parameter describing the effective compressibility of an elastic sheet to a change in its projected area is the area compressibility, $K_A$ defined as $$K_A=A_p{\frac{\partial \Sigma}{\partial A_p}} .$$ By integrating this last equation, one obtains $$\Sigma=K_A\ln{\left({\frac{A_p}{A^\dagger_p}}\right)}\simeq K_A\frac{A_p-A^\dagger_p}{A^\dagger_p}, \label{SigKA}$$ where the last equality holds for values of $A_p$ close to $A^\dagger_p$, which is the area at which $\Sigma$ vanishes. Upon integration of eq. (\[SigKA\]), we obtain the classical expression for the free energy of a stretched (or compressed) elastic sheet as a a function of $A_p$ around the equilibrium value $A^\dagger_p$ $$F=\frac{K_A}{2 A^\dagger_p} {\left({A_p-A^\dagger_p}\right)}^2,$$ where the reference free energy at $A=A^\dagger_p$ has been chosen to be equal to zero. Inspection of figure \[compare\_S\] indicates that the surface tension $\Sigma$ obtained by simulations follows the behaviour predicted by eq. (\[SigKA\]): this quantity is linear within a range of values around the equilibrium area $A^\dagger_p$. We find that for $A_p\ll A^\dagger_p$, ($a_p\simeq2.065$) the surface tension $\Sigma$ is no longer a monotonous function of $a_p$: this is probably due to the fact that, for such small projected area per molecule, the system exhibits buckling, and the membrane cannot be considered flat on average. In this case, the basic hypothesis that the membrane is isotropic and homogeneous along the the $z$-axis is not fulfilled. However, such hypothesis is required for measuring $\Sigma$ with the method discussed in this section, and thus the results obtained for very small values of $a_p$ must be inaccurate. Thus in the following we will consider values of $a_p$ such that $\Sigma$ is a monotonic function of $a_p$, i.e. $a_p>2.065$. The values of the equilibrium area, of the area compressibility and of the slope of $\Sigma$ as a function of $a_p$, for the three system sizes here considered, are listed in table \[tabS\]. In the same table, we also report the results for a smaller system which was considered in a previous work [@GL1]. [|c|c|c|c|]{} $N$& $a^\dagger_p\, (\ell^2)$& $K_A\, (\epsilon/\ell^2)$& $\partial \Sigma/ \partial a_p\, (\epsilon/\ell^3)$\ 128 & $2.15\pm0.02$ & $11.8\pm1.5$& $5.5\pm 0.7$\ 512 & $2.12\pm0.01$ & $11.7\pm1.5$& $5.5 \pm 0.7$\ 768 & $ 2.12 \pm 0.01$ & $9.4 \pm 1.1$ & $4.5 \pm 0.5$\ 1152 & $2.12\pm0.01$& $8.8\pm0.8$& $4.1 \pm 0.4$\ The results listed in table \[tabS\] suggests that the equilibrium area $A^\dagger_p$ does not depend on the system size, but is an intrinsic properties of the model amphiphilic molecule used here. However, the area compressibility decreases as a function of the system size, as already found in [@MM]: this decrease is due to the fact that a larger number of oscillation modes are introduced in the system as its projected area is increased. Thus, the bilayer becomes easier to compress or to stretch if its total projected area $A_p$ increases while the projected area per amphiphile $a_p$ is kept constant. Note that the values of the area compressibility found for the present model range between 263 and 360 $mJ/m^2$, see table \[tabS\]. Such values are slightly greater than those observed for real lipid bilayers, that are in the range 140-240 $mJ/m^2$ [@Ma], but are very similar to those values found in other works on computer simulations of amphiphilic bilayer with atomic resolution [@MM; @LE2] Elasticity Hamiltonian and the fluctuation surface tension. {#mis_s} =========================================================== In the Monge representation, the shape of the bilayer is described by the height function $h(x,y)$ which measures the distance of its midsurface from the reference $(x,y)$ plane. The classical elasticity Hamiltonian of fluctuating membranes, for small fluctuations, reads [@Can; @Helf] $$\begin{aligned} H&=&\sigma A_p +\int_{A_p} {{\mathrm d}}x {{\mathrm d}}y{\left[{ \frac 1 2 \sigma (\nabla h)^2 + \frac 1 2 \kappa (\nabla^2 h)^2}\right]}\nonumber \\ &=&\sigma A_p + \frac{1} {2 L^2} \sum _{{\mathbf{q}}}{\left[{\sigma q^2+ \kappa q^4}\right]} |{\tilde h_{\mathbf{q}}}|^2 \, , \label{fapp}\end{aligned}$$ where ${\tilde h_{\mathbf{q}}}$ define the Fourier coefficients of $h(x,y)$ $${\tilde h_{\mathbf{q}}}=\frac {1} {A_p}\int_{A_p} {{\mathrm d}}x {{\mathrm d}}y\, \exp{\left[{\mathrm{i} {\mathbf{r}}\cdot {\mathbf{q}}}\right]} h(x,y).$$ The two parameters appearing in eq. (\[fapp\]) are the [*bending rigidity*]{} $\kappa$, describing the resistance of the system to bending, and the surface tension $\sigma$, which takes into account the contribution of the bilayer total area $A$ to the system energy. In fact, eq. (\[fapp\]) can be rewritten as $$H=\sigma A+ \int {{\mathrm d}}x {{\mathrm d}}y \frac 1 2 \kappa (\nabla^2 h)^2.$$ The membrane fluctuation spectrum, defined as $S(q)\equiv|{\tilde h_{\mathbf{q}}}|^2$, depends only on $q=|{\mathbf{q}}|$ and exhibits the functional form [@LB] $$S(q)=\frac{k_B T}{A_p {\left({\sigma q^2 +\kappa q^4}\right)}}. \label{fluspe}$$ Equation (\[fluspe\]) has been obtained under the hypothesis that the bilayer can be modelled as a geometrical surface, neglecting its thickness. Thus eq. (\[fluspe\]) holds only for wavenumbers smaller than an upper bond ${{q^*}}$, which is usually taken to be ${{q^*}}\simeq 2 \pi /d_0$, where $d_0$ is the bilayer thickness. We find that the model bilayer thickness is $d_0\simeq 6 \ell$, independently of the system size. For $q>q^*$, the shape of the bilayer is characterized by the local protrusions of the amphiphiles, and thus the fluctuation spectrum is dominated by a local surface tension term, see refs [@Ste; @GL2; @alb2]. However these sub-optical modes are not accessible in experiments, and thus, in the present work, we will focus on the small wavenumber, and on the effective values of $\kappa$ and $\sigma$ that can be obtained by eq. (\[fluspe\]). By sampling the height field $h(x,y)$ during the simulations, one can obtain the mean fluctuation spectrum (\[fluspe\]). By fitting $S(q)$ to the rhs of eq. (\[fluspe\]), for $q\lesssim q^*$, the values of the bending rigidity $\kappa$ and of the surface tension $\sigma$ can thus be estimated. The fluctuation spectrum shape depends on the projected area: and thus both $\kappa$ and $\sigma$ depend on this parameter. By changing the value of $A_p$ one can achieve the tensionless ($\sigma=0$) state [@GL2; @alb2]. The surface tension has been measured for the three system sizes here considered $N=521, \, 768,\, 1152$, for different projected area, and keeping constant the overall volume, see table \[tab1\]. For each value of the projected area, we run $10\times 10^5$ MD steps: a run of $6\times 10^6$ MC steps has been interposed after each run of $10^5$ steps. The fluctuation spectrum has been sampled during the MD simulations, every $5000$ MD steps. In fig.\[sq2\] the fluctuation spectrum of the larger system here considered is plotted as a function of the wavenumber $q$ for different value of the projected area per amphiphile $a_p$: the surface tension $\sigma$ vanishes for a given values of $a_p=a^*_p$, which depends on the microscopic details of the system (on the model parameters in the case of simulations), while for $a_p>a_p^*$ ($a_p<a_p^*$) it is greater (smaller) than zero. This is clearly shown in figure \[compare\_s\], where the surface tension $\sigma$ is plotted as a function of $a_p$ for the three values of $N$ here considered. By linear fit of the data in figure \[compare\_s\], we obtain the tensionless area $a^*_p$ and the slope of the surface tension $\sigma$ as a function of $a_p$, the results are listed in table \[tabsig\]. As in the case of the mechanical surface tension $\Sigma$, the slope of the curve $\sigma(a_p)$ decreases as $N$ is increased. Comparison of tables \[tabS\] and \[tabsig\] indicates that the slopes of $\Sigma$ and $\sigma$ as functions of $a_p$ are different for a given system size. [|c|c|c|]{} $N$& $a^\dagger_p\, (\ell^2)$& $\partial \sigma/ \partial a_p\, (\epsilon/\ell^3)$\ 512 & $2.085\pm0.005$ & $5.4 \pm 0.7$\ 768 & $ 2.09 \pm 0.01$ & $4.0 \pm 0.4$\ 1152 & $2.09\pm0.01$& $3.6 \pm 0.4$\ Comparison between the two surface tension $\sigma$ and $\Sigma$ {#compare_sec} ================================================================ Comparison of figure \[compare\_s\] with figure \[compare\_S\] shows clearly that the projected area per amphiphile $a_p^*\simeq2.09$ where the surface tension $\sigma$ vanishes is different from the projected area per amphiphile $a_p^\dagger\simeq2.12$, at which the mechanical tension $\Sigma$ vanishes. Since these two quantities have been measured independently, for [*each*]{} value of the projected area, this last result suggests that one is dealing with two different quantities. In addition, the two vanishing areas $a_p^*$ and $a_p^\dagger$ do not exhibit any dependence on the system size $N$, and thus one can argue that they are two independent intrinsic properties of the amphiphilic molecule. Comparison of tables \[tabS\] and \[tabsig\], and inspection of figure \[compare\_sS\], clearly indicate that the curves $\Sigma(a_p)$ and $\sigma(a_p)$ exhibit different slopes. Furthermore, if one plots on the same figure $\sigma$ and $\Sigma$ as functions of $a_p$, in the range where $\Sigma$ is linear with respect to $a_p$, the two data sets appear clearly shifted the one respect to the other, see figure \[compare\_sS\], where these two quantities have been plotted for the $N=1152$ case. In order to relate the two surface tension $\sigma$ and $\Sigma$, we need first to define the statistical ensemble which characterizes the system considered here, i.e., the minimal set of thermodynamical variables which fully describe the system state. The classical argument that the system area readjusts to its preferred value (see, e.g., [@Saf]) cannot be invoked here because of the constrain represented by the frame area (simulation box area). The effective bilayer area $A$ can be evaluated using a discrete version of the formula $$A=\int_{A_p} {{\mathrm d}}x{{\mathrm d}}y \sqrt{1+{\left({\nabla h}\right)}^2}, \label{defA}$$ where the field $h(x,y)$ is sampled during the MD simulations. The effective area per amphiphile $a$ is plotted in fig. \[effa\] as a function of $a_p$. Inspection of this figure clearly indicates that the effective area $a$ is not independent of the projected area $a_p$, but is rather a function of $a_p$. Thus, the thermodynamical ensemble characterizing the framed bilayer here considered is the $(T,A_p=L^2, N)$ ensemble, where $L$ is the size of the simulation box face parallel to the bilayer. The free energy in the $(T,A_p=L^2,N)$ ensemble for a bilayer membrane system is $$\begin{aligned} F&=&-k_B T\ln Z=-k_B T\ln{\left({\int {\mathcal D} h\, e^{-\beta H(\{h\})} }\right)}\nonumber\\ &\simeq& -k_B T\ln{\left({ \prod_{i=0}^{N-1}\int \frac{ d h_i}{\lambda} e^{-\beta H(\{h_i\})} }\right)}\nonumber \\ &=&-k_B T\ln {\left({ \prod_{{\mathbf{q}}}\int \frac{d h_{{\mathbf{q}}}}{\sqrt N a_p\lambda} e^{-\beta H(\{h_q\})} }\right)}\nonumber \\ &=&\sigma A_p-k_B T\ln {\left[{ \prod_{{\mathbf{q}}}\int \frac{d h_{{\mathbf{q}}}}{\sqrt N a_p\lambda} e^{- \frac \beta{2 L^2} \sum_{{\mathbf{q}}}(\sigma q^2+ \kappa q^4) |h_{\mathbf{q}}|^2 } }\right]}\nonumber \\ &=&\sigma A_p-k_B T \ln {\left[{ \prod_{{\mathbf{q}}} \frac{2\pi}{\lambda^2 a_p \beta (\sigma q^2+ \kappa q^4)} }\right]}^{\frac 1 2}\nonumber \\ &=&\sigma A_p+{\frac{1}{2}}k_B T\sum_{\mathbf{q}}\ln {\left[{\frac{\lambda^2 a_p (\sigma q^2+ \kappa q^4)}{2\pi k_B T} }\right]}\, . \label{free}\end{aligned}$$ The parameter $\lambda$ has the dimension of a length and is the analogous of the De Broglie wave-length in the ideal gas partition function. Without introducing this parameter, the partition function would be dimensionful. The equation of state for the intrinsic area is $$\begin{aligned} {\left\langle A \right\rangle}&=&{\frac{\partial F(T,A_p)}{\partial \sigma}} =A_p+\frac 1 2 k_B T \sum_{\mathbf{q}}\frac 1 {\sigma+\kappa q^2}, \label{area} $$ which is a well known result, see, e.g, [@Rou]. Let us now introduce the quantity $N'$, which represents the number of membrane patches which fluctuate independently ($N'\le N/2$): the summations in equations (\[free\]) and (\[area\]) run over these $N'$ wavemodes, and thus we have $\sum_{{\mathbf{q}}}=N'$. Note that, in the large $A_p=L^2$ limit, $N'$ can be estimated as follows $$\begin{aligned} N'&=& \sum_{{\mathbf{q}}}\simeq {\left({\frac{L}{2 \pi}}\right)}^2 \int {{\mathrm d}}q^2 =\frac{L^2}{2 \pi} \int^{{q^*}}_{{q_{min}}}q {{\mathrm d}}q\nonumber \\ &=& \frac{L^2}{4 \pi} {\left({{{q^*}}^2-{{q_{min}}}^2}\right)} \simeq \frac{A_p}{4 \pi}, \label{evaln1}\end{aligned}$$ where we have taken ${{q^*}}\simeq 1$ (see discussion is section \[mis\_s\]), and ${{q^*}}\gg{{q_{min}}}\simeq 2 \pi/L\sim 0$, in the large system size limit. Taking into account that $q^2=(2 \pi)^2 (n_x^2+n_y^2)/L^2$, and since the surface tension $\Sigma$ is the thermodynamic conjugate of $A_p=L^2$, as defined by eq. (\[defSigma\]), we have $$\begin{aligned} \Sigma(T,A_p)&=&{\frac{\partial F}{\partial A_p}}=\sigma -{\frac{1}{2}}\frac{k_B T}{A_p} \sum_{{\mathbf{q}}} \frac{\kappa q^4}{\kappa q^4+\sigma q^2}\\ &=&\sigma +{\frac{1}{2}}\frac{k_B T}{A_p} \sum_{{\mathbf{q}}} \frac{\sigma }{\kappa q^2+\sigma } -\frac{k_B T}{2} \frac{N'}{A_p}\label{Sigma2}\\ &=& \frac{{\left\langle A \right\rangle}}{A_p} \sigma -\frac{k_B T}{2} \frac{N'}{A_p}\label{Sigma4}\, ,\end{aligned}$$ where we substituted eq. (\[area\]) into eq. (\[Sigma2\]). Equation (\[Sigma4\]) relates thus the frame surface tension $\Sigma$ with two of the variables which define the actual ensemble $T,A_p$, and with the fluctuation surface tension $\sigma$. Comparison of tables \[tabS\] and \[tabsig\] indicates that, for a given system size, the slope of $\Sigma(a_p)$ is larger than the slope of $\sigma(a_p)$: since $A>A_p$ by definition of intrinsic area, the first term on the rhs of eq. (\[Sigma4\]) accounts for the difference of the slope between the two curves. Note that the second term on the rhs of eq. (\[Sigma4\]) takes into account the effect of the entropic elasticity on the system surface tension. This term is proportional to the system temperature, and inversely proportional to the system effective area per amphiphile $a_p'=A_p/N'$: the analogy with the case of polymers can be immediately drawn. It is worth noting that a similar results was obtained by Farago and Pincus in ref. [@FP], where a continuous expression for the free energy (\[free\]) was used, and where the surface tension $\sigma$ was taken to depend explicitly on the effective area $A$. A first rapid check of equation (\[Sigma4\]) can be done by noting that it predicts that, if the surface tension $\sigma$ vanishes, the frame surface tension $\Sigma$ is negative. This is confirmed by inspection of figures \[compare\_S\] and \[compare\_s\]. In equation (\[Sigma4\]) the only adjustable parameter is $N'$, while $\sigma$ can be estimated as described in section \[mis\_s\], and $A$ can be sampled during the MD simulations using eq. (\[defA\]). One can thus compare the values for $\Sigma$ predicted by eq.(\[Sigma4\]) with those directly measured as described in section \[mis\_S\]. In the following, the optimal value of the parameter $N'$ will be determined by fitting eq. (\[Sigma4\]) to the measured values plotted in fig. \[compare\_S\], for the three values of $N$ here considered. With this fit we find $N'=155$ for $N=512$, $N'=181$ for $N=768$ and $N'=213$ for $N=1152$. Using this values for $N'$, we plot the measured and the calculated values for $\Sigma$, as a function of $a_p$, for $N=512$ fig. \[simul1\], for $N=768$ fig. \[simulm\], and for $N=1152$ fig. \[simul2\]. Inspection of these figures indicates a good agreement between the values of $\Sigma$ predicted by eq.(\[Sigma4\]) and those calculated as described in section \[mis\_S\]. We now consider the scaling behaviour of the quantity $N'$ as a function of the system size. In fig. \[n1\], $N'$ is plotted as a function of the tensionless projected area $A^*_p$ (projected area at which $\sigma=0$), and as a function of the number of molecules $N$. Inspection of this figure, suggests that $N'$ is a linear function of $A^*_p$, and the slope of such function is in good agreement with that predicted by eq. (\[evaln1\]). It is worth noting that, as discussed is section \[mis\_s\], the projected area per molecule $a^*_p=A^*_p/(N/2)$ is independent of the system size, and thus we have $N'\sim A^*_p/(4 \pi)\sim N$, as one would expect in the large $N$ (or large $A_p$) limit. Note that the intercept of the line plotted in fig. \[n1\] is non-zero, while eq. (\[evaln1\]) predicts a vanishing intercept in the large projected area limit. However the argument used to obtain eq. (\[evaln1\]) is no longer valid for small value of $A_p$ ($N$), and thus in this limit such an equation is incorrect. Conclusion {#concl} ========== In the present paper we have measured both the surface tension $\sigma$ appearing in the Hamiltonian which governs the bilayer shape fluctuations (\[fapp\]), and the surface tension $\Sigma$ which governs the elastic response of the system to a change in its projected area $A_p$. Using computer simulations, we have measured independently these two quantities for bilayer with different projected area. Our results indicate that the two surface tensions have different values for a given value of $A_p$. In particular the two projected area per amphiphile $a^*_p$ and $a^\dagger_p$ are different, where $a^*_p$ is the projected area per amphiphile at which the surface tension $\sigma$ vanishes, while $a^\dagger_p$ is the projected area per amphiphile where the surface tension $\Sigma$ vanishes. Furthermore, the two surface tensions are found to exhibit different slopes as functions of the projected area per molecule $a_p$, for a given system size. Using a simple thermodynamic argument, we manage to relate the two quantities $\sigma$ and $\Sigma$, and the relation between them which we found, eq. (\[Sigma4\]), nicely fits the data that we obtain from simulations. This is the most important result of the present work: it indicates that eq. (\[Sigma4\]) succeeds to capture the basic relation between the two quantities. Note that in a previous work [@Ste], where computer simulations of model membrane were considered, the two surface tensions were found to be proportional the one respect to the other, but no argument was introduced to explain such an effect. The difference between the two quantities, does not appear to be due to some size effect, since the vanishing areas $a^*_p$ and $a^\dagger_p$ do not depend on the system size. The scaling behaviour of the effective wavemode number $N'$ as a function of the system size is a rather delicate issue. In the present work we find that $N'$ scales linearly with the number of molecule $N$ (the tensionless area $A^*_p$), but in a restricted range of values of $N$ ($A_p$). We were limited in the choice of the values of $N$ by the considerable amount of computation time required by the simulations. For larger values of $N$ one would expect that $N'$ grows either as $N$ or more slowly than $N$. If this were the case, since $A_p\sim N$, the second term of the rhs of eq. (\[Sigma4\]) would vanish in the large $N$ limit. In this case eq. (\[Sigma4\]) would read $\Sigma\simeq A/A_p \cdot \sigma$. We plan to consider larger system size in a future work. In conclusion, the results contained in this paper puts in evidence that in a bilayer membrane, the surface tension $\sigma$ appearing in the elasticity Hamiltonian (\[fapp\]) and the mechanical surface tension $\Sigma$ are two distinct thermodynamic quantities, which have to be measured independently for a full characterization of the system elastic properties, although for a constrained system, the geometrical constraints impose a relation between them. It would be interesting to measure these two quantities in a real system, e.g, a lipid vesicle manipulated with a micropipette. On the basis of the results presented in this paper, one would expect that the geometrical constrains imposed by the micropipette would lead to a difference between the two surface tensions. The author is indebted to R. Lipowsky for introducing him to this topic, for many interesting discussions and for a critical reading of this manuscript. I am grateful to J.B. Fournier for the long and interesting discussions on the two tension issue. I also thank L. Peliti for many discussions, and for his encouragements. This work was partially supported by MIUR-PRIN 2004. Finally, I thank the CRdC AMRA for the use of its computational resources. [99]{} J. Israelachvili, Intermolecular and Surface Forces, (Academic Press, London, 1992). B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, [*Molecular Biology of the Cell*]{}, (Garland, New York, 2002). R. Lipowsky, Vesicles and biomembranes in [*Encyclopedia of Applied Physics*]{} Vol. 23, pages 199-222, (VCH Publishers, Weinheim and New York 1998). Safran S. A., Statistical Thermodynamics of Surfaces, Interfaces and Membranes (Addison-Wesley, Reading, Mass. 1994). S.J. Marrink and A.E. Mark, [*J. Phys. Chem. B*]{} [**105**]{}:6122-6127 (2001). R. Goetz and R. Lipowsky, [*J. Chem. Phys.*]{} [**108**]{}:7397-7409 (1998). E. Lindhal and O. Edholm, [*J. Chem. Phys.*]{} [**113**]{}:3882 (20 00). F. David and S. Leibler, [*J. Phys. France*]{} [**1**]{}:959-976 (1991). O. Farago and P. Pincus, [*J. Chem. Phys.*]{} [**120**]{}: 2934–2950 (2004). J. Stecki, [*J. Chem. Phys.*]{} [**120**]{}: 3508–3516 (2004). R. Goetz, G. Gompper and R. Lipowsky, [*Phys. Rev. Lett.*]{} [**82**]{}:221-224 (1999). A. Imparato, J.C. Shillcock and R. Lipowsky, [*Eur. Phys. J. E* ]{} [**11**]{}:21-28 (2003). A. Imparato, J.C. Shillcock and R. Lipowsky, [*Europhys. Lett.*]{} , [**69**]{}: 650-656 (2005). E. Egberts and H.J.C. Berendsen [*J. Chem. Phys.* ]{} [**89**]{}:3718-3732 (1988). M.P. Allen and D.J. Tildesley, [*Computer Simulations of Liquids*]{} (Oxford, Oxford, 1987). J.G. Kirkwood and F.P. Buff, [*J. Chem. Phys.*]{}, [**17**]{}:338 (1 949). P. Schofield , J.R. Henderson, Proc. R. Soc Lond. A MAT [**379**]{} ( 1776): 231-246 1982. D. Marsh , CRC handbook of lipid bilayers, Boca Raton, CRC Press, (1990). P.B. Canham, [*J. Theor. Biol*]{} [**26**]{}:61-81 (1970). W. Helfrich, [*Z. Naturforsch.*]{}, [**28c**]{}:693 (1973). J.F. Lennon and F. Brochrard, [*J. Phys. (Paris)*]{} [**36**]{}, 1035 (1975). D. Roux, [*Physica A*]{} [**172**]{}, 242–257 (1991). O.Farago and P. Pincus, [*Eur. Phys. J. E*]{}, [**11**]{}:399 (2003). \[.9\][$\phi_i$]{} \[.9\][$i-1$]{} \[.9\][$i$]{} \[.9\][$i+1$]{} ![Cartoon of the model amphiphile molecule used in this paper.[]{data-label="mol"}](molecules.eps "fig:"){width="3cm"} \[bl\]\[bl\]\[.8\][$N=1152$]{} \[bl\]\[bl\]\[.8\][$N=768$]{} \[bl\]\[bl\]\[.8\][$N=512$]{} \[br\]\[br\]\[1.\]\[-90\][$\Sigma$]{} \[ct\]\[ct\]\[1.\][$a_p$]{} ![Plot of $\Sigma$ as a function of the projected area per amphiphile $a_p$ for the three values of $N$ here considered, $N=512$, $N=768$ and $N=1152$.[]{data-label="compare_S"}](confronto_Sigma.eps "fig:"){width="12cm"} \[bl\]\[bl\]\[1.\]\[-90\][$S$]{} \[br\]\[br\]\[1.\][$q$]{} \[tr\]\[tr\]\[.8\][$a_p=2.066$]{} \[tr\]\[tr\]\[.8\][$a_p=2.095$]{} \[tr\]\[tr\]\[.8\][$a_p=2.127$]{} \[tr\]\[tr\]\[.8\][$a_p=2.157$]{} \[tr\]\[tr\]\[.8\][$a_p=2.203$]{} ![Fluctuation spectrum $S$ as a function of the wavenumber $q$, for different values of the projected area per amphiphile $a_p$, for bilayers with $N=1152$. The tensionless state ($\sigma=0$) corresponds to the projected area per amphiphile $a_p^*=2.095$. The dashed line is obtained by fitting the fluctuation spectrum of the tensionless system to eq. (\[fluspe\]), with $\sigma=0$, for $q\lesssim 1$.[]{data-label="sq2"}](sq.eps "fig:"){width="12cm"} \[bl\]\[bl\]\[.8\][$N=1152$]{} \[bl\]\[bl\]\[.8\][$N=768$]{} \[bl\]\[bl\]\[.8\][$N=512$]{} \[br\]\[br\]\[1.\]\[-90\][$\sigma$]{} \[ct\]\[ct\]\[1.\][$a_p$]{} ![Plot of $\sigma$ as a function of the projected area per amphiphile $a_p$ for the three values of $N$ here considered.[]{data-label="compare_s"}](confronto_sigma.eps "fig:"){width="12cm"} \[ct\]\[ct\]\[1.\][$a_p$]{} \[ct\]\[ct\]\[.8\][$\sigma$]{} \[ct\]\[ct\]\[.8\][$\Sigma$]{} ![Plot of the fluctuation surface tension $\sigma$ and of the mechanical surface tension $\Sigma$ as functions of the projected area per amphiphile $a_p$, for the $N=1152$ system. The lines are linear fits to the two sets of data. The two sets of data appear to be clearly shifted and tilted with respect to each other.[]{data-label="compare_sS"}](confr_sS_1152.eps "fig:"){width="12cm"} \[ct\]\[ct\]\[1.\][$a_p$]{} \[bl\]\[bl\]\[1.\]\[-90\][$a$]{} \[br\]\[br\]\[.8\][$N=1152$]{} \[br\]\[br\]\[.8\][$N=768$]{} \[br\]\[br\]\[.8\][$N=512$]{} ![Plot of the measured effective area per amphiphile $a$, as a function of the projected area per amphiphile $a_p$ (simulation box area), for the three values of $N$ here considered. []{data-label="effa"}](a_ap.eps "fig:"){width="12cm"} \[br\]\[br\]\[1.\][$a_p$ ]{} \[br\]\[br\]\[.8\][$\Sigma$ ]{} \[br\]\[br\]\[.8\][$\Sigma(T,A_p,\sigma)$]{} ![Plot of the measured surface tension $\Sigma$ and of the estimate of the same quantity, as given by eq. (\[Sigma4\]), as functions of $a_p$, for a system with $N=512$ amphiphiles.[]{data-label="simul1"}](confr_2_sigma "fig:"){width="12cm"} \[br\]\[br\]\[1.\][$a_p$ ]{} \[br\]\[br\]\[.8\][$\Sigma$ ]{} \[br\]\[br\]\[.8\][$\Sigma(T,A_p,\sigma)$]{} ![Plot of the measured surface tension $\Sigma$ and of the estimate of the same quantity, as given by eq. (\[Sigma4\]), as functions of $a_p$, for a system with $N=768$ amphiphiles.[]{data-label="simulm"}](confr_2_sigma_768.eps "fig:"){width="12cm"} \[br\]\[br\]\[1.\][$a_p$ ]{} \[br\]\[br\]\[.8\][$\Sigma$ ]{} \[br\]\[br\]\[.8\][$\Sigma(T,A_p,\sigma)$]{} ![Plot of the measured surface tension $\Sigma$ and of the estimate of the same quantity, as given by eq. (\[Sigma4\]), as functions of $a_p$, for a system with $N=1152$ amphiphiles.[]{data-label="simul2"}](confr_2_sigma_1152 "fig:"){width="12cm"} \[ct\]\[ct\]\[1.\][$N$]{} \[br\]\[br\]\[1.\]\[-90\][$N'$]{} \[ct\]\[ct\]\[1.\][$A^*_p$]{} ![Plot of the effective fluctuation modes $N'$ as a function of the vanishing tension projected area $A^*_p$ ($\sigma=0$) and as a function of the number of molecules $N$ (upper $x$-axis). The line is a linear fit of the data, whose slope, as a function of $A^*_p$, is $0.086\pm 0.003\simeq 1/(4 \pi)$, as predicted by eq. (\[evaln1\]). []{data-label="n1"}](n.eps "fig:"){width="12cm"}

--- author: - | Bernard F Schutz\ Max Planck Institute for Gravitational Physics\ (Albert Einstein Institute), 14476 Potsdam/Golm, Germany,\ and\ Department of Physics and Astronomy,\ Cardiff University, Cardiff, Wales, UK.\ [Bernard.Schutz@aei.mpg.de]{} title: 'Thoughts About a Conceptual Framework for Relativistic Gravity [^1]' --- [**1. Introduction**]{}\ Mine is one of several talks at this meeting that consider the revival of relativity and its integration into the mainstream of physics, beginning in the 1950s. Ted Newman has described the physics problems that created confusion during the slow period 1930–1950, and how eventually a new generation of young physicists pulled the theory out of its mire. Silvio Bergia has emphasized the changes of thinking that were required, and the importance of the physical insight and especially the geometrical perspective that John Wheeler, among others, brought to the subject. I want to focus on the gulf that opened up during the slow period between relativists and the rest of what I will call mainstream theoretical physics. This gulf is important not just for the negative influence it exerted on the development of relativity. It also has much to teach us about what physicists expect from a theory of physics, and especially about the role of heuristic concepts in physicists’ communication with one another. My thesis is that general relativity, despite its essential [*mathematical*]{} completeness in 1916, did not become a complete theory of [*physics*]{} until the 1970s. In order to understand this period, scholars of relativity need to look, not just at progress in understanding the mathematical theory, but at the slow development of heuristic concepts that were needed to enable relativists to talk to other physicists in a common language. Today we have a fairly secure set of heuristic concepts: for example, we know what a black hole is, we know what gravitational waves do, we know how gravitational lenses work. These concepts – black holes, gravitational waves, gravitational lenses – have gained a kind of concrete physical reality, even though if you take them apart they are just ideas that rest ultimately on rather complex (and usually approximate) solutions of Einstein’s field equations. Very importantly, they are concepts that relativists can communicate to nonrelativists who may need them (astronomers, experimental physicists, historians, the general public) without needing to pass on all their mathematical underpinnings. The key accomplishment of the generation of physicists who revived relativity is that they created a wide range of useful concepts like these out of the confusions that plagued the previous generation. This took a huge amount of work, but the work was not done at random. Rather, a handful of creative and senior physicists, many of whom came to relativity from other branches of physics, very deliberately shaped the directions of research toward developing these paradigmatic concepts, thereby adding the physics to the mathematical skeleton of the theory. In my view, the absence of such a vision of how to make relativity into a working theory of physics was what, in the dark period, led to the increasing isolation of relativity from the mainstream. [**2. The Gulf of Relativity**]{}\ The gulf between mainstream physics and relativity between 1930 and 1960 is remarkable for how huge it was (Eisenstaedt 1989, 2006). Rarely has an important sub-field of physics enjoyed such a poor reputation. Very few physicists moved back and forth across the gulf or even made an effort to communicate across the divide. Equally remarkable has been the subsequent huge turn-around. Gravitational physics is mainstream physics today. Massive amounts of money fund gravitational wave experiments; the holy grail of theoretical particle physics is to unify the nuclear and electromagnetic forces with gravity; a course in general relativity is standard for physics graduate students. A few short anecdotes serve to illustrate the depths to which relativity sank and the heights to which it has subsequently risen. [*Anecdote 1.*]{} The Nobel-Prize-winning astrophysicist Subrahmanyan Chandrasekhar kept a remarkable scientific diary, in which at the end of each year he summarized his scientific work and decisions of that year. Shortly after Chandra’s death in 1995, Norman Lebovitz (private communication) showed me some of the entries. Very interestingly, Chandra writes that, during the 1930s, he considered starting to do research in relativity, in order to explore what would happen to a compact star that exceeded the maximum white-dwarf mass that Chandra himself had recently established. He consulted other physicists, who strongly advised him against doing this. General relativity, one told him, had proved to be a “graveyard of many theoretical astronomers”. Chandra particularly mentions that Niels Bohr discouraged him from making a move into relativity. (Considering Bohr’s exchanges with Einstein on the interpretation of quantum mechanics, this is a tantalizing remark!) Chandra’s reputation and career were by no means secure in the 1930s, and so he looked (very productively) elsewhere for research problems. It was not until after 1960 that he felt confident enough of his reputation that he finally indulged his long-postponed wish to work on general relativity. \[Kip Thorne, one of the dominant figures in modern relativity research, reports (Thorne 1994) that he had similarly negative advice when he was contemplating doing graduate work in relativity in the early 1960s.\] [*Anecdote 2.*]{} It is worth looking here at Richard Feynman’s famous reaction (in a letter to his wife) to the 1962 Warsaw relativity meeting (Feynman 1988): > I am not getting anything out of the meeting. I am learning nothing. Because there are no experiments this field is not an active one, so few of the best men are doing work in it. The result is that there are hosts of dopes here and it is not good for my blood pressure: such inane things are said and seriously discussed that I get into arguments outside the formal sessions (say, at lunch) whenever anyone asks me a question or starts to tell me about his “work.” The “work” is always: (1) completely un-understandable, (2) vague and indefinite, (3) something correct that is obvious and self-evident, but worked out by a long and difficult analysis, and presented as an important discovery, or (4) a claim based on the stupidity of the author that some obvious and correct fact, accepted and checked for years, is, in fact, false (these are the worst: no argument will convince the idiot), (5) an attempt to do something probably impossible, but certainly of no utility, which, it is finally revealed at the end, fails, or (6) just plain wrong. There is a great deal of “activity in the field” these days, but this “activity” is mainly in showing that the previous “activity” of somebody else resulted in an error or in nothing useful or in something promising. It is like a lot of worms trying to get out of a bottle by crawling all over each other. It is not that the subject is hard; it is that the good men are occupied elsewhere. Remind me not to come to any more gravity conferences! This is, of course, typical Feynman hyperbole. We know that at that meeting (Infeld 1964) a core group of relativists was already coming to grips with issues like energy, black holes, and the reality of gravitational waves. And ironically it took place just a year before the first Texas Symposium in Relativistic Astrophysics (Robinson, et al, 1965), which is often regarded as the moment that relativity began to have real interest to astrophysicists. Nevertheless Feynman’s remarks show why it would still be another couple of decades before mainstream theoretical physics would completely drop its prejudices against the relativity community. The two sides were not communicating. [*Anecdote 3.*]{} I vividly remember my own personal experiences as a young relativist. In the 1970s if I mentioned black holes to an astronomer, the best I could usually hope for was a patronizing smile. And this was after the discovery of what we now know was the first black hole in a binary system, Cyg X-1, by the Uhuru satellite (which led to the award of the 2002 Nobel Prize to Riccardo Giacconi). Later, during the 1980s, when I moved into gravitational wave detection, many astronomers told me I was throwing my career away. And they were the sympathetic ones; others just saw me as a misguided threat to their own research funding! [*Anecdote 4.*]{} If the low point of relativity was very low, the current high point is indeed very high. Nothing illustrates the dramatic nature of this turn-around better than money. By 2020 at least 3-4 billion dollars will have been invested by a dozen national and international scientific organizations in building gravitational wave detectors on the ground and in space. Most of this money has already been committed, at least in a planning sense, and this has all happened even before the first direct detection of a gravitational wave! Where has today’s immense faith in general relativity come from? How did relativity establish such strong credentials after being in such disrepute? It seems to me that to answer this question we need to do more than simply catalog the details of what happened in relativity and astrophysics to get us where we are today. We have to understand how physicists judge the credibility of other physicists. The tortured development of gravitational physics is a good case study of how physicists decide that other people are really doing physics, even though they may not understand the mathematical and technical details. [**3. Heuristics in General Relativity**]{}\ I won’t attempt to give anything like a complete set of answers to the questions I have just posed, but I think a key to answering them lies in the fact that physicists have a characteristic way of thinking, which they call physical intuition. Physicists think in terms of models, of heuristic concepts that they connect up using this physical intuition. Physicists’ models must of course be founded on the mathematical expression of a theory, but physicists are typically not happy if all they have are mathematical links between their models. They want concepts that enable them to understand essential parts of theories, even if they have not developed a facility with the mathematics of those theories. They need to have models they can exchange with physicists in other specialties, which allow those physicists to work with the concepts without being expert in their underlying theory. I will illustrate the conceptual changes in relativity between the “dark ages” and the modern era by considering two key issues that were also listed by Ted Newman in his talk as key problems that were not solved during the dark years of relativity. The first is the meaning of the Schwarzschild solution. In the 1930s people talked about the “Schwarzschild singularity” (by which they meant the horizon, not the crunch at the center). Today we use the term “black hole”. There is a world of difference between the ideas behind these different terminologies. If you think you have a singularity then you can’t use it in a physical model. You don’t know how to include such an object in a physical system, either as the outcome of gravitational collapse or as an object that might affect other objects with its gravitational field. On the other hand, the term “black hole” is a shorthand description of a real object, one which you can confidently include in models for some physical systems: as the constituent of a model for an X-ray binary system, for example, or as a gravitating center in the middle of a galaxy. In the first case you are paralyzed by incomprehension. In the second you can hide away all the nonlinear general relativity, if you wish, and treat the object as just another member of the vast zoo of objects that makes up our fascinating universe. My second example is gravitational radiation. In the low period, people worried about the reality of the radiation itself. Doubting that waves could remove energy from sources and/or deposit it in detectors, relativists were unable to draw the clear parallels with electromagnetic radiation that would have emphasized the natural place that general relativity has in theoretical physics. By resolving these issues, relativists were finally in the position by 1980 to take advantage of the discovery of the Hulse-Taylor binary pulsar to show that observations supported the dynamical sector of Einstein’s theory. Not all the mathematical problems associated with gravitational waves are yet solved even today, but the field has enough confidence in its approximation methods and its control over the remaining outstanding issues that it has been able to develop a thoroughly convincing physical picture of gravitational waves. [**4. Why Did the Gulf Drift Open?**]{}\ So why did relativists find themselves excluded from the rest of theoretical physics in the 1930s to 1950s? Apart from a few notable exceptions, such as J. Robert Oppenheimer and Lev Landau, hardly anyone worked in relativity and other areas of theoretical physics between 1930 and 1960. And Oppenheimer and Landau were largely ignored by relativists (Thorne 1994). Let me list some explanations that are often offered and indicate why I don’t find them adequate. 1. General relativity is mathematically very difficult. The combination of nonlinearity and coordinate freedom made it difficult to make definite statements. This certainly underlay the problem that relativists had, and it explains why progress on understanding the theory was slow. But it does not explain the low regard that “real” physicists had for relativists. Indeed, one might have expected them to have gained respect from the rest of physics for making even small progress with such a difficult theory. 2. As Feynman remarked, there was little experimental data. This meant that progress relied especially strongly on the ability to ask and resolve the right kinds of theoretical questions. But one might have expected the field to have exploited the few observational hints that did exist. Chandrasekhar’s upper limit on the mass of white dwarfs, coupled with Fritz Zwicky’s suggestion that supernova explosions led to neutron stars, were a clear invitation to explore gravitational collapse and the Schwarzschild solution. But only Oppenheimer and Landau seem to have found this interesting. Importantly, they were physicists who approached relativity from outside, from the point of view of the mainstream theoretical community. Moreover, it is significant that the revival of relativity started during the 1950s without the stimulus of any new experimental or observational data. So, while an abundance of data would certainly have driven the field in the right direction had it been available, I don’t think that its absence explains why the field slipped into such a low state. 3. Relativity had to compete with quantum theory for good people. As Feynman says, “few of the best men are doing work in it”. The competition was certainly there, but I don’t believe that physics was that compartmentalized in the 1930s to 1950s. Leading quantum theorists had a deep interest in general relativity; Wolfgang Pauli wrote a beautiful textbook on it. The theory was widely regarded as the supreme achievement of 20th century theoretical physics. One would think that if relativists had made their own work interesting to mainstream physicists then they would not have worked in such isolation. There might have been many more Oppenheimers and Landaus crossing the gulf if the relativity community had welcomed them and worked with them, or even been able to communicate with them. 4. The Second World War got in the way. There is no doubt that this seriously retarded research, removing young people from research and inhibiting international scientific communication. The cold war afterwards did not help. Nuclear physics had proved so useful to the military that it (including particle physics) was well-funded after the war, whereas relativity fell into a theoretical backwater. But I am not convinced that this should have caused relativists to lose their way. Attacking the key problems of this period did not require a lot of money. A small field can still earn the respect of the majority. And the relativity community seems to have suffered less than other fields from the divisions of the cold war. It seems clear to me that, once the revival started, it went significantly more rapidly because of the relatively free intellectual interchange between Western and Soviet-bloc scientists working in relativity.[^2] I believe that the gulf opened between relativity and mainstream physics, not directly because of the problems listed above, but because the relativity community’s response to at least the first two problems was to ask the wrong questions. For example, one of the serious mathematical challenges that they faced was coordinate freedom. Ted Newman in his talk at this meeting cataloged the way the community clearly missed opportunities to understand that the so-called Schwarzschild singularity is just a coordinate effect. To us today this episode is baffling. Relativists do not seem to have understood the importance of controlling the effects of coordinates on their results, despite Einstein’s emphasis that the physics should be coordinate-invariant. They even had coordinate systems available at that time (from the work of Sir Arthur Eddington and Georges Lemaître) that went across the horizon in a non-singular way. In the same years, quantum physicists were (at times painfully) revolutionizing their physical thinking, agreeing that they should only concern themselves with the results of measurements, which they called observables, and that they should not try to create physical models for what can’t be measured, such as the “paths” of quantum particles. Special relativity already had a similar tradition, going back to Einstein’s gedanken experiments, which were designed to focus attention on the outcome of experimental measurements rather than phrase the predictions of special relativity in terms of observer-dependent notions of time and space. Yet this trend did not seem to influence research in general relativity in the period 1930–1950 as much as it should have. Ted Newman also mentioned another example, the deep confusion over the concept of energy in space-times containing gravitational waves. The resolution of this issue only began when Hermann Bondi and his successors, who included Roger Penrose and Ted himself, discovered how to treat radiated energy far from its source. It is easy to understand why relativists felt that they needed to clarify the idea of energy: energy is one of the key heuristics of mainstream physics. However, I confess that I don’t understand why relativists allowed the genuine difficulties of defining gravitational wave energy to stop their developing a physical understanding of gravitational waves themselves. It appears that, because it was difficult to define the energy of a radiating system or to localize the energy carried by waves, relativists during this period were unable to develop any kind of useful physical model for gravitational waves. We know today that it is perfectly possible to describe the generation of gravitational waves and their action on a simple detector without once referring to energy; the quadruple formula for the generation of the waves and the geodesic equation for their action on a simple detector are all one needs, and these tools were available from 1918. It is also possible to show that gravitational waves certainly deposit energy in some kinds of detectors, without having a full global energy conservation law. Indeed, Feynman at the earlier relativity meeting in Chapel Hill in 1957 (Bergmann 1957) presented a simple argument to show how a gravitational wave would heat a detector that has internal friction. The argument is so direct that I used a version of it myself in my undergraduate-level relativity textbook (Schutz 2009), and I extended it there to derive the standard expression (first put on a firm foundation by Isaacson 1968) for the local average energy flux in gravitational waves. Feynman was I think right to be disappointed that his argument at Chapel Hill seemed to impress no one and was not taken up and developed by relativists at the time. I think this example goes to the heart of the question. Feynman was asking a physicist’s question, about how gravitational waves act. All he wanted was a convincing intuitive argument that the waves were real and that he could treat them as part of the rest of physics, for example by extracting thermal energy from them. The relativists of his day, on the other hand, were not interested in this kind of physicist’s answer, not even apparently as a first step toward a more complete understanding of gravitational waves. Instead, they seemed to want to transplant as much of the apparatus of energy conservation as they could from the rest of classical physics. Energy conservation is of course a key concept in theoretical physics. But the work of Emmy Noether had shown long before that one should not expect exact energy conservation in the absence of time invariance, e.g. in a space-time containing gravitational waves. In relativity energy will always be a subsidiary concept, valid in some circumstances and useless in others. I believe Feynman found it intensely frustrating that relativists seemed more interested in the pure-radiation energy concept – in other words, relativity for its own sake – than in exploring the interaction of gravitational waves with material systems – gravitational waves as part of physics. Feynman’s example was more than just symptomatic of the way mainstream physics reacted to relativity. Feynman was one of the few mainstream physicists who attempted to cross over the gulf in the 1950s, and he was a prominent opinion-former. Relativity might have been accepted back into the mainstream physics community much earlier if relativists had succeeded in establishing a fruitful dialog with Feynman. Instead, his well-publicized scorn surely damaged the standing of the relativity community materially. [**5. Einstein and the Gulf**]{}\ It is hard to escape the conclusion that Einstein himself was one of the main reasons that the relativity community found itself excluded from mainstream physics. His influence on relativity research was naturally enormous. He appears to have rejected the idea of gravitational collapse, for reasons that today are hard to understand. He also appears not to have been comfortable with gravitational waves, troubled by the coordinate problems. Coordinates were a particular issue, as Silvio Bergia emphasized at this meeting in connection with the issue of general covariance, a principle that seems to have inhibited the development of heuristic concepts until Wheeler began emphasizing a more explicitly geometrical perspective on gravity. Perhaps most importantly, Einstein was focused mainly on finding a unified field theory. He does not seem to have been interested in the importance that general relativity had in classical theoretical physics, still less in its potential in astronomy. Einstein’s key bridge to mainstream physics was the unified field theory. Its failure seems to have left relativity without any other bridges. One further aspect of Einstein’s position that I believe may have been important was his rejection of the standard interpretation of quantum mechanics. Naturally, this isolated him from mainstream physics thinking. Perhaps Bohr’s advice to Chandrasekhar not to go into relativity had at least something to do with this. But I think there was a more profound way in which Einstein’s rejection of quantum heuristics hurt relativity. As I mentioned earlier, quantum physics changed the philosophy of theoretical physics. The key objective of quantum theory became the observable: don’t try to describe or understand something that you cannot measure. Relativity could have benefited in the period 1930–1950 from this imperative to focus only on what is – at least in principle – measurable. It is paradoxical that quantum physicists focused on the importance of observables long before relativists did. After all, coordinate-invariance was a key tenet of general relativity. The difference between quantum theorists and relativists is that in the quantum field the principle was practiced, while in relativity there seems to have been no systematic effort to focus on measurables as a way to solve coordinate difficulties until the “revival” began. I remember, as a graduate student, my supervisor Kip Thorne emphasizing that if coordinate confusion threatened, then one should construct a thought experiment and worry only about what the experimenter could in principle measure; and he made it clear that his own supervisor, John Archibald Wheeler, had emphasized this to him. Wheeler, of course, had worked extensively on quantum physics before taking up relativity in the mid-1950s. The idea of focussing on observables was natural to him and to other physicists of his generation. It had unfortunately not developed sufficiently in relativity, and it seems clear to me that introducing the strict discipline of observability was essential to ending relativity’s isolation. [**6. The Gulf Closes**]{}\ It is arguable that a key reason that relativity pulled out of its doldrums was that new blood entered the field with this maxim from quantum theory deeply ingrained in their physical thinking. For people like Bondi, Pascual Jordan, Wheeler, and Yakov Zel’dovich, among others, it was natural to test any question about general relativity with the demand that it be phrased in terms of observables. Is there something you can measure, at least an experiment in principle? At a stroke this way of thinking forces you, for example, to look for other physical features of the black-hole horizon than just the bad behavior of some metric components. Do the local tidal stretching forces near the Schwarzschild “singularity” remain finite? Can a real body reach and cross this surface in a finite amount of its own time? Regarding gravitational waves, this perspective leads you to ask whether a radiating body experiences a back reaction that changes its observable behavior, and whether the radiated gravitational waves in turn produce a measurable effect in the detector. It is then natural to ask under what circumstances it is reasonable to expect that a definition of energy exists that plays a role in self-gravitating systems analogous to what physicists are used to in nonrelativistic physics; but the energy question does not stop you from answering the questions about observable physical effects of gravitational waves. It may unfortunately not be a coincidence that relativity began climbing out of the doldrums at about the same time that Einstein died. His disappearance left the subject open for people to come in who had a background in mainstream physics and who were asking different kinds of questions. A large number of people working actively in classical theoretical relativity today (leaving aside the quantum gravity and string theory communities) can trace their lineage back to a handful of key physicists who entered relativity between about 1950 and 1960. These physicists reinvigorated the subject by asking the right kinds of questions, and they answered these questions with new heuristic notions that enabled relativity to communicate with and fit into the rest of physics. Nothing illustrates this change better than the evolution of the black hole concept, to which I referred earlier. The term “black hole” was coined as late as 1967 by Wheeler to describe something whose reality he initially also doubted, but which he finally came to understand was the likely endpoint for the evolution of a large range of massive systems. Today we talk about black holes, not just the Kerr metric or the Schwarzschild solution. That is because the concept of a black hole is wider than just these time-independent exact solutions of Einstein’s vacuum field equations. Wheeler himself took a major step toward our present picture by showing, with Tullio Regge, that the Schwarzschild horizon and exterior are stable against small perturbations. Immediately this meant that the idealized Schwarzschild solution was robust enough to include in models of more complicated physical systems: it would retain its essential properties even when disturbed. This robust object is what we call the black hole. Once the new generation of mathematical physicists recognized that their job was to develop a heuristic understanding of this object, they set to work. An immense number of research papers between 1960 and 1990, including some remarkably elegant mathematics, led to the modern concept of a black hole. This concept is far wider than the exact Kerr solution of Einstein’s equations. Black holes can have accretion disks around them, in which case they are not Kerr. They can have matter falling into them, so they need not even be time-independent. They can convert matter into energy, as Penrose showed. They radiate thermal radiation, as Stephen Hawking showed. They even obey the laws of thermodynamics. When an astronomer and a relativist talk about black holes, they need this common heuristic concept. In order to use black holes in models for astronomical systems, the astronomer needs to regard the black hole as a kind of black box, an object whose inputs and outputs are known but whose inner workings can be ignored. The astronomer wants to feel safe that he can put a black hole into a binary system without worrying about the details of the horizon or the curvature singularity inside. Relativists today are able to provide astronomers with this black-hole black box. [**7. General Relativity Is Part of Physics**]{}\ This is absolutely typical of physical thinking in other fields. Astronomers talk about stars, by which they mean a synthesis of a huge amount of physics. Nobody can even write down the complete mathematics needed to give an adequate description of a star. Nevertheless an astrophysicist knows pretty well what a star is. The same could be said about a laser, a superconductor, the plasma in a tokamak, or even about relatively simple composite systems like atoms, protons, neutrons. Even in front-line research, where such concepts are not settled, physicists work hard to develop them. The string theory community uses very visual and geometrical heuristics to describe their work. The extension of strings to multi-dimensional branes has opened up a rich source of possible phenomenology, and it is striking to me that, when I listen to talks given by theorists about the applications of brane theory to cosmology and to gravitation theory, the speakers often skip completely over the mathematics in favor of drawings that condense the mathematics into visual relationships. I believe that this is a basic aspect of the way physicists think about physics. The mathematical representation of the laws of physics is their foundation, but physicists would generally be paralyzed if they could not package up physical systems into heuristic black boxes, confident that they know (or at least someone knows!) enough about their internal complexity to understand how they will interact with each other. General relativity has a reasonably well-developed set of physical constructs today. This was the reason, for example, that Ted Newman could give his talk without showing any equations: when he talked about black holes and gravitational waves, we all knew what he meant. Or at least, those of you who are not specialists in general relativity knew something about what he meant, and you had faith that those of us who are specialists knew sufficiently more about what he meant for it to be safe for us all to talk about these concepts as physical reality! Without that faith, physics would simply not be possible. In the 1930s relativity had few such heuristic concepts to offer, and it did not look like it was moving toward constructing many more of them. I suggest that this is what led to the big gulf between relativists and mainstream theoretical physicists between 1930 and 1950. If this picture is right, then general relativity emerged mathematically complete in 1916, but as a theory of physics it was not completed until the 1980s. This must be one of the most gradual of Kuhnian revolutions ever! [**References**]{}\ DeWitt, CŽcile M., and Rickles, Dean (eds.), (2011) [*The Role of Gravitation in Physics: Report from the 1957 Chapel Hill Conference*]{}, Max Planck Research Library for the History and Development of Knowledge (Sources 5) (Max Planck Institute for the History of Science, Berlin, Germany).\ Eisenstaedt, Jean (1989), “The low water mark of general relativity, 1925-1955”, in [*Einstein and the History of General Relativity*]{}, eds. Howard, Don, and Stachel, John, Birkhäuser, Boston, MA, 277-292.\ Eisenstaedt, Jean (2006), [*The Curious History of Relativity : how Einstein’s Theory of Gravity was Lost and Found Again*]{}, Princeton University Press, Princeton.\ Feynman, Richard P. (1988), [*What Do You Care What Other People Think?*]{}, W.W. Norton, New York.\ Infeld, Leopold (1964), [*Relativistic Theories of Gravitation*]{}, Pergamon Press, Oxford.\ Isaacson, Richard (1968), “Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Effective Stress Tensor”, [*Physical Review*]{} [**166**]{}, 1272–1280.\ Robinson, Ivor, Schild, Alfred, and Schücking, Engelbert L. (1965), [*Quasi-stellar sources and gravitational collapse*]{}, University of Chicago Press, Chicago.\ Schutz, Bernard F. (2009), [*A First Course in General Relativity*]{} (2nd ed.), Cambridge University Press, Cambridge.\ Thorne, Kip S. (1994), [*Black Holes and Time Warps*]{}, W.W. Norton, New York. [^1]: To be published, without the abstract and with small editorial changes, in [*Einstein and the Changing Worldviews of Physics*]{} ([*Einstein Studies*]{}, vol. 12). ed C Lehner, J Renn, M Schemmel. Boston: Birkhäuser (2011). Based on a talk delivered at the Seventh International Conference on the History of General Relativity, Tenerife, Canary Islands, March 2005. [^2]: The International Society for General Relativity and Gravitation (known as the GRG Society), which is today the main professional society for relativists worldwide, is one of the few societies adhering directly to IUPAP which has individual scientists as members, not national organizations. During the cold war this structure enabled it and its predecessor (the International Committee for General Relativity and Gravitation) to organize relatively apolitical meetings that scientists from both sides of the Iron Curtain attended. An example was the famous meeting in Warsaw that Feynman criticised.

--- abstract: 'Congruence theory has many applications in physical, social, biological and technological systems. Congruence arithmetic has been a fundamental tool for data security and computer algebra. However, much less attention was devoted to the topological features of congruence relations among natural numbers. Here, we explore the congruence relations in the setting of a multiplex network and unveil some unique and outstanding properties of the multiplex congruence network. Analytical results show that every layer therein is a sparse and heterogeneous subnetwork with a scale-free topology. Counterintuitively, every layer has an extremely strong controllability in spite of its scale-free structure that is usually difficult to control. Another amazing feature is that the controllability is robust against targeted attacks to critical nodes but vulnerable to random failures, which also differs from normal scale-free networks. The multi-chain structure with a small number of chain roots arising from each layer accounts for the strong controllability and the abnormal feature. The multiplex congruence network offers a graphical solution to the simultaneous congruences problem, which may have implication in cryptography based on simultaneous congruences. Our work also gains insight into the design of networks integrating advantages of both heterogeneous and homogeneous networks without inheriting their limitations.' author: - 'Xiao-Yong Yan$^{1,5}$, Wen-Xu Wang$^{2,6,}$' - 'Guan-Rong Chen$^{3}$' - 'Ding-Hua Shi$^{4,}$' title: Multiplex congruence network of natural numbers --- Introduction {#sec:1 .unnumbered} ============ Congruence is a fundamental concept in number theory. Two integers $a$ and $r$ are said to be congruent modulo a positive integer $m$ if their difference $a-r$ is integrally divisible by $m$, written as $a \equiv r \ ({\rm mod} \ m)$ [@hardy79]. Congruence theory has been widely used in physics, biology, chemistry, computer science, and even music and business [@n2p92; @guterl94; @ding96; @yan02; @schr08]. Because of the limited computational and storage ability of computers, congruence arithmetic is particularly useful and applicable to computing with numbers of infinite length [@yan02]. Significant and representative applications include generating random numbers [@knuth97], designing hash functions [@preneel94] and checksumming in error detections [@yan02; @wang05]. As a cornerstone of modern cryptography, congruence arithmetic has been successfully used in public-key encryption [@salo13], secret sharing [@adi79], digital authentication, and many other data security applications [@ding96; @yan02; @schr08]. Despite the well-established congruence theory with a broad spectrum of applications, a comprehensive understanding of the congruence relation among natural numbers is still lacking. Our purpose is to uncover some intrinsic properties of the network consisting of natural numbers with congruence relations. A link in the congruence network is defined in terms of the congruence relation $j \equiv r \ ({\rm mod} \ i)$, where $r$ is the reminder of $j$ divided by $i$. For a fixed value of $r$, we discern an infinite set of integer pairs $(i,j)$. For each pair of such integers, a directed link from $i$ to $j$ (suppose $i<j$) characterizes the congruence relation between $i$ and $j$, giving rise to a congruence network for a given reminder $r$. Let $G(r,N)$ denote a congruence network, where $N$ is the largest integer considered. Note that congruence networks associated with different values of $r$ share the same set of nodes (integers), thereby a multiplex network [@yuan14] with a number of layers is formed, as shown in Fig. \[fig1\](a). To our knowledge, the multiplex congruence network (MCN) has not been explored in spite of some effort dedicated to complex networks associated with natural numbers [@corso04; @zhou06; @luque08; @shi10; @garcia14; @sheka15]. We will demonstrate several unique and prominent properties of the MCN regarding some typical dynamical processes. Specifically, analytical results will show that all layers of the MCN are sparse with the same power-law degree distribution. A counterintuitive property of the MCN is that every layer of the MCN has an extremely strong controllability, which significantly differs from ordinary scale-free networks requiring a large fraction of driver nodes. To steer the network in a layer, the minimum number of driver nodes is nothing but the reminder $r$ that is negligible as compared to the network size. The controllability of the MCN is also very robust against targeted removal of nodes but relatively vulnerable to random failures, which is also in sharp contrast to ordinary scale-free networks. This amazing robustness against attacks can be interpreted in terms of the multi-chain structure in MCN. The MCN therefore sheds light on the design of heterogeneous networks with high searching efficiency [@klainberg] and strong controllability simultaneously. Another application of the MCN is that it can graphically solve the simultaneous congruences problem in a more intuitive way than currently used methods, such as the Garner’s algorithm. The solution of the simultaneous congruences problem is to locate common neighbors of relevant numbers in different layers. This alternative approach by virtue of the MCN may have implication in cryptography based on simultaneous congruences. {width="15.0cm"} To our knowledge, the multiplex congruence network (MCN) has not been explored in spite of some effort dedicated to complex networks associated with natural numbers [@corso04; @zhou06; @luque08; @shi10; @garcia14; @sheka15]. We will demonstrate several unique and prominent properties of the MCN regarding some typical dynamical processes. Specifically, analytical results will show that all layers of the MCN are sparse with the same power-law degree distribution. A counterintuitive property of the MCN is that every layer of the MCN has an extremely strong controllability, which significantly differs from ordinary scale-free networks requiring a large fraction of driver nodes. To steer the network in a layer, the minimum number of driver nodes is nothing but the number of reminder $r$ that is negligible as compared to the network size. The controllability of the MCN is also very robust against targeted removal of nodes but relatively vulnerable to random failures, which is also in sharp contrast to ordinary scale-free networks. This amazing robustness against attacks can be interpreted in terms of the multi-chain structure in MCN. The MCN therefore sheds light on the design of heterogeneous networks with high searching efficiency [@klainberg] and strong controllability simultaneously. One of the most important applications of the MCN is that it allows to solve the simultaneous congruences problem in a much more efficient way than existing methods using the Gauss algorithm. To tackle the problem with high computational efficiency, the network of each layer should be stored in advance, e.g., in a distributed storage system. The solution of the simultaneous congruences problem is to locate common neighbors of relevant numbers in different layers. This new efficient approach by virtue of the MCN offers deeper and broader insight into the modern cryptography and has potential applications in many other fields, such as communications and computer science. Results {#sec:3 .unnumbered} ======= Topology of MCN {#topology-of-mcn .unnumbered} --------------- MCN consists of a number of congrence networks (layers) $G(r>0,N)$, as shown in Fig. \[fig1\](a). Each layer contains all the natural numbers larger than $r$ but less than or equal to $N$, so the size (number of nodes) of a layer is $N-r$. The remainder $r$ is a parameter that determines the structure of congruence network. When $r=0$, the congruence network reduces to a divisibility network [@shi10], in which the dividend links to all of its divisors except itself. The out-degrees of nodes are heterogeneous in each layer of the MCN. We have analytically derived the distribution of the out-degrees in the thermodynamic limit (see details in the section of **Methods**): $$P(k)= \frac{1}{k(k+1)}.$$ For large $k$, the out-degree distribution becomes $P(k) \sim k^{-2}$, thus $G(r,N)$ is a typical scale-free network. All $G(r>0,N)$ have similar out-degree distributions, as shown in Fig. \[fig1\](b), but the divisibility network $G(0,N)$ has a different out-degree distribution. For small $k$, $P(k)$ of $G(0,N)$ deviates from the other networks $G(r>0,N)$. The main factor that accounts for the difference lies in that half of the nodes in $G(0,N)$ have no outgoing links, but in $G(r>0,N)$ there are only $r$ nodes without outgoing links. {width="17cm"} Analytical results demonstrate that the average degree of any layer increases logarithmically with the network size (see Fig. \[fig1\](b) and the section of **Methods** for details), and a larger value of $r$ corresponds to a sparser layer. These results indicate that $G(r,N)$ is always a sparse network. Hence, the MCN is compatible with a sparse storage, which is important for applying the MCN to solve real-world problems. According to the definition of MCN, the numbers in each layer $G(r > 0,N)$ can be classified into $r$ arithmetic sequences: $$a_{n}^{i} =i +nr, \label{eq:seq}$$ where $1\leq i \leq r, n =1, 2, \cdots, \lfloor \frac{N-i}{r} \rfloor $ ($\lfloor x \rfloor $ denotes the largest integer not greater than $x$). The consecutive numbers in the sequence are linked from small to large, resulting in $r$ chains traversing all nodes in a layer, as shown in Fig. \[fig2\](a). The root node of a chain is the minimum number in the chain. There are totally $r$ root nodes associated with $r$ chains. The end of a chain is always the maximum number in the chain. Note that $r=0$ is a special case, because the arithmetic sequence does not exist in the layer $G(0,N)$, rendering the absence of the multi-chain structure in the divisibility network. The above results indicate that although the divisibility network $G(0,N)$ is a special case of $G(r,N)$, it has some fundamental differences from $G(r>0,N)$. Only when $r>0$ the multi-chain structure emerges, and the number of nodes without outgoing links is negligible. Some evidence has suggested that the multi-chain structure and the absence of nodes with low out-degrees play an important role in the controllability of complex networks [@wang12; @giulia14]. In the next section, we will further investigate the controllability properties of the MCN. Controllability of MCN {#controllability-of-mcn .unnumbered} ---------------------- In principle, the MCN composed of natural numbers is not a dynamical system, such that it cannot be controlled. However, because of the multi-chain structure, the MCN provides significant insight into the design of heterogeneous networked systems with strong controllability. Thus, we treat the MCN as a dynamical system and explore its unique and outstanding controllability properties. The central problem of controlling complex networks is to discern a minimum set of driver nodes, on which external input signals are imposed to fully control the whole system. Let $N_{\rm D}$ denote the minimum number of driver nodes and $n_{\rm D}$ denote the fraction of driver nodes in a network. In general, a network with a smaller value of $n_{\rm D}$ is said to be more controllable. According to the exact controllability theory for complex networks [@yuan13] and the sparsity of MCN, we can prove that (see in the section of **Methods**) $$N_{\rm D}=r,$$ and $$n_{\rm D}=\frac{r}{N-r}.$$ For large $N$ (namely $r \ll N$), $n_{\rm D} \rightarrow 0$ and $G(r>0,N)$ is considered as highly controllable. Furthermore, according to both the exact controllability theory [@yuan13] and the structural controllability theory [@liu11], the driver nodes are the $r$ root nodes of the chains in $G(r>0,N)$ (see details in the section of **Methods**). Meanwhile, the driver nodes are the hub nodes with the maximum degree. In comparison, due to the absence of chains, the divisibility network $G(0,N)$ with $N_D= \lceil N/2 \rceil$ ($\lceil x \rceil$ denotes the smallest integer not less than $x$) is hard to control for large $N$. It has been recognized that scale-free networks are often difficult to control [@liu11]. In particular, Liu et al. [@liu11] have analytically found that when the network size $N \to \infty$, one must control almost all nodes in order to fully control a scale-free network with scaling index $\gamma \to 2$. The MCN is a scale-free network with scaling exponent $\gamma = 2$, but one just needs to control $r$ root nodes to achieve full control. Such a strong controllability stems from the inherent multi-chain structure in MCN. From the perspective of structural controllability, all nodes in the chains in MCN are matched except the $r$ roots, which need to be controlled. Thus, the MCN is valuable for designing heterogeneous networks with strong controllability. We also found that the MCN is strongly structurally controllable (SSC) because of the multi-chain structure (see [**Methods**]{}), which provides significant insight into the design of heterogeneous and controllable networks without exact link weights. A network is said to be SSC if and only if its controllability will not be affected by the link weights in its adjacency matrix [@liu11], or equivalently, for any distribution of link weights, the network will be fully controllable from the same set of driver nodes. The SSC property implies that the MCN is robust against the fluctuation and uncertainty of link weights. This is an outstanding feature with practical significance since sometimes link weights are hard to be exactly measured and they are sometimes time-varying in real situations. The robustness against attacks is also a significant problem for the design of a controllable networked system [@pu12]. We explore the robustness of the controllability of the MCN against attacks on nodes and find some unique properties, which is useful for the design of practical networks. On the one hand, due to the existence of chains rooted in $r$ driver nodes in MCN, targeted attacks to driver nodes will not destroy the multi-chain structure nor increase $n_{\rm D}$. Here, nodes critical for targeted attacks can be identified based on the rank of node degrees or their hierarchical structure [@liu12]. In MCN, driver nodes (the $r$ root nodes) become such critical nodes. In this regard, MCN is robust against targeted attacks. On the other hand, random attacks to nodes may cut some chains. As a result, an additional driver is required to control each new breakpoint, leading to an increase of $n_{\rm D}$. Thus, the controllability of MCN is unusual in resisting attacks in the sense that it is robust against intentional attacks but vulnerable to random attacks, which significantly differs from general scale-free networks. The results in comparing the MCN and SF networks generated by using static model [@goh01] are shown in Fig. \[fig2\](b). To make an unbiased comparison, a scale-free network with the same scaling exponent $\gamma$ and $\langle k\rangle$ as the MCN is necessary. However, because of the graphicality constraint, it is not possible to generate a random SF network with $\gamma=2$ [@del11; @baek12]. Thus, we slightly release the requirement by using static model to generate SF networks with the same $\langle k\rangle$ but with $\gamma=2.001$. Indeed, one can see that $n_{\rm D}$ remains nearly unchanged under targeted attacks to driver nodes; whereas random attacks to node causes clear increase of $n_{\rm D}$. This phenomenon is consistent with our analysis in terms of the multi-chain structure. Moreover, for the presence of random attacks, in a wide range of the fractions of failed nodes, the controllability of MCN is still better than SF networks in general. Solving the simultaneous congruences problem {#solving-the-simultaneous-congruences-problem .unnumbered} -------------------------------------------- One of the applications of MCN is that one can graphically solve the simultaneous congruences problem, which has implication in communication security and computer science. In particular, one can find that MCN is exactly a topological representation of the system of simultaneous congruences [@schr08]. A system of simultaneous congruences is a set of congruence equations: $$\left\{ \begin{array}{c} x \equiv r_{1} \ ({\rm mod} \ m_{1})\\ x \equiv r_{2} \ ({\rm mod} \ m_{2})\\ \cdots \\ x \equiv r_{k} \ ({\rm mod} \ m_{k}) \end{array} \right. \label{eq:sim}$$ If the moduli $m_{1},m_{2}\dots m_{k}$ are pairwise coprime, then a unique solution modulo $m_{1} m_{2} \dots m_{k}$ exists. This is the *Chinese remainder theorem* (CRT), which has many applications in computing, coding and cryptography [@ding96; @yan02; @schr08; @sorin07]. A well-known algorithm to solve the simultaneous congruences in CRT is the Gaussian algorithm [@menezes96], also known as *Dayanshu* [@lib06] in ancient China. Here, we present an intuitive approach based on MCN to solve the simultaneous congruences in Eq. (\[eq:sim\]). Firstly, we construct an MCN containing $k$ subnetworks with remainders $r_{1},r_2 \dots r_k$, respectively. To focus on the minimum solution of Eq. (\[eq:sim\]), we set the maximum number in the network as $N=m_{1} m_{2} \dots m_{k}$. Then, we find the common successor neighbor of the nodes $m_{1},m_{2}\dots m_{k}$ in this MCN, which is precisely the solution $x$. We use a well-known example of CRT, recorded in *Sunzi Suanjing* [@lam04], to demonstrate our approach. The problem in this example is: ‘Suppose we have an unknown number of objects. When grouped in threes, 2 are left out, when grouped in fives, 3 are left out, and when grouped in sevens, 2 are left out. How many objects are there?’ This problem is equivalent to the following simultaneous congruences $$\left\{ \begin{array}{c} x \equiv 2 \ ({\rm mod} \ 3)\\ x \equiv 3 \ ({\rm mod} \ 5)\\ x \equiv 2 \ ({\rm mod} \ 7) \end{array} \right. \label{eq:sz}$$ To solve the problem, we first construct an MCN of two layers, $G(2,105)$ and $G(3,105)$, as shown in Fig. \[fig3\]. Then, we find the common successor neighbor of the three moduli, 3, 5 and 7, and finally get the result $x=23$. It is noteworthy that the traditional algorithm for solving the simultaneous congruences problem, e.g., the Garner’s algorithm [@menezes96], is more efficient than our algorithm based on the MCN that is essentially a brute-force search. Thus, it is infeasible to immediately use the graphical approach in data security. However, the graphical algorithm offers new routes to the simultaneous congruences problem in the viewpoint of a complex network, which may be useful to improve the currently used algorithm. ![ [**Solving simultaneous congruences using MCN.**]{} For visualization, we only show a part of nodes in the MCN. In the upper layer, the set of successor neighbors of node $5$ is $S_{m=5}=\{8,13,18,23\}$, and similarly $S_{m=3}=\{5,8,11,14,17,20,23\}$ and $S_{m=7}=\{9,16,23\}$ in the lower layer. Thus the common successor neighbor of the three nodes is $23$, which is the solution of the simultaneous congruences problem described by in Eq. (\[eq:sz\]).[]{data-label="fig3"}](fig3.eps){width="9cm"} Discussion {#sec:3 .unnumbered} ========== We have defined a multiplex congruence network composed of natural numbers and uncovered its unique topological features. Analytical results demonstrate that every layer of the multiplex network is a sparse and scale-free subnetwork with the same degree distribution. Counterintuitively, every layer with a scale-free structure has an extremely strong controllability, which significantly differs from ordinary scale-free networks. In general, a scale-free network with power-law degree distribution is harder to control than homogeneous networks. This is attributed to the presence of hub nodes, at which dilation arises according to the structural control theory [@liu11]. As a result, downstream neighbor nodes of hubs are difficult to control. Moreover, due to a large number of nodes connecting to hubs, scale-free networks are usually of weak controllability with a large fraction of driver nodes. In contrast, in spite of the scale-free structure of the congruence network, the long chains in each layer considerably inhibit dilation and reduce the number driver nodes. Furthermore, an interesting finding is that every layer is also strong structurally controllable in that link weights have no effect on the controllability. This indicates that the controllability of the multiplex congruence network is extremely robust against the inherent limit to precisely accessing link weights in the real situation. To our knowledge, a scale-free network with strong structural controllability has not been reported prior to our congruence network. An unusual controllability property is that the controllability of each layer is robust against targeted attacks to driver nodes, but relatively fragile to random failures of nodes, which is also different from common scale-free networks. Previously reported results [@pu12; @liu12] demonstrate that targeted removal of high degree nodes and nodes in the top level of a hierarchical structure causes maximum damage to the network controllability. Under the two kinds of intentional removals, a network is easier to break to pieces, such that more driver nodes are required to achieve full control. Thus, targeted attacks are defined in terms of the two types of node removals. In the congruence network, high degree nodes and high level nodes are exactly identical, leading to the combination of the targeted attacks. Strikingly, the congruence network is robust against the targeted attacks, because of the existence of the chains. Targeted attacks will not destroy the chains in the downstream of the attacked node. As a result, the number of driver nodes nearly does not increase, even when a large fraction of nodes has been targeted attacked. The outstanding structural and controllability properties of the multiplex congruence network are valuable for designing heterogeneous networks with strong controllability and high searching efficiency rooted in the scale-free structure. Another application of the multiplex congruent network is to solve the simultaneous congruences problem in a graphical and intuitive manner. The multiplex congruence network by converting the algebraic problem of solving simultaneous congruences equations to be a graphical problem of finding common neighbors in a graph, offers an alternative route to the traditional approaches. Despite this property, the traditional algorithms, such as Gaussian algorithm and Garner’s algorithm, outperform the graphical method in computational efficiency. Hence, the graphical method is not applicable in data security at the present. Nevertheless, the graphical approach may inspire the combination of the graphical and algebraic method to improve the current algorithm, which is potentially valuable in communication security, computer science and many fields relevant to cryptography. Our work may also stimulate further effort toward studying of networks arising from natural relationships among numbers, with outstanding features and applied values. Many topological insights can be expected from complex networks consisting of natural numbers. Methods {#methods .unnumbered} ======= Deriving the out-degree distribution of MCN {#deriving-the-out-degree-distribution-of-mcn .unnumbered} ------------------------------------------- For a sub-network $G(r>0,N)$ in MCN, the total number of nodes is $N-r$ and the number of nodes without out-links is $r$. The out-degree of a node labelled $m$ in the range of $(\frac{N-r}{2},N-r]$ is 1, because node $m$ can only link to one node i.e. node $m+r$ in the network; similarly, the out-degree of a node labelled $m$ in the range of $(\frac{N-r}{3},\frac{N-r}{2}]$ is 2, because node $m$ can only link to two nodes i.e. nodes $m+r$ and $2m+r$; similar scenarios appear to the other nodes. Thus, we can derive the distribution of out-degrees in the thermodynamic limit, as follows: $$P(k) =\frac{\frac{N-r}{k}-\frac{N-r}{k+1}}{N-r} = \frac{1}{k(k+1)}, \ k \geq 1. \label{eq:deg}$$ In $G(0,N)$, the numbers larger than $N/2$ have no out-links, i.e. $P(0)=\frac{N- \lfloor N/2 \rfloor}{N}$, and the numbers in the range of $(N/3,N/2]$ have only one out-link, i.e. $P(1)=\frac{\lfloor N/2 \rfloor-\lfloor N/3 \rfloor}{N}$. Analogously, as $N \to \infty$, $$P(k) =\frac{\frac{N}{k+1}-\frac{N}{k+2}}{N} = \frac{1}{(k+1)(k+2)}, \ k \geq 0. \label{eq:deg0}$$ This is the same as the in-degree distribution of a growing network with copying [@ksky05], which can be regarded as a random version of the divisibility network [@shi10]. Calculating the average degree of sparse MCN {#calculating-the-average-degree-of-sparse-mcn .unnumbered} -------------------------------------------- Note that the minimum number $r+1$ in $G(r>0,N)$ has the maximum out-degree $ \lfloor \frac{N-r} {r+1} \rfloor $ and the second minimum number $r+2$ has the out-degree $\lfloor \frac{N-r} {r+2} \rfloor $, and so on. Thus, the average degree $\bar{k}$ of $G(r>0,N)$ is $$\begin{aligned} \bar{k} &=\frac{ \sum\limits_{i=r+1}^{N-r}{ \lfloor \frac{N-r}{i} \rfloor } }{N-r} \\ & \approx \frac{ (N-r) (\ln {(N-r)} +2C-1)-\sum\limits_{i=1}^{r}{ \lfloor \frac{N-r}{i} \rfloor }}{N-r} \\ & \approx \ln(N-r) +H, \end{aligned} \label{eq:av}$$ where $C$ is the Euler constant and $H = 2C-1 -\frac{\sum\limits_{i=1}^{r}{ \lfloor \frac{N-r}{i} \rfloor }}{N-r}$ is a constant, which is approximately $C-1-\ln(r)$ when $r$ is very large. Eq. (\[eq:av\]) is not valid for the divisibility network $G(0,N)$. In $G(0,N)$, number 1 has the maximum out-degree $N-1$, and 2 has the second maximum out-degree $ \lfloor N/2 \rfloor -1$, and so on. In a similar way to Eq. (\[eq:av\]), we can obtain the average degree of $G(0,N)$, as $$\bar{k} =\frac{\sum\limits_{i=1}^{N}{ \lfloor \frac{N}{i} -1 \rfloor }}{N} \approx \ln(N) +2C-2, \label{eq:av0}$$ which is consistent with the analytical results of the undirected divisibility network [@sheka15]. Controllability of MCN and identification of driver nodes {#controllability-of-mcn-and-identification-of-driver-nodes .unnumbered} --------------------------------------------------------- An arbitrary network with linear time-invariant dynamics under control can be described by $$\dot{\mathbf{x}} = A\mathbf{x} + B\mathbf{u}, \label{eq:ct}$$ where the vector $\mathbf{x} = (x_1 , \dots , x_N )^{\mathrm{T}}$ stands for the states of $N$ nodes, $A$ denotes the coupling matrix (transpose of the adjacency matrix) of a network, $\mathbf{u} = (u_1 , \dots , u_m )^{\mathrm{T}}$ is the vector of $m$ input signals, and $B \in \mathbb{R}^{N \times m}$ is the input matrix. System (\[eq:ct\]) is said to be (state) controllable if the input signal $\mathbf{u}$ imposed on a minimum number $N_{\rm D}$ of driver nodes specified by control matrix $B$ can steer the state $\mathbf{x}$ from any initial state to any target state in finite time. The level of controllability of the networked system (\[eq:ct\]) is defined by the fraction $n_{\rm D}$ of driver nodes in the sense that a complex network is more controllable if a smaller fraction of driver nodes is needed to achieve full control. According to Liu [*et al*]{}. [@liu11], the key is to find a matrix $B$ associated with the minimum number of controllers to ensure full control of system (\[eq:ct\]). Because of the sparsity of MCN, $N_{\rm D}$ of a layer $G(r,N)$ is determined by [@yuan13] $$N_{D} = \max \{1,N-r- \mathrm{rank} (A) \}. \label{eq:pbh}$$ Because in $G(r>0,N)$ each node-labelled number can only link to the node-labelled numbers larger than itself, the coupling matrix $A$ is a strictly lower-triangular matrix. Moreover, $A$ of $G(r>0,N)$ is in a column echelon form because the minimum number linked from node $m$ is always $m+r$. In other words, the leading coefficient of the $i$th column in $A$ is precisely in the $(i+r)$th row. On the other hand, the last $r$ columns of $A$ are all zeros because the maximum $r$ nodes in $G(r>0,N)$ have no out-links, so the rank of $A$ is exactly $N-2r$. An example of matrix $A$ of $G(1,9)$ is $$A_{G(1,9)}= \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ \end{pmatrix}. \label{eq:mtx1}$$ According to Eq. (\[eq:pbh\]), we can show that the minimum number of driver nodes $N_{\rm D}$ for $G(r>0,N)$ is $r$ and the value of non-zero elements in $A$ does not affect ${\rm rank}(A)$, which indicate that $G(r>0,N)$ is SSC. According to the exact controllability theory [@yuan13], the control matrix $B$ to ensure full control of the congruence network $G(r>0,N)$ should satisfy the following condition $$\mathrm{rank} [-A,B] = N-r. \label{eq:cond}$$ Notice that the rank of the matrix $[-A, B]$ is contributed by the number of linearly independent rows, hence the input signals specified via $B$ should be imposed on the linearly dependence rows in $A$ so as to eliminate all linear correlations in Eq. (\[eq:cond\]). Apparently, the first $r$ rows in the coupling matrix $A$ of $G(r>0,N)$ are all zero rows (see Eq.(\[eq:mtx1\])), hence the $r$ driver nodes that need to be controlled to maintain full control are just the minimum $r$ nodes of the congruence network, i.e. the $r$ roots of the chains in the congruence network (see Fig. \[fig2\](a)). The coupling matrix of the divisibility network $G(0,N)$ is also a strictly lower-triangular matrix and in a column echelon form, but the rank of the matrix is $ \lfloor N/2 \rfloor $, because in $G(0,N)$ the node with labelled number larger than $ \lfloor N/2 \rfloor $ has no out-links, namely, the last $ \lceil N/2 \rceil $ columns of the matrix are all zeros. An example of $G(0,9)$ is $$A_{G(0,9)}= \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} .\label{eq:mtx0}$$ Therefore, according to Eq. (\[eq:pbh\]), we finally obtain the minimum number of driver nodes of $G(0,N)$ as $N_D=\lceil N/2 \rceil$, indicating that one must control half of the nodes in order to control the whole divisibility network. Moreover, the value of non-zero elements in $A$ does not affect ${\rm rank}(A)$, which indicate that $G(0,N)$ is SSC. Thus, the MCN composed of $G(0,N)$ and $G(r>0,N)$ is SSC. [**Acknowledgments:**]{} We thank Dr. Zhengzhong Yuan for useful discussions. X.-Y.Y. was supported by NSFC under Grant Nos. 61304177, 71525002 and the Fundamental Research Funds of BJTU under Grant No. 2015RC042. W.-X.W. was supported by NSFC under Grant No. 61573064. G.-R.C. was supported by the Hong Kong Research Grants Council under the GRF Grant CityU11208515. D.-H.S. was supported by NSFC under Grant No. 61174160. [10]{} url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} & ** (, ). , , & ** (, ). . ** (). ** (, ). ** (, ). ** (, ). **, vol.  (, ). . ** ****, (). & . In **, (, ). ** (, ). . ** ****, (). , , , & . ** ****, (). . ** ****, (). , , & . ** ****, (). , & . ** ****, (). & . ** ****, (). , & . ** ****, (). , & . ** ****, (). . ** ****, (). , , & . ** ****, (). , & . ** ****, (). , , , & . ** ****, (). , & . ** ****, (). , & . ** ****, (). , & . ** ****, (). , & . ** ****, (). , & . ** ****, (). , , & . ** ****, (). . ** ****, (). , & ** (, ). **, vol.  (, ). & ** (, ). & . ** ****, ().

--- abstract: 'We study the effect of electron interaction in an electronic system with a high-order Van Hove singularity, where the density of states shows a power-law divergence. Owing to scale invariance, we perform a renormalization group (RG) analysis to find a nontrivial metallic behavior where various divergent susceptibilities coexist but no long-range order appears. We term such a metallic state as a *supermetal*. Our RG analysis reveals the Gaussian fixed point and a nontrivial interacting fixed point, which draws an analogy to the $\phi^4$ theory. We further present a finite anomalous dimension at the interacting fixed point by a controlled RG analysis, thus establishing an interacting supermetal as a non-Fermi liquid.' author: - Hiroki Isobe - Liang Fu title: Supermetal --- Introduction ============ A Bloch electron in a crystal is described by the energy dispersion $E_{\bm{k}}$ that relates the energy with its wave vector $\bm{k}$. For metals, the energy dispersion determines the density of states (DOS) at the Fermi level, which to a large extent governs various thermodynamic properties such as charge compressibility, spin susceptibility and specific heat. Van Hove’s seminal work [@VanHove] revealed that the DOS exhibits non-analyticity at an extremum or a saddle point of the energy dispersion, where $\nabla_{\bm{k}} E_{\bm{k}}=0$. Importantly, Van Hove singularities (VHS) are guaranteed to exist in every energy band by the continuity and the periodicity of $E_{\bm{k}}$ over the Brillouin zone. The behavior of the DOS at a VHS depends on whether it is at an energy extremum or a saddle point, and also on the dimensionality of the system. For example, at a saddle point in two dimensions with $E_{\bm{k}} = k_x^2-k_y^2$, the DOS diverges logarithmically. As the chemical potential crosses the VHS, the topology of Fermi surface changes from electron to hole type, known as electronic topological transition. Recently, we have extended the notion of VHS to high-order saddle points, where, besides $\nabla_{\bm{k}}E_{\bm{k}}=0$, the Hessian matrix $D_{ij} =\partial_{k_i} \partial_{k_j} E$ satisfies $\det D(\bm{k})=0$ [@high-order]. These high-order saddle points occur where two Fermi surfaces touch [*tangentially*]{}, or at the common intersection of three or more Fermi surfaces [@Fu2011; @multicritical1]. An example of the former is $E_{\bm{k}}= k_x^2 - k_y^4$, and of the latter is $E_{\bm{k}}=k_x^3 - 3 k_x k_y^2$. Generally speaking, high-order saddle points can be realized by tuning the energy dispersion with one or more control parameter. At high-order saddle points in two dimensions, the DOS shows a power-law divergence $|E|^{-\epsilon}$, much stronger than a logarithmic one at ordinary VHS [@high-order; @multicritical1]. The existence of high-order VHS has recently been identified in a variety of materials including twisted bilayer graphene near a magic angle, trilayer graphene-hexagonal BN heterostructure [@high-order], and Sr$_3$Ru$_2$O$_7$ [@multicritical2]. In particular, scanning tunneling spectroscopy measurements [@Pasupathy] on twisted bilayer graphene found power-law divergent DOS at the magic twist angle, providing direct evidence for high-order VHS [@high-order]. In the presence of electron-electron interaction, a large DOS near the Fermi level may have important consequences. On the one hand, it may trigger Stoner instability to ferromagnetism. On the other hand, a large DOS may result in strong screening of repulsive interaction, so that a Fermi liquid description remains valid at low energy. In this work, we study interacting electron systems with a high-order saddle point near the Fermi level. Assuming that electron interaction is weak, dominant contributions to low-energy thermodynamic properties of the system come from those states in the vicinity of the saddle point, from which the DOS divergence originates. This allows us to formulate a [*continuum*]{} field theory of interacting fermions by taking the leading-order energy dispersion relation $E_{\bm{k}}$ near the saddle point and extending the range of momentum to infinity. In this field theory, when the high-order VHS is right at the Fermi level, the Fermi surface in $\bm{k}$-space becomes [*scale-invariant*]{}. As the VHS approaches the Fermi level, charge and spin susceptibilities exhibit power-law divergence, reminiscent of critical phenomena. Motivated by these observations, we develop a renormalization group (RG) theory for interacting fermions near high-order VHS, which parallels Wilson–Fisher RG approach to $\phi^4$ theory [@Wilson-Fisher; @Wilson]. By introducing a small parameter $\epsilon$ associated with the DOS divergence, we present a [*controlled*]{} RG analysis and find that short-range repulsive interaction is relevant at the noninteracting fixed point and drives the system into a nontrivial $T=0$ interacting fixed point. The former is the analog of Gaussian fixed point in Fermi system, and the latter the analog of the Wilson–Fisher fixed point. The metallic state at the interacting fixed point exhibits scale-invariance in space/time and power-law divergent charge and spin susceptibility, but finite pairing susceptibility. In other words, this is a metal on the verge of becoming charge-ordered and ferromagnetic. We call such a metallic state with various power-law divergent susceptibilities but without any long-range order, a *supermetal*. We further show by a two-loop RG calculation that the fermion field acquires a finite anomalous dimension. Hence the interacting supermetal we found is a non-Fermi liquid. The outline of the paper is as follows: In Sec. \[sec:model\], we introduce a model with high-order VHS and calculate the power-law divergent DOS, whose exponent is determined from the scaling property of energy dispersion. In Sec. \[sec:energy-shell\], we perform the energy-shell RG analysis step by step. We first define the energy shell as a region of momentum space. Then, the tree-level and one-loop RG equations for the chemical potential and interaction strength are derived in sequence, which resembles the case of $\phi^4$ theory. We identify the noninteracting Gaussian fixed point and the nontrivial interacting fixed point which is the analog of the Wilson–Fisher fixed point in Fermi system. We next consider other relevant perturbations to the system, including Zeeman and pairing field as well as additional symmetry-allowed terms in the energy dispersion. The discussion about higher-loop renormalization in the energy-shell RG analysis follows, though an actual two-loop calculation appears in the later section. In Sec. \[sec:analysis\], we perform the scaling analysis for thermodynamic quantities and correlation functions. The generic formalism is first presented, followed by the one-loop result for various exponents of divergent susceptibilities. We also discuss the Ward identity in this section, which results from charge conservation and imposes a constraint on the field renormalization and the charge compressibility. In Sec. \[sec:field\_theory\], another RG scheme, the field theory approach, is introduced. We briefly discuss the two RG schemes before presenting the RG analysis from the field theory approach. The field theory approach has an advantage in calculating higher-order perturbative corrections compared to the energy-shell RG analysis. The UV regulator with a soft energy cutoff is then introduced, which is confirmed to satisfy the Ward identity. The one-loop calculation reproduces the energy-shell RG analysis in Sec. \[sec:energy-shell\]. Furthermore, the two-loop calculation shows the finite anomalous dimension at a high-order VHS realized at a saddle point of an energy dispersion. It proves the non-Fermi liquid nature of interacting supermetal. In Sec. \[sec:lifetime\], we evaluate the quasiparticle lifetime at finite temperature due to electron interaction. From a perturbative calculation at two-loop order, we find an unusual temperature dependence in the quasiparticle lifetime. In Sec. \[sec:discussions\], we summarize the results and discuss their significance in the broad context of Van Hove physics, RG approach to Fermi systems, and non-Fermi liquids. We also discuss possible material realizations of supermetal. Model {#sec:model} ===== An example of high-order VHS in two dimensions {#sec:tight-binding} ---------------------------------------------- We consider a tight-binding model on an anisotropic square lattice $$\begin{aligned} H = - \sum_j \left( t_x c_{j+\hat{x}}^\dagger c_j + t_y c_{j+\hat{y}}^\dagger c_j + t'_y c_{j+2\hat{y}}^\dagger c_j \right) + \text{H.c.}\end{aligned}$$ $t_x$ and $t_y$ are the nearest-neighbor hopping amplitudes along the $x$ and $y$ directions, respectively, and $t'_y$ is the second-nearest neighbor hopping along the $y$ direction. The energy dispersion is obtained as $$\label{eq:tb_dispersion_full} E_{\bm{k}} = -2t_x \cos(k_xa) -2t_y \cos(k_ya) -2t'_y \cos(2k_ya),$$ with the lattice constant $a$. For $|t_y|\geq|t'_y|/4$, there are four VHS points in the Brillouin zone at the high symmetry points: $\Gamma=(0,0)$, $X=(\pi/a,0)$, $Y=(0,\pi/a)$, and $M=(\pi/a,\pi/a)$. With $t_x$, $t_y$, $t'_y>0$, the energy minimum and maximum are located at $\Gamma$ and $M$ points, respectively, and $X$ and $Y$ points are the saddle points \[Fig. \[fig:tb\](a)\]. At the special value $t'_y=t_y /4$, the energy dispersions takes the form $$E_{\bm{k}} = k_x^2 - k_y^4 \label{x2y4}$$ near $Y$ and $k_x^2 + k_y^4$ near $M$, where coefficients have been eliminated by rescaling $k_x$ and $k_y$. The former is a high-order saddle point, while the latter is a high-order energy extremum. ![ Lattice model for a high-order VHS. (a) Energy contour plot with $t_y/t_x=0.8$ and $t'_y/t_x=0.2$. We find the energy minimum at $\Gamma$, the maximum at $M$, and the two saddle points at $X$ and $Y$. $Y$ and $M$ are high-order VHS points. At $Y$, we see that the two Fermi surfaces touch tangentially while they cross linearly at $X$. (b) DOS for the energy dispersion in (a). The four VHS points give rise to analytic singularities in the DOS, where the corresponding points are labeled in the figure. The two peaks at $Y$ and $M$ correspond to high-order VHS, fitted by the analytic formula for the continuum theory Eq. . []{data-label="fig:tb"}](tb_c.pdf){width="\hsize"} A VHS manifests itself as an analytic singularity in the DOS $$\label{eq:DOS_formula} D(E) = \int_{\bm{k}} \delta(E-E_{\bm{k}}),$$ where $\int_{\bm{k}}=\int\frac{d^dk}{(2\pi)^d}$ stands for the momentum integration in $d$ dimensions. The DOS for the present model $(d=2)$ is depicted in Fig. \[fig:tb\](b). We find four singularities in the DOS and each of them tied to the individual VHS of the model. The band bottom at $\Gamma$ gives rise to a discontinuity in the DOS and the saddle point at $X$ shows a logarithmic divergence in the DOS. Those two are conventional VHS, known since the original work of Van Hove [@VanHove]. Here we focus on the high-order VHS at $Y$ and $M$. They exhibit distinct behavior: the DOS has a power-law divergence as $|E|^{-1/4}$ instead of a logarithm. In addition, the divergence at $Y$ is stronger on the electron side by the factor $\sqrt{2}$ than on the hole side. Such an asymmetry is not seen for a conventional VHS with a logarithmic divergence at $X$. The two Fermi surfaces touch tangentially at $Y$ at the Van Hove energy \[Fig. \[fig:contour\](a)\]. When the chemical potential $\mu$ crosses the Van Hove energy, the Fermi surface topology changes from being closed to open in the $k_y$ direction. In Fig. \[fig:contour\](b), the DOS peaks at the two high-order VHS in our tight-binding model are fitted by the analytical expressions of the DOS calculated from the continuum model (\[x2y4\]). The calculation will be shown in the next subsection. We can see a close fit within a finite energy range. Since the divergent DOS and hence susceptibilities originate from the vicinity of the high-order VHS, the continuum model is expected to capture universal features at low energy. Using the continuum model has the advantage of removing non-universal aspects associated with high-energy regions away from the high-order VHS in the tight-binding model. We will show that infrared (IR) scaling properties are not indeed affected by the UV cutoff in the continuum model. Before proceeding, we briefly mention the Fermi surfaces in strained Sr$_2$RuO$_4$ [@ruthenate1; @ruthenate2; @ruthenate3]. It has a quasi-two-dimensional electronic structure with a layered perovskite structure. Under uniaxial pressure, a Lifshitz transition occurs on the Brillouin zone boundary [@ruthenate2]. At the transition point, there is one VHS in the Brillouin zone at the Fermi energy. The Fermi surface of the band of interest resembles the one obtained from Eq. . Generalization {#sec:generalization} -------------- From now on, we study a continuum model of fermions with a high-order energy dispersion. For the purpose of controlled RG analysis later, here we consider the generalized energy dispersion in the $d$-dimensional $k$-space $$\label{eq:model} E_{\bm{k}} = A_+ k_+^{n_+}- A_- k_-^{n_-}.$$ The momentum is denoted by $$\bm{k} = (\bm{k}_+, \bm{k}_-),$$ where $\bm{k}_\pm$ are $d_\pm$-dimensional vectors with $d_++d_-=d$, and $k_\pm = |\bm{k}_\pm|$. Analyticity of the energy dispersion requires $n_\pm$ to be positive integers. We consider the case of even $n_\pm$, so that $E_{\bm{k}} = E_{-\bm{k}}$ satisfies time-reversal symmetry. When at least one of $n_\pm$ is greater than two, this energy dispersion has a high-order VHS at $k=0$, which is defined as a point where the Hessian matrix $D_{ij} =\partial_{k_i} \partial_{k_j} E_{\bm{k}}$ fulfills $\det D_{ij}=0$. The energy dispersion Eq.  follows the scaling relation $$E_{\bm{k}} = b E_{\bm{k}'} \text{ with } \bm{k}'=(\bm{k}_+/b^{1/n_+},\bm{k}_-/b^{1/n_-}). \label{scalingE}$$ It then follows from Eq.  and Eq.  that the DOS satisfies $$\begin{gathered} \label{eq:DOS} D(E) = \begin{cases} D_+ E^{-\epsilon} & (E>0) \\ D_- (-E)^{-\epsilon} & (E<0), \end{cases}\end{gathered}$$ where the DOS singularity exponent $\epsilon$ is $$\epsilon = 1-\frac{d_+}{n_+}-\frac{d_-}{n_-}.$$ Throughout this work, we consider the case $\epsilon>0$. For example, the high-order VHS introduced in the preceding section corresponds to the case of $d_+ = d_-=1, n_+=2, n_-=4$, so that $\epsilon = 1/4$. We calculate the prefactors $D_\pm$ for the dispersion (\[eq:model\]) explicitly and find \[eq:DOS\_prefactor\] $$\begin{gathered} D_s = D_0 \sin\left(\frac{\pi d_s}{n_s}\right) \quad (s=\pm),\end{gathered}$$ with the common factor $$D_0 = \frac{4\Gamma(\epsilon)}{\pi(4\pi)^{d/2}} \prod_{s=\pm} \frac{\Gamma\left(d_s/n_s\right)}{n_s A_s^{d_s/n_s} \Gamma\left(d_s/2\right)}.$$ We note that in calculating DOS, the $d$-dimensional momentum integral over $\bm k\in (-\infty, \infty) $ is convergent for all $E \neq 0$. Note that $D_+ \neq D_-$ for $d_+/n_+ \neq d_-/n_-$. It describes the asymmetry in the DOS above and below $E=0$. This is a feature of the high-order saddle points defined by Eq.(\[eq:model\]), distinct from conventional saddle points in two dimensions where the logarithmically divergent DOS peak is symmetric. ![ Energy contour plot for $E_{\bm{k}} = k_x^2 - k_y^4$. The thick line is the Fermi surface at the Van Hove energy, which is scale-invariant. The colored region has the energy inside the cutoff $\Lambda$, where the red (blue) area corresponds to $E>0$ ($E<0$). At every RG step of the energy-shell RG scheme, high-energy modes within the energy shell shown in darker colors are integrated out. In the field theory approach, all states below the cutoff $\Lambda$ are integrated over at once. []{data-label="fig:contour"}](contour_a_2.pdf){width="\hsize"} The nontrivial fixed point to be shown later is controlled by the smallness of $\epsilon$. For the model defined by Eq. , the exponent can be any rational number between $0<\epsilon<1$. By choosing positive integers $n_\pm$ and $d_\pm$ judiciously, we can make $\epsilon$ arbitrarily small in high-dimensional crystals, while keeping the energy-momentum dispersion an analytic function. We now introduce our model of interacting electrons near a high-order VHS: $$\begin{aligned} \label{eq:action} S =&\int_0^{1/T} d\tau \int d^dr \Big[ \bar{\psi}_{\sigma} (\partial_\tau + E_{-i\partial_{\bm{r}}} -\mu) \psi_{\sigma} + g \bar{\psi}_{\uparrow} \bar{\psi}_{\downarrow} \psi_{\downarrow} \psi_{\uparrow} \Big].\end{aligned}$$ We set $k_B=\hbar=1$ throughout the paper. Here we formulate the model at temperature $T$. Temperature $T$ is regarded as the system size $L_\beta \equiv 1/T$ in the imaginary time direction. $\psi_\sigma$ is the fermionic field and the summation over the spin index $\sigma=\uparrow (+)$, $\downarrow (-)$ is implicit. We consider the local contact interaction between electrons with opposite spins. Later, we shall consider the effects of other interactions and external fields. From the action, we define the noninteracting Green’s function $$G_0(\bm{k},\omega_n)= \frac{1}{i\omega_n - E_{\bm{k}}},$$ with the fermionic Matsubara frequency $\omega_n=(2n+1)\pi T$ ($n$: integer). The partition function $\mathcal{Z}$ is expressed as $$\mathcal{Z} = \int D\bar{\psi}D\psi e^{-S}. \label{Z}$$ We are interested in thermodynamic quantities such as specific heat. These are obtained from the free energy density $$\label{eq:free_energy} F=-\frac{T}{V}\ln \mathcal{Z}$$ where $V$ is the volume of the system. Energy-shell RG analysis {#sec:energy-shell} ======================== Formalism at zero temperature ----------------------------- In this section, we adopt the Wilsonian approach to the RG equations for the action Eq. . For clarity, we consider first the action at $T=0$, where we will find fixed points. Then, the Matsubara frequency becomes continuous $\omega_n \to \omega$, and the action is written with frequency $\omega$ and momentum $\bm{k}$ as $$\begin{aligned} S &= \int\frac{d\omega}{2\pi} \int_{\bm{k}} \bigg[ \bar{\psi}_\sigma(k) (-i\omega + E_{\bm{k}} -\mu) \psi_\sigma(k) \nonumber\\ &\quad + g \bigg( \prod_{j=1}^{4} \int\frac{d\omega_j}{2\pi} \int_{\bm{k}_j} \bigg) (2\pi)^{d+1} \delta (k_1+k_2-k_3-k_4) \nonumber\\ &\quad\ \times \bar{\psi}_{\uparrow}(k_1) \bar{\psi}_{\downarrow}(k_2) \psi_{\downarrow}(k_3) \psi_{\uparrow}(k_4).\end{aligned}$$ We introduce the shorthand notation $k = (\bm{k},\omega)$. We impose a UV energy cutoff $\Lambda$ on this action to remove unphysical UV divergences that appear in electron density of the ground state, etc. We note that the UV cutoff here is imposed on energy, but not on momentum directly. The region in $\bm{k}$-space with $|E_{\bm{k}}| \leq \Lambda$ still extends to infinity. Importantly, this UV cutoff does not affect universal scaling properties of IR fixed points in the analysis of high-order VHS, as we shall show. The UV cutoff merely appears in the prefactors of IR scaling functions. We use two different energy cutoff schemes in this paper: an energy shell with a hard cutoff and a soft energy cutoff. The former scheme allows the Wilsonian RG approach, which offers a rather simple analysis and understanding. The latter requires a field theoretical analysis, which is apparently complicated, but high order perturbative corrections become more tractable. This section focuses on the energy-shell RG scheme, which imposes a constraint on momentum integrals. By converting the momentum integral to an energy integral with the help of the DOS, we write the momentum integral with the cutoff $\Lambda$ as $$\begin{aligned} \int_{\bm{k}}^\Lambda \mathcal{F}(E_{\bm{k}}) = \int_{-\Lambda}^{\Lambda} dE D(E) \mathcal{F}(E),\end{aligned}$$ for an arbitrary function $\mathcal{F}$. We denote the action with the energy cutoff $\Lambda$ as $S_\Lambda$, obtained by replacing the momentum integral $\int_{\bm{k}}$ by $\int_{\bm{k}}^\Lambda$. The UV energy cutoff designates an unbounded region in $\bm{k}$-space, reflecting the extended Fermi surface with scale invariance (Fig. \[fig:contour\]). Note that frequency integrals still range from $-\infty$ to $+\infty$. We now sketch how an RG transformation works with the energy-shell RG scheme. To access the IR behavior, we progressively eliminate UV modes and focus more on remaining modes. In the energy-shell RG scheme, we first split the energy range into two parts; one corresponds to lower energies $E_{\bm{k}} \in [-\Lambda/b, \Lambda/b]$ and the other to higher energies $E_{\bm{k}} \in[-\Lambda,-\Lambda/b)$, $(\Lambda/b,\Lambda]$ $(b>1)$. Accordingly, the fermion field $\psi$ is decomposed as $$\begin{aligned} \psi_\sigma(k) = \psi_\sigma^<(k) + \psi_\sigma^>(k),\end{aligned}$$ where $\psi_\sigma^<$ represents the low-energy modes and $\psi_\sigma^>$ the high-energy modes. We write a momentum integral in the same way: $$\int_{\bm{k}}^\Lambda = \int_{\bm{k}}^< + \int_{\bm{k}}^>.$$ Due to this division, the action is decomposed into the three parts as $$S_\Lambda[\psi] = S^<[\psi^<] + S^>[\psi^>] + S^{<>}[\psi^<,\psi^>].$$ The first term $S^<[\psi^<]$ consists only of the low-energy modes $\psi^>$ and the second term $S^>[\psi^>]$ of the high-energy modes $\psi^>$. The last term $S^{<>}[\psi^<,\psi^>]$ describes the coupling of the low- and high-energy modes, which arises when the interaction is finite $(g \neq 0)$. To obtain the effective action without the high-energy modes, we need to integrate them out: $$\begin{aligned} \label{eq:effective} &\quad S_{\Lambda/b}[\psi^<] \nonumber\\ &= S^<[\psi^<] - \ln \left( \int D\bar{\psi}^> D\psi^> e^{-S^>[\psi^>] -S^{<>}[\psi^<,\psi^>]} \right) \nonumber\\ &= S^<[\psi^<] - \ln \left( \int D\bar{\psi}^> D\psi^> e^{-S^{<>}[\psi^<,\psi^>]} \right) + \text{const}.\end{aligned}$$ Now the high-energy modes are eliminated and the new action has the smaller cutoff $\Lambda/b$. One may be tempted to compare $S_\Lambda[\psi]$ and $S_{\Lambda/b}[\psi^<]$ to look into low-energy properties. However, it is like “comparing apples to oranges” [@Shankar] as the two actions are defined in different domains. For a fair comparison, we should make a change of variables ($\bm{k}$, $\omega$ and $\psi$) to restore the cutoff $\Lambda$. This procedure, called rescaling, completes the RG step. It results in the change of parameters in the model, which is described by RG equations. The RG equations describe the flow of the parameters under a scale transformation. When the parameters do not change under a scale transformation, the system reaches an RG fixed point and exhibits scale-invariant properties. Away from a fixed point, the parameters flow. If the flow converges to a fixed point in its vicinity, then the fixed point is called a stable fixed point. If the parameters flow away from a fixed point, then it is an unstable fixed point. The RG equations also tell us how various susceptibilities and correlation lengths diverge as the critical point is approached, and the scaling properties of correlation functions at the critical point. Tree-level analysis ------------------- The mixing term $S^{<>}$ can be calculated by expanding the logarithm in powers of the coupling constant $g$. We first consider the zeroth-order contribution in $g$. Since the remaining terms are described by tree diagrams without loops, the approximation is referred to as the tree-level analysis. At tree-level, the effective action with the cutoff $\Lambda/b$ becomes $S_{\Lambda/b}[\psi^<] = S^<[\psi^<]$. To compare with $S_\Lambda[\psi]$, we need to change the variables to put the cutoff $\Lambda/b$ to $\Lambda$. Now we change the variables so that the energy satisfies the relation $$\label{eq:rescaling_energy} E_{\bm{k}'} = bE_{\bm{k}}.$$ For the energy dispersion given by Eq. , this immediately leads to rescaling of the momentum $$\label{eq:rescaling_momentum} k_+' = b^{1/n_+} k_+, \quad k_-' = b^{1/n_-} k_-,$$ while the coefficients do not change: $$\begin{gathered} A'_+ = A_+, \quad A'_-=A_-.\end{gathered}$$ To retain the form of the action, we also need to rescale the field $\psi$, frequency $\omega$, chemical potential $\mu$, and coupling constant $g$ to be $$\begin{gathered} \psi' = b^{-(3-\epsilon)/2} \psi^<, \\ \label{eq:omega_tree} \omega' = b \omega, \\ \label{eq:mu_tree} \mu' = b \mu, \\ \label{eq:g_tree} g' = b^{\epsilon} g.\end{gathered}$$ When we look at the parameters of the model, the chemical potential $\mu$ and the coupling constant $g$ change after an RG step, whereas the coefficients of the energy dispersion $A_\pm$ do not. The flow of an parameter under an infinitesimal scale transformation $(b\to 1)$ is described by a differential equation, namely the RG equation. For $\mu$ and $g$, the RG equations are obtained from Eqs.  and : $$\begin{gathered} \label{eq:RG_tree} \frac{d\mu}{dl} = \mu, \quad \frac{dg}{dl} = \epsilon g.\end{gathered}$$ with $l = \ln b$. In the present case, we find the Gaussian fixed point at $\mu=g=0$ in Eq. , where the system becomes noninteracting and the partition function takes a functional form of the Gaussian integral. If the parameters are away from the fixed point, they grow as $l$ increases i.e., in low energies, and flow away from the fixed point. Therefore, the fixed point at $\mu=g=0$ is unstable and both $\mu$ and $g$ are relevant perturbations to the unstable fixed point. So far we have only considered the contact interaction. However, electron-electron interactions can take a more complicated form. Other types of interactions will be generated under RG even if not present initially, and thus their effects should be considered as well. In general, a finite-range interaction can be expanded in powers of spatial derivatives, with contact interaction being the lowest order term. The next leading term $g_- (\bar{\psi}_\uparrow \partial_{r_-} \bar{\psi}_\downarrow) (\psi_\downarrow \partial_{r_-} \psi_\uparrow)$ contains two spatial derivatives, and has a different scaling relation: $g_-' = b^{\epsilon-2/n_-} g_-$, which has a much smaller exponent than $\epsilon$ for the contact interaction. As an example, for the energy dispersion (\[x2y4\]) in two dimensions, we have $\epsilon=1/4$ and $n_-=4$, so that $g_-$ is irrelevant. It is therefore legitimate to retain only the contact interaction in RG analysis. One-loop analysis {#sec:energy-shell_one-loop} ----------------- In the presence of interaction, elimination of the high-energy modes gives rise to corrections in the effective action through the mixing of low- and high-energy modes in $S^{<>}[\psi^<,\psi^>]$. When depicted diagrammatically, $S^{<>}[\psi^<,\psi^>]$ involves diagrams with loops, corresponding to integrations of the high-energy modes. We here consider perturbative corrections to one-loop order. The effective action Eq.  can be calculated perturbatively with respect to the coupling constant $g$ when it is small. Here we also treat the chemical potential $\mu$ as a perturbation as we are interested in critical phenomena where there is no characteristic scale in the system. Including perturbative corrections, we write down the action in the form $$\begin{aligned} \label{eq:action_one-loop} &\quad S_{\Lambda/b}[\psi^<] \nonumber\\ &= \int\frac{d\omega}{2\pi} \int_{\bm{k}}^< \bar{\psi}^<_\sigma(k) [-i\omega + E_{\bm{k}} -\mu +\Sigma] \psi^<_\sigma(k) \nonumber\\ &\quad + (g+\delta g) \bigg( \prod_{j} \int\frac{d\omega_j}{2\pi} \int_{\bm{k}_j}^< \bigg) \nonumber\\ &\quad\ \times (2\pi)^{d+1} \delta (k_1+k_2-k_3-k_4) \nonumber\\ &\quad\ \times \bar{\psi}^<_{\uparrow}(k_1) \bar{\psi}^<_{\downarrow}(k_2) \psi^<_{\downarrow}(k_3) \psi^<_{\uparrow}(k_4) \nonumber\\ &\quad+ \cdots,\end{aligned}$$ where $\delta g$ is a correction to the coupling constant and $(\cdots)$ consists of interactions with derivatives that may be generated after integrating out the high-energy modes. As we have discussed above, finite-range interactions are irrelevant, so that we can safely neglect them. ![ Diagrammatic representation of perturbative corrections. The solid lines with arrows are the noninteracting electron propagators $G_0$. Each vertex corresponds to the contact interaction with the coupling constant $g$. (a) Self-energy $\Sigma$ to one-loop order. The first term represents the Hartree term and the second shows the one-loop correction linear in the chemical potential $\mu$. (b) Correction to the coupling constant $\delta g$. There are particle-particle (left) and particle-hole (right) contributions. (c) One-loop correction to the pairing field $\Delta$. (d) Two-loop correction to the self-energy, which gives rise to the finite field renormalization, and thus to the anomalous dimension. []{data-label="fig:loop"}](fig_loop_8-3.pdf){width="\hsize"} Perturbative corrections to the lowest order, namely to one-loop order, are diagrammatically depicted in Fig. \[fig:loop\](a) and (b), corresponding to $\Sigma$ and $\delta g$, respectively. We find that the one-loop corrections to the self-energy $\Sigma$ and coupling constant $\delta g$ can be written as $$\begin{gathered} \label{eq:sigma_t0} \Sigma = -g\Sigma_\text{H} + g\mu \Pi_\text{ph}, \\ \label{eq:delta_g_t0} \delta g = -g^2 (\Pi_\text{pp}+\Pi_\text{ph}).\end{gathered}$$ We emphasize that the all loop corrections should be evaluated at zero external frequency and momentum. The one-loop corrections are obtained to $O(l)$ $(l = \ln b)$ as \[eq:one-loop\_T0\] $$\begin{gathered} \Sigma_\text{H} = \int\frac{d\omega}{2\pi} \int_{\bm{k}}^> G_0(\bm{k},\omega) \simeq l c_\text{H} \Lambda D(\Lambda) , \\ \label{eq:one-loop_T0_pp} \Pi_{\text{pp}} = \int\frac{d\omega}{2\pi} \int_{\bm{k}}^> G_0(\bm{k},\omega) G_0(-\bm{k},-\omega) \simeq l c_\text{pp} D(\Lambda) , \\ \Pi_\text{ph} = \int\frac{d\omega}{2\pi} \int_{\bm{k}}^> G_0(\bm{k},\omega) G_0(\bm{k},\omega) =0,\end{gathered}$$ where $D(\Lambda)$ is the DOS at the cutoff energy and the dimensionless constants $c_\text{H}$ and $c_\text{pp}$ are $$\begin{gathered} \label{eq:c_H_0} c_\text{H} = \frac{1}{2} \left( 1-\frac{D_-}{D_+} \right), \\ \label{eq:c_pp_0} c_\text{pp} = \frac{1}{2} \left( 1+\frac{D_-}{D_+} \right).\end{gathered}$$ We can see that the particle-hole contribution vanishes identically after the frequency integration, i.e., at $T=0$ there is no particle-hole screening coming from states near the cutoff energy $\Lambda$. On the other hand, the particle-particle loop has a finite contribution. The Hartree contribution $\Sigma_\text{H}$ can be finite only when the DOS is asymmetric on the electron and hole side: $D_+ \neq D_-$. There is no frequency or momentum dependence in the self-energy to one-loop order, so that the self-energy only renormalizes the chemical potential $\mu$. The field renormalization or renormalization of the energy dispersion does not appear at one-loop order. They appear at two-loop order from the diagram shown in Fig. \[fig:loop\](d), which will be examined with the field theory approach in Sec. \[sec:field\_theory\]. With the one-loop corrections obtained, the new parameters $\mu'$ and $g'$ after rescaling are $$\begin{gathered} \mu' = b (\mu-\Sigma) \simeq b [\mu - lc_\text{H}\Lambda gD(\Lambda)], \\ g' = b^\epsilon (g + \delta g) \simeq b^\epsilon g [ g- l c_\text{pp} g^2 D(\Lambda) ],\end{gathered}$$ which lead to the RG equations for the chemical potential $\mu$ and the coupling constant $g$. It is convenient to define the dimensionless chemical potential $\bar{\mu}$ and coupling constant $\bar{g}$ as $$\begin{gathered} \label{eq:dimensionless_mu_g} \bar{\mu} = \frac{\mu}{\Lambda}, \quad \bar{g} = g D(\Lambda).\end{gathered}$$ Then, we obtain RG equations for $\bar{\mu}$ and $\bar{g}$ as \[eq:RG\_mu\_g\_0\] $$\begin{gathered} \label{eq:RG_mu_0} \frac{d\bar{\mu}}{dl} = \bar{\mu} + c_\text{H} \bar{g}, \\ \label{eq:RG_g_0} \frac{d\bar{g}}{dl} = \epsilon \bar{g} - c_\text{pp} \bar{g}^2.\end{gathered}$$ ![ RG flow of the coupling constant $\bar{g}$. There are two fixed points: $\bar{g}_1^*=0$ corresponds to the Gaussian fixed point and $\bar{g}_2^*$ to the nontrivial interacting fixed point. On the $\bar{g}$ line, the nontrivial fixed point $\bar{g}_2^*$ is stable whereas the Gaussian fixed point $\bar{g}_1^*$ is unstable. $\bar{g}_2^*$ is a positive number of order $\epsilon$, i.e., the stable fixed point has weak repulsive interaction with its strength controlled by the DOS singularity exponent $\epsilon$. []{data-label="fig:flow_g"}](flow_g.pdf){width="0.9\hsize"} Since we are interested in the low-energy behavior, we consider the RG flow by increasing $l$. In the RG equation for the coupling constant $\bar{g}$, we find two fixed points $$\begin{gathered} \label{eq:fixed_points} \bar{g}_1^* = 0, \quad \bar{g}_2^* = \frac{\epsilon}{c_\text{pp}}(>0).\end{gathered}$$ The RG flow is shown in Fig. \[fig:flow\_g\]. $\bar{g}_1^*$ corresponds to the Gaussian (noninteracting) fixed point as it has already been seen in the tree-level analysis. The new fixed point $\bar{g}_2^*$ is the nontrivial interacting fixed point with finite repulsive interaction, whose strength is of order $\epsilon$. The smallness of the coupling constant allows a controlled analysis by the DOS singularity exponent $\epsilon$ about the nontrivial fixed point. We can find the similarity to the $\phi^4$ theory in the structure of the RG equation : the coefficient of the quadratic term $r \phi^2$ corresponds to the chemical potential $\bar{\mu}$ and the quartic interaction term $\phi^4$ to the coupling constant $\bar{g}$. From this viewpoint, our theory can be regarded as the fermionic analog of the $\phi^4$ theory. The nontrivial fixed point in the $\phi^4$ theory is referred to as the Wilson–Fisher fixed point. One can perform a controlled analysis around the Wilson–Fisher fixed point because of the smallness of the coupling constant of order $\epsilon$. We emphasize that the small parameter $\epsilon$ has different meanings in the $\phi^4$ theory and the present model. In the $\phi^4$ theory, $\epsilon$ describes the dimension of the system measured from the upper critical dimension of four, i.e, the dimension of the system is $4-\epsilon$. On the other hand, $\epsilon$ is intrinsic to the energy dispersion for a high-order VHS and it is not restricted to an integer value but can take any rational value as discussed in Sec. \[sec:generalization\]. In the $\phi^4$ theory, the RG flow of $r$ describes the phase transition between ordered and disordered states: the RG flow to $r \gg 0$ corresponds to the disordered state and $r \ll 0$ to the ordered state, where the field $\phi$ has a finite expectation value associated with spontaneous symmetry breaking. The parameter $r$ is analogous to the chemical potential $\mu$ in the present fermionic model, where $\mu \gg 0$ yields the electron Fermi surface and $\mu \ll 0$ the hole Fermi surface. The sign change of $\mu$ describes the electronic topological transition at which the Fermi surface changes the topology. Note that the electron and hole Fermi liquids are indistinguishable by symmetry. Let us now look into the RG equation for $\bar{\mu}$ \[Eq. \] in detail. We have $\bar{\mu} \neq 0$ at the nontrivial interacting fixed point with $\bar{g}_2^*\neq 0$, whereas $\bar{\mu}=0$ at the Gaussian fixed point at $\bar{g}_1^*=0$. The shift of the chemical potential is in parallel with the shift of $r$ at the Wilson–Fisher fixed point in the $\phi^4$ theory. This states that the Wilson–Fisher fixed point relocates $r$ from the mean-field value and that the critical properties are observed at the displaced $r$. As we have mentioned, the contribution from the Hartree term $c_\text{H}$ is finite when the DOS is asymmetric on the electron and hole sides: $D_+ \neq D_-$. The RG equation for $\bar{\mu}$ asserts that a finite chemical potential is generated by the interaction when $c_\text{H}\neq 0$. This is consistent with $D_+ \neq D_-$, since a finite chemical potential $\bar{\mu}$ breaks electron-hole symmetry and hence the contact interaction itself cannot generate $\bar{\mu}$ unless the symmetry does not exist from the beginning. Relevant perturbations ---------------------- We have identified the two fixed points: the Gaussian fixed point and the nontrivial interacting fixed point. With the chemical potential tuned at $\bar{\mu}=0$, the Gaussian fixed point $\bar{g}_1^*$ is an unstable fixed point and the nontrivial fixed point $\bar{g}_2^*$ is a stable fixed point. The chemical potential is a relevant perturbation around both fixed points. We have included the chemical potential even in the analysis of the simplest case above as it can be generated by interaction in the absence of particle-hole symmetry. In addition to the chemical potential, we consider other relevant perturbations to the fixed points, including the magnetic field $h$ and the $s$-wave pairing field $\Delta$. Those relevant perturbations adds the following terms to the action at criticality: $$\begin{gathered} -\mu \bar{\psi} \psi, \quad h (\bar{\psi_\uparrow} \psi_\uparrow - \bar{\psi_\downarrow} \psi_\downarrow ), \quad \Delta \bar{\psi}_\uparrow \bar{\psi}_\downarrow + \Delta^* \psi_\downarrow \psi_\uparrow.\end{gathered}$$ Finite temperature is also a relevant perturbation. Its effect is taken account of via Matsubara frequencies. We further consider other relevant perturbations. For an energy dispersion, e.g., $E_{\bm{k}}=k_x^2-k_y^4$, the fermion bilinear terms with derivatives $\partial_{k_x}$, $\partial_{k_y}$, $\partial_{k_y}^2$, $\partial_{k_y}^3$, $\partial_{k_x} \partial_{k_y}$ are also relevant perturbations. Perturbations to the system are subject to symmetry constraints: Particle conservation forbids the pairing term, spin-rotational symmetry nonzero $h$, and reflection symmetry odd-derivative terms in $x$ or $y$. With all three symmetries present, only two terms $\mu\bar{\psi}\psi$ and $\bar{\psi}\partial_{k_y}^2\psi$ are allowed as perturbations to the system with $E_{\bm{k}}=k_x^2-k_y^4$. This means that we need to tune two parameters to reach the critical point. To one-loop order, the term $\bar{\psi}\partial_{k_y}^2\psi$ does not receive a correction from the interaction since the self-energy $\Sigma$ is independent of momentum. We assume that the pairing term is induced by a proximity effect, or it might be regarded as a test field for the $s$-wave pairing fluctuation, because it must vanish in the presence of particle number conservation. Likewise, the magnetic field $h$ can be thought of as an external field or a test field for the spin susceptibility. In this viewpoint, the chemical potential is conjugate to the particle number, and hence it is related to the charge compressibility. Corrections to the perturbations $h$ and $\Delta$ are calculated similarly as those for $\mu$ and $g$ at $T=0$. To consider a correction to the pairing field $\Delta$, we include the particle-particle loop diagram, where the one-loop diagram is shown in Fig. \[fig:loop\](c). We include the corrections to write the magnetic field $h + \delta h$ and the pairing field $\Delta + \delta\Delta$. Integrating out the high-energy modes is followed by rescaling. The parameters of the model should be rescaled at tree level as $h' = bh$ and $\Delta' = b\Delta$. Those parameters are relevant and thus their values increase as we proceed with RG steps. When the perturbative corrections are included, the new parameters after an RG step are $$\begin{gathered} h' = b (h + \delta h), \quad \Delta' = b (\Delta + \delta\Delta).\end{gathered}$$ To one-loop order, the correction terms are expressed as $$\begin{gathered} \delta h = 0, \quad \delta\Delta = -g\Pi_\text{pp}.\end{gathered}$$ $\delta h=0$ follows from spin-rotational symmetry of the model. The one-loop correction $\Pi_\text{pp}$ is obtained in Eq. . Then, the parameters change as $$\begin{gathered} h' = b h, \quad \Delta' \simeq b [\Delta - lc_\text{pp}gD(\Lambda)].\end{gathered}$$ With the dimensionless quantities $$\begin{gathered} \bar{h}=\frac{h}{\Lambda}, \quad \bar{\Delta}=\frac{\Delta}{\Lambda},\end{gathered}$$ we reach the RG equations \[eq:RG\_h\_delta\] $$\begin{gathered} \label{eq:RG_h} \frac{d\bar{h}}{dl} = \bar{h}, \\ \label{eq:RG_Delta} \frac{d\bar{\Delta}}{dl} = ( 1- c_\text{pp}\bar{g} ) \bar{\Delta}. \end{gathered}$$ We confirm that the perturbations $h$ and $\Delta$ are relevant around the two fixed point, given in Eq. . Finite temperature is also a relevant perturbation, which scales in the same manner as energy and frequency. All low-energy fixed points are found at $T=0$, and thus we focus on zero temperature in the main part. The one-loop RG equations at finite temperature are presented in Appendix \[sec:finite-temperature\]. The physical consequences, i.e., scaling properties of thermodynamic quantities, are discussed in the next section. Discussion for higher-order corrections {#sec:RG_Wilson_discussion} --------------------------------------- So far, we have made the energy-shell RG analysis to one-loop order. We now illustrate how it works in the case with higher-order corrections. Again, we consider here the minimal case with $\mu$ and $g$ at $T=0$ for clarity. Inclusion of other relevant contributions such as $T$, $h$, and $\Delta$ is straightforward. Higher-order perturbative corrections give rise to the frequency and momentum dependence in the self-energy $\Sigma$ in Eq. . We expand the self-energy with respect to the frequency and momentum to find corrections to the field, energy dispersion, and chemical potential. In the following discussion, we assume that the momentum-dependent part of the self-energy involves terms proportional to $k_+^{n_+}$ and $k_-^{n_-}$ and that other relevant terms are not generated or eliminated by symmetry or fine-tuning the system to criticality. Then, the expansion of the self-energy is given by $$\begin{aligned} \label{eq:sigma_expansion} \Sigma &= \Sigma_0 + (i\omega) \Sigma_\omega + \mu \Sigma_\mu + A_+ k_+^{n_+} \Sigma_+ - A_- k_-^{n_-} \Sigma_- \nonumber\\ &\quad + \text{(high order terms)},\end{aligned}$$ where irrelevant high order terms are safely neglected. After integrating out the high-energy modes within the energy shell, we obtain the effective action $$\begin{aligned} S_{\Lambda/b}[\psi^<] &= \int\frac{d\omega}{2\pi} \int_{\bm{k}}^< \bar{\psi}^<_\sigma(k) \{ -i\omega (1-\Sigma_\omega) + [A_+ k_+^{n_+}(1+\Sigma_+) -A_- k_-^{n_-}(1+\Sigma_-)] -\mu(1-\Sigma_\mu) \} \psi^<_\sigma(k) \nonumber\\ &\quad + (g+\delta g) \bigg( \prod_{j} \int\frac{d\omega_j}{2\pi} \int_{\bm{k}_j}^< \bigg) (2\pi)^{d+1} \delta (k_1+k_2-k_3-k_4) \bar{\psi}^<_{\uparrow}(k_1) \bar{\psi}^<_{\downarrow}(k_2) \psi^<_{\downarrow}(k_3) \psi^<_{\uparrow}(k_4).\end{aligned}$$ The next step in the energy-shell RG analysis is to rescale the momentum and restore the energy cutoff $\Lambda/b$ to $\Lambda$; see Eqs.  and . However, the effective action $S_{\Lambda/b}$ still evidently has a different form from $S_\Lambda$. To recover the form of the action, we rescale the other quantities as follows: $$\begin{gathered} \label{eq:scaling_omega} \omega' = b\omega , \\ \label{eq:scaling_A} A'_\pm = (1+\Sigma_\pm) (1-\Sigma_\omega)^{-1} A_\pm \equiv b^{\gamma_{A_\pm}} A_\pm, \\ \label{eq:scaling_mu} \mu' = b (1-\Sigma_\mu) (1-\Sigma_\omega)^{-1} \mu \equiv b^{\gamma_\mu} \mu, \\ g' = b^\epsilon (1-\Sigma_\omega)^2 (g+\delta g), \\ \psi' = b^{-(3-\epsilon)/2} (1-\Sigma_\omega)^{-1/2} \psi^< \equiv b^{-(3-\epsilon)/2} b^{\gamma_\psi/2} \psi^<.\end{gathered}$$ Here we also introduce the scaling exponents $\gamma_{A_\pm}$, $\gamma_\mu$, and $\gamma_\psi$. Note that there is an ambiguity in defining $\omega'$ and $\psi'$ as the factor $(1-\Sigma_\omega)$ can be imposed on either $\omega'$ or $\psi'$. We choose to scale $\omega$ linearly in $b$ and hence the factor $(1-\Sigma_\omega)$ contributes to the field renormalization. We shall show later that the Ward identity requires $\gamma_\mu = 1$. For $\gamma_{A_\pm} \neq 0$, if we continue to rescale momentum according to Eq.  and the coefficients $A_\pm$ according to Eq. , the cutoff energy $\Lambda/b$ is not mapped to $\Lambda$. To remedy this issue, we rescale momentum as $$\label{eq:scaling_k_tilde} k'_\pm = b^{1/\tilde{n}_\pm} k_\pm \text{ with } \tilde{n}_\pm = \frac{n_\pm}{1+\gamma_{A_\pm}},$$ so that $E_{\bm{k}'} = bE_{\bm{k}}$ is satisfied. In this way, the coefficients $A_\pm$ do not change under rescaling. Rescaling of the magnetic field $h$ and the pairing field $\Delta$ can be considered similarly. Including the field renormalization, we obtain $$\begin{gathered} \label{eq:scaling_h} h' = b (h+\delta h) (1-\Sigma_\omega)^{-1} \equiv b^{\gamma_h} h, \\ \label{eq:scaling_Delta} \Delta' = b (\Delta + \delta\Delta) (1-\Sigma_\omega)^{-1} \equiv b^{\gamma_\Delta} \Delta,\end{gathered}$$ where we define the exponents $\gamma_h$ and $\gamma_\Delta$. Analysis {#sec:analysis} ======== Scaling analysis ---------------- ### Generic case {#sec:scaling_generic} Scale invariance at the fixed points enables us to extract various scaling relations. Since the partition function $\mathcal{Z}$ is invariant under the scale transformation, the free energy density $F$, defined in Eq. , reflects the scaling of the factor $T/V$: $$F' = b^{1 + d_+/\tilde{n}_+ + d_-/\tilde{n}_-} F,$$ where the volume $V$ scales according to Eq.  and temperature scales the same manner as energy and frequency. For convenience, we rewrite the exponent as $$\begin{aligned} 1 + \frac{d_+}{\tilde{n}_+} + \frac{d_-}{\tilde{n}_-} &= 2 - \left( \epsilon - \frac{d_+ \gamma_{A_+}}{\tilde{n}_+} - \frac{d_- \gamma_{A_-}}{\tilde{n}_-} \right) \nonumber\\ &\equiv 2-\tilde{\epsilon}.\end{aligned}$$ By explicitly showing the parameters of $F$, we obtain the scaling relation of the free energy density $$\label{eq:scaling} F(\mu,h,\Delta;T) = b^{-2+\tilde{\epsilon}} F(b \mu , b^{\gamma_h}h, b^{\gamma_\Delta}\Delta; b T).$$ Here, the exponents $\gamma_h$ and $\gamma_\Delta$ correspond to the values at a fixed point, $\gamma_h(\bar{g}^*)$ and $\gamma_h(\bar{g}^*)$, respectively. In deriving this relation, we use the Ward identity $\gamma_\mu = 1$. The coupling constant $g$ itself does not appear in the scaling relation of the free energy density $F$, but the effect is imprinted on $\gamma_h$, $\gamma_\Delta$, and $\tilde{\epsilon}$ as the fixed point properties. We shall see that $\gamma_{A_\pm}$ are at most of order $\epsilon^2$ at the nontrivial interacting fixed point and thus $\tilde{\epsilon}$ is also a small positive quantity. We then consider the critical exponents of the charge compressibility $\kappa$, magnetic susceptibility $\chi$, heat capacity per unit volume $C_V$, and $s$-wave pairing susceptibility $\chi_\text{BCS}$. From Eq. , we find $$\begin{gathered} \label{eq:scaling_compressibility} \kappa = \left(\frac{\partial n}{\partial\mu}\right)_T \sim \begin{cases} T^{-\tilde{\epsilon}} \\ |\mu|^{-\tilde{\epsilon}}, \end{cases} \\ \label{eq:scaling_susceptibility} \chi = -\lim_{h\to0}\left(\frac{\partial^2 F}{\partial h^2}\right)_T \sim \begin{cases} T^{-(\tilde{\epsilon}+2\gamma_h-2)} \\ |\mu|^{-(\tilde{\epsilon}+2\gamma_h-2)}, \end{cases} \\ \label{eq:scaling_heat-capacity} \frac{C_V}{T} = -\left(\frac{\partial^2 F}{\partial T^2}\right)_V \sim \begin{cases} T^{-\tilde{\epsilon}} \\ |\mu|^{-\tilde{\epsilon}} \\ |h|^{-\tilde{\epsilon}/\gamma_h} \\ |\Delta|^{-\tilde{\epsilon}/\gamma_\Delta}, \end{cases} \\ \label{eq:scaling_pairing} \chi_\text{BCS} = \left(\frac{\partial^2 F}{\partial\Delta \partial\Delta^*}\right)_T \sim \begin{cases} T^{-(\tilde{\epsilon}+2\gamma_\Delta-2)} \\ |\Delta|^{-2+(2-\tilde{\epsilon})/\gamma_\Delta}. \end{cases}\end{gathered}$$ We also examine the pair correlation function $$\begin{aligned} C(\bm{r},\tau) &= \langle (\psi_\uparrow \psi_\downarrow) (\bm{r},\tau) (\bar{\psi}_\downarrow \bar{\psi}_\uparrow) (0,0) \rangle \nonumber\\ &\quad - \langle (\psi_\uparrow \psi_\downarrow) (\bm{r},\tau) \rangle \langle (\bar{\psi}_\downarrow \bar{\psi}_\uparrow) (0,0)\rangle,\end{aligned}$$ with $\langle \mathcal{O} \rangle = \int D\bar{\psi} D\psi \mathcal{O} e^{-S}/\mathcal{Z}$. From the comparison between $\chi_\text{BCS}$ and $C(\bm{r},\tau)$, we obtain the scaling form $$\begin{aligned} C(r_+,r_-,\tau) = \kappa^{2(2-\tilde{\epsilon}-\gamma_\Delta)} \hat{c}\left( r_+ \kappa^{1/\tilde{n}_+}, r_- \kappa^{1/\tilde{n}_-}, \tau \kappa \right),\end{aligned}$$ where $\kappa$ is an arbitrary energy scale and $\hat{c}$ is a scaling function. The field renormalization with the exponent $\gamma_\psi$ appears in the two-point correlation function $G(\bm{r},\tau)$. We shall show the derivation later with the field theory approach. In the critical region, the exponent $\gamma_\psi$ can be replaced with a constant $\eta = \gamma_\psi(\bar{g}^*)$; the scaling form is given by \[eq:correlation\_scaling\] $$\begin{aligned} \label{eq:correlation_scaling_real} G(r_+,r_-,\tau) = \kappa^{1-\tilde{\epsilon}+\eta} \hat{g}\left( r_+ \kappa^{1/\tilde{n}_+}, r_- \kappa^{1/\tilde{n}_-}, \tau \kappa \right),\end{aligned}$$ or its Fourier transform is $$\begin{aligned} \label{eq:correlation_scaling_frequency} G(k_+,k_-,\omega) = \kappa^{-(1-\eta)} \hat{g}' \left( \frac{k_+}{\kappa^{1/\tilde{n}_+}}, \frac{k_-}{\kappa^{1/\tilde{n}_-}}, \frac{\omega}{\kappa} \right),\end{aligned}$$ where $\hat{g}$ and $\hat{g}'$ are scaling functions. Particularly, we see the frequency dependence $G(\omega) \propto 1/|\omega|^{1-\eta}$, which differs from the noninteracting correlation function $G(\omega)\propto 1/|\omega|$ with finite $\eta$. $\eta$ corresponds to the anomalous dimension and specifies the non-Fermi liquid behavior. ### One-loop results To one-loop order, we find from the RG equation the exponents at the fixed points $$\begin{gathered} \label{eq:exponents_one-loop_1} \gamma_h=1, \\ \gamma_\Delta=1-c_\text{pp}(0)\bar{g}_j^* = \begin{cases} 1 & \text{(Gaussian)} \\ 1-\epsilon & \text{(Nontrivial)}. \end{cases}\end{gathered}$$ with $\tilde{\epsilon} = \epsilon$. Most exponents in Eq. – are the same at the Gaussian and nontrivial fixed points, which is identical to that of the DOS in the noninteracting state. The difference is found when the pairing field $\Delta$ is involved. The exponent for the pairing field $\gamma_\Delta$ renders different exponents for the pairing susceptibility $\chi_\text{BCS}$: $$\chi_\text{BCS} \sim \begin{cases} T^{-\epsilon},\ |\Delta|^{-\epsilon} & \text{(Gaussian)} \\ T^{+\epsilon},\ |\Delta|^{+\epsilon} & \text{(Nontrivial)}. \end{cases}$$ The $s$-wave pairing susceptibility remains finite at the nontrivial fixed point whereas it diverges at the Gaussian fixed point. We also find a difference in the pair correlation function $$\begin{aligned} &\quad C(r_+,r_-,\tau) \nonumber\\ &= \hat{c} \left( r_+ \kappa^{1/\tilde{n}_+}, r_- \kappa^{1/\tilde{n}_-}, \tau \kappa \right) \times \begin{cases} \kappa^{-2(1-\epsilon)} & \text{(Gaussian)} \\ \kappa^{-2} & \text{(Nontrivial)}. \end{cases}\end{aligned}$$ It shows a faster decay at the nontrivial fixed point, reflecting the suppressed pairing susceptibility. Supermetal ---------- We coin a term, *supermetal*, to describe a state where the system resists ordering and remains a quantum critical state with power-law divergent susceptibilities. In this regard, the Gaussian fixed point is a noninteracting supermetal, and the nontrivial interacting fixed point is an interacting supermetal. Various susceptibilities in a noninteracting supermetal are determined by the DOS; they diverge with the DOS singularity exponent. In contrast, susceptibilities can have distinct exponents in an interacting supermetal. The one-loop analysis shows that the $s$-wave pairing susceptibility remains finite whereas the charge compressibility and spin susceptibility still diverge with the same exponent as that of the DOS. One may wonder whether an interacting supermetal is a Fermi liquid and whether the charge compressibility and spin susceptibility have the same exponents even with higher-order corrections. Those questions await an RG analysis to two-loop order in Sec. \[sec:field\_theory\]. The criticality of supermetals relies on scale invariance. Any relevant perturbations, such as $\mu$, $h$, and $\Delta$ considered above, introduce energy scales to the system, and they potentially drive the system toward ordering instabilities with broken symmetries. Here we outline possible consequences of relevant perturbations. Suppose that we add a finite chemical potential to force the system away from the critical point. At first, it drives the system to an electron or hole Fermi liquid state. However, a $2k_F$ singularity in the charge susceptibility arises and the Kohn–Luttinger mechanism for superconductivity [@Kohn-Luttinger] eventually takes effect. When a finite magnetic field is applied as a perturbation instead, it would trigger a phase transition to the ferromagnetic state. Ward identity ------------- In Sec. \[sec:scaling\_generic\], we derived the scaling relations for thermodynamic quantities and correlation functions. In the derivations, we used the fact that the chemical potential and frequency (or temperature) should have the same scaling exponent. This is a consequence of charge conservation. The Ward identity (more generally the Ward–Takahashi identity) describes the conservation law [@Ward; @Takahashi]. The identity is regarded as the quantum analog to Noether’s theorem. We show how it works in our present analysis. Also, the identity should hold even after an RG analysis, and thus it can be used to check the validity of an RG scheme, or specifically a choice of a cutoff. Now we investigate the structure of the self-energy $\Sigma$. To be concrete, we look into the expansion of the self-energy Eq.  to find a relation between $\Sigma_\omega$ and $\Sigma_\mu$. The Ward identity concludes that $\Sigma_\omega$ and $\Sigma_\mu$ are equal at $T=0$: $$\label{eq:Ward_Sigma} \Sigma_\omega = \Sigma_\mu.$$ The identity is based on charge conservation or the U(1) gauge invariance; the action and correlation functions are invariant under the transformations $\psi \mapsto e^{i\alpha(\bm{r},\tau)} \psi$ and $\bar{\psi} \mapsto \bar{\psi} e^{-i\alpha(\bm{r},\tau)}$ with a smooth scalar function $\alpha(\bm{r},\tau)$. In the following, we consider the Ward identity from the diagrammatic point of view. We can relate the $\mu$-derivative of the self-energy to the vertex function corresponding to the coupling $\varphi\bar{\psi}\psi$. We note that the scalar field $\varphi$ is associated with the chemical potential $\mu$. We write the vertex function as $\Gamma_\mu (\omega+\omega',\omega)$, where we focus only on the frequency dependence in the following discussion. The vertex function modifies the coupling term $\bar{\psi}(\omega+\omega') \varphi(\omega') \psi(\omega)$ to be $\Gamma_\mu(\omega+\omega',\omega) \bar{\psi}(\omega+\omega') \varphi(\omega') \psi(\omega)$. ![ (a) Relation between the bare vertex and the noninteracting Green’s function. (b) Diagrammatic representation of the Ward–Takahashi identity. []{data-label="fig:WT"}](WT.pdf){width="0.9\hsize"} To illustrate how Eq.  is derived, we make use of the equality $$G_{0}^{-1}(\bm{k},\omega+\omega') - G_{0}^{-1}(\bm{k},\omega) = i\omega',$$ and equivalently $$\label{eq:Ward_Green} G_0(\bm{k},\omega+\omega') (i\omega') G_0(\bm{k},\omega) = G_0(\bm{k},\omega) - G_0(\bm{k},\omega+\omega').$$ This equation is diagrammatically shown in Fig. \[fig:WT\](a). It relates the noninteracting vertex function $\Gamma_\mu = 1$ and the noninteracting Green’s function $G_0$. Now we add corrections to the self-energy, as depicted in Fig. \[fig:WT\](b) as shaded blobs. The dressed vertex function is obtained from the dressed self-energy by attaching the external scalar field $\varphi$ to every internal fermion line. Thus, considering Eq. , we find the Ward–Takahashi identity $$\begin{aligned} &\quad G(\bm{k},\omega+\omega') (i\omega') \Gamma_\mu(\omega+\omega',\omega) G(\bm{k},\omega) \nonumber\\ &= G(\bm{k},\omega) - G(\bm{k},\omega+\omega').\end{aligned}$$ The full Green’s function $G(\bm{k},\omega)$ is given by $$G(\bm{k},\omega) = \frac{1}{i\omega - E_{\bm{k}} - \Sigma(\bm{k},\omega)},$$ with the full self-energy $\Sigma$. Taking the zero frequency limit $\omega' \to 0$, we obtain the Ward identity $$\Gamma_\mu(\omega,\omega) = 1-\frac{\partial\Sigma(\omega)}{\partial(i\omega)}.$$ The vertex function $\Gamma_\mu(\omega,\omega)$ is equivalent to the correction to the chemical potential; the chemical potential dressed by quantum corrections is $\mu\Gamma_\mu(\omega,\omega) = \mu (1-\Sigma_\mu)$. Therefore, the Ward identity confirms the relation Eq. . It immediately concludes $$\gamma_\mu = 1$$ from Eq. . The relation holds to all orders of perturbative calculations at $T=0$. The result of the energy-shell RG analysis to one-loop order in Sec. \[sec:energy-shell\] satisfies the Ward identity. We notice that a frequency shell instead of the energy shell violates the Ward identity. Field theory approach {#sec:field_theory} ===================== This section focuses on the RG analysis from the field theory approach. To begin with, we briefly argue the two RG schemes: the energy-shell RG analysis and the field theory approach. We then confirm that it gives the same result as that from the energy-shell RG analysis at one-loop order. It is followed by the two-loop calculations to show the anomalous dimension and the correction to the energy dispersion. RG schemes {#sec:schemes} ---------- An objective of RG analyses is to track the flow of parameters in a theory under a scale transformation. Here, we illustrate two different RG schemes: the Wilsonian approach, including the preceding energy-shell RG analysis, and the field theory approach. The common feature is to divide the integration manifold (frequency and momentum in the present case) into two parts and integrate out modes belonging to one of them. The two schemes differ in intervals of integrations. The first scheme involves an integration within a hard shell. In the energy-shell RG analysis, fluctuations inside the thin energy shell $E\in[-\Lambda,-\Lambda/b)$, $(\Lambda/b,\Lambda]$ are eliminated. This mode elimination followed by rescaling enables us to keep track of the change of parameters under a scale transformation. On the other hand, in the field theory approach, we integrate out all low-energy fluctuations below the cutoff $\Lambda$. Then, we deduce the RG flow of parameters by comparing results at different cutoffs $\Lambda$ and $\Lambda'$. The two schemes have advantages in different aspects. In the Wilsonian approach, the frequency-momentum space is progressively integrated over, so the interpretation of the RG procedure is rather simple. The inclusion of low-energy modes results in a theory at low energies with different parameters. In spite of its simple interpretation, higher-loop calculations are not easy with the Wilsonian approach. In a one-loop calculation, we have only one shell to be concerned about. However, higher-loop diagrams consist of many internal lines (virtual states), so that we have to take care of shells for each of them. On the other hand, the field theory approach does not require such error-prone steps as it deals with all modes below the cutoff at once. This makes higher-loop calculations more tractable. Although not as intuitive as the Wilsonian approach, the field theory approach leads to the same results on critical phenomena. More descriptions about the comparison between the two schemes can be found in e.g. Ref. [@Shankar]. A brief review of the field theory approach is given in Appendix \[sec:field-theory\_review\]. Soft cutoff ----------- In the field theory approach, we calculate the connected $N$-point correlation function $G^{(N)}$ or the one-particle irreducible $N$-point function $\Gamma^{(N)}$. If we face a UV divergence in calculating them, we need to cure the divergence to obtain physically meaningful results. There are several ways to do so; we here choose to employ the UV energy cutoff $\Lambda$ to make a comparison to the preceding energy-shell RG analysis. The functions $G^{(N)}$ and $\Gamma^{(N)}$ can be obtained perturbatively with the noninteracting Green’s function $G_0$. We introduce the UV energy cutoff by suppressing the high-energy contributions in $G_0$. We define the noninteracting Green’s function with the energy cutoff $G_{0\Lambda}(\bm{k},\omega_n)$ as $$\begin{aligned} G_{0\Lambda}(\bm{k},\omega_n) &= G_0(\bm{k},\omega_n) K_\Lambda(E_{\bm{k}}) \nonumber\\ &= \frac{K_\Lambda(E_{\bm{k}})}{i\omega_n - E_{\bm{k}}},\end{aligned}$$ with the UV energy cutoff factor $$K_\Lambda(E) = \frac{\Lambda^2}{\Lambda^2+E^2}.$$ Note that the cutoff factor smoothly varies from 0 to 1 and thus works as a soft energy cutoff. This is in contrast to the energy-shell RG analysis, where the interval of an energy integration is cut off abruptly at $\Lambda$ and $\Lambda/b$. We can interpret the modified Green’s function as a Green’s function with an energy-dependent quasiparticle weight $K_\Lambda(E)$. The weight fades away in the high-energy limit $E\to\pm\infty$ to eliminate UV divergences, while $K_\Lambda(E) \to 1$ for energies much lower than the cutoff $\Lambda$. One may be tempted to see the modified Green’s function in a different way. For example, it can be rewritten as $$\begin{aligned} G_{0\Lambda}(\bm{k},\omega_n) &= \frac{1}{i\omega_n-E_{\bm{k}}} - \frac{1}{i\omega_n-E_{\bm{k}}} \frac{E_{\bm{k}}^2}{\Lambda^2+E_{\bm{k}}^2}. \end{aligned}$$ It may be viewed as a variation of the Pauli–Villars regularization, where the additional term cures a UV divergence but vanishes in the limit $\Lambda\to\infty$. However, we cannot think of it as a propagator with a large mass term since we cannot add a mass term for the electronic energy dispersion which is continuous and unbounded. It should be noted that the cutoff factor $K_\Lambda(E)$ does not depend on frequency. It potentially causes a violation of the Ward identity, which would result in wrong conclusions. For example, if one chooses a cutoff factor of the form $\Lambda^2/(\Lambda^2+E^2+\omega_n^2)$, it invalidates the Ward identity. The absence of the frequency in the cutoff factor ensures the Ward identity. Formalities ----------- ### Structure of the RG analysis To derive RG equations and see scaling properties, we calculate the one-particle irreducible $N$-point function $\Gamma^{(N)}_\Lambda$ with the cutoff $\Lambda$ and examine its cutoff dependence. The cutoff dependence is seen by comparing two $N$-point functions at different cutoffs; see Eq. . Specifically, we compare $\Gamma^{(N)}_\Lambda$ to one at a reference point $\Gamma^{(N)}_R$. The energy scale at the reference point is referred to as the renormalization scale. The procedure of fixing the model to the reference is equivalent to setting the initial parameters in the Wilsonian approach. We first analyze the case with $T=h=\Delta=0$. We impose the renormalization conditions $$\begin{gathered} \label{eq:condition_2} \Gamma_R^{(2)}(k) = -i\omega_n + E_{\bm{k},0} -\mu_0, \\ \label{eq:condition_4} \Gamma_R^{(4)}(k_1,k_2;k_3,k_4) = g_0,\end{gathered}$$ where the condition for $\Gamma^{(4)}$ should be considered at $k_1+k_2 = k_1+k_3 = k_1 +k_4 = 0$. The subscript $0$ denotes quantities at the renormalization scale. The interaction dresses the two-point and four-point functions and they acquire cutoff-dependent corrections. We here use the energy dispersion Eq.  for analysis. We assume that there is no additional term to the energy dispersion generated under the RG analysis, as we have discussed in Sec. \[sec:RG\_Wilson\_discussion\]. Then, the two-point and four-point functions at the cutoff $\Lambda$ can be expressed as $$\begin{gathered} \label{eq:Gamma_2_Lambda} \Gamma_\Lambda^{(2)} = -i\omega_n Z_\psi^{-1} + Z_{A_+}^{-1} A_+ k_+^{n_+} - Z_{A_-}^{-1} A_- k^{n_-} - Z_\mu^{-1} \mu, \\ \label{eq:Gamma_4_Lambda} \Gamma_\Lambda^{(4)} = Z_g^{-1} g,\end{gathered}$$ where the corrections $Z_\psi$, $Z_{A_\pm}$, $Z_\mu$, and $Z_g$ are calculated perturbatively. The $N$-point functions at the renormalization scale and the cutoff $\Lambda$ are related by $$\begin{gathered} \Gamma_R^{(N)} = Z_\psi^{N/2} \Gamma_\Lambda^{(N)}.\end{gathered}$$ The last equation leads to the RG equations. Since the left-hand side does not depend on the cutoff $\Lambda$, we obtain the differential equation $$\begin{aligned} \Lambda \frac{d}{d\Lambda} \Gamma_R^{(N)} = 0.\end{aligned}$$ It leads to the Callan–Symanzik equation. We obtain for the one-particle irreducible $N$-point function $$\begin{gathered} \label{eq:CS} \left[ \Lambda\frac{\partial}{\partial\Lambda} -\beta(\bar{g})\frac{\partial}{\partial\bar{g}} - \beta_\mu(\bar{g},\bar{\mu}) \frac{\partial}{\partial\bar{\mu}} - \beta_{A_+}(\bar{g},A_\pm) \frac{\partial}{\partial A_+} - \beta_{A_-}(\bar{g},A_\pm) \frac{\partial}{\partial A_-} - \frac{N}{2} \gamma_\psi(\bar{g}) \right] \Gamma_\Lambda^{(N)} = 0,\end{gathered}$$ where we use the dimensionless parameters defined in Eq. . The beta functions and $\gamma_\psi$ are defined by $$\begin{gathered} \label{eq:beta_g} \beta(\bar{g}) = -\left( \Lambda\frac{\partial\bar{g}}{\partial\Lambda} \right)_{\bar{g}_0,\bar{\mu}_0,A_{\pm,0}}, \\ \label{eq:beta_mu} \beta_\mu(\bar{g},\bar{\mu}) = -\left( \Lambda\frac{\partial\bar{\mu}}{\partial\Lambda} \right)_{\bar{g}_0,\bar{\mu}_0,A_{\pm,0}}, \\ \label{eq:beta_A} \beta_{A_\pm}(\bar{g},A_\pm) = -\left( \Lambda\frac{\partial A_\pm}{\partial\Lambda} \right)_{\bar{g}_0,\bar{\mu}_0,A_{\pm,0}}, \\ \label{eq:gamma_def} \gamma_\psi = -\left( \Lambda\frac{\partial}{\partial\Lambda} \ln Z_\psi \right)_{\bar{g}_0,\bar{\mu}_0,A_{\pm,0}}.\end{gathered}$$ We can rewrite the beta functions as $$\begin{gathered} \beta = \bar{g} \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_g -2\gamma_\psi \right) , \\ \beta_\mu = \bar{\mu} \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_\mu -\gamma_\psi \right) , \\ \beta_{A_\pm} = A_\pm \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_{A_\pm} -\gamma_\psi \right),\end{gathered}$$ since the renormalized values are given by $$\begin{gathered} \bar{g} = Z_g Z_\psi^{-2} \bar{g}_0, \\ \bar{\mu} = Z_\mu Z_\psi^{-1} \bar{\mu}_0, \\ A_\pm = Z_{A_\pm} Z_\psi^{-1} A_{\pm,0}.\end{gathered}$$ Those equations show that the field renormalization gives additional effects to the beta functions and hence the scaling properties. ### Solutions The Callan–Symanzik equation can be solved by the method of characteristics; see Appendix \[sec:field-theory\_review\]. The beta functions describe the RG flows of the parameters: $$\begin{gathered} \label{eq:RG_g_beta} \frac{d\bar{g}}{dl} = \beta(\bar{g}), \\ \label{eq:RG_mu_beta} \frac{d\bar{\mu}}{dl} = \beta_{\mu} (\bar{g}, \bar{\mu}), \\ \label{eq:RG_A_beta} \frac{dA_\pm}{dl} = \beta_{A_\pm} (\bar{g},A_\pm).\end{gathered}$$ $l=\ln{\Lambda_0/\Lambda}$ denotes the RG scale, measured relative to the renormalization scale $\Lambda_0$. Those RG equations are to be compared with those obtained by the energy-shell RG analysis in Sec. \[sec:energy-shell\]. In general, they are coupled differential equations and zeros of the beta functions determine fixed points. When we expand the beta functions $\beta_\mu$ and $\beta_{A_\pm}$ around a fixed point with $\bar{g}^*$ as $$\begin{gathered} \beta_\mu(\bar{g}^*,\bar{\mu}) \approx \beta_\mu(\bar{g}^*,0) + \gamma_\mu(\bar{g}^*) \bar{\mu}, \\ \beta_{A_\pm}(\bar{g}^*,A_\pm) \approx \gamma_{A_\pm}(\bar{g}^*) A_{\pm},\end{gathered}$$ $\gamma_\mu(\bar{g}^*)$ and $\gamma_{A_\pm}(\bar{g}^*)$ give the exponents in the scaling region. Recall that $\gamma_\mu=1$ is required by the Ward identity, regardless of $\bar{g}$. We note that $\beta_\mu(\bar{g}^*,0)$ does not affect the scaling behavior apart from a shift of the chemical potential and that it can be neglected, as we have discusses in Sec. \[sec:energy-shell\_one-loop\]. With constant $\gamma_\mu(\bar{g}^*)$ and $\gamma_{A_\pm}(\bar{g}^*)$, we observe the scaling properties $$\begin{gathered} \bar{\mu}(l) = \bar{\mu}_0 e^{l}, \quad \label{eq:scaling_A_2} A_\pm(l) = A_{\pm,0} e^{\gamma_{A_\pm}(\bar{g}^*)l}.\end{gathered}$$ The function $\gamma_\psi$ is ascribed to the anomalous dimension $\eta$ when it is computed at a fixed point. To see this, we solve the Callan–Symanzik equation ; see Appendix \[sec:field-theory\_review\] for details. The solution of the two-point function is given by $$\begin{aligned} \label{eq:two-point_solution} &\quad \Gamma^{(2)}_\Lambda \left(e^{-l/n_\pm}k_{\pm,0}, e^{-l}\omega_0; \bar{g}(0), \bar{\mu}(0), A_\pm(0) \right) \nonumber\\ &= e^{-l} \Gamma_\Lambda^{(2)} \left( k_{\pm,0}, \omega_0; \bar{g}(l), \bar{\mu}(l), A_\pm(l) \right) \exp \left[ \int_0^l dl' \gamma_\psi(\bar{g}(l')) \right].\end{aligned}$$ We now examine the behavior in the critical region as a function of $\omega$, $k_+$, and $k_-$. Close to a fixed point, we use the values at the fixed point for $\bar{g}$, $\bar{\mu}$, and $\gamma_\psi$: $\bar{g}(0)=\bar{g}(l)=\bar{g}^*$, $\bar{\mu}(0)=\bar{\mu}(l)=0$, and $\gamma_\psi(\bar{g}^*)=\eta$. However, we have to keep the $l$ dependence of $A_\pm$ as they appear together with $k_\pm$. We assume the two-point function is a function of $A_+ k_+^{n_+}$, $A_- k_-^{n_-}$, $\omega$. Since those three quantities and $\Lambda$ have the dimension of energy, we can write the two-point function in the scaling region as $$\begin{aligned} \Gamma^{(2)}_\Lambda \left( k_+, k_-, \omega; A_+, A_- \right) = \Lambda \hat{\Gamma}^{(2)}_\Lambda \left( \frac{A_+ k_+^{n_+}}{\Lambda}, \frac{A_- k_-^{n_-}}{\Lambda}, \frac{\omega}{\Lambda} \right),\end{aligned}$$ where $\hat{\Gamma}^{(2)}_\Lambda$ is a scaling function. Here, we do not assume homogeneity for $\hat{\Gamma}^{(2)}_\Lambda$ and determine the exponents for $A_+ k_+^{n_+}/\Lambda$, $A_- k_-^{n_-}/\Lambda$, and $\omega/\Lambda$, separately. In Eq. , $l$ is an arbitrary quantity; to inspect the scaling behavior in terms of $\omega$, we set $l=\ln (\omega_0/\omega)$ and $k_+=k_-=0$. The momentum dependence is considered in the same manner with $l=\ln (k_{\pm,0}/k_\pm)^{n_\pm}$ and Eq. . We then find $$\begin{aligned} &\quad \Gamma^{(2)}_\Lambda (k_+,k_-,\omega) \nonumber\\ &\propto \begin{cases} \left(k_+^{n_+/[1+\gamma_{A_+}(\bar{g}^*)]}\right)^{1-\eta} & (k_-=\omega=0) \\ \left(k_-^{n_-/[1+\gamma_{A_-}(\bar{g}^*)]}\right)^{1-\eta} & (k_+=\omega=0) \\ \omega^{1-\eta} & (k_+=k_-=0). \end{cases}\end{aligned}$$ It confirms the scaling relation of the two-point correlation function Eq.  along with the relation $G = [\Gamma^{(2)}]^{-1}$. {width="\hsize"} {width="0.95\hsize"} One-loop calculations $\bm{(h=\Delta=0)}$ ----------------------------------------- We calculate the two-point and four-point functions to obtain the beta functions and $\gamma_\psi(\bar{g})$. This is accomplished by evaluating the perturbative corrections to the two-point and four-point functions (Figs. \[fig:loop\_two-point\] and \[fig:stu\]). As the corrections to the coupling constant $g$, there are three possible one-loop diagrams shown in Fig. \[fig:stu\]. To determine the perturbative correction $\delta g$, all diagrams should be evaluated with zero momentum transfer $q=0$, which is required by the renormalization condition Eq. . The three one-loop diagrams in Fig. \[fig:stu\](a) correspond to the BCS, density-density, and exchange channels (from left to right). Out of the three, the density-density channel does not contribute at one-loop order because of the Pauli exclusion principle for the contact interaction. This contribution is allowed when we assume the density-density interaction $\bar{\psi}_\sigma\bar{\psi}_{\sigma'} \psi_{\sigma'} \psi_\sigma$ with arbitrary spins $\sigma$, $\sigma'$. (For reference, we note that the three channels are referred to as the BCS, ZS (zero sound), and ZS$'$ in Ref. [@Shankar]; or $s$-, $t$-, and $u$-channels with the Mandelstam variables.) To one-loop order, the two-point and four-point functions acquire corrections to the chemical potential and the coupling constant, but not to field or the energy dispersion as we have seen in the energy-shell RG analysis. One-loop diagrams include $\Sigma_\text{H}$, $\Pi_\text{pp}$, and $\Pi_\text{ph}$ like Eq.  and , so the two-point and four-point functions become $$\begin{gathered} \label{eq:Gamma_2_oneloop} \Gamma^{(2)}_\Lambda = -i\omega_n + E_{\bm{k}} -\mu -g\Sigma_\text{H} +g\mu\Pi_\text{ph}, \\ \label{eq:Gamma_4_oneloop} \Gamma^{(4)}_\Lambda = g - g^2 (\Pi_\text{pp}+\Pi_\text{ph}).\end{gathered}$$ These equations lead to $$\begin{gathered} Z_\psi^{-1} = 1, \quad Z_{A_\pm}^{-1} = 1, \\ Z_\mu^{-1} = 1 + \frac{g}{\mu}\Sigma_\text{H} - g\mu\Pi_\text{ph}, \\ Z_g^{-1} = 1 - g(\Pi_\text{pp}+\Pi_\text{ph}).\end{gathered}$$ Here we calculate the perturbative corrections with the soft cutoff $K_\Lambda$. The actual calculations for the beta functions require the $\Lambda$-derivatives instead of the corrections themselves. We thus obtain the one-loop corrections as follows: $$\begin{aligned} &\quad \Lambda\frac{\partial}{\partial\Lambda} \Sigma_\text{H} \nonumber\\ &= T\sum_n \int_{\bm{k}} G_0(\bm{k},\omega_n) \Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{k}}) \nonumber\\ &= T\sum_{n\geq0} \int dE D(E) \frac{-2E}{\omega_n^2 + E^2} \frac{2\Lambda^2 E^2}{(\Lambda^2 + E^2)^2} \nonumber\\ &= -\Lambda^2 \int dE D(E) \tanh\left(\frac{E}{2T}\right) \frac{E^2}{(\Lambda^2 + E^2)^2} \nonumber\\ &= -[D(\Lambda)-D(-\Lambda)] \Lambda \int_0^\infty dx \frac{x^{2-\epsilon}}{(1+x^2)^2} \tanh\left(\frac{\Lambda}{2T}x\right) \nonumber\\ &\equiv -\Lambda D(\Lambda) \tilde{c}_\text{H}(\bar{T}) ,\end{aligned}$$ $$\begin{aligned} &\quad \Lambda\frac{\partial}{\partial\Lambda} \Pi_\text{pp} \nonumber\\ &= T\sum_n \int_{\bm{k}} G_0(\bm{k},\omega_n) G_0(-\bm{k},-\omega_n) \Lambda\frac{\partial}{\partial\Lambda} K^2_\Lambda(E_{\bm{k}}) \nonumber\\ &= T\sum_{n\geq0} \int dE D(E) \frac{2}{\omega_n^2 + E^2} \frac{4\Lambda^4 E^2}{(\Lambda^2+E^2)^3} \nonumber\\ &= 2\Lambda^4 \int dE D(E) \frac{E}{(\Lambda^2+E^2)^3} \tanh\left(\frac{E}{2T}\right) \nonumber\\ &= 2[D(\Lambda)+D(-\Lambda)] \int_0^\infty dx \frac{x^{1-\epsilon}}{(1+x^2)^3} \tanh\left(\frac{\Lambda}{2T}x\right) \nonumber\\ &\equiv D(\Lambda) \tilde{c}_\text{pp}(\bar{T}),\end{aligned}$$ $$\begin{aligned} &\quad \Lambda\frac{\partial}{\partial\Lambda} \Pi_\text{ph} \nonumber\\ &= T\sum_{\omega_n} \int_{\bm{k}} G_0^2(\bm{k},\omega_n) \Lambda\frac{\partial}{\partial\Lambda} K^2_\Lambda(E_{\bm{k}}) \nonumber\\ &= T\sum_{n\geq0} \int dE D(E) \frac{-2(\omega_n^2-E^2)}{(\omega_n^2+E^2)^2} \frac{4\Lambda^4 E^2}{(\Lambda^2+E^2)^3} \nonumber\\ &= -\frac{\Lambda^4}{T} \int dE D(E) \frac{E^2}{(\Lambda^2+E^2)^3} \frac{1}{\cosh^2\left(\dfrac{E}{2T}\right)} \nonumber\\ &= -[D(\Lambda)+D(-\Lambda)] \frac{\Lambda}{T} \int_0^\infty dx \frac{x^{2-\epsilon}}{(1+x^2)^3} \frac{1}{\cosh^2\left(\dfrac{\Lambda}{2T}x\right)} \nonumber\\ &\equiv -D(\Lambda) \tilde{c}_\text{ph}(\bar{T}).\end{aligned}$$ As a result, we obtain the beta functions Eq.  and $$\begin{gathered} \beta(\bar{g}) = \epsilon\bar{g} - \bar{g}^2 \left[ \tilde{c}_\text{pp}(\bar{T})-\tilde{c}_\text{ph}(\bar{T}) \right], \\ \beta_\mu(\bar{g},\bar{\mu}) = \left[ 1-\tilde{c}_\text{ph}(\bar{T}) \bar{g} \right]\bar{\mu} + \tilde{c}_\text{H}(\bar{T}) \bar{g}.\end{gathered}$$ Note that the tree-level scaling terms appear from the definitions of the dimensionless parameters $\bar{g}_0=g_0 D(\Lambda)$ and $\bar{\mu}_0 = \mu/\Lambda$. The beta functions are to be compared with the result from the energy-shell RG analysis Eq. . To confirm, we first evaluate the coefficients $\tilde{c}_\text{H}$, $\tilde{c}_\text{pp}$, $\tilde{c}_\text{ph}$ at $T=0$: $$\begin{gathered} \tilde{c}_\text{H}(0) = \left( 1-\frac{D_-}{D_+} \right) \frac{\pi}{4} (1-\epsilon) \frac{1}{\cos\left(\dfrac{\pi\epsilon}{2}\right)}, \\ \tilde{c}_\text{pp}(0) = \left( 1+\frac{D_-}{D_+} \right) \frac{\pi}{4} \left( 1+\frac{\epsilon}{2} \right) \frac{\epsilon}{\sin\left(\dfrac{\pi\epsilon}{2}\right)}, \\ \tilde{c}_\text{ph}(0) = 0.\end{gathered}$$ The zeros of the beta function $\beta(\bar{g})$ give the two fixed points $$\label{eq:fixed_points_field} \bar{g}_1^* = 0, \quad \bar{g}_2^* = \frac{\epsilon}{\tilde{c}_\text{pp}(0)} (>0).$$ We now find the Gaussian fixed point and the nontrivial interacting fixed point from the field theory approach. Although the value of $\bar{g}_2^*$ differs in the two schemes, resulting exponents for the thermodynamic quantities are not suffered from the difference since the exponents are not directly dependent on the coupling constant $\bar{g}$ at fixed point. We explicitly confirm this in the next subsection by calculating the beta functions for the magnetic field $\Delta$ and pairing field $\Delta$. RG equations for $\bm{h}$ and $\bm{\Delta}$ ------------------------------------------- The beta functions for the magnetic field $h$ and pairing field $\Delta$ can be obtained from the corresponding vertex functions $\Gamma^{(2,h)}$ and $\Gamma^{(2,\Delta)}$, respectively. Perturbative corrections to them are depicted in Figs. \[fig:loop\_two-point\](b) and \[fig:loop\_two-point\](c). We impose the renormalization conditions $$\begin{gathered} \Gamma^{(2,h)}_R = Z_\psi \Gamma^{(2,h)}_\Lambda = h_0, \\ \Gamma^{(2,\Delta)}_R = Z_\psi \Gamma^{(2,\Delta)}_\Lambda = \Delta_0,\end{gathered}$$ where the vertex functions with the cutoff $\Lambda$ are expressed as $$\begin{gathered} \Gamma^{(2,h)}_\Lambda = Z_h^{-1} h, \quad \Gamma^{(2,\Delta)}_\Lambda = Z_\Delta^{-1} \Delta.\end{gathered}$$ To obtain the beta functions to one-loop order, it is sufficient to consider the Callan–Symnanzik equations without corrections to the energy dispersion and the chemical potential: $$\begin{gathered} \left[ \Lambda\frac{\partial}{\partial\Lambda} - \beta(\bar{g})\frac{\partial}{\partial \bar{g}} - \beta_h(\bar{g},\bar{h})\frac{\partial}{\partial\bar{h}} - \gamma_\psi(\bar{g}) \right] \Gamma^{(2;h)} = 0, \\ \left[ \Lambda\frac{\partial}{\partial\Lambda} - \beta(\bar{g})\frac{\partial}{\partial \bar{g}} - \beta_\Delta(\bar{g},\bar{\Delta})\frac{\partial}{\partial\bar{\Delta}} - \gamma_\psi(\bar{g}) \right] \Gamma^{(2;\Delta)} = 0,\end{gathered}$$ where the beta functions for the magnetic field and pairing field are defined by $$\begin{gathered} \label{eq:beta_h} \beta_h(\bar{h}) = -\left( \Lambda\frac{\partial\bar{h}}{\partial\Lambda} \right)_{\bar{g}_0}, \\ \label{eq:beta_Delta} \beta_{\bar{\Delta}}(\bar{g}) = -\left( \Lambda\frac{\partial\bar{\Delta}}{\partial\Lambda} \right)_{\bar{g}_0}.\end{gathered}$$ Using the relations $$\begin{gathered} \bar{h} = Z_h Z_\psi^{-1} \bar{h}_0, \quad \bar{\Delta} = Z_\Delta Z_\psi^{-1} \bar{\Delta}_0,\end{gathered}$$ the beta functions become $$\begin{gathered} \beta_h(\bar{g},\bar{h}) = \bar{h} \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_h -\gamma_\psi \right), \\ \beta_\Delta(\bar{g},\bar{\Delta}) = \bar{\Delta} \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_\Delta -\gamma_\psi \right).\end{gathered}$$ They are related to the exponents $\gamma_h$ and $\gamma_\Delta$ when evaluated at a fixed point: $$\begin{gathered} \beta_h(\bar{g}^*,\bar{h}) = \gamma_h(\bar{g}^*) \bar{h}, \quad \beta_\Delta(\bar{g}^*,\bar{\Delta}) = \gamma_\Delta(\bar{g}^*) \bar{\Delta}.\end{gathered}$$ We calculate the vertex functions for $h$ and $\Delta$ to one-loop order and find $$\begin{gathered} \Gamma^{(2,h)}_\Lambda = h, \quad \Gamma^{(2,\Delta)}_\Lambda = \Delta - g\Delta\Pi_\text{pp}.\end{gathered}$$ We note that there is no perturbative correction to $\Gamma^{(2,h)}_\Lambda$ because of the spin-rotational symmetry of the model. The vertex functions lead to the beta functions $$\begin{gathered} \beta_h(\bar{g},\bar{h}) = \bar{h}, \\ \beta_\Delta(\bar{g},\bar{\Delta}) = \left[ 1-\tilde{c}_\text{pp}(\bar{T})\bar{g} \right] \bar{\Delta}.\end{gathered}$$ Now we confirm that the exponent for the pairing field $\Delta$ is the same independent of the RG schemes. Particularly at the nontrivial fixed point, we obtain $\beta_\Delta(\bar{g}_2^*) = (1-\epsilon)\bar{\Delta}$. This is consistent with the result from the energy-shell RG analysis. The coefficient $\tilde{c}_\text{pp}$, which determines the value of the coupling constant at the nontrivial fixed point, does not appear to the exponent of the pairing field. Two-loop calculation and anomalous dimension {#sec:anomalous_dimension} -------------------------------------------- An advantage of the field theory approach is considerable when we deal with higher-order corrections. In the following, we consider the two-loop corrections at $T=0$ for the anomalous dimension and the correction to the energy dispersion. The field renormalization is seen from the frequency dependence of the self-energy. The linear term $\Sigma_\omega$ in Eq.  is given by $$\begin{aligned} \Sigma_\omega &= \frac{\partial}{\partial(i\omega)} \Sigma(\bm{k}=\bm{0},\omega) \bigg|_{\omega=0} \equiv \sum_{j\geq 2} g^j \Sigma_\omega^{(j)}.\end{aligned}$$ We expand $\Sigma$ with respect to the coupling constant $g$. Similarly, the renormalization of the energy dispersion arises from the momentum dependence of the self-energy $$\begin{aligned} \Sigma_{\bm{k}} \equiv \Sigma(\bm{k},\omega=0) \equiv \sum_{j\geq 2} g^j \Sigma_{\bm{k}}^{(j)},\end{aligned}$$ and $\Sigma_\pm$ is obtained as $$\begin{gathered} \Sigma_\pm = \frac{\partial \Sigma_{\bm{k}}}{\partial k_\pm^{n_\pm}} \bigg|_{k=0}.\end{gathered}$$ We have used that the fact that the one-loop correction, i.e., the Hartree contribution, does not yield the frequency or momentum dependence, so that the expansions with respect to the coupling constant begin at second order. The renormalization condition Eq.  reads $$\begin{gathered} Z_\psi^{-1} = 1- \Sigma_\omega, \quad Z_{A_\pm}^{-1} = 1 + \Sigma_{k_\pm}.\end{gathered}$$ Then, the field renormalization $\gamma_\psi$ is expressed as $$\begin{aligned} \label{eq:gamma_two-loop} \gamma_\psi &= \Lambda\frac{\partial}{\partial\Lambda} \ln \left( 1 -\sum_{j\geq2} g^j \Sigma_\omega^{(j)} \right) \nonumber\\ &= -g^2 \Lambda\frac{\partial}{\partial\Lambda} \Sigma_\omega^{(2)} + O(g^3).\end{aligned}$$ Again, assuming that no additional term than the corrections to $A_\pm$ is generated from $\Sigma_{\bm{k}}$, we obtain the beta function for the coefficient of the energy dispersion $A_\pm$ as $$\begin{aligned} \label{eq:beta_A_two-loop} \beta_{A_\pm} &= -\Lambda\frac{\partial A_\pm}{\partial\Lambda} \nonumber\\ &= A \left[ g^2 \Lambda\frac{\partial}{\partial\Lambda} (\Sigma_{\pm}^{(2)} + \Sigma_\omega^{(2)}) + O(g^3) \right] .\end{aligned}$$ We now calculate the two-loop correction to the self-energy $\Sigma^{(2)}$. The frequency and momentum dependent contribution can appear in the sunrise diagram, shown in Fig. \[fig:loop\](d) and \[fig:loop\_two-point\](a) as the rightmost term. It is calculated from $$\begin{aligned} \label{eq:two-loop_function} \Sigma^{(2)}(k) &= -\int_{pql} G_{0\Lambda}(p) G_{0\Lambda}(q) G_{0\Lambda}(l) \nonumber\\ &\qquad\times (2\pi)^{d+1} \delta(p+q-l-k).\end{aligned}$$ We use the shorthand notations $p=(\bm{p},\omega_p)$ and $\int_p = \int\frac{d\omega_p}{(2\pi)}\int\frac{d^dp}{(2\pi)^d}$. Then, we obtain the $\omega$-linear contribution $$\begin{aligned} \label{eq:two-loop_frequency} &\quad -\Lambda\frac{\partial}{\partial\Lambda} \Sigma_\omega^{(2)} \nonumber\\ &= \frac{\partial}{\partial(i\omega_k)} \int_{pql} (2\pi)^{d+1} \delta(p+q-l-k) \frac{1}{i\omega_p-E_{\bm{p}}} \frac{1}{i\omega_q-E_{\bm{q}}} \frac{1}{i\omega_l-E_{\bm{l}}} \Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{p}}) K_\Lambda(E_{\bm{q}}) K_\Lambda(E_{\bm{l}}) \bigg|_{k=0} \nonumber\\ &= \left( \int_{\bm{p}\bm{q}}^+ \int_{\bm{l}}^- + \int_{\bm{p}\bm{q}}^- \int_{\bm{l}}^+ \right) (2\pi)^{d} \delta(\bm{p}+\bm{q}-\bm{l}) \frac{1}{(E_{\bm{p}}+E_{\bm{q}}-E_{\bm{l}})^2} \Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{p}}) K_\Lambda(E_{\bm{q}}) K_\Lambda(E_{\bm{l}}) \nonumber\\ &= 2\Lambda^{-2\epsilon} \left( \int_{\bar{\bm{p}}\bar{\bm{q}}}^+ \int_{\bar{\bm{l}}}^- + \int_{\bar{\bm{p}}\bar{\bm{q}}}^- \int_{\bar{\bm{l}}}^+ \right) \frac{(2\pi)^{d} \delta(\bar{\bm{p}}+\bar{\bm{q}}-\bar{\bm{l}})}{(E_{\bar{\bm{p}}}+E_{\bar{\bm{q}}}-E_{\bar{\bm{l}}})^2} \frac{3E_{\bar{\bm{p}}}^2E_{\bar{\bm{q}}}^2E_{\bar{\bm{l}}}^2 + 2(E_{\bar{\bm{q}}}^2E_{\bar{\bm{l}}}^2+E_{\bar{\bm{p}}}^2E_{\bar{\bm{l}}}^2+E_{\bar{\bm{p}}}^2E_{\bar{\bm{q}}}^2) + (E_{\bar{\bm{p}}}^2+E_{\bar{\bm{q}}}^2+E_{\bar{\bm{l}}}^2)}{(1+E_{\bar{\bm{p}}}^2)^2 (1+E_{\bar{\bm{q}}}^2)^2 (1+E_{\bar{\bm{l}}}^2)^2} \nonumber\\ &\equiv D^2(\Lambda) C^{(2)},\end{aligned}$$ and the momentum-dependent part $$\begin{aligned} \label{eq:two-loop_momentum} &\quad \Lambda\frac{\partial}{\partial\Lambda} \Sigma^{(2)}(\bm{k},\omega_k=0) \nonumber\\ &= -\int_{pql} (2\pi)^{d+1} \delta(p+q-l-k) \frac{1}{i\omega_p-E_{\bm{p}}} \frac{1}{i\omega_q-E_{\bm{q}}} \frac{1}{i\omega_l-E_{\bm{l}}} \Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{p}}) K_\Lambda(E_{\bm{q}}) K_\Lambda(E_{\bm{l}}) \bigg|_{\omega_k=0} \nonumber\\ &= -\left( \int_{\bm{p}\bm{q}}^+ \int_{\bm{l}}^- + \int_{\bm{p}\bm{q}}^- \int_{\bm{l}}^+ \right) (2\pi)^{d} \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}) \frac{1}{E_{\bm{p}}+E_{\bm{q}}-E_{\bm{l}}} \Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{p}}) K_\Lambda(E_{\bm{q}}) K_\Lambda(E_{\bm{l}}) \nonumber\\ &= -2\Lambda^{-2\epsilon} \left( \int_{\bar{\bm{p}}\bar{\bm{q}}}^+ \int_{\bar{\bm{l}}}^- - \int_{\bar{\bm{p}}\bar{\bm{q}}}^- \int_{\bar{\bm{l}}}^+ \right) \frac{(2\pi)^{d} \delta(\bar{\bm{p}}+\bar{\bm{q}}-\bar{\bm{l}}-\bar{\bm{k}})}{|E_{\bar{\bm{p}}}+E_{\bar{\bm{q}}}-E_{\bar{\bm{l}}}|} \frac{3E_{\bar{\bm{p}}}^2E_{\bar{\bm{q}}}^2E_{\bar{\bm{l}}}^2 + 2(E_{\bar{\bm{q}}}^2E_{\bar{\bm{l}}}^2+E_{\bar{\bm{p}}}^2E_{\bar{\bm{l}}}^2+E_{\bar{\bm{p}}}^2E_{\bar{\bm{q}}}^2) + (E_{\bar{\bm{p}}}^2+E_{\bar{\bm{q}}}^2+E_{\bar{\bm{l}}}^2)}{(1+E_{\bar{\bm{p}}}^2)^2 (1+E_{\bar{\bm{q}}}^2)^2 (1+E_{\bar{\bm{l}}}^2)^2} \nonumber\\ &\equiv D^2(\Lambda) C^{(2)}_{\bm{k}}.\end{aligned}$$ Here we denote the dimensionless quantities by adding bars; we define $\bar{\omega} = \omega/\Lambda$, $\bar{p}_+ = p_+/\Lambda^{1/n_1}$, and $\bar{p}_- = p_-/\Lambda^{1/n_2}$. The momentum is scaled by $\Lambda$ so that the energy becomes dimensionless: $E_{\bar{\bm{k}}} = E_{\bm{k}}/\Lambda$. $\int^{\pm}_{\bm{p}} = \int_{\bm{p}} \Theta (\pm E_{\bm{p}})$ stands for the momentum integral within the positive (negative) energy domain. The constraints on the momentum integrals emerge after the frequency integrals. They can be evaluated by identifying the position of poles on the complex plane, leading to the restricted regions of the momentum integrals. We expect finite results for the two-loop results Eqs.  and at a saddle point of an energy dispersion because of the constraints on the momentum integrals $\int_{\bm{p}\bm{q}}^\pm \int_{\bm{l}}^\mp$. The two-loop contributions vanish at a band edge since there is no sign change in the energy dispersion. Now we scrutinize the frequency-dependent part $\Sigma_\omega^{(2)}$, which is responsible to the field renormalization and hence the anomalous dimension. As we have discussed, the contribution vanishes at band edges and thus an anomalous dimension does not arise. It can be finite only at an energy saddle point. In addition, it is worth pointing out that the integrand of Eq.  is guaranteed to be positive. Therefore, if there exists a finite volume that satisfies the constraint of the momentum integrals, we find a finite result: $C^{(2)}>0$. The constraints on the momentum integrals can be rephrased as follows: There exists a momentum $\bm{l}=\bm{p}+\bm{q}$ such that $\operatorname{sgn}(E_{\bm{l}}) = -\operatorname{sgn}(E_{\bm{p}}) = -\operatorname{sgn}(E_{\bm{q}})$. Such a momentum $\bm{l}$ in general exists near a saddle point because the energy dispersion near a saddle point comprises two or more filled Fermi seas and the area is not convex. We do not further evaluate the expression of the two-loop correction as its value depends on the explicit form of the energy dispersion. Equation  has a numerical factor $C^{(2)}$, which is independent of the cutoff $\Lambda$. From Eq. , we find the field renormalization $$\gamma_\psi = C^{(2)} \bar{g}^2 + O(\bar{g}^3).$$ This quantity gives the anomalous dimension when evaluated at a fixed point. It can be finite at the nontrivial fixed point to become $$\eta = C^{(2)} \bar{g}_2^{*2} + O(\bar{g}_2^{*3}) (>0).$$ A finite anomalous dimension concludes a non-Fermi liquid behavior at the nontrivial fixed point. This happens at a saddle point of an energy dispersion with a power-law DOS singularity. The momentum-dependent part Eq.  can be considered in the same way. It becomes finite only at a saddle point of an energy dispersion, but not at a band edge. A finite $C_{\bm{k}}^{(2)}$ leads to the beta functions for $A_\pm$ $$\beta_{A_\pm} = A_\pm \left[ \bar{g}^2 \left( \left.\frac{\partial C_{\bm{k}}^{(2)}}{\partial k_\pm^{n_\pm}}\right|_{\bm{k}=0} - C^{(2)} \right) +O(\bar{g}^3) \right].$$ The quantity inside the square brackets is identified as $\gamma_{A_\pm}$ when evaluated at a fixed point. Quasiparticle decay rate {#sec:lifetime} ======================== The preceding RG analyses focused on the real part of the self-energy or equivalently the two-point function. They give rise to the corrections to the action, which are captured through the RG equations. On the other hand, the imaginary part of the self-energy describes the damping of the quasiparticle, which is the focus of this section. It is generated by the interaction in the present model. Unlike the real part of the self-energy, the imaginary part can be calculated without a cutoff; we do not employ an RG method in this section, but integrate over the entire frequency and momentum space at once. We calculate the quasiparticle decay rate $\Gamma(\bm{k},\omega)$, obtained from the retarded self-energy as $$\Gamma(\bm{k},\omega) = -\operatorname{Im}\Sigma^R(\bm{k},\omega).$$ The retarded self-energy $\Sigma^R(\bm{k},\omega)$ is calculated from the self-energy $\Sigma(\bm{k},\omega_n)$, with the analytic continuation of the Matsubara frequency to the real frequency: $i\omega_n = \omega + i\delta$ ($\delta$: infinitesimal positive quantity). In the presence of the contact interaction, a finite imaginary part of the self-energy $\Sigma^R$ emerges at two-loop order and higher. The one-loop correction, or the Hartree term $\Sigma_\text{H}$, does not yield a finite imaginary component. Here we consider the two-loop diagram (the sunrise diagram) \[Fig. \[fig:loop\](d)\] to calculate the quasiparticle decay rate $\Gamma$. Like Eq. , it is given by $$\begin{aligned} \label{eq:two-loop_damping} &\quad \Sigma^{(2)}(\bm{k},\omega_n) \nonumber\\ &= -T\sum_{\omega_p} T\sum_{\omega_q} T\sum_{\omega_l} \int_{\bm{p}\bm{q}\bm{l}} G_0(\bm{p},\omega_p) G_0(\bm{q},\omega_q) G_0(\bm{l},\omega_l) \nonumber\\ &\quad\times \frac{(2\pi)^d}{T} \delta(\omega_p+\omega_q-\omega_l-\omega_n) \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}),\end{aligned}$$ but we do not need a cutoff for the imaginary part. The calculation of $\Sigma^{(2)}$ is standard and can be found in e.g. Ref. [@AGD]; we also show the derivation in Appendix \[sec:two-loop\_lifetime\] and just present the result here. The quasiparticle decay rate to two-loop order is given by $\Sigma^{(2)}$ after the analytic continuation: $$\begin{aligned} &\quad \Gamma(\bm{k},\omega) \nonumber\\ &= -g^2 \operatorname{Im}\Sigma^{(2)R}(\bm{k},\omega) \nonumber\\ &= \frac{\pi}{4} g^2 \cosh\left(\frac{\omega}{2T}\right) \nonumber\\ &\quad \times \int_{\bm{p}\bm{q}} \frac{\delta(E_{\bm{p}}+E_{\bm{q}}-E_{\bm{p}+\bm{q}-\bm{k}})}{\cosh\left(\dfrac{E_{\bm{p}}}{2T}\right) \cosh\left(\dfrac{E_{\bm{q}}}{2T}\right) \cosh\left(\dfrac{E_{\bm{p}+\bm{q}-\bm{k}}}{2T}\right)} .\end{aligned}$$ This relation holds for an arbitrary energy dispersion $E_{\bm{k}}$. We extract the temperature dependence by introducing dimensionless quantities in terms of temperature $T$: we define $\tilde{p}_\pm = p_\pm/T^{1/n_\pm}$, so that the energy dispersion satisfies $E_{\tilde{\bm{p}}}=E_{\bm{p}}/T$. Here we are interested in the low-frequency limit with $\omega \ll T$. By substituting $\bm{k}=\bm{0}$ and $\omega=0$, we obtain $$\begin{aligned} \label{eq:decay_rate_0} \Gamma &= \frac{\pi}{4} g^2 T^{1-2\epsilon} \int_{\tilde{\bm{p}}\tilde{\bm{q}}} \frac{\delta(E_{\tilde{\bm{p}}}+E_{\tilde{\bm{q}}}-E_{\tilde{\bm{p}}+\tilde{\bm{q}}})}{\cosh\left(\dfrac{E_{\tilde{\bm{p}}}}{2}\right) \cosh\left(\dfrac{E_{\tilde{\bm{q}}}}{2}\right) \cosh\left(\dfrac{E_{\tilde{\bm{p}}+\tilde{\bm{q}}}}{2}\right)} \nonumber\\ &\propto \bar{g}^2 T^{1-2\epsilon}.\end{aligned}$$ At the nontrivial fixed point $\bar{g} = \bar{g}_2^*$, the quasiparticle decay rate exhibits the temperature dependence $\Gamma\propto T^{1-2\epsilon}$. In the Fermi liquid theory, when the temperature is much larger than the Fermi energy $\epsilon_F \ll T$, the decay rate is proportional to $T^2$. This result relies on the existence of the Fermi surface with finite DOS. On the other hand, the decay rate Eq.  is distinct from the Fermi liquid results, which reflects the divergent DOS at $\mu=0$. Also, we note that the result does not depend on whether the power-law divergent DOS is located at a saddle point or a band edge of the energy dispersion. This is in contrast to the anomalous dimension, which can only be found at a saddle point as we have discussed in Sec. \[sec:anomalous\_dimension\]. Summary and discussions {#sec:discussions} ======================= We have analyzed electron interaction effects near a high-order VHS with a scale-invariant Fermi surface with a power-law divergent DOS. Scale invariance of the system allows an RG analysis to search for fixed points and a scaling analysis of thermodynamic quantities and correlation functions around the fixed points. The one-loop RG analysis finds that electron interaction around high-order VHS offers a fermionic analog of the $\phi^4$ theory. We have identified the two RG fixed points: the Gaussian fixed point and the nontrivial interacting fixed point. The latter is an analog of the Wilson–Fisher fixed point in the $\phi^4$ theory. Like the $\phi^4$ theory, the Gaussian fixed point is unstable and the interacting fixed point is stable in terms of the RG flow of the coupling constant. We performed a controlled RG analysis about the interacting fixed point owing to the smallness of the DOS singularity exponent $\epsilon$. The two-loop RG analysis reveals that the nontrivial interacting fixed point manifests a non-Fermi liquid behavior when the high-order VHS is realized at a saddle point of the energy dispersion. We diagnosed it with a finite anomalous dimension. Importantly, the DOS singularity exponent $\epsilon$ can be any rational number, associated with an analytic energy dispersion. This is in contrast to $\epsilon$ in the $\phi^4$ theory, where $\epsilon$ denotes the dimensionality of the system. The system remains metallic at the interacting fixed point with power-law divergent susceptibilities, while the pairing susceptibility turns to be finite unlike the Gaussian fixed point. We term such a metallic state with power-law divergent susceptibilities but yet without a long-range order as a supermetal. In this sense, the Gaussian fixed point can be viewed as a noninteracting supermetal and the nontrivial fixed point as an interacting supermetal. The scaling exponents of thermodynamic quantities and correlation functions are calculated around the two fixed points. It is worth drawing a comparison between a supermetal and a normal metal. Finite density in a normal metal invalidates scale invariance as it defines an innate length scale. The RG analysis with a closed Fermi surface is commonly referred to as Shankar’s RG [@Shankar]. It requires a judicious choice of RG transformations; Shankar’s RG takes the small cutoff limit to perform the RG analysis with the quartic interaction. As a result, only a very thin shell around the Fermi surface is considered. In addition, we should be aware of the non-analyticity of the charge susceptibility due to the innate length scale. It causes the BCS instability in a higher angular momentum channel at very low temperature through the Kohn–Luttinger mechanism [@Kohn-Luttinger]. The relation between the pairing susceptibility and Shankar’s RG is discussed in Ref. [@Parameswaran]; finite repulsion suppresses the $s$-wave pairing susceptibility. Throughout the analysis, we assume that the DOS singularity exponent $\epsilon$ is a small positive quantity $(\epsilon>0)$. The formula for the DOS at a high-order VHS Eq.  can be used in the limit $\epsilon \to +0$. At a saddle point of an energy dispersion, the DOS has a logarithmic divergence instead of a power-law divergence. It then necessitates a UV cutoff in the momentum integral for the DOS, which is equivalent to subtract the DOS at a large energy: $$\begin{aligned} D(E) - D(s\Lambda) &= D_{s} \left( |E|^{-\epsilon} -\Lambda^{-\epsilon} \right) \nonumber\\ &\to \epsilon D_s \ln \left(\frac{\Lambda}{|E|}\right) \quad (\epsilon\to+0)\end{aligned}$$ with $s=\operatorname{sgn}(E)$. Note that $\epsilon D_s$ is constant in the limit $\epsilon\to+0$, reflecting $1/\epsilon$ singularities in the coefficients $D_\pm$ for $d_\pm>0$. (At a band edge, we instead find a discontinuity without a divergence.) In two dimensions, we obtain a conventional VHS $E_{\bm{k}}=k_x^2-k_y^2$ as a saddle point of an energy dispersion. This is also a scale-invariant system; however, its logarithmically-divergent DOS $D(E)\propto \ln (\Lambda/E)$ raises technical issues in an RG analysis. The non-analyticity of the DOS hinders the RG analysis and the UV cutoff $\Lambda$ cannot be eliminated from RG equations [@Kallin; @Kapustin]. It occurs as a sequel that the low-energy physics is affected by the UV scale $\Lambda$. One of the other theories for a non-Fermi liquid is the Hertz–Millis–Moriya theory for the quantum critical phenomenon in itinerant magnets [@Hertz; @Millis; @Moriya]. It describes the coupling between electrons with a Fermi surface and bosonic fluctuations near the magnetic transition. In this theory, low-energy modes of electrons are integrated out to yield a nonlocal effective action for the bosonic modes, which requires a justification. See reviews e.g. Refs. [@NFL1; @NFL2; @NFL3] for details and various other theories for non-Fermi liquids. Our present analysis does not suffer from those three difficulties related to a closed Fermi surface, a logarithmic DOS, and the coexistence of gapless fermionic and bosonic modes. A scale-invariant Fermi surface with a power-law divergent DOS removes non-analyticity due to the logarithmic DOS and susceptibilities, and a small DOS singularity exponent guarantees that only the short-range interaction is relevant. Our analysis relies on scale invariance of the Fermi surface at a high-order VHS. To realize it in materials, there must be a single high-order VHS in the Brillouin zone at the energy range in focus. Otherwise, an energy or a length scale appears, which violates scale invariance. In reality, materials could have multiple VHS points at the same energy in the Brillouin zone, related by symmetry. Even though each VHS point is separately scale invariant, the distance among the VHS points set the length scale, potentially leading to instabilities. However, for a certain parameter range before an ordering instability takes place, there could exist a scaling region where thermodynamic or transport quantities follow scaling properties. For example, when temperature $T$ and the carrier density $n$ are control parameters, a physical quantity $Q$ follows the scaling relation $$Q(T,n) = T^a \hat{\mathcal{F}}\left( n T^{-(1-\epsilon)} \right)$$ around the Gaussian fixed point, where the exponent $a$ is determined by a dimensional analysis of $Q$ and $\hat{\mathcal{F}}$ is a scaling function. The connection between VHS and superconductivity was proposed in relation to an electronic mechanism for superconductivity [@Hirsch; @Newns], and electronic systems with multiple conventional VHS (i.e., logarithmically divergent DOS) in the Brillouin zone are first studied in the context of copper oxide superconductors [@Dzyaloshinskii; @Schulz; @Lederer; @Furukawa; @LeHur; @Raghu]. There are two VHS points at the high symmetry points $(\pi,0)$ and $(0,\pi)$. As the spectral weight is peaked at those points, the VHS points are also called hot spots. The large DOS and Fermi surface nesting at the wave vector that connects the hot spots invoke $d$-wave superconductivity or a density-wave state. Similar analyses are applied for doped monolayer graphene [@Nandkishore; @Gonzalez] and twisted bilayer graphene [@Isobe]. The existence of multiple VHS requires a set of coupling constants to describe interactions as scattering processes among the hot spots. Also, the overlap of the Fermi surfaces around the hot spots is quantified by so-called nesting parameters in the particle-hole and particle-particle channels. The nesting parameters control the RG flow of the coupling constants, and hence determine the resultant instability if exists. When there are multiple high-order VHS points (or high-order and conventional VHS points) in the Brillouin zone, the formal structure of the coupled RG equations for the coupling constants resembles that for the conventional VHS case. However, we could find the differences between the two cases. First, the coupling constants have finite scaling dimensions for high-order VHS due to the power-law DOS. Recall that for the contact interaction, the product of the coupling constant and the DOS becomes dimensionless. It results in the tree-level scaling of the coupling constants, and the RG equations acquire linear terms with respect to the coupling constants. As we have seen, the coexistence of the linear and quadratic terms give rise to fixed points in the weak coupling region. An RG flow around those fixed points would describe the weak-coupling behavior. Nevertheless, quadratic terms would be dominant in identifying instabilities, where the coupling constants becomes large; see also Ref. [@multicritical1]. Another difference appears in nesting parameters. With conventional VHS, Fermi surface nesting yields an additional logarithm to that of the DOS, including the double logarithm in the BCS susceptibility. However, a high-order VHS does not alter the analytic properties but only the prefactors of susceptibilities are affected; cf. Appendix \[sec:susceptibilities\] for the comparison between the particle-hole and particle-particle susceptibilities at a high-order VHS. It relaxes nesting conditions for instabilities with high-order VHS. In addition, from a theoretical point of view, the absence of double logarithms retrieves the analytic tractability of RG equations. We acknowledge helpful discussions with A. V. Chubukov, E. Fradkin, S.-S. Lee, and especially E. Berg. This work was supported by DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0018945. LF was supported in part by a Simons Investigator Award from the Simons Foundation. One-loop RG analysis at finite temperature {#sec:finite-temperature} ========================================== We work on the RG equations at finite temperature $T \neq 0$. Temperature has a dimension of energy, so that it is one of relevant perturbations. Here we consider the one-loop RG equations from the energy-shell RG analysis. At finite temperature, the one-loop corrections $\Sigma_\text{H}$, $\Pi_\text{pp}$, and $\Pi_\text{ph}$ should be evaluated with $T\neq 0$. To order $l$, we obtain $$\begin{gathered} \Sigma_\text{H} = T\sum_{\omega_n} \int_{\bm{k}}^> G_0(\bm{k},\omega_n) \simeq l c_\text{H}(\bar{T}) \Lambda D(\Lambda) , \\ \Pi_{\text{pp}} = T\sum_{\omega_n} \int_{\bm{k}}^> G_0(\bm{k},\omega_n) G_0(-\bm{k},-\omega_n) \simeq l c_\text{pp}(\bar{T}) D(\Lambda) , \\ \Pi_\text{ph} = T\sum_{\omega_n} \int_{\bm{k}}^> G_0(\bm{k},\omega_n) G_0(\bm{k},\omega_n) \simeq l c_\text{ph}(\bar{T}) D(\Lambda) ,\end{gathered}$$ where we introduce the dimensionless temperature $\bar{T}=T/\Lambda$. Again, all quantities are evaluated at zero external frequency and momentum, so that the results depend only on the DOS. The temperature-dependent dimensionless coefficients $c_{\mu}$, $c_\text{pp}$, and $c_\text{ph}$ are $$\begin{gathered} \label{eq:c_mu} c_\text{H}(\bar{T}) = \frac{1}{2} \left( 1-\frac{D_-}{D_+} \right) \tanh\left(\frac{1}{2\bar{T}}\right), \\ \label{eq:c_pp} c_\text{pp}(\bar{T}) = \frac{1}{2} \left( 1+\frac{D_-}{D_+} \right) \tanh\left(\frac{1}{2\bar{T}}\right), \\ \label{eq:c_ph} c_\text{ph}(\bar{T}) = \frac{1}{2\bar{T}} \left( 1+\frac{D_-}{D_+} \right) \frac{1}{2\cosh^2\left(\dfrac{1}{2\bar{T}}\right)}.\end{gathered}$$ Unlike the calculation at $T=0$, $\Pi_\text{ph}$ becomes finite for $T\neq 0$, while the correction to the field or the energy dispersion remains absent to one-loop order. In the analysis at zero temperature, we rescale the frequency in Eq. . At finite temperature, rescaling of the Matsubara frequency leads to rescaling of temperature [@Millis]. Temperature obeys the same scaling relation as that for the frequency: $T' = bT$. Including the temperature-dependent factors $c_{\mu}(\bar{T})$, $c_\text{pp}(\bar{T})$, and $c_\text{ph}(\bar{T})$, we obtain the changes of the parameters at an RG step as $$\begin{gathered} T' = b T, \\ \mu' \simeq b [\mu + lc_\text{H}(\bar{T})gD(\Lambda) - lc_\text{ph}(\bar{T})\mu gD(\Lambda) ], \\ h' = b h, \\ \Delta' \simeq b [\Delta - lc_\text{pp}(\bar{T})gD(\Lambda)], \\ g' \simeq b^{\epsilon} \{ g - l [c_\text{pp}(\bar{T}) + c_\text{ph}(\bar{T})] g^2 D(\Lambda) \}.\end{gathered}$$ Then, we reach the RG equations $$\begin{gathered} \frac{d\bar{T}}{dl} = \bar{T}, \\ \frac{d\bar{\mu}}{dl} = [ 1- c_\text{ph} (\bar{T})\bar{g} ] \bar{\mu} + c_\mu(\bar{T}) \bar{g}, \\ \frac{d\bar{h}}{dl} = \bar{h}, \\ \frac{d\bar{\Delta}}{dl} = [ 1- c_\text{pp}(\bar{T})\bar{g} ] \bar{\Delta}, \\ \frac{d\bar{g}}{dl} = \epsilon \bar{g} - \left[ c_\text{pp}(\bar{T})+c_\text{ph}(\bar{T}) \right] \bar{g}^2.\end{gathered}$$ Since temperature $T$ is relevant and its fixed point is located at $T=0$, the fixed points of the parameters are found at $T=0$, as we discussed in the main part. Brief review of the field theory approach to RG analyses {#sec:field-theory_review} ======================================================== Here we describe a field theory approach to RG equations, in light of the Wilsonian approach. We derive the Callan–Symanzik equation and the beta functions, which show how scale-dependent parameters affect physical quantities. The following descriptions are partly motivated by the references [@Zinn-Justin; @Peskin; @Amit; @Nair]. For the sake of clarity, we consider a theory with a scalar field $\phi$ and a set of dimensionless parameters $\{\bar{g}_\nu\}$, where we write the action as $S[\phi;\bar{g}]$. The partition function is given by $$\mathcal{Z} = \int D\phi e^{-S[\phi;\bar{g}]}.$$ When the model suffers from UV divergences, i.e., perturbative loop corrections have UV divergences, we need to cure them to extract meaningful information. Those UV divergences can be regularized by removing UV modes from the model. To this end, we decompose the field $\phi$ depending on the energy range to which they contribute: $\phi_{\Lambda'}^{\Lambda}$ accounts for the energy between $\Lambda'$ and $\Lambda$. Then, we redefine the partition function as $$\mathcal{Z}_{\Lambda_0}[\bar{g}_{0}] = \int D\phi_0^{\Lambda_0} e^{-S_{\Lambda_0}[\phi_0^{\Lambda_0};\bar{g}_{0}]}.$$ It does not obviously have a UV divergence because no UV modes are included. Here the energy scale $\Lambda_0$ works as a UV energy cutoff. (This is equivalent to impose the effective action to be finite at $\Lambda_0$ without a cutoff. From this viewpoint, counterterms are introduced to cure UV divergences. See below for the effective action.) The scale $\Lambda_0$ is an arbitrary energy scale to regularize UV divergences. The next thing we should check is how a change of the characteristic energy scale affects the theory. To see it, we define the effective action at an energy scale $\Lambda(<\Lambda_0)$ as $$\begin{aligned} S_\Lambda^\text{eff}[Z_\Lambda^{1/2} \phi_0^\Lambda;\bar{g}(\Lambda)] = -\ln \left[ \int D\phi_\Lambda^{\Lambda_0} e^{-S_{\Lambda_0}[\phi_0^\Lambda + \phi_\Lambda^{\Lambda_0};\bar{g}_{0}]} \right].\end{aligned}$$ We require the effective action $S^\text{eff}$ to have the same form as the action $S$. Then, the partition function can be written as $$\begin{aligned} \mathcal{Z}_{\Lambda_0} [\bar{g}_{0}] &= \int D\phi_0^\Lambda D\phi_\Lambda^{\Lambda_0} e^{-S_{\Lambda_0}[\phi_0^\Lambda + \phi_\Lambda^{\Lambda_0};\bar{g}_{0}]} \nonumber\\ &= \int D\phi_0^{\Lambda} e^{-S^\text{eff}_\Lambda [Z_\Lambda^{1/2} \phi_0^\Lambda;\bar{g}(\Lambda)]} \nonumber\\ &\equiv \mathcal{Z}_\Lambda [Z_\Lambda^{1/2} \phi_0^\Lambda;\bar{g}(\Lambda)].\end{aligned}$$ This is simply rewriting of the partition function with the effective action at the scale $\Lambda$. We relate the partition functions at different scales to find $$\begin{aligned} \mathcal{Z}_\Lambda [Z_\Lambda^{1/2} \phi_0^\Lambda;\bar{g}(\Lambda)] = \mathcal{Z}_{\Lambda'} [Z_{\Lambda'}^{1/2} \phi_0^{\Lambda'};\bar{g}(\Lambda')].\end{aligned}$$ This equality tells us that we have the same partition function defined at different energy scales $\Lambda$ and $\Lambda'$, together with the changes of the weight $Z$ and the parameters $\bar{g}_a$. We then aim to calculate the $N$-point correlation function with the cutoff $\Lambda$ and the parameters $\bar{g}_a(\Lambda)$: $$\begin{aligned} &\quad \langle \phi_0^{\Lambda}(k_1) \cdots \phi_0^{\Lambda}(k_N) \rangle_{\Lambda;\bar{g}(\Lambda)} \nonumber\\ &= \frac{1}{\mathcal{Z}_{\Lambda}[\phi_0^\Lambda;\bar{g}(\Lambda)]} \int D\phi_0^{\Lambda} \phi_0^{\Lambda}(k_1) \cdots \phi_0^{\Lambda}(k_N) e^{-S^\text{eff}_{\Lambda}[\phi_0^{\Lambda};\bar{g}(\Lambda)]}.\end{aligned}$$ When all momenta $k_a$ correspond to energies below $\Lambda$ and $\Lambda'$, we find $\phi_0^{\Lambda}(k_a) = \phi_0^{\Lambda'}(k_a)$, which enables us to relate the $N$-point correlation functions at different scales as $$\begin{aligned} &\quad Z_{\Lambda}^{-N/2} \langle \phi_0^{\Lambda}(k_1) \cdots \phi_0^{\Lambda}(k_N) \rangle_{\Lambda;\bar{g}(\Lambda)} \nonumber\\ &= Z_{\Lambda'}^{-N/2} \langle \phi_0^{\Lambda'}(k_1) \cdots \phi_0^{\Lambda'}(k_N) \rangle_{\Lambda';\bar{g}(\Lambda')}.\end{aligned}$$ We now write this relation with the connected $N$-point correlation function $G^{(N)}$: $$\begin{aligned} Z_{\Lambda}^{-N/2} G_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{k_a\}) = Z_{\Lambda'}^{-N/2} G_{\Lambda';\bar{g}(\Lambda')}^{(N)}(\{k_a\}).\end{aligned}$$ The scale dependence of this equality can be written in the form of a differential equation: $$\begin{aligned} \left[ \Lambda\frac{\partial}{\partial\Lambda} -\beta_\nu(\bar{g})\frac{\partial}{\partial\bar{g}_\nu} + \frac{N}{2}\gamma(\bar{g}) \right] G_{\Lambda;\bar{g}_\nu(\Lambda)}^{(N)}(\{k_a\}) = 0.\end{aligned}$$ Note that the repeated index is summed over. This equation is called the Callan–Symanzik equation [@Callan; @Symanzik1; @Symanzik2] for the connected $N$-point correlation function $G^{(N)}$ with the beta functions $\beta_\nu$ and the field renormalization $\gamma$ defined by $$\begin{gathered} \beta_\nu(\bar{g}) = -\left(\Lambda\frac{\partial\bar{g}_\nu}{\partial\Lambda}\right)_{\bar{g}_{0}}, \\ \gamma(\bar{g}) = -\left(\Lambda\frac{\partial}{\partial\Lambda} \ln Z_{\Lambda}\right)_{\bar{g}_{0}}.\end{gathered}$$ The correlation functions are obtained from perturbative calculations. Actually, it is more straightforward to obtain the one-particle irreducible $N$-point function $\Gamma^{(N)}$ instead of the $N$-point correlation function $G^{(N)}$. When a model involves a quartic interaction $\phi^4$ without a cubic term $\phi^3$, $\Gamma^{(2)}$ and $\Gamma^{(4)}$ are given by $$\begin{gathered} \Gamma^{(2)}(k) = [G^{(2)}(k)]^{-1}, \\ \Gamma^{(4)}(k_1,k_2,k_3,k_4) = \frac{G^{(4)}(k_1,k_2,k_3,k_4)}{G^{(2)}(k_1) G^{(2)}(k_2) G^{(2)}(k_3) G^{(2)}(k_4)}.\end{gathered}$$ For the definition of $\Gamma^{(N)}$ from the effective action, see the references [@Zinn-Justin; @Peskin; @Amit; @Nair]. Roughly speaking, $\Gamma^{(N)}$ corresponds to the coefficient of the $\phi^N$ term in the effective action. Again, using the fact that $\phi_0^{\Lambda}(k_a) = \phi_0^{\Lambda'}(k_a)$ holds when the energy corresponding to the momentum $k_a$ is smaller than $\Lambda$ and $\Lambda'$, we find the relation $$\begin{aligned} \label{eq:Gamma_scale} Z_{\Lambda}^{N/2} \Gamma_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{k_a\}) = Z_{\Lambda'}^{N/2} \Gamma_{\Lambda';\bar{g}(\Lambda')}^{(N)}(\{k_a\}).\end{aligned}$$ It results in the Callan–Symanzik equation for $\Gamma^{(N)}$: $$\begin{aligned} \left[ \Lambda\frac{\partial}{\partial\Lambda} -\beta_\nu(\bar{g})\frac{\partial}{\partial\bar{g}_\nu} - \frac{N}{2}\gamma(\bar{g}) \right] \Gamma_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{k_a\}) = 0.\end{aligned}$$ So far, we have compared $G^{(N)}$ or $\Gamma^{(N)}$ at different cutoffs $\Lambda$ and $\Lambda'$, so that the dependence on $\Lambda$ is explicit. However, this comparison is still theoretical; our aim is to compare the two theories with the same cutoff $\Lambda$. For this sake, we rescale the coordinate to change the cutoff. Suppose we have the scaling relations $$\begin{gathered} \Lambda' = b \Lambda, \\ k'_j = b^{d_{k_j}} k_j, \\ \phi_0^{\Lambda'}(k') = b^{d_\phi} \phi_0^{\Lambda}(k),\end{gathered}$$ where $k_j$ is a component of $k=(\bm{k},\omega)$ and $d_{\mathcal{O}}$ denotes the scaling (energy) dimension of $\mathcal{O}$. Those relations lead to $$\begin{aligned} \label{eq:Gamma_scale_2} \Gamma_{\Lambda';\bar{g}(\Lambda')}^{(N)}(\{k_a'\}) = b^{d_{\Gamma^{(N)}}} \Gamma_{\Lambda;\bar{g}(\Lambda')}^{(N)}(\{k_a\}),\end{aligned}$$ Rescaling the momentum forces the cutoff $\Lambda'$ back to $\Lambda$ with the overall factor $b^{d_{\Gamma^{(N)}}}$, but this process does not alter the dimensionless parameters $\bar{g}_\nu$. From Eqs.  and , we find the relation $$\begin{aligned} \label{eq:scale_Wilsonian} \Gamma_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{ b^{-d_{k_j}} k_{a,j} \}) = \frac{Z_{\Lambda/b;\bar{g}(\Lambda/b)}^{N/2}}{Z_{\Lambda;\bar{g}(\Lambda)}^{N/2}} b^{-d_{\Gamma^{(N)}}} \Gamma_{\Lambda;\bar{g}(\Lambda/b)}^{(N)}(\{ k_{a,j} \}).\end{aligned}$$ Importantly, this equation compares the $N$-point function $\Gamma^{(N)}$ with the same cutoff $\Lambda$ but at different momenta and parameters. The interpretation of the scale-dependent parameters can be found from this equation: The parameters $\bar{g}_\nu(\Lambda/b)$ describe the physics at scale $k_j/b^{d_{k_j}}$. This procedure actually illustrates rescaling in the Wilsonian RG scheme. We rewrite Eq.  as $$\begin{aligned} \Gamma_{\Lambda;\bar{g}(\Lambda/b)}^{(N)}(\{ k_{a,j} \}) = \frac{Z_{\Lambda;\bar{g}(\Lambda)}^{N/2}}{Z_{\Lambda/b;\bar{g}(\Lambda/b)}^{N/2}} b^{d_{\Gamma^{(N)}}} \Gamma_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{ b^{-d_{k_j}} k_{a,j} \}),\end{aligned}$$ which can be interpreted in the following way. We integrate out fluctuations between the range of $(\Lambda/b,\Lambda]$ (corresponding to $Z_{\Lambda;\bar{g}(\Lambda)}^{N/2}/Z_{\Lambda/b;\bar{g}(\Lambda/b)}^{N/2}$) and rescale the field and parameters (multiplying the factor $b^{d_{\Gamma^{(N)}}}$) to obtain the new action with a different coupling constant but with the same cutoff $\Lambda$. We can also write down the Callan-Symanzik equation to describe the momentum dependence, instead of the cutoff $\Lambda$. We differentiate Eq.  with respect to $b$ and then set $b=1$ to obtain $$\begin{aligned} \left[ d_{k_j} k_{a,j} \frac{\partial}{\partial k_{a,j}} + \beta_\nu(\bar{g})\frac{\partial}{\partial\bar{g}_\nu} - d_{\Gamma^{(N)}} + \frac{N}{2}\gamma(\bar{g}) \right] \Gamma_{\Lambda;\bar{g}}^{(N)}(\{k_{a,j}\}) = 0.\end{aligned}$$ Equivalently, we can introduce a factor $b$ to scale all momenta $b^{d_{k_j}} k_{a,j}$ at once, so the Callan–Symanzik equation becomes $$\begin{aligned} \left[ b\frac{\partial}{\partial b} + \beta_\nu(\bar{g})\frac{\partial}{\partial\bar{g}_\nu} - d_{\Gamma^{(N)}} + \frac{N}{2}\gamma(\bar{g}) \right] \Gamma_{\Lambda;\bar{g}}^{(N)}(\{ b^{d_{k_j}} k_{a,j} \}) = 0.\end{aligned}$$ The Callan–Symanzik equation can be solved by the method of characteristics. With a parameter $l$, we obtain the differential equations $$\begin{gathered} \frac{db}{dl} = b, \\ \frac{d\bar{g}_\nu}{dl} = \beta_\nu(\bar{g}), \\ \frac{d\Gamma_{\Lambda;\bar{g}}^{(N)}(\{ b^{d_{k_j}}k_{a,j} \})}{dl} = \left[ d_{\Gamma^{(N)}} -\frac{N}{2}\gamma(\bar{g}) \right] \Gamma_{\Lambda;\bar{g}}^{(N)}(\{ b^{d_{k_j}}k_{a,j} \}).\end{gathered}$$ The solution to the first equation is straightforward: $$b(l) = e^l,$$ with the initial condition $b(0)=1$. We write the solution to the second formally as $$\bar{g}_\nu(l) = \int_0^l dl' \beta_\nu(\bar{g}(l')).$$ Also, the formal solution to the third equation is $$\begin{aligned} &\quad \Gamma_{\Lambda;\bar{g}(l)}^{(N)}(\{ e^{d_{k_j} l} k_{a,j} \}) \nonumber\\ &= e^{d_{\Gamma^{(N)}}l} \Gamma_{\Lambda;\bar{g}(0)}^{(N)}(\{ k_{a,j} \}) \exp \left[ -\frac{N}{2}\int_{0}^{l} dl' \gamma(\bar{g}(l')) \right].\end{aligned}$$ Changing $k_{a,j}$ to $e^{-d_{k_j} l} k_{a,j}$, we can express the $N$-point function as $$\begin{aligned} &\quad \Gamma_{\Lambda;\bar{g}(0)}^{(N)}(\{ e^{-d_{k_j} l} k_{a,j} \}) \nonumber\\ &= e^{-d_{\Gamma^{(N)}}l} \Gamma_{\Lambda;\bar{g}(l)}^{(N)}(\{ k_{a,j} \}) \exp \left[ \frac{N}{2}\int_{0}^{l} dl' \gamma(\bar{g}(l')) \right].\end{aligned}$$ This equation describes the parameters $\bar{g}_\nu$ effectively behaves as if they are $\bar{g}_\nu(l)$ with small momenta $e^{-d_{k_j} l} k_{a,j}$ $(l>0)$. Before concluding the section, we note that $\gamma$ corresponds to the anomalous dimension. Let us consider the two-point function $\Gamma^{(2)}$, which corresponds to the inverse of the two-point correlation function. At a fixed point, the beta function vanishes, and hence both $\bar{g}$ and $\gamma$ are constant; we set $\bar{g}^*$ and $\eta$, respectively. Here we consider the frequency dependence, so we choose $k_j = \omega_0$. Since $d_\omega=1$, we obtain $$\begin{aligned} \Gamma_{\Lambda;\bar{g}^*}^{(2)}(e^{-l} \omega_0) = e^{-d_{\Gamma^{(2)}} l} \Gamma_{\Lambda;\bar{g}^*}^{(2)}(\omega_0) e^{\eta l}.\end{aligned}$$ As $l$ is arbitrary, we set $l=\ln(\omega_0/\omega)$ to find $$\begin{aligned} \Gamma_{\Lambda;\bar{g}^*}^{(2)}(\omega) = \omega^{d_{\Gamma^{(2)}}-\eta} \omega_0^\eta \Gamma_{\Lambda;\bar{g}^*}^{(2)}(\omega_0) \propto \omega^{d_{\Gamma^{(2)}}-\eta}.\end{aligned}$$ A naive power counting predicts $\Gamma^{(2)}\propto \omega^{d_{\Gamma^{(2)}}}$, but actually it behaves differently with the exponent $d_{\Gamma^{(2)}}-\eta$. The deviation $\eta$ corresponds to the anomalous dimension. Two-loop self-energy for the quasiparticle lifetime {#sec:two-loop_lifetime} =================================================== The quasiparticle damping is captured by a finite imaginary part of the self-energy $\Sigma$. In a series of perturbative expansions, the lowest order correction appears at second order, which is diagrammatically shown in Fig. \[fig:loop\](d). Its contribution is given in Eq.  and here we calculate it explicitly. We first calculate the Matsubara summations. With the standard procedure, we can convert the summations into the contour integrals on the complex plane, to obtain $$\begin{aligned} &\quad \Sigma^{(2)}(\bm{k},\omega_n) \nonumber\\ &= -\frac{1}{(2\pi)^2} \int_{\bm{p}\bm{q}\bm{l}} (2\pi)^d \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}) \int_{-\infty}^{\infty} d\omega_p d\omega_l \nonumber\\ &\quad \times \bigg\{ G_0(\bm{p},\omega_n-i\omega_p) [ \operatorname{Im}G_0^R(\bm{q},\omega_l) \operatorname{Im}G_0^R(\bm{l},\omega_p+\omega_l) -\operatorname{Im}G_0^R(\bm{q},\omega_l-\omega_p) \operatorname{Im}G_0^R(\bm{l},\omega_l) ] \coth\left(\frac{\omega_p}{2T}\right) \tanh\left(\frac{\omega_l}{2T}\right) \nonumber\\ &\qquad + \operatorname{Im}G_0^R(\bm{p},\omega_p) [ G_0(\bm{q},\omega_n+i\omega_p-i\omega_l) \operatorname{Im}G_0^R(\bm{l},\omega_l) + \operatorname{Im}G_0^R(\bm{q},\omega_l) \cdot G_0(\bm{l},-\omega_n-i\omega_p-i\omega_l) ] \tanh\left(\frac{\omega_p}{2T}\right) \tanh\left(\frac{\omega_l}{2T}\right) \bigg\},\end{aligned}$$ We define the noninteracting retarded (advanced) Green’s function $G_0^R$ $(G_0^A)$ as $$G_0^{R/A}(\bm{k},\omega) = \frac{1}{\omega - E_{\bm{k}} \pm i\delta}.$$ The retarded function is obtained by the analytic continuation $i\omega_n = \omega + i\delta$. We insert $\int d\omega_q \delta(\omega_p+\omega_q-\omega_l-\omega)$ to write the self-energy $\Sigma^{(2)R}$ in a symmetric form: $$\begin{aligned} &\quad \Sigma^{(2)R}(\bm{k},\omega) \nonumber\\ &= -\frac{1}{(2\pi)^2} \int_{\bm{p}\bm{q}\bm{l}} (2\pi)^d \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}) \int_{-\infty}^{\infty} d\omega_p d\omega_q d\omega_l \delta(\omega_p+\omega_q-\omega_l-\omega) \nonumber\\ &\quad\times \bigg\{ G_0^R(\bm{p},\omega_p) \operatorname{Im}G_0^R(\bm{q},\omega_q) \operatorname{Im}G_0^R(\bm{l},\omega_l) \coth\left(\frac{\omega_l-\omega_q}{2T}\right) \left[ \tanh\left(\frac{\omega_q}{2T}\right) - \tanh\left(\frac{\omega_l}{2T}\right) \right] \nonumber\\ &\qquad + \operatorname{Im}G_0^R(\bm{p},\omega_p) \cdot G_0^R(\bm{q},\omega_q) \operatorname{Im}G_0^R(\bm{l},\omega_l) \tanh\left(\frac{\omega_p}{2T}\right) \tanh\left(\frac{\omega_l}{2T}\right) \nonumber\\ &\qquad + \operatorname{Im}G_0^R(\bm{p},\omega_p) \operatorname{Im}G_0^R(\bm{q},\omega_q) \cdot G_0^A(\bm{l},\omega_l) \tanh\left(\frac{\omega_p}{2T}\right) \tanh\left(\frac{\omega_q}{2T}\right) \bigg\}.\end{aligned}$$ Now we take the imaginary part to obtain $$\begin{aligned} &\quad \operatorname{Im}\Sigma^{(2)R}(\bm{k},\omega) \nonumber\\ &= \frac{1}{(2\pi)^2} \cosh\left(\frac{\omega}{2T}\right) \int_{\bm{p}\bm{q}\bm{l}} (2\pi)^d \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}) \int_{-\infty}^{\infty} d\omega_p d\omega_q d\omega_l \delta(\omega_p+\omega_q-\omega_l-\omega) \nonumber\\ &\quad\times \operatorname{Im}G_0^R(\bm{p},\omega_p) \operatorname{Im}G_0^R(\bm{q},\omega_q) \operatorname{Im}G_0^R(\bm{l},\omega_l) \frac{1}{\cosh\left(\dfrac{\omega_p}{2T}\right) \cosh\left(\dfrac{\omega_q}{2T}\right) \cosh\left(\dfrac{\omega_l}{2T}\right)} \nonumber\\ &= -\frac{\pi}{4} \cosh\left(\frac{\omega}{2T}\right) \int_{\bm{p}\bm{q}} \frac{\delta(\omega-E_{\bm{p}}-E_{\bm{q}}+E_{\bm{p}+\bm{q}-\bm{k}})}{\cosh\left(\dfrac{E_{\bm{p}}}{2T}\right) \cosh\left(\dfrac{E_{\bm{q}}}{2T}\right) \cosh\left(\dfrac{E_{\bm{p}+\bm{q}-\bm{k}}}{2T}\right)} ,\end{aligned}$$ where we use the relation $\operatorname{Im}G_0^R(\bm{k},\omega) = -\operatorname{Im}G_0^A(\bm{k},\omega) = -\pi \delta(\omega-E_{\bm{k}})$. Susceptibilities at a high-order VHS {#sec:susceptibilities} ==================================== A divergent DOS $D(E)$ accompanies divergent susceptibilities. Here we consider the noninteracting susceptibilities in the particle-hole and particle-particle channels, $\chi_\text{ph}$ and $\chi_\text{pp}$, respectively: $$\begin{gathered} \chi_\text{ph} (\bm{q},\omega;T) = \int_{\bm{p}} \frac{f(\xi_{\bm{p}+\bm{q}})-f(\xi_{\bm{p}})}{\omega+i\delta-\xi_{\bm{p}+\bm{q}}+\xi_{\bm{p}}}, \\ \chi_\text{pp} (\bm{q},\omega;T) = \int_{\bm{p}} \frac{f(\xi_{\bm{p}+\bm{q}})-f(-\xi_{-\bm{p}})}{\omega+i\delta-\xi_{\bm{p}+\bm{q}}-\xi_{\bm{p}}},\end{gathered}$$ where $f(\xi)=(e^{\xi/T}+1)^{-1}$ is the Fermi–Dirac distribution and $\xi_{\bm{p}} = E_{\bm{p}}-\mu$. In the following, we focus on the static susceptibilities $(\omega=0)$. At $\mu = 0$, the noninteracting susceptibilities follow the scaling relations for momentum $\bm{q}$ and temperature $T$, described by $$\begin{gathered} \chi_\text{ph}(\bm{q},\omega=0;T) = \kappa^{-\epsilon} \hat{\chi}_\text{ph} \left( \frac{q_+^{n_+}}{\kappa}, \frac{q_-^{n_-}}{\kappa}, \frac{T}{\kappa} \right), \\ \chi_\text{pp}(\bm{q},\omega=0;T) = \kappa^{-\epsilon} \hat{\chi}_\text{pp} \left( \frac{q_+^{n_+}}{\kappa}, \frac{q_-^{n_-}}{\kappa}, \frac{T}{\kappa} \right),\end{gathered}$$ where $\hat{\chi}_\text{ph}$ and $\hat{\chi}_\text{pp}$ are the scaling functions. Those scaling behaviors of the noninteracting susceptibilities lead to the approximate relations $$\begin{gathered} \chi_\text{ph}(\bm{q},\omega=0;T) \sim \max(T^{-\epsilon}, q_+^{-\epsilon n_+}, q_-^{-\epsilon n_-}), \\ \chi_\text{pp}(\bm{q},\omega=0;T) \sim \max(T^{-\epsilon}, q_+^{-\epsilon n_+}, q_-^{-\epsilon n_-}).\end{gathered}$$ Although the explicit forms depend on the specific form of the energy dispersion, the temperature dependence reflects only the DOS Eq. : $$\begin{gathered} \label{eq:diff_ph} \chi_\text{ph}(T) = T^{-\epsilon} (D_++D_-) (2^{1+\epsilon}-1) \Gamma(1-\epsilon) [-\zeta(-\epsilon)], \\ \label{eq:diff_pp} \chi_\text{pp}(T) = \frac{1}{\epsilon} T^{-\epsilon} (D_++D_-) (2^{1+\epsilon}-1) \Gamma(1-\epsilon) [-\zeta(-\epsilon)],\end{gathered}$$ where $\zeta(s)$ is the Riemann zeta function. We see that $\chi_\text{pp}$ is larger than $\chi_\text{ph}$ by the numerical factor $1/\epsilon$. For the case of a high-order VHS with a power-law divergent DOS, differences between $\chi_\text{ph}$ and $\chi_\text{pp}$ appear as the prefactors of the terms $T^{-\epsilon}$, $q_+^{-\epsilon n_+}$, $q_-^{-\epsilon n_-}$ as we have shown in Eqs.  and . This is in contrast to the conventional VHS with a logarithmically divergent DOS; while $\chi_\text{ph}$ has a logarithmic divergence reflecting the DOS, $\chi_\text{pp}$ exhibits a double logarithmic divergence. The additional logarithm in the BCS channel appears in the presence of time-reversal symmetry, which can be regarded as the Fermi surface nesting with itself. [99]{} L. Van Hove, *The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal*, Phys. Rev. **89**, 1189 (1953). N. F. Q. Yuan, H. Isobe, and L. Fu, *Magic of high order van Hove singularity*, arXiv:1901.05432. R. R. Biswas, L. Fu, C. R. Laumann, and S. Sachdev, *SU(2)-invariant spin liquids on the triangular lattice with spinful Majorana excitations*, Phys. Rev. B **83**, 245131 (2011). A. Shtyk, G. Goldstein, and C. Chamon, *Electrons at the monkey saddle: A multicritical Lifshitz point*, Phys. Rev. B **95**, 035137 (2017). D. V. Efremov, A. Shtyk, A. W. Rost, C. Chamon, A. P. Mackenzie, and J. J. Betouras, *Multicritical Fermi surface topological transitions*, arXiv:1810.13392. A. Kerelsky, L. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, C. Dean, A. Rubio, and A. N. Pasupathy, *Magic Angle Spectroscopy*, arXiv:1812.08776. K. G. Wilson and M. E. Fisher, *Critical Exponents in 3.99 Dimensions*, Phys. Rev. Lett. **28**, 240 (1972). K. Wilson and J. Kogut, *The renormalization group and the $\epsilon$ expansion*, Phys. Rep. **12**, 75–199 (1974). B. Burganov, C. Adamo, A. Mulder, M. Uchida, P. D. C. King, J. W. Harter, D. E. Shai, A. S. Gibbs, A. P. Mackenzie, R. Uecker, M. Bruetzam, M. R. Beasley, C. J. Fennie, D. G. Schlom, and K. M. Shen, *Strain Control of Fermiology and Many-Body Interactions in Two-Dimensional Ruthenates*, Phys. Rev. Lett. **116**, 197003 (2016). A. Steppke, L. Zhao, M. E. Barber, T. Scaffidi, F. Jerzembeck, H. Rosner, A. S. Gibbs, Y. Maeno, S. H. Simon, A. P. Mackenzie, C. W. Hicks, *Strong peak in $T_c$ of Sr$_2$RuO$_4$ under uniaxial pressure*, Science **355**, eaaf9398 (2017). Y. Luo, P. Guzman, A. P. Dioguardi, A. Pustogow, S. M. Thomas, F. Ronning, N. Kikugawa, D. Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks, E. D. Bauer, I. I. Mazin, and S. E. Brown, *Normal state $^{17}$O NMR studies of Sr$_2$RuO$_4$ under uniaxial stress*, arXiv:1810.01209. R. Shankar, *Renormalization-group approach to interacting fermions*, Rev. Mod. Phys. **66**, 129 (1994). W. Kohn and J. M. Luttinger, *New Mechanism for Superconductivity*, Phys. Rev. Lett. **15**, 524 (1965). J. C. Ward, Phys. Rev. **78**, 182 (1950) Y. Takahashi, Nuovo Cimento **6**, 371 (1957). A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, *Methods Of Quantum Field Theory In Statistical Physics* (Dover, New York, 1975). S. A. Parameswaran, R. Shankar, and S. L. Sondhi, *Renormalization group and the superconducting susceptibility of a Fermi liquid*, Phys. Rev. B **82**, 195104 (2010). S. Ghamari, S.-S. Lee, and C. Kallin, *Renormalization group analysis of a neck-narrowing Lifshitz transition in the presence of weak short-range interactions in two dimensions*, Phys. Rev. B **92**, 085112 (2015). A. Kapustin, T. McKinney, and I. Z. Rothstein, *Wilsonian effective field theory of two-dimensional Van Hove singularities*, Phys. Rev. B **98**, 035122 (2018). J. A. Hertz, *Quantum critical phenomena*, Phys. Rev. B **14**, 1165–1184 (1976). A. J. Millis, *Effect of a nonzero temperature on quantum critical points in itinerant fermion systems*, Phys. Rev. B **48**, 7183 (1993). T. Moriya, *Spin Fluctuations in Itinerant Electron Magnetism* (Springer, Berlin, 1985). G. R. Stewart, *Non-Fermi-liquid behavior in $d$- and $f$-electron metals*, Rev. Mod. Phys. **73**, 797 (2001). H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, *Fermi-liquid instabilities at magnetic quantum phase transitions*, Rev. Mod. Phys. **79**, 1015 (2007). S.-S. Lee, *Recent Developments in Non-Fermi Liquid Theory*, Annu. Rev. Condens. Matter Phys. **9**, 227 (2018). J. E. Hirsch and D. J. Scalapino, *Enhanced Superconductivity in Quasi Two-Dimensional Systems*, Phys. Rev. Lett. **56**, 2732 (1986). D. M. Newns, H. R. Krishnamurthy, P. C. Pattnaik, C. C. Tsuei, and C. L. Kane, *Saddle-point pairing: An electronic mechanism for superconductivity*, Phys. Rev. Lett. **69**, 1264 (1992). I. E. Dzyaloshinskii, *Superconducting transitions due to Van Hove singularities in the electron spectrum*, Sov. Phys. JETP **66**, 848 (1987). H. J. Schulz, *Superconductivity and Antiferromagnetism in the Two-Dimensional Hubbard Model: Scaling Theory*, Europhys. Lett. **4**, 609 (1987). P. Lederer, G. Montambaux, and D. Poilblanc, *Antiferromagnetism and superconductivity in a quasi two-dimensional electron gas. Scaling theory of a generic Hubbard model*, J. Phys. **48**, 1613 (1987). N. Furukawa, T. M. Rice, and M. Salmhofer, *Truncation of a Two-Dimensional Fermi Surface due to Quasiparticle Gap Formation at the Saddle Points*, Phys. Rev. Lett. **81**, 3195 (1998). K. Le Hur and T. M. Rice, *Superconductivity close to the Mott state: From condensed-matter systems to superfluidity in optical lattices*, Ann. Phys. **324**, 1452 (2009). S. Raghu, S. A. Kivelson, and D. J. Scalapino, *Superconductivity in the repulsive Hubbard model: An asymptotically exact weak-coupling solution*, Phys. Rev. B **81**, 224505 (2010). R. Nandkishore, L. S. Levitov, and A. V. Chubukov, *Chiral superconductivity from repulsive interactions in doped graphene*, Nat. Phys. **8**, 158 (2012). J. González, *Kohn-Luttinger superconductivity in graphene*, Phys. Rev. B **78**, 205431 (2008). H. Isobe, N. F. Q. Yuan, and L. Fu, *Unconventional Superconductivity and Density Waves in Twisted Bilayer Graphene*, Phys. Rev. X **8**, 041041 (2018). J. Zinn-Justin, *Quantum Field Theory and Critical Phenomena*, Fourth ed. (Oxford University Press, New York, 2002). M. E. Peskin and D. V. Schroeder, *An Introduction To Quantum Field Theory* (Addison-Wesley, Reading, 1995). D. J. Amit, *Field Theory, the Renormalization Group, and Critical Phenomena*, Second ed. (World Scientific, Singapore, 1984). V. P. Nair, *Quantum Field Theory* (Springer, New York, 2005). C. G. Callan, Jr., *Broken Scale Invariance in Scalar Field Theory*, Phys. Rev. D **2**, 1541 (1970). K. Symanzik, *Small distance behaviour in field theory and power counting*, Commun. Math. Phys. **18**, 227 (1970). K. Symanzik, *Small-distance-behaviour analysis and Wilson expansions*, Commun. Math. Phys. **23**, 49 (1971).

--- abstract: 'We propose a new algorithm for the distributed computation of Wasserstein Barycenters over networks. Assuming that each node in a graph has a probability distribution, we prove that every node can reach the barycenter of all distributions held in the network by using local interactions compliant with the topology of the graph. We provide an estimate for the minimum number of communication rounds required for the proposed method to achieve arbitrary relative precision both in the optimality of the solution and the consensus among all agents for undirected fixed networks.' author: - 'César A. Uribe, Darina Dvinskikh, Pavel Dvurechensky, Alexander Gasnikov and Angelia Nedić [^1]' bibliography: - 'IEEEfull.bib' - 'wass.bib' - 'opt\_dec2.bib' title: '**Distributed Computation of Wasserstein Barycenters over Networks**' --- Introduction ============ Optimal Transport (OT) distances (also known as *earth mover’s distances* or *Wasserstein distances*) design an optimal plan to move “mass" from one probability distribution to another. This problem can be traced back to the early work of Monge [@Monge1781] and Kantorovich [@Kantorovich1942] and has been of constant interest for allowing natural formulations to the problems of comparing, interpolating, and measuring distances of functions [@Levy2017]. On the other hand, computational OT has gained popularity for its applications in learning theory [@Frogner2015], computer vision [@Rabin2011], computer graphics [@Solomon2015], statistical inference [@Srivastava2015a], information fusion [@Bishop2014a]; and its relative complexity advantages with respect to classical methods [@Dvurechensky2018]. . Comprehensive accounts of the OT problem and its computational aspects can be found in [@Villani2008; @Solomon2017; @Peyre2017; @Levy2017]. One of the common uses of the Wasserstein distance is the aggregation of distributions by considering their barycenter [@Agueh2011], which itself is another distribution [@Cuturi2014]. Wasserstein Barycenters (WB) have been shown superior to traditional Euclidean means in a range of application such as image processing [@Agueh2011], economics and finance [@Beiglbock2013], and condensed matter physics [@Buttazzo2012]. Figure \[fig:sevens\] shows a sample of $30$ images of the digit $7$ from the MNIST dataset [@LeCun1998], and their respective Euclidean mean and Wasserstein mean. The WB better captures the structural features of the input images. For discrete and finite distributions, the WB can be efficiently computed by solving a large linear program [@Anderes2016] or using regularization to approximate a solution efficiently and exploit its convenient algebraic properties [@Agueh2011; @Cuturi2014; @Cuturi2016]. Here, we consider the problem of computation of WB over a network. The flexibilities induced by the distributed setup make it suitable for problems involving large quantities of data with no centralized storage [@boy11; @ned16w; @ned17e; @Nedic2017a]. Mainly, we assume a group of agents is connected over a network, and each agent locally holds a probability distribution with finite support. The group seeks to compute the WB of all distributions in the network cooperatively. Figure \[fig:erdos\_sevens\] shows an Erdős-Rényi random graph with $160$ agents where each agent holds a sample of the digit $7$ from the MNIST dataset. Distributed consensus with the Wasserstein metric was introduced in [@Bishop2014a; @Bishop2014]. In [@Bishop2014], the authors showed asymptotic converge to the WB of the initial distributions given some weak connectivity assumptions on the graph over which agents exchange information. Nevertheless, the proposed algorithm requires that each agent computes an exact WB of local distributions at each iteration. Although one can have closed-form solutions for some families of continuous distributions, in general, the problem can be intractable. On the other hand, a recent approach [@Staib2017] explores the computational advantages of a dual formulation of the WB and exploits the parallelizable structure of the problem to propose a scalable, and communication-efficient algorithm for its computation on arbitrary continuous distributions. Nevertheless, it requires a central fusion center that coordinates the actions of the parallel machines. In contrast with existing literature [@Bishop2014; @Staib2017], propose a first-order algorithm that can be executed distributedly over a network with unknown topology. We derive an explicit convergence rate of the order $O(1/k^2)$ with an additional cost that depends on the condition number of the graph over which the agents interact. Additionally, we present two numerical examples to illustrate and validate our results. First, we show some basic properties of the algorithm for the problem of computing WB of univariate, discrete and truncated Gaussian distributions. Then, we show the result of applying our algorithm to a subset of the MNIST digit database on a large-scale network of $1000$ agents. This paper is organized as follows. Section \[sec:problem\] presents basic definitions for the problem of computation of WB over networks. Section \[sec:results\] states auxiliary results, introduces the proposed algorithm and states our main results on its convergence rate and dependency on the problem parameters. Section \[sec:numerical\] shows two numerical examples that experimentally verify the theoretical properties of the algorithm. Conclusions and future work are presented in Section \[sec:conclusions\]. **Notation:** We assume that the agents are indexed from $1$ through $m$. The enumeration is not needed in the execution of the proposed algorithm. It is only used in the analysis. Superscripts $i$ or $j$ denote agent indices and subscript $k$ iteration indices. $[A]_{ij}$ denotes the $i$-th row and $j$-th column entry of a matrix $A$. $I_{n}$ is the identity matrix of size $n$. . Problem Statement {#sec:problem} ================= In this section, we recall some basic definitions of the optimal transport problem. We describe the Wasserstein distance and the WB of a set of discrete probability distributions with finite support. Finally, we introduce the problem of distributed computation of WB over networks. Entropy Regularized Optimal Transport ------------------------------------- Consider two probability distributions $p,q \in S_1(n)$ with support on a finite set of points $\{x_i \in \mathbb{R}^d \}_{i=1}^n$ such that $p(x_i) = p_i$ and $q(x_i) = q_i$, where $S_1(n) = \{ p \in \mathbb{R}_+^n \mid p^T \boldsymbol1 =1 \}$. Moreover, consider a non-negative symmetric matrix , where $[M]_{ij}\in \mathbb{R}_+$ accounts for the cost of moving mass from $p_i$ to bin $q_j$. Without loss of generality, in the numerical example we will consider the Euclidean costs where $[M]_{ij} = \|x_i - x_j\|_2^2$. Additionally, define the set of couplings or *transportation polytope* $U(p,q)$ as The entropy-regularized OT problem [@Cuturi2013] seeks to minimize the transportation costs while maximizing the entropy (maximum-entropy principle) and is defined as where $\gamma>0$, and and $\forall x>0, h(x) \triangleq x\log x$ and $h(0) \triangleq 0$. A solution $\mathcal{W}_0(p,q)$ is called the Wasserstein distance between $p$ and $q$ , problem  is convex and admits a unique solution . $$\begin{aligned} \mathcal{W}_{\gamma,q}(p) \triangleq \mathcal{W}_{\gamma}(p,q). \end{aligned}$$ One particular advantage of entropy-regularizing the Wasserstein distance is that there exists closed-form representations for the dual problem and its gradients [@Agueh2011; @Cuturi2016] where the Fenchel-Legendre transform of  is defined as \[thm:cuturi\] For $\gamma >0$, the Fenchel-Legendre dual function $\mathcal{W}^*_{\gamma,q}(y)$ is differentiable and its gradient $\nabla \mathcal{W}^*_{\gamma,q}(y)$ is $1/\gamma$-Lipschitz with where $y \in \mathbb{R}^n$, $\alpha = \exp( {y}/{\gamma}) $ and . We will use the result Theorem \[thm:cuturi\] to design an algorithm for the computation of the WB on graphs based recent ideas of dual approaches for convex optimization problems with affine constrains [@all17; @Gasnikov2016] and optimal algorithms for distributed optimization [@Uribe2017; @uribe2018dual]. Computation of a Wasserstein Barycenter over a Network ------------------------------------------------------ The uniform WB [@Agueh2011; @Cuturi2014] of a family of discrete distributions $q_i\in S_1(n)$, for $i=1,\ldots,m$; is defined as the solution to the following optimization problem The WB is an extension of the Euclidean barycenter to nonlinear metric spaces to the empirical Fréchet mean [@Frechet1948]. The existence and uniqueness of WB has been studied in the literature [@Bigot2016]. Problem  is strictly convex and admits a unique solution, denoted by $p^*$ [@Cuturi2016]. ${{{\mathtt{p}} = [p_1^T,\cdots,p_m^T]^T}}$ and ${{{\mathtt{q}} = [q_1^T,\cdots,q_m^T]^T}}$, where , and rewrite the problem in an equivalent form We denote the unique solution of  by . We seek to solve problem  in a distributed manner over a network, where each distribution $q_i$ is held by an agent $i$ on a network. We model such a network as a fixed *connected undirected graph* , where $V$ is the set of $m$ nodes, and $E$ is a set of edges. We assume that the graph $\mathcal{G}$ does not have self-loops. The network structure imposes information constraints; specifically, each node $i$ has access to $q_i$ only and a node can exchange information only with its immediate neighbors, i.e., a node $i$ can communicate with node $j$ if and only if $(i,j)\in E$. We can represent the communication constraints imposed by the network by introducing a set equivalent to the constraints in . To do so, we define the Laplacian matrix $\bar W{\in \mathbb{R}^{m\times m}}$ of the graph $\mathcal{G}$ by [$$\begin{aligned} [\bar W]_{ij} = \begin{cases} -1, & \text{if } (i,j) \in E,\\ \text{deg}(i), &\text{if } i= j, \\ 0, & \text{otherwise,} \end{cases} \end{aligned}$$]{} where $\text{deg}(i)$ is the degree of the node $i$, i.e., the number of neighbors of the node. Finally, define the communication matrix (also referred to as an interaction matrix) by , where $\otimes$ indicates the Kronecker product. Throughout the paper, [*we assume that graph $\mathcal{G} = (V,E)$ is undirected and connected*]{}. Under this assumption, the Laplacian matrix $\bar W$ is symmetric and positive . Furthermore, the vector $\boldsymbol{1}$ is the unique (up to a scaling factor) eigenvector associated with the eigenvalue $\lambda=0$. $W$ inherits the properties of $\bar W$, including symmetry and positive semidefiniteness. Moreover, $W{x} = 0$ if and only if [$x_1 = \cdots = x_m$]{}, and $\sqrt{W}{x} = 0$ if and only if [$x_1 = \cdots = x_m$]{}. Therefore, one can equivalently rewrite problem  as Note that the constraint set is the same as the set , since due to the connectivity of the graph $\mathcal{G}$. In the next section, we state the proposed algorithm for solving  and analyze its convergence rate. Algorithm and Results {#sec:results} ===================== In this section, we build on recent results on dual approaches for optimal distributed optimization [@Uribe2017; @ani17] to construct an algorithm that solves  over a network. Moreover, we analyze its convergence rates and provide explicit dependencies on the problem parameters and the network topology. Dual Approach for Strongly Convex Functions and Affine Constraints {#S:DualApprMain} ------------------------------------------------------------------ In [@ani17], the authors proposed a novel analysis for the minimization of strongly convex functions with affine constraints of the form $$\begin{aligned} \label{eq:linear} \min_{Ax=0}f(x), \end{aligned}$$ where $f(x)$ is $1$-strongly convex with respect to the with the corresponding dual problem defined as $$\begin{aligned} \label{eq:dual_to_linear} \min_y g(y) \ \ \ \text{where} \ \ g(y) = \max_x \{\left\langle A^Ty,x \right\rangle -f(x)\}. \end{aligned}$$ The dual function $g(y)$ is $L$-smooth with , where $A_i$ is the $i$-th column of $A$. Thus, one can use accelerated first order methods such as Nesterov’s Fast Gradient [@nes83] or one of its recent reformulations [@all14] to obtain an approximate solution. The novelty in [@ani17] lies in the statement of the convergence rate of the accelerated methods in terms of the duality gap and the constraint violation. Additionally, it was shown that for the linear coupling accelerated algorithm [@all17] one can guarantee that the solutions will remain in a closed ball around the optimal solution, with a radius proportional to the distance between the initial point of the algorithm and the optimal solution. Next, we state a technical result, based on [@ani17], that will help us in the design and analysis of our proposed algorithm for the distributed computation of the WB. \[thm:gasnikov\] The fast gradient method based on linear coupling proposed in [@all14] with the change $y_k$ to $w_k$ and $x_k$ to $y_k$ and applied to , has the following properties: $ \dd{\forall ~} k \geq N$ and $\varepsilon >0$, it holds that $$\begin{aligned} g(\pd{w_k}) + f(\breve x_k) \leq \varepsilon \qquad \text{and} \qquad \|A\breve x_k\|_{\pd{2}}\leq \varepsilon / R, \end{aligned}$$ where , $N \triangleq \sqrt{\pd{16} \cu{L} R^2/\varepsilon}$, and $y^*$ is the optimal point of $g(\cdot)$ with minimal norm. \[S:DualWasser\] Additionally, from Theorem \[thm:cuturi\] the gradient can be expressed in closed form as where $p^*_j(\tilde y_j) = \alpha(\tilde y_j) \circ ( K \cdot { q_i}/{(K \alpha(\tilde y_j) )} )$. Moreover, it holds that one can recover the solution $\mathtt{p}^*$ to the primal problem  from a solution $\mathtt{y}^*$ to the dual problem  as $ \mathtt{p}^* = \mathtt{p}^*(\sqrt{W}\mathtt{y}^* )$. The optimality relation between the dual and the primal problem follows from Theorem $3.1$ in [@Cuturi2016]. In general, the dual problem  can have multiple solutions of the form $\mathtt{y}^* + \ker(\sqrt{W})$ when the matrix $\sqrt{W}$ does not have a full row rank. When the solution is not unique, we [*will use $\mathtt{y}^*$ to denote the smallest norm solution*]{}, and we let $R$ be its norm, i.e. $R = \|\mathtt{y}^*\|_2$. . As a consequence, the computation of the WB of a set of discrete probability distributions $\{q_i\}_{i=1}^m$ is equivalent to solving the dual decomposable $L$-smooth (with respect to the $2$-norm) optimization problem  with  [@Kakade2009]. Specifically, in this setup it holds that \[S:AlgAndMain\] We can explicitly write the Nesterov’s Accelerated Gradient Method (FGM) [@nes13] for smooth functions. Particularly, we use follow the linear coupling approach recently proposed in [@all14]. Setting $\hat{\mathtt{w}}_{k} = \hat{\mathtt{z}}_{k}=\hat{\mathtt{y}}_{k} = \boldsymbol{0}$, the FGM generates iterates according to: where $\alpha_{k+1} = (k+2)/(2L)$ and $\tau_k = 2/(k+2)$. Unfortunately, algorithm  cannot be executed in a distributed manner. Although the entries of local gradient vectors can be computed independently by each node, the sparsity pattern of the matrix $\sqrt{W}$ need not be the same as the communication constraints induced by the graph $\mathcal{G}$. Thus, the variables $\hat {\mathtt{w}}_{k}$ and $\hat {\mathtt{z}}_{k}$ cannot be computed on the network. This problem is solved by a change of variables such that $\tilde{ \mathtt{y}} = \sqrt{W}\hat{\mathtt{y}}$, $\tilde{ \mathtt{w}} = \sqrt{W}\hat{\mathtt{w}}$ and $\tilde{ \mathtt{z}} = \sqrt{W}\hat{\mathtt{z}}$. Algorithm \[alg:main\] presents the resulting distributed accelerated gradient method for the dual problem of the WB problem. . Now, we are ready to state our main result that provides a convergence rate for Algorithm \[alg:main\] with explicit dependencies on the problem parameters and the topology of the network. \[thm:main\] Let $\varepsilon>0$ and assume that $\|\nabla \mathcal{W}^*_{\gamma,\mathtt{q}}(\tilde{\mathtt{y}})\|_2 \leq \dd{G} $ on a ball $B_R(0)$. Then, it holds that that after iterations, the outputs of Algorithm \[alg:main\], i.e. $\mathtt{p}^*_N = [(p_N^*)_1^T,\cdots,(p_N^*)_m^T]^T$ and $\mathtt{y}^*_N = [(y_N^*)_1^T,\cdots,(y_N^*)_m^T]^T$ have the following properties: $$\begin{aligned} \mathcal{W}_{\gamma,\mathtt{q}}(\mathtt{p}^*_N)+ \mathcal{W}^*_{\gamma,\mathtt{q}}(\mathtt{y}^*_N) \leq \varepsilon \ \ \text{and} \ \ \|\sqrt{W}\mathtt{p}^*_N\|_2 \leq {\varepsilon}/{R}. \end{aligned}$$ The dual function $\mathcal{W}^*_{\gamma,\mathtt{q}}(\mathtt{y})$ is $(\cu{d_{\max}}/\gamma)$-smooth. Thus, from Theorem \[thm:gasnikov\] it follows that $$\begin{aligned} \mathcal{W}_{\gamma,\mathtt{q}}(\mathtt{p}^*_N)+ \mathcal{W}^*_{\gamma,\mathtt{q}}(\mathtt{y}^*_N) \leq \varepsilon \ \ \text{and} \ \ \|\sqrt{W}\mathtt{p}^*_N\|_2 \leq {\varepsilon}/{R}. \end{aligned}$$ holds for $k \geq \sqrt{16 \cu{d_{\max} }R^2/(\gamma \varepsilon)}$.\ Moreover, Thus, we require $ k \geq {\scriptscriptstyle \sqrt{\frac{16 \cu{G}^2}{\gamma \cdot \varepsilon}\cu{\frac{d_{\max}}{d_{\min}} }}}$ and the desired result follows. Theorem \[thm:main\] provides an estimate of the minimum number of iterations required for the proposed algorithm to reach some arbitrary relative accuracy in the solution of the distributed WB problem. The convergence rate is shown to be of the order $O(1/k^2)$ which has been established to be optimal for smooth convex optimization problems [@nes13] with an additional cost proportional to the . In general, one might be interested in finding a WB for the original Wasserstein distance with no regularization term. That is, to solve problem  with $\gamma=0$. The next theorem explains a choice of $\gamma$ that provides a convergence rate results with respect to the non-regularized optimal transport based on the iterates of Algorithm . Let $\varepsilon>0$, and assume that $\|\nabla \mathcal{W}^*_{\gamma,\mathtt{q}}(\tilde{\mathtt{y}})\|_2 \leq \dd{G} $ on a ball $B_R(0)$. Moreover, set $\gamma = \varepsilon/(4\pd{ m} \log n)$. iterations, the outputs of Algorithm \[alg:main\], i.e. $\mathtt{p}^*_N = [(p_N^*)_1^T,\cdots,(p_N^*)_m^T]^T$ and $\mathtt{y}^*_N = [(y_N^*)_1^T,\cdots,(y_N^*)_m^T]^T$ have the following properties [ $$\begin{aligned} \mathcal{W}_{0,\mathtt{q}}(\mathtt{p}^*_N)- \pd{\mathcal{W}_{0,\mathtt{q}}(\mathtt{p}^*)} \leq \varepsilon \ \ \text{and} \ \ \|\sqrt{W}\mathtt{p}^*_N\|_2 \leq {\varepsilon}/{(\pd{2}R)}. \end{aligned}$$ ]{} Numerical Experiments {#sec:numerical} ===================== In this section, we show two numerical experiments to validate the results of Theorem \[thm:main\]. We explore the problem of computing WB of univariate, discrete and truncated Gaussian densities and the computation of the WB of a subsample of $1000$ digit images from the MNIST dataset. Barycenter of Gaussian Distributions ------------------------------------ Initially, we explore the computation of WB for sets of univariate, discretized and truncated Gaussian densities [@Agueh2011]. We consider a network of agents where each agent $i$ holds a univariate, discretized and truncated Gaussian distribution, with mean $\mu_i \in [-5,5]$, standard deviation $\sigma \in [0.1,2]$ and equally spaced support of $100$ points in $[-5,5]$. The entropy regularization parameter is set to $\gamma = 0.1$. Figure \[fig:results\] shows the distance to optimality and the distance to consensus of Algorithm \[alg:main\] for the star, cycle, complete, and Erdős-Rényi random graph graphs with a fixed size of $50$ nodes. Also, Figure \[fig:results\] shows the scalability of the algorithm, i.e., the number of iterations required to reach an $\varepsilon$ accuracy in the distance to optimality and consensus for networks of increasing size. MNIST Dataset ------------- We randomly sample $1000$ images for each digit of the MNIST dataset [@LeCun1998; @LeCun1998a]. Each image has $28\times28$ pixels and is scaled uniformly at random between $0.5$ and $2$ of its size and randomly located on a larger $56\times56$ blank image. The pixel values of the image are normalized to add up to $1$. We assign one sample from each digit to each agent on a group of $1000$ agents, and the objective is to jointly compute the WB for each digit of the $1000$ samples present in the network. The agents are connected over an Erdős-Rényi random graph with $1000$ nodes and connectivity parameter $4/1000$. The entropy regularization parameter is set to $\gamma = 0.01$. Figure \[fig:mnist\] shows the local barycenter of the $9$ digits for a subset of $3$ agents in the network for $0$, $60$, and $300$ iterations. As the number of iteration increases, all agents converge to a common image, namely, the WB of the images held by the agents in the network. at (0,0) [ $\ \ \ \ N=0$]{}; \ at (0,0) [ $\ \ \ N=60$]{}; \ at (0,0) [ $\ \ N=300$]{}; Discussion and Conclusions {#sec:conclusions} ========================== We developed a novel algorithm for the distributed computation of WB over networks where a group of agents connected over a network and each agent holds some local probability distribution with finite support. Our results provably guarantee that all agents in the network will converge to the WB of all distribution held by the agents in the network. We provide an explicit and non-asymptotic convergence rate of the order $O(1/k^2)$ with an additional cost proportional to the ratio between the maximum and minimum degree among the nodes in the graph over which the agents exchange information. The case where spectral information of the network is not available or when the graphs are directed or change with time require further study. The use of second-order information can also be exploited to get better performance. Also, recently proposed stochastic approaches can provide more efficient algorithms [@Claici2018; @Staib2017; @dvurechensky2018decentralize; @rogozin2018optimal]. [^1]: C.A. Uribe (*cauribe@mit.edu*) is with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology. A. Gasnikov (*gasnikov@yandex.ru*) is with the Moscow Institute of Physics and Technology, and the Institute for Information Transmission Problems RAS. A. Nedić (*angelia.nedich@asu.edu*) is with the ECEE Department, Arizona State University, and Moscow Institute of Physics and Technology. The work of A. Gasnikov in Section III-C was supported by the grant of the president of Russian Federation no. MD-1320.2018.1. The work of A. Nedić and C.A. Uribe is supported by the National Science Foundation under grant no. CPS 15-44953.

--- abstract: '“Theorem proving is similar to the game of Go. So, we can probably improve our provers using deep learning, like DeepMind built the super-human computer Go program, AlphaGo [@alphaGo].” Such optimism has been observed among participants of AITP2017. But is theorem proving really similar to Go? In this paper, we first identify the similarities and differences between them and then propose a system in which various provers keep competing against each other and changing themselves until they prove conjectures provided by users.' author: - Yutaka Nagashima bibliography: - 'easychair.bib' title: Designing Game of Theorems --- The Game of Go and Theorem Proving ================================== #### Formally defined rules \[similarity\]. Both the games of Go and theorem proving have algorithms to evaluate the results. In the game of Go, one can judge the result of each game when it is over by counting the stones and spaces for each player, and no ambiguity is left in deciding the result of a game. In theorem proving, when one finds a proof, others can systematically check if the alleged proof is a valid proof or not. #### Expressive power of the system \[difference\]. Even though both systems are based on a set of simple rules, the expressive power of these systems differ. Depending on the underlying logics, a theorem proving task can involve advanced concepts such as abstraction, universal quantification, existential quantification, and polymorphism, which Go scores cannot express natively. This is especially true for more expressive logics such as classical higher-order logics or variants of calculus of constructions, where stronger proof automation is needed. #### Amount of available training data \[difference\]. Some theorem proving researchers boast that they have large proof corpora. For example, the Isabelle community has the Archive of Formal Proofs (AFPs) [@AFP], consisting of more than 1.5 millions of lines of code and 100 thousands lemmas. Unfortunately, even though these proof corpora are large for the small community of theorem provers, they are small compared to the data deployed in other domains. #### Preference towards small data \[difference\]. The size of the community is not the only reason of small data available in the theorem proving community: logicians and mathematicians have developed expressive logics to describe general ideas in a concise manner. Combined with the trade-off between proof automation and expressive power of underlying logic, this is doubly unfortunate: the more expressive logic we use, the less proof automation we have, but the more expressive the logic is, the less training data we can expect, which makes it hard to improve the proof automation for expressive logics using machine learning techniques. #### Self-playability \[similarity/difference\]. One might suspect that large data are not necessary to develop a powerful proof automation tool using machine learning. After all, DeepMind has made AlphaGo Zero [@alphaGoZero] stronger than any previous versions of AlphaGo via self-play without using data from human games. Unfortunately, even though both Go and theorem proving are based on clearly defined rules, theorem proving is not a two-player game by default. In the rest of this paper, we propose an approach to introducing self-playability to theorem proving. The Design of Self-playable Games of Theorem Proving ==================================================== One straightforward design of self-playable games of theorem proving is as follows: (1) prepare a set of proof obligations from existing proof corpora, (2) let two competing provers try to prove these proof obligations, (3) count how many obligations each prover discharges, (4) consider the prover that solves more obligations as the winner, and the other one as the loser. We can use this naive approach as a part of reinforcement learning or evolutionary computation to optimize our provers for proof obligations that have already been proved. However, this approach is probably not powerful enough to improve provers for conjectures that are significantly different from the theorems in the training data For example, let us assume that we enhance our prover, say `P`, via self-play using 100 theorems and their proofs in the AFPs. Since we already know how to prove these theorems, we can improve `P`, so that `P` can prove all of the 100 theorems within a reasonable time-out. However, when we try to improve `P` to discharge a new conjecture, say Goldbach’s conjecture, we will find ourselves at a loss of training data: Currently, there is no known proofs of Goldbach’s conjecture or auxiliary lemmas that are verified to be useful to prove Goldbach’s conjecture. If we add Goldbach’s conjecture to the above dataset, the improvement via self-play would saturate after producing a prover that can discharge the 100 theorems from the AFP but not Goldbach’s conjecture: since the gap between the theorems from the AFPs and Goldbach’s conjecture is too large, minor mutations to `P`’s variants cannot produce a useful observable difference in the result of the game. What we need here is a mechanism to produce conjectures that we can reasonably expect to be useful to prove our target conjecture (Goldbach’s conjecture in this example) but not too difficult for our current prover `P`. Therefore, we propose to *treat conjecturing and proof search as one problem*. Of course, we cannot be 100% sure which conjecture is useful to train our prover for Goldbach’s conjecture, since nobody has proved it yet. But if we consider a heuristic proof search as the exploration of an and-or tree, we can estimate how important each node in the tree is from the search heuristics of the prover. Furthermore, given a long time-out, we can expect that the prover can discharge some of emerging subgoals, even if the prover cannot discharge the root-node, which corresponds to the target conjecture (Goldbach’s conjecture, in this example). Our idea is to *use these proved subgoals to judge the competence of other versions of prover* `P`. First, we produce two versions of our prover `P` by mutation. Let us call them `Pa` and `Pb`, respectively. Using the approach explained above, we let `Pa` produce a dataset `Da` and let `Pb` produce `Db`. Now, we let `Pb` try to prove the theorems in `Da`, and let `Pa` try to prove the theorems in `Db`. When `Pa` and `Pb` run out of time, we sum up the estimated values of theorems proved by each prover (`Pa`, for example). Note that it was the opponent prover (`Pb` in this example) that has decided the value of each theorem in each dataset (`Db` in this case) when finding proofs of these subgoals for the first time. The prover that has gained more value is the winner of the game, and the other is the loser. Then, we keep running this game by mutating the winner until we produce a prover that can discharge the target conjecture. Since this process generates new conjectures tagged with their estimated values from the target conjecture in each iteration, we expect this approach continues producing conjectures useful to prove the target conjecture. We are still in the early stage of the design. We might generalize this idea to n-player games to avoid over-fitting. For the moment, we prefer the design of the game to be irrelevant to any of underlying logics, ML algorithms for search heuristics, and mutation algorithms. Acknowledgement {#acknowledgement .unnumbered} =============== This work was supported by the European Regional Development Fund under the project AI&Reasoning (reg. no. CZ.02.1.01/0.0/0.0/15\_003/0000466). \[sect:bib\]

--- abstract: 'In [@HMo theorem 2.5] Harris and Morrison construct semistable families $f\colon F\to Y$ of $k$-gonal curves of genus $g$ such that for every $k$ the corresponding modular curves give a sweeping family in the $k$-gonal locus $\overline{{\mathcal{M}}^{k}_{g}}$. Their construction depends on the choice of a smooth curve $X$. We show that if the genus $g(X)$ is sufficiently high with respect to $g$ then the ratio $\frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}$ is $8$ asymptotically with respect to $g(X)$. Moreover, if the conjectured estimates given in [@HMo p. 351-352] hold, we show that if $g$ is big enough, then $F$ is a surface of positive index.' address: - | Dipartimento di Matematica e Geoscienze, Università degli studi di Trieste,\ via Valerio 12/b, 34127 Trieste, Italy\ `beorchia@units.it` - | Dipartimento di Matematica e Informatica\ via delle Scienze 206, Università degli studi di Udine\ Udine, 33100 Italy\ `Francesco.Zucconi@uniud.it` author: - Beorchia Valentina and Zucconi Francesco title: A note on Harris Morrison sweeping families of maximal gonality --- Introduction. ============= Let $\overline{\mathcal{M}}_{g}$ be the moduli space of stable curves of genus $g$. A family ${\mathcal{B}}$ of curves $Z\subset {\overline {\mathcal M}_g}$ is said to be a sweeping family if $\cup{\mathcal{B}}$ is a Zariski dense subset of $\overline{\mathcal{M}}_{g}$. The invariants of sweeping families can furnish important information on the geometry of $\overline{\mathcal{M}}_{g}$; see: [@CFM]. In this paper we focus on the geometry of the families of curves constructed in [@HMo Theorem 2.5]. We revise this construction in Section 2; here we point out only that it depends on choosing a smooth curve $X$ of genus $g(X)$. For the aim of [@HMo] $g(X)$ can be taken to be zero, but Harris Morrison construction applies then to any curve $X$. We compute the invariants of the smooth surface $F$ supporting the Harris Morrison family. We denote by $e(F)$, $\chi({\mathcal{O}}_{F})$, $K^{2}_{F}$ respectively the topological characteristic of $F$, its cohomological characteristic and the self-intersection of the canonical divisor. We also denote by $q(F)$ the irregularity of $F$. We have: [**[Theorem]{}**]{}[*[ Let $g,k\in\mathbb N$ such that $k$ can occur as the gonality of a smooth curve of genus $g$. There exist Harris Morrison sweeping $k$-gonal semistable genus-$g$ fibrations $f \colon F\to Y$ such that the ratio $\frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}$ is $8$ asymptotically with respect to $g(X)>0$.]{}*]{} See Theorem \[secondaadifferenza\] for a refined statement. We stress that our Theorem is not obtained by a base change over a family as in [@HMo Theorem 2.5] starting from ${\mathbb{P}}^{1}$. We remark that the above result applies to any gonality $k$. If we consider only those families with maximal gonality, then we show that (see Proposition \[secondo\]): [**Proposition**]{} [*[Let $f\colon F\to Y$ be an Harris Morrison genus-$g$ fibration starting from any plane curve $X$ of genus $g(X)>>0$. Assume that the gonality $k$ is maximal and that the conjectured estimates in [@HMo p. 351-352] are true. If $g$ is big enough, then $F$ is a surface of positive index i.e. $K^{2}_{F}> 8\chi({\mathcal{O}}_{F})$. Moreover if $f\colon F\to Y$ is general in its class then the irregularity of $F$ is $g(Y)$. In particular, $f\colon F\to Y$ is the Albanese morphism of $F$.]{}*]{} We recall that any surface of general type satisfies the Miyaoka-Yau inequality $K^{2}_{F}\leq 9\chi({\mathcal{O}}_{F})$, and equality holds if and only if $F$ is a ball quotient. Surfaces of positive index satisfy the inequalities $8\chi({\mathcal{O}}_{F})< K^{2}_{F}{\bf < }\ 9\chi({\mathcal{O}}_{F})$. These surfaces are still quite mysterious objects [@Re], [@MT], [@My]. The statement of the Proposition above is close to the recent results of Urz[ú]{}a. He has found in [@Ur1] simply connected projective surfaces of general type with $\frac{K_{F}^{2}}{e(F)}$ arbitrarily close to $\frac{71}{26}$; this improves previous estimates by Persson-Peters-Xiao [@PPX]. By his method, based on a generalisation of the method of line arrangements on ${\mathbb{P}}^{2}$, he finds interesting surfaces of positive index; see: [@Ur2]. Thanks to the Kapranov construction of the moduli space ${\overline{M}}_{0,d+1}$ of rational curves with $d+1$ marked points, see [@Ka1], [@Ka2], line arrangements correspond to curves inside ${\overline{M}}_{0,d+1}$. In this sense Urz[ú]{}a construction is related to the Harris Morrison one. Hence the fact that our results on Harris Morrison construction of $k$-gonal sweeping families match those obtained in [@Ur2] is an evidence for the conjectural results stated in [@HM]. The authors would like to thank D. Chen for his comments on the first version of the paper. This research is supported by MIUR funds, PRIN project [*Geometria delle varietà algebriche*]{} (2010), coordinator A. Verra. The first author is also supported by funds of the Università degli Studi di Trieste - Finanziamento di Ateneo per progetti di ricerca scientifica - FRA 2011. The basic construction. ======================== Harris-Morrison families. ------------------------- We review and we explain some features of the basic construction of [@HMo]. We could construct a slight refinement of the basic construction of [@HMo Section 2] using not a product surface $X\times{\mathbb{P}}^{1}$ but a ruled surface ${\mathbb{P}}({\mathcal{E}})\to X$ over $X$, but we have decided to follow [@HMo Section 2] for simplicity. We denote by $[N]$ and $N$ a finite set and its order respectively. If $[M]$ is a subset of $[N]$ and is invariant under the action of $\mathfrak{S}_k$ we will denote the quotient $[M]/\mathfrak{S}_k$ by $[\widetilde M]$. Let ${\overline{M_{0, b}}}$ be the compactification of the moduli space of $b$-pointed stable curves of genus $0$ and let ${\overline\beta}\colon{\overline{H_{k, b}}}\to {\overline{M_{0, b}}}$ be the natural morphism on the Hurwitz scheme of admissible $k$-covers of stable $b$-pointed curves of genus $0$ constructed in [@HM]. Then the morphism $\beta\colon H_{k,b}\to{\mathcal{M}}_{0,b}$ is a $\widetilde N(k,b)$ sheeted unramified covering, where $\widetilde N(k,b)$ counts the $k$-sheeted [*[connected]{}*]{} covers of ${\mathbb{P}}^{1}$ simply branched at $b$ fixed points. From now on we set $N:=N(k,b)$, where $N(k,b)$ is defined in [@HMo page 334]. Concerning the subset $[N]$ we recall that $\mathfrak{S}_k$ acts on $[N]$ and by [@HMo Lemma 1.24] this action is free if $k\geq 3$ and trivial for $k=2$, so $N=k!\ \widetilde N$ if $k\geq 3$ and $N=\widetilde N$ if $k=2$. Let $X$ be any smooth complete curve and let $\pi_{X}\colon X\times{\mathbb{P}}^{1}\to X$ be the projection. By [@Ha] ${\rm{NS}}(X\times{\mathbb{P}}^{1})=[s]\mathbb Z\oplus [f]\mathbb Z$ where $[s]$ is the numerical class of a $\pi_{X}$-section $s$ and $[f]$ is the numerical class of a $\pi_{X}$-fiber $f$. In the sequel we will not distinguish between $s$ and its class $[s]$ if no danger of confusion can arise. Let $s$ be a $\pi_{X}$-section with $s^{2}=0$. Let $b\in \mathbb N$ and let $C_{1},\ldots, C_{b}$ be effective divisors on $X$ of degrees $c_i >0$, such that ${\mathcal{O}}_{X\times{\mathbb{P}}^{1}}(s+\pi_X ^\star C_i)$ is very ample, $i=1,\ldots ,b$. Let $\sigma_{i}\in |s+\pi_X ^\star C_i|$ be a smooth curve for every $i=1,\ldots, b$, such that the sections $\sigma_{i}$ meet transversely everywhere and such that each $\pi_{X}$-fiber contains at most one of these intersections. We set $$\sigma:=\sum \sigma_{i}$$ and we point out that $[\sigma]= b[s]+[\pi_X ^\star C]=b[s]+c[f]$ where $C=\sum C_{i}$ and $c=\sum c_{i}$. The reader is warned that if $X={\mathbb{P}}^{1}$ the above costruction easily works. If $X$ is a curve of positive genus then it is sufficient to assume that for every $i=1,\ldots, b$ the number $c_{i}$ is at least equal to the degree of a very ample divisor on $X$. Let $[I_{X}]$ be the set of nodes of $\sigma$ or equivalently the set of intersections of the $\sigma_{i}$’s. Then $$I_{X}=(b-1)c;$$ see: [@HMo Formula 2.1 on page 338]. Let $B_{X}$ be the blow-up of $X\times{\mathbb{P}}^{1}$ at $[I_{X}]$ and denote by $B_{x}$ its fiber over $x\in X$ and $\widetilde \sigma_{i}$ the strict transform of $\sigma_{i}$ on $B_{X}$. Let $\alpha\colon X\to{\overline{{\mathcal{M}}_{0,b}}}$ be the map which sends a point $x\in X$ to the class of the fiber $B_{x}$ marked by its $b$ points of intersections with the $\widetilde\sigma_{i}$ where $i=1,\ldots, b$. Let $Y:=X\times_{{\overline{{\mathcal{M}}_{0,b}}}}{\overline{H_{k, b}}}$. By the same argument of [@HMo pages 338-339] the induced map $\mu\colon Y\to X$ is a covering of degree $\widetilde N$. Let $\pi_{Y}\colon Y\times{\mathbb{P}}^{1}\to Y$ be natural projection. If we set $\nu:=\mu\times{\rm{id}}\colon Y\times{\mathbb{P}}^{1}\to X\times{\mathbb{P}}^{1}$ we see that $[I_{Y}]:=\nu^{\star}([I_{X}])$ is the scheme of singularities of the divisor $$\tau:=\nu^{\star}\sigma$$ on $Y\times{\mathbb{P}}^{1}$. Let $[J_{Y}]\subset Y$ be the $\pi_{Y}$-projection of $[I_{Y}]$ and let $[J_{X}]\subset X$ be the $\pi_{X}$-projection of $[I_{X}]$. By construction we have a morphism from the complement of $[J_{Y}]$ to ${\overline{{\mathcal{M}}_{g}}}$, and we extend it to a morphism $\rho\colon Y\to {\overline{{\mathcal{M}}_{g}}}$ following the argument of [@HMo Theorem 2.15]. Here we recall first that $\mu\colon Y\setminus [J_{Y}]\to X\setminus [J_{X}]$ is unramified and that $[J_{Y}]$ is partitioned into three classes: $$[J_{Y}]=[J_{Y, (1)}]\cup [J_{Y, (2,2)}]\cup [J_{Y, (3)}].$$ The topological description given in [@HMo Page 340] yields that $\mu$ is not ramified on $[J_{Y, (1)}]\cup [J_{Y, (2,2)}]$ and it is triply branched at points of $[J_{Y, (3)}]$. In particular $$I_{Y}=\left({\widetilde{N_{1}}}+{\widetilde{N_{2,2}}}+ \frac{{\widetilde{N_{3}}}}{3}\right) (b-1)c;$$ where ${\widetilde N}={\widetilde{N_{1}}} +{\widetilde{N_{ 2,2}}}+{\widetilde{N_{3}}}$ and a combinatorical definition of ${\widetilde{N_{1}}}, {\widetilde{N_{2,2}}}, {\widetilde{N_{3}}}$ is given in [@HMo Proposition 2.9]. Following [@HMo] we blow-up $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$ at the points of the set $[I_{Y,(1)}]$ and we construct a finite cover $\pi\colon G\to A$ and a semistable fibration $f\colon F\to Y$ see [@HMo Diagram (2.2), page 338] such that over the complement $Y'$ of $[J_{Y}]$ in $Y$ the surface $G$ is obtainable by the pull-back of the smooth universal admissible cover $\theta\colon{\mathcal{U}}_{{\mathcal{H}}_{k,b}}\subset{\mathbb{P}}^{1}\times {\mathcal{H}}_{k,b}\to{\mathcal{H}}_{k,b}$ whose existence is shown in [@HM]. We recall that over $Y'$, $G$ is a stable fibration $G_{Y'}\to Y'\subset Y$, it coincides with $f\colon F\setminus f^{-1}([J_{Y}])=F_{Y'}\to Y'$. Moreover $A$ coincides with $Y\times{\mathbb{P}}^{1}$ over $Y'$ and there exists a [*[finite]{}*]{} morphism $G_{Y'}=F_{Y'}\to A_{Y'}={\mathbb{P}}({\mathcal{O}}_{Y'}\oplus{\mathcal{O}}_{Y'})$. By the very explicit local analytic description of the [*[finite]{}*]{} cover $\pi\colon G\to A$ and of the blow-up $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$ done in [@HMo from page 343 to the top of page 347] we obtain an explicit description of the smooth surface $F$ and of the semistable fibration $f\colon F\to Y$ inducing the desired moduli map $\rho\colon Y\to{\overline{{\mathcal{M}}_{g}}}$. We set $Z:=\rho(Y)$. We sum up the above construction in the following diagram: $$\label{diag} \xymatrix{ S \ar[dr]^h \ar@{<-}[d]^u & &&\\ G \ar[r]^\zeta \ar[d]_\pi^{k:1} & F\ar[dr]^f \ar@{-->}[d]&& \qquad Z \subset {{\overline {\mathcal M}}_{g}}\ar@{<-}[dl]^\rho\\ A \ar[r]^{\epsilon } & Y\times{\mathbb{P}}^{1}\ar[r]^{\pi_Y} \ar[d] _\nu^{\widetilde N:1} & Y \ar[d]_\mu ^{\widetilde N:1}\ar[r]& \overline{H}_{k,b}\ar[d]^{\overline\beta}\\ & X\times{\mathbb{P}}^{1} \ar[r]^{\pi_X} & X \ar[r]^\alpha &\overline {{\mathcal{M}}_{0,b}}\\ }$$ What makes this families very interesting is that thanks to the work done in [@HMo] the dominant morphism $\zeta\colon G\to F$ is very explicit and hence the nature of the fibers of $f\colon F\to Y$ over $[J_{Y}]$ is evident. If $y\in Y$ we denote by $F_{y}$ and $G_{y}$ respectively the $f$-fiber over $y$ and the $(f\circ \zeta)$-fiber over $y$. The analysis of fibers of type $(1)$, that is $y\in [J_{Y, (1)}]$, splits into four cases: $(1_{0})$, $(1_{j, 0})$ $(1_{j, g})$, $(1_{j, i})$ where $j>0$ and $1\leq i\leq g/2$; see [@HMo Diagram (2.12)] and with obvious notation we write $y\in [J_{Y,\Pi}]$ if $y\in [J_{Y, (1)}]$ and $y$ is of type $\Pi$. If $y\in [J_{Y, (1_{0})}]$ then $F_{y}=F'_{y}\cup E$ where $F'_{y}$ is a genus $g-1$ curve and $E$ is a rational $(-2)$ curve; see the top of diagram [@HMo Diagram (2.12)]. The corresponding fiber $G_{y}$ on $G$ is $$G_{y}=G''_{y}\cup E'\cup\bigcup_{j=1}^{k-2}E_{j,y}$$ where $G''_{y}$ is a genus $g-1$ curve, $E_{j,y}^{2}=-1$ and $E'$ is a -$2$ curve. Over a neighbouhood of $y$ the morphism $\zeta\colon G\to F$ consists on the contraction of the $k-2$ rational curves $E_{j,y}$. If $y\in [J_{Y, (1_{j, 0})}]$ or $y\in [J_{Y, (1_{j, g})}]$ then $F_{y}$ is a [*[smooth]{}*]{} semistable fiber and over a neighbouhood of $y$ the morphism $\zeta\colon G\to F$ consists on a [*[ordered contraction]{}*]{} of $k$ rational curves. More precisely if $y\in [J_{Y, (1_{j, 0})}]\cup [J_{Y, (1_{j, g})}]$ then $$G_{y}=G'_{y}\cup E'_{y}\cup G''_{y}\cup \bigcup_{i=1}^{k-2}E_{i,y}$$ where $G'_{y}$ is a rational curve, $E'_{y}$ is a $(-2)$-rational curve, $E'_{y}\cdot G'_{y}=1$, $G''_{y}$ is a smooth curve of genus $g$, $E'_{y}\cdot G''_{y}=1$, $G''_{y}$ is a smooth curve of genus $g$ and $E_{i,y}^{2}=-1$. The map $\zeta$ first contracts $E_{i,y}$ where $i=1,\ldots ,k-2$. After this contraction process the image of $G'_{y}$ is a $-1$ curve which intersects the image of $E'_{y}$ in a unique point. Hence to get the semistable reduction $F$ we need [*[first]{}*]{} to contract the image of $G'_{y}$ and then the image of $E'_{y}$. If $y\in [J_{Y, (1_{j, i})}]$ where $j>0$ and $1\leq i\leq [g/2]$ then $F_{y}=F_{y,i}\cup E_{y}\cup F_{y,g-i}$ where $F_{y,i}\cap F_{y,g-i}=\emptyset$, $E_{y}^{2}=-2$, $F_{y,i}\cdot E_{y}= F_{y,g-i}\cdot E_{y}=1$ and $$G_{y}= G'_{y}\cup E'_{y}\cup G''_{y}\cup\bigcup_{i=1}^{k-2}E_{i,y},$$ where $E_{i,y}$ are $-1$-rational curves, $E_{y}$ is a $(-2)$ curve. Moreover $F_{y,i}$ is smooth of genus $i$ and admits a $j$ covering over the fiber ${\mathbb{P}}^{1}_{y}\subset Y\times{\mathbb{P}}^{1}$ and $F_{y,g-i}$ is smooth of genus $g-i$ and admits a $k-j$ covering over ${\mathbb{P}}^{1}_{y}$. Note that if $e_{y}\subset A$ is the exceptional curve over $y$ arised in the blow-up $A\to Y\times{\mathbb{P}}^{1}$ then the restriction of $\pi\colon G\to A$ to $E'_{y}\cup\left(\bigcup_{i=1}^{k-2}E_{i,y}\right)$ gives the $k$-cover of $e_{y}$ contained in $G$. Over a neighbouhood of $y$ the morphism $\zeta\colon G\to F$ consists on the contraction of the $k-2$ rational curves $E_{i,y}$. \[tipodueduetre\] If $y\in [J_{Y, (2,2)}]\cup [J_{Y, (3)}]$ then the fiber $F_{y}$ of $f\colon F\to Y$ over $y$ is a smooth semistable curve. The rational map $F\dashrightarrow Y\times{\mathbb{P}}^{1}$ is finite in a neighbourhood of $F_{y}$. Moreover the surfaces $G$ and $F$ coincide over an analytic neighboorhood of $y\in Y$. See [@HMo Pages 343-346]. \[daGaS\] Over each of the points of $[I_{X}]=\pi_{X}([J_{X}])\subset X$ there are ${\widetilde{N}}_{{\rm{sing}}}$ points $y\in Y$ such that the fiber $F_{y}\subset F$ is singular. Any singular fiber of $f\colon F\to Y$ contains a unique $(-2)$ rational curve. This is a restatement of [@HMo Theorem 3.1] and it easily follows by the above description of the singularities of the fibration $f\colon F\to Y$. A semistable fibration $f\colon F\to Y$ as the one of Proposition \[daGaS\] is called a Harris-Morrison family. Note that by Harris Morrison construction it follows \[harris\] Let $X$ be any smooth complete curve of genus $g(X)$. Let $g\geq 3$ be any natural number. If $g$ is odd let $k=\frac{g+3}{2}$ and set $k=\frac{g+2}{2}$ if $g$ is even. Then the family of curves $Z\subset {\overline {\mathcal M}_g}$ constructed by Harris and Morrison varying the curve $X$ forms a sweeping family. The chosen number $k$ is the maximal gonality for a curve of genus $g$. Hence the claim follows by a standard result of Brill-Noether theory. See also [@HMo page 350]. Notice that, even without the assumption of maximal gonality, the families of curves $Z\subset {\overline {\mathcal M}_g}$ constructed by Harris and Morrison varying the curve $X$ depend freely on $g(X)$. In particular we will study those families such that the parameter $g(X)>>0$. Numerical invariants. --------------------- We recall that $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$ is the blow-up of $[I_{Y, (1)}]$ which is a reduced scheme of length $(b-1)c{\widetilde{N_{1}}}$. Let $E_{A}$ be the exceptional divisor of $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$. Hence $$\label{autointe} E_{A}^{2}=-I_{Y, (1)}=-(b-1)c{\widetilde{N_{1}}}.$$ Let $\widetilde\tau$ be the strict transform of $\tau$, that is $\widetilde\tau=\epsilon_{A}^{\star}(\tau)-2E_{A}$. Since $\tau=\nu^{\star}(\sigma)=\nu^{\star}(bs+cf)$ then $$\label{tautildequadro} (\widetilde\tau)^{2}=2bc{\widetilde{N}}-4(b-1)c{\widetilde{N_{1}}}.$$ Let $R$ be the ramification divisor of $\pi\colon G\to A$. The following relation holds: $$\label{pistarR} \pi_{\star}R=\widetilde\tau.$$ Since we have simple branch on the fibers the local analysis shows that $$\label{pistabassopistaralto} \pi^{\star}\pi_{\star}R=2R+\widetilde R$$ and that: $$\label{erreerretilde} R\cdot\widetilde R=(b-1)c\left(2{\widetilde{N}}_{2,2}+\frac{4}{3}{\widetilde{N}}_{3}\right).$$ \[erreerre\] The ramification divisor $R$ of the finite cover $\pi\colon G\to A$ satisfies: $$R^{2}=\left({\widetilde N}+(b-1)\left(\frac{1}{3}{\widetilde N_{3}}- {\widetilde N_{1}}\right)\right)c$$ It is as in [@HMo Theorem 2.15]. By construction we have seen that over any point $y\in [J_{Y,(1)}]$ there exist $k-2$ rational curves $E_{i,y}\subset G_{y}$ such that $E_{i,y}^{2}=-1$. In particular there are $(k-2){\widetilde{N_{1}}}(b-1)c$ exceptional curves contained inside fibers of $f_{G}\colon G\to Y$ over $J_{Y,(1)}$ . \[laesse\] Let $u\colon G\to S$ be the contraction of these $(k-2){\widetilde{N_{1}}}(b-1)c$ rational curves. Then $S$ is a smooth surface. Moreover $K_{S}^{2}=K_{G}^{2}+ (k-2){\widetilde{N_{1}}}(b-1)c$. It follows straightly from Castelnuovo contraction theorem. Now we want to compute the invariants of $S$. \[eccez=0\] Let $E$ be any $-1$ rational curve which is contracted by $u\colon G\to S$. Then $R\cdot E=0$. We use the ramification divisor formula: $K_{G|Y}=\pi^{\star}K_{A|Y}+R$. Let $E$ be any $-1$ rational curve which is contracted by $u\colon G\to S$ then the finite morphism $\pi\colon G\to A$ restricts to an isomorphism $\pi_{|E}\colon E\to L\subset A$ where $L$ is an exceptional curve produced via the blow-up $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$. In particular $\pi_{\star}E=L$. Then $R\cdot E=-1-\pi^{\star}(K_{A|Y})E=-1-K_{A|Y}\cdot L=-1+1=0$. \[erretildeconmenouno\] Let $E$ be any $-1$ rational curve which is contracted by $u\colon G\to S$. Then $E\cdot{\widetilde R}=2$. By Equation \[pistabassopistaralto\] and by Proposition \[eccez=0\] we have: $0=E\cdot R=E\cdot\frac{1}{2}(\pi^{\star}\pi_{\star}R-\widetilde R)= \frac{1}{2}(L\cdot\widetilde\tau-E\cdot\widetilde R)$. Since $\widetilde\tau=\epsilon^{\star}(\tau)-2E_{A}$ and $E_{A}\cdot L=-1$ then the claim follows. We now consider the $(-2)$ rational curves of $G$. We notice that if $y\in[J_{Y,{1_{(2,0)}}}]\cup [J_{Y,{1_{(2k-2,g)}}}]$ then there is a $(-2)$ rational curve $G'_{y}\subset G_{y}$ which is mapped to a fiber of the canonical projection $\pi_{Y}\colon Y\times{\mathbb{P}}^{1}\to Y$; that is: $\epsilon\circ\pi(G'_{y})=\pi_{Y}^{-1}(y)\subset Y\times{\mathbb{P}}^{1}$. Next lemma deals with the $(-2)$ curves which are mapped to the exceptional curves of the morphism $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$. \[erretildeconmenodue\] Let $E$ be any $(-2)$ rational curve which is contained in a fiber of $f_{G}\colon G\to Y$ and which is mapped $2$-to-$1$ to a cuve $L\subset A$ such that $\epsilon(L)\in [I_{Y}]$. Then $R\cdot E=2$. Moreover $E\cdot\widetilde R=0$ If $E$ is a $(-2)$ curve contained in a fiber of $f_{G}\colon G\to Y$ which is mapped to an exceptional curve $L$ of the morphism $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$ then $\pi_{|_E}\colon E\to L$ is a $2$-to-$1$ covering. Hence $\pi_{\star}E=2L$. Then $0=E\cdot K_{G|Y}=E\cdot (\pi^{\star}K_{A|Y}+R)=2L^{2}+E\cdot R$. This implies $R\cdot E=2$. Then $E\cdot{\widetilde{R}}=E\cdot\pi^{\star}\pi_{\star}R-4= 2L\cdot\widetilde\tau-4=0$. By the local description of $f\colon S\to Y$ we know that there are $(b-1)c\left[{\widetilde{N}}_{{\rm{sing}}} +\sum_{j=0}^{[\frac{k}{2}]}({\widetilde{M_{j,0}}}+{\widetilde{M_{j,g}}})\right]$ mutually disjoint $(-2)$ rational curves, where the $[M_{j,i}]$ are given in [@HMo Formula 1.35]. \[combinatorics\] $\sum_{j=0}^{[\frac{k}{2}]}({\widetilde{M_{j,0}}} +{\widetilde{M_{j,g}}})\leq {\widetilde{N_{1}}}$. By [@HMo Formula 1.30] $[N_{1}]=\cup_{i=0}^{[\frac{k}{2}]}[M_{j}]$ where $[M_{j}]$ is defined in [@HMo Formula 1.28]. Since by [@HMo Formula 1.36] $[M_{j}]=\cup_{i=1}^{g}[M_{j,i}]$ the claim follows. Moreover let $$\label{discrepanza} e:=\sum_{j=0}^{[\frac{k}{2}]}(\widetilde{M_{j,0}}+{\widetilde{M_{j,g}}}).$$ Obviously $(b-1)ce=\sum _{j=0}^{[\frac{k}{2}]}(J_{Y,1_{j,0}}+J_{Y,1_{j,g}})$. We set $$\bigcup _{j=0}^{[\frac{k}{2}]}([J_{Y,1_{(j,0)}}]\cup [J_{Y,1_{(j,g)}}])=:\{a_{1},\ldots ,a_{(b-1)ce}\}\subset Y.$$ We recall that the morphism $h\colon S\to F$ contracts two rational curves of any fiber of $f_{S}\colon S\to Y$ over $\{a_{1},\ldots ,a_{(b-1)ce}\}\subset Y$. Moreover the fiber $F_{a_{i}}$ $i=1,\ldots ,(b-1)ce$ of $f\colon F\to Y $is smooth, but over a neighbourhood $U_{i}$ of $a_{i}\in Y$ the map $\rho_{F}\colon F_{U}\dashrightarrow U\times{\mathbb{P}}^{1}$ is [*[only a rational one]{}*]{}. We know that the fiber $f_{S}^{\star}(a_{i}):=S_{a_{i}}= F'_{a_{i}}+E_{a_{i}}+E'_{a_{i}}\subset S$ where $F'_{a_{i}}$ is a smooth curve of genus $g$, $E_{a_{i}}$ is a $(-2)$ rational curve such that $E_{a_{i}}\cdot F'_{a_{i}}=1$. Moreover we know that $E'_{a_{i}}$ is a $-1$-rational curve and $E_{a_{i}}\cdot E'_{a_{i}}=1$. \[calcolodiscrepanza\] Let $h\colon S\to F$ be the morphism which factorises the morphism $\zeta\colon G\to F$. Then $K_{F}^{2}=K_{S}^{2}+2(b-1)ce$. By Harris Morrison construction the contraction of $k-2$ rational curves in each one of the $(b-1)c{\widetilde N}_{1}$ singular fibers of $f_{G}\colon G\to Y$ gives the morphism $u\colon G\to S$. To reach $F$ we need to contract first $E'_{a_{i}}$ and then the image of $E_{a_{i}}$ for every $i=1,\ldots, (b-1)ce$. Hence $\zeta\colon G\to F$ is given by $h\zeta=h\circ u$ where $h\colon S\to F$ is a composition of $2(b-1)ce$ simple contractions. Hence $K_{F}^{2}=K_{S}^{2}+2(b-1)ce$. By Lemma \[tipodueduetre\] the points of type $(2,2)$ and the points of type $(3)$ give a smooth semistable fiber of $f\colon F\to Y$. By Proposition \[daGaS\] there are ${\widetilde{N}}_{{\rm{sing}}}(b-1)c$ singular fibers of $f\colon F\to Y$. Set $$r:={\widetilde{N}}_{{\rm{sing}}}(b-1)c.$$ Let $\{y_{i}\}_{i=1}^{r}={\rm{Sing}}(f)$ be the subset of $Y$ given by the points $y\in Y$ such that $f^{-1}(y)=F_{y}$ is a singular fiber. By the local description of the singular fibers of $f\colon F\to Y$ it follows that for any $l=1,\ldots, r$, $f^{-1}(y_{l})=F_{y_{l}}=F_{y_{l},i}\cup E_{y_{l}}\cup F_{y_{l},g-i}$ where $E_{y_{l}}$ is a $(-2)$ curve, $F_{y_{l},i}\cdot E_{y_{l}}= F_{y_{l},g-i}\cdot E_{y_{l}}=1$ and obviously $F_{y_{l},i}\cdot F_{y_{l},g-i}=0$, or $F_{y_{l}}=F'_{y_{l}}\cup E$ where $F'_{y_{l}}$ is a genus $g-1$ curve and $E$ is a rational $(-2)$ curve. We study the invariants of the surface $F$. \[eulerodif\]Let $f\colon F\to Y$ be a Harris Morrison family. Then the topological Euler characteristic of $F$ is: $$e(F)=4(g-1)(g(Y)-1)+2r.$$ By a standard topological argument it follows that $e(F)=4(g-1)(g(Y)-1)+\sum_{i=1}^{r} (e(F_{y_{i}})-(2-2g))$. By the analysis of singular fibers, see Proposition \[daGaS\], it follows that for every $i=1,\ldots, r$ $e(F_{y_{i}})-(2-2g)=2$. Then $e(F)=4(g-1)(g(Y)-1)+2r$. We point out the reader the very important relation: $$\label{tuttoli} {\widetilde{N}}_{1}=e+{\widetilde{N}}_{{\rm{sing}}}$$where $e$ is defined in Equation \[discrepanza\]. \[kappaquadroeffe\] Let $f\colon F\to Y$ be an Harris Morrison semistable fibration. Then $$K^{2}_{F}=c\left[(b-1)\left[2e +{\widetilde{N}}_{1}+(8g-7)\frac{{\widetilde{N}}_{3}}{3} \right]-3{\widetilde{N}}\right] +8{\widetilde{N}}(g(X)-1)(g-1)$$ We notice that by Corollary \[calcolodiscrepanza\] $K^{2}_{F}=2e(b-1)c+K^{2}_{S}$. Then by Lemma \[laesse\] $K^{2}_{F}=2e(b-1)c+(k-2)(b-1)c{\widetilde{N}}_{1}+K^{2}_{G}$. By definition of $R$ we have $K_{G}^{2}=R^{2}+2R\cdot\pi^{\star}K_{A}+(\pi^{\star}K_{A})^{2}$. By Equation \[autointe\] we have $$(\pi^{\star}K_{A})^{2}=kK^{2}_{A}=-k\left[8(g(Y)-1)+(b-1)c{\widetilde{N}}_{1}\right].$$ By projection formula and by Equation \[pistarR\] we have: $$2R\cdot\pi^{\star}K_{A}=2{\widetilde{\tau}}K_{A}= \left[ 2b(g(X)-1)-2c\right]{\widetilde{N}}+b(b-1)c\frac{2{\widetilde{N}}_{3}}{3}.$$ Now the claim follows by Proposition \[erreerre\] taking into account that $b=2g+2k-2$ and $$\label{generedi Y} 2g(Y)-2={\widetilde{N}}(2g(X)-2)+\frac{2}{3}(b-1)c{\widetilde{N}}_{3}.$$ since Riemann-Hurwitz formula. We put $$\label{alpha} \alpha:=(b-1)\left[2e +{\widetilde{N}}_{1}+(8g-7)\frac{{\widetilde{N}}_{3}}{3} \right]-3{\widetilde{N}}$$ \[notdependence\] For every fibration constructed as in [@HMo Theorem 2.5] we can write by Proposition \[kappaquadroeffe\] and by Equation \[alpha\]: $$\label{facile} K^{2}_{F}=c\alpha+8{\widetilde{N}}(g(X)-1)(g-1)$$ where the coefficient $\alpha$ does not depend on $g(X)$. \[kappaquadroeffehaarris\] If $f\colon F\to Y$ is the fibration constructed in [@HMo Theorem 2.5] then $$\chi({\mathcal{O}}_{F})=\frac{c}{12}\left[(b-1)\left[ +3{\widetilde{N}}_{1}+(12g-11)\frac{{\widetilde{N}}_{3}}{3} \right]-3{\widetilde{N}}\right] +{\widetilde{N}}(g(X)-1)(g-1) .$$ By Noether Identity and by Proposition \[eulerodif\] we have $$12\chi({\mathcal{O}}_{F})=K^{2}_{F}+4(g-1)(g(Y)-1) +2{\widetilde{N}}_{{\rm{sing}}}(b-1)c.$$ By Proposition \[kappaquadroeffe\] we can write $$12\chi({\mathcal{O}}_{F})=c\left[(b-1)\left(2e +{\widetilde{N}}_{1}+(8g-7)\frac{{\widetilde{N}}_{3}}{3} \right)-3{\widetilde{N}}\right] +$$ $$\qquad \qquad \qquad \qquad +8{\widetilde{N}}(g(X)-1)(g-1)+4(g-1)(g(Y)-1) +2{\widetilde{N}}_{{\rm{sing}}}(b-1)c.$$ By Equation \[tuttoli\] and by Equation \[generedi Y\] we next obtain: $$12\chi({\mathcal{O}}_{F})=c\left[(b-1)\left( 3{\widetilde{N}}_{1}+(12g-11)\frac{{\widetilde{N}}_{3}}{3} \right)-3{\widetilde{N}}\right] +12{\widetilde{N}}(g(X)-1)(g-1)$$ and the claim follows. We put $$\label{alphaprimo} \alpha':=\frac{1}{12}\left[(b-1)\left( 3{\widetilde{N}}_{1}+(12g-11)\frac{{\widetilde{N}}_{3}}{3} \right)-3{\widetilde{N}}\right].$$ In particular we have: \[notdependencebis\] For every fibration constructed as in [@HMo Theorem 2.5] we can write by Proposition \[kappaquadroeffehaarris\] and by Equation \[alphaprimo\]: $$\label{facilefacile} \chi({\mathcal{O}}_{F})=\alpha'c+{\widetilde{N}} (g(X)-1)(g-1).$$ where the coefficient $\alpha'$ does not depend on $g(X)$. \[primadifferenza\] Let $X$ be any smooth complete curve with genus $g(X)$. Let $f\colon F\to Y$ be the fibration constructed in [@HMo Theorem 2.5] then $$K^{2}_{F}-8\chi({\mathcal{O}}_{F})=c\left[(b-1) \left[2e-{\widetilde{N}}_{1}+\frac{{\widetilde{N}}_{3}}{9}\right] -{\widetilde{N}}\right].$$ It follows by Corollary \[kappaquadroeffehaarris\] and by Proposition \[kappaquadroeffe\]. The proof of the Theorem ------------------------ Now we want to analyse the relation between the fiber degrees $c_{i}$ for the divisors $\sigma_{i}\in |s+\pi_X ^\star C_i|$ and the genus $g(X)$ in order that a family $f\colon F\to Y$ as in [@HMo Theorem 2.5] exists if we start from a curve $X$. Since we are working over a product surface $X\times{\mathbb{P}}^{1}$ a divisor $L$ which is numerically equivalent to $s+c_{i}f$ is very ample iff $L_{|(s=0)}$ is very ample; hence a necessary condition is that $c_{i}$ is the degree of a very ample divisor on $X$. On the other hand if ${\mathcal{O}}_{X}(l)$ is a very ample sheaf on $X$ then $s+\pi_{X}^{\star}(l)$ is a very ample divisor on $X\times{\mathbb{P}}^{1}$. For a general $X$ if ${\mathcal{O}}_{X}(l)$ is a nonspecial very ample divisor then by Halphen theorem [@Ha Proposition 6.1] it follows that $c_{i}\geq g(X)+3$. Then if we want to construct families $f\colon F\to Y$ as in [@HMo Theorem 2.5] starting from a curve $X$ where $c$ is small with respect to $g(X)$, definitely we need to consider curves $X$ with very ample special divisors. \[secondaadifferenza\] Let $g,k\in\mathbb N$ with $3\le k \le \lfloor {(g+3)\over 2} \rfloor$. For every real number $\epsilon >0$, there exists a real number $\Delta(\epsilon)\geq 0$ such that there are families $f\colon F\to Y$ obtained by the Harris Morrison construction starting from any plane curve $X$ of genus $g(X)\geq \Delta(\epsilon)$ such that the following holds: $$8-\epsilon\leq \frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}\leq 8+\epsilon.$$ Since $X$ is a plane curve of genus $g(X)$ then by Clebsh formula its degree is $d(X)=\frac{3+\sqrt{8g(X)+1}}{2}$. We can consider $c_{i}=d(X)$ where $i=1,\ldots, b$ where $b=2(g+k-1)$. Then $c$ can be taken equal to $bd(X)$. For every Harris-Morrison family we can write: $$\label{facilefacilefacile} \frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}= \frac{c\alpha+8{\widetilde{N}}(g(X)-1)(g-1)}{c\alpha'+{\widetilde{N}} (g(X)-1)(g-1)}.$$ Since the parameters $\alpha$, $\alpha'$ [*[do not depend]{}*]{} on $g(X)$ since $g,k$ are fixed and since $d(X)=\frac{3+\sqrt{8g(X)+1}}{2}$ then we can find $\frac{2g(X)-2}{3+\sqrt{8g(X)+1}}\geq \frac{b||\alpha-8\alpha'|-\epsilon\alpha'|}{\epsilon(g-1)}$ to obtain $|\frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}-8|\le \epsilon$. \[pianodifferenza\] Theorem \[secondaadifferenza\] can be extended easily to subcanonical curves $X$ where the subcanonical degree is sufficiently small with respect to $g(X)$. We point out the reader that the invariants of surfaces of general type which supports fibrations as those of Proposition \[pianodifferenza\] are strongly influenced by the base $Y$ of the fibration, in a way which is quite new for the theory of surfaces of general type, as far as we know. We have shown Theorem stated in the Introduction. Maximal gonality and surfaces of positive index. ================================================ In [@HMo] the genus $g(X)$ plays no role, see: [@HMo Corollary 3.15]. In this work it plays an essential role due to the Equations \[facile\] and Equation \[facilefacile\]. We consider the expressions of $K^{2}_{F}$ and $\chi({\mathcal{O}}_{F})$ as (linear) polynomials in the variable $g(X)$. In this section we consider the Harris Morrison families obtained in [@HM] and with [*[maximal]{}*]{} gonality. In particular in this section we have: $$\label{casodisparo} g=2n+1, k=n+2, b=6n+4, m=(6n+3)c$$ or $$\label{casoparo} g=2n, k=n+1, b=6n, m=(6n-1)c$$ Let us see how $k$ influences $K^{2}_{F}$ and $8\chi({\mathcal{O}}_{F})$. We will use the following \[tecnico\] Let $f\colon F\to Y$ be an Harris Morrison genus-$g$ fibration as in [@HMo Theorem 2.5]. Assume that the gonality $k$ is maximal, that is $k=\frac{g+3}{2}$ if $g$ is odd and $k=\frac{g+2}{2}$ if $g\geq 4$ is even. Assume that the conjectured estimate in [@HMo p. 351-352] is true. Then if $g>>0$, we have $\alpha> 8\alpha'$. By Equation \[facile\], by Equation \[facilefacile\] and by Corollary \[primadifferenza\] we have $$\alpha-8\alpha'=(b-1)\left[2e-{\widetilde{N}}_{1}+ \frac{{\widetilde{N}}_{3}}{9}\right]-{\widetilde{N}}.$$ Since $e\geq 0$ and $b=2(g+k-1)>0$ it is sufficient to show that $(b-1)\left[ -{\widetilde{N}}_{1}+\frac{{\widetilde{N}}_{3}}{9}\right]-{\widetilde{N}}\geq 0$. In the case of maximal gonality up to the first factor by [@HM bottom of the page 351] if $k>>0$ it is conjectured that ${\widetilde{N}}_{3}\simeq (k-2){\widetilde{N}}_{1}$, ${\widetilde{N}}\simeq (k)(k-1)\frac {{\widetilde{N}}_{1}} {2}$. Finally assume that $g=2n+1$. By Equation \[casodisparo\] we have $b-1=6n+3$ then up to the first order $(b-1)\left[ -{\widetilde{N}}_{1}+\frac{{\widetilde{N}}_{3}}{9}\right]-{\widetilde{N}}\simeq {\widetilde{N}}_{1}\left[ (6n+3)\left[ -1+\frac{n}{9}\right]-(n+2)(n+1)\frac{1}{2}\right]= {\widetilde{N}}_{1}\frac{n^{2}-43n-18}{6}>0$ if $n\geq 44$, that is $k\ge 46$. By the same argument if $g=2n$ we have that $\alpha\geq 8\alpha'$ if $3n^{2}-131n+20\geq 0$ that is if $n\geq 43$ and then $k\ge 44$. \[secondo\] Let $f\colon F\to Y$ be an Harris Morrison genus-$g$ fibration starting from any plane curve $X$ of genus $g(X)>>0$. Assume that the gonality $k$ is maximal and that the conjectured estimates in [@HMo p. 351-352] are true. If $g$ is big enough, then $F$ is a surface of positive index i.e. $K^{2}_{F}> 8\chi({\mathcal{O}}_{F})$. Moreover if $f\colon F\to Y$ is general in its class then the irregularity of $F$ is $g(Y)$. In particular, $f\colon F\to Y$ is the Albanese morphism of $F$. For every Harris Morrison family as in the statment: $$\label{nonfacilefacilefacile} \frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}= \frac{c\alpha+8{\widetilde{N}}(g-1)(g(X)-1)}{c\alpha'+{\widetilde{N}} (g-1)(g(X)-1)}$$ since Equation \[facile\] and Equation \[facilefacile\]. Then the first claim is equivalent to show that $\alpha\geq 8\alpha'$ and under our assumption this follows by Lemma \[tecnico\]. Let $q(F)$ be the irregularity of $Y$. By contradiction assume that $q(Y)>g(Y)$. Then by universal property of the Albanese morphism it follows that the Jacobian of the general fiber of $f\colon F\to Y$ contains an Abelian subvariety; but the locus of such curves in $\overline{\mathcal{M}}_{g}$ is a proper closed. Hence the family is not a sweeping one: a contradiction to Proposition \[harris\]. We have shown the Proposition stated in the Introduction. We conclude by observing that if the conjectured estimates of Harris Morrison hold, then the families $F$ of Proposition \[secondo\] furnish an intriguing example of surfaces with ratio $\frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}$ asymptotically $8$, which are minimal as semistable models, but with slope of the supported fibration aymptotically equal to $12$. We think that this kind of divergence between the two fundamental ratios among the invariants of a fibered surface which is minimal as a semistable model is worthy to be studied in the light of the recent results of Urz[ú]{}a quoted in the Introduction of this paper. [Muk04]{} D. Chen, G. Farkas, I. Morrison, *Effective divisors on moduli spaces of curves*, preprint (2012), arXiv:1205.6138v1. J. Harris and I. Morrison, *Slopes of effective divisors on the moduli space of stable curves*, Invent. Math. 99 (1990), 321-355. J. Harris and D. Mumford, *On the Kodaira dimension of the moduli space of curves*, Invent. Math. 67 (1982), 23-86. R. Hartshorne, *Algebraic geometry*, Grad. Texts Math., Vol. 52, Springer Verlag, New York-Heidelberg-Berlin, (1977). M. M. Kapranov, *Chow quotients of Greassmannians I*, Adv. Soviet Math. 16, part 2, A.M.S. (1993) 29-110. M. M. Kapranov, *Veronese curves and Grothendieck-Knudsen moduli space ${\overline{M_{0,n}}}$*, J. Algebraic Geom. [**[2]{}**]{} (1993), no. 2, 335-365. B. Moishezon and M. Teicher, *Simply-connected algebraic surfaces of positive index*, Invent. Math. 89 (1987), no. 3, 601-643. Y. Miyaoka, *Algebraic surfaces with positive indices. Classification of algebraic and analytic manifolds (Katata, 1982)*, 281-301, Progr. Math., 39, BirkhŠuser Boston, Boston, MA, 1983. D. Mumford, *Stability of Projective Varieties*, L�Ens. Math. 23 (1977) 39-110. R. Pandharipande *Descendent bounds for effective divisors on ${{\overline {\mathcal M}}_{g}}$*, J. Algebraic Geometry, vol. [**21**]{}, (2012), 299-303. U. Persson, C. Peters, G. Xiao, *Geography of spin surfaces*, Topology [**35**]{} (1996), no. 4, 845-862. I. Reider *Geography and the number of moduli of surfaces of general type*, Asian J. Math. [**[9]{}**]{} (2005), no. 3, 407-448. L. Szpiro, *Séminaire sur les pinceaux de courbes de genre au moins deux*, Astérisque 86 (1981). G. Urz[ú]{}a, *Arrangements of curves and algebraic surfaces*, J. Algebraic Geom. [**[19]{}**]{} (2010), no. 2, 335-365. G. Urz[ú]{}a, *Arrangements of rational sections over curves and the varieties they define*. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. [**[22]{}**]{} (2011), no. 4, 453-486. G. Xiao, *Irregularity of surfaces with a linear pencil*, Duke Math. J. vol. [**55**]{}, no. 3 (1987), 597-602.

--- abstract: 'Microring resonators are critical photonic components used in filtering, sensing and nonlinear applications. To date, the development of high performance microring resonators in LNOI has been limited by the sidewall angle, roughness and etch depth of fabricated rib waveguides. We present large free spectral range microring resonators patterned via electron beam lithography in high-index contrast $Z$-cut LNOI. Our microring resonators achieve an FSR greater than 5 nm for ring radius of 30 $\mu$m and a large 3 dB resonance bandwidth. We demonstrate 3 pm/V electro-optic tuning of a 70 $\mu$m-radius ring. This work will enable efficient on-chip filtering in LNOI and precede future, more complex, microring resonator networks and nonlinear field enhancement applications.' author: - Inna Krasnokutska - 'Jean-Luc J. Tambasco' - Alberto Peruzzo title: Tunable large free spectral range microring resonators in lithium niobate on insulator --- [^1] [^2] [^3] Introduction ============ Microring resonators are fundamental components in any high-index contrast photonic platform [@Bogaerts:2012; @Vahala:03]. They are a highly sought after cavity component, as they enable on-chip field enhancement as well as spectral filtering and fast modulation of optical signals [@Bogaerts:2012; @Xia:07; @Baba:13; @Levy:11; @Hu:12]. In the past decade, microring resonators have been demonstrated in a multitude of platforms including silicon (Si) [@Xu:08; @Shi:12], silicon nitride (SiN) [@Popovic:06], aluminium nitride (AlN) [@Pernice:12; @Pernice2:12], galium arsenide (GaAs) [@Absil:00; @Ibrahim:02] and indium phospide (InP) [@Ciminelli:13]. The applications of microring resonators are vast, ranging from sensing biological samples [@Guo:06], to filtering and demultiplexing telecommunication lines [@Xia:07; @Ibrahim:02], and generating frequency combs for spectroscopy [@Jung:13]. Microring resonators are challenging photonic components to fabricate, as losses incurred in the cavity are greatly amplified. To achieve a large free spectral range (FSR) for telecommunication applications and sensing, small, single-mode high-index contrast waveguides are required. Microring resonators can also be cascaded to increase the spectral enhancement, or create various types of filters [@Kim:16] and this requires very precise and careful control of the 3 dB resonance bandwidth and FSR [@Popovic:06]. In general, a ring performance is limited either by the material properties, such as 2-photon absorption in the C-band of Si, or the ability to nanostructure the material to produce small waveguides with smooth sidewalls. Lithium Niobate (LN) could greatly benefit from microring resonators to enhance its nonlinear and electro-optic (EO) properties, as well as prepare it for telecommunication use. Traditionally, LN has only supported prohibitive low-index contrast waveguides made from Ti:LN [@Janner:2009] and PE:LN [@Jackel:82; @Stepanenko:16]. With the commercialization of Lithium Niobate On Insulator wafers (LNOI) [@Poberaj:2012kf], high-index contrast waveguides in LN can now be achieved. Small radius, multimode waveguide rings have been reported in LNOI, but suffered from high propagation loss due to fabrication imperfection [@Poberaj:2012kf; @7353164]. Recent improvement in the fabrication process has led to reduced sidewall roughness, enabling the fabrication of ultra-low loss waveguides [@Zhang:17; @Krasnokutska:18] and microring resonators with extremely high $Q$-factors [@Zhang:17]. Due to the complex nature of processing LNOI, low-loss, single-mode, compact rings (down to 30[$\mu$m ]{}of radius) in LNOI with a high FSR are yet to be reported. Due to the complex chemistry of etching LNOI, most etching processes result in $\sim50^\circ$ sidewall angle waveguide [@Wang:2017ul; @Liang:17], hampering the ability to produce small gaps, which are critical in the fabrication of grating couplers, compact directional couplers and multistage microring resonator filters; however, this problem was recently solved where $\sim75^\circ$ sidewall angle waveguides were reported [@Krasnokutska:18]. The tuning and reconfigurability of photonic components is a necessity for many practical applications. Tunable rings resonators have been reported in several photonic platforms, including Si, which achieve a high-speed modulation via carrier-depletion and thermal wavelength tuning via resistive heaters [@Li:11]. However, carrier depletion suffers from increased optical absorption and a limited response time, restricting the performance of high-speed switches. Electro-optics offers a solution to these challenges, and has been demonstrated in Si on LN [@Chen:13; @Chen:14], AlN [@Jung:14] and LNOI [@Wang:18; @Guarino:2007bv; @Siew:cx]. The devices reported to date in $Z$-cut LNOI have either required two-step laser lithography [@Guarino:2007bv] or had performance challenges due to the waveguides being multimode and having limiting propagation losses [@Siew:cx]. In this work, we present a detailed study of all-pass microring resonators fabricated monolithically in $Z$-cut LNOI from small, low-loss, high-index contrast and single mode C-band waveguides. We analyze the performance of multiple rings with varying radii from 30 [$\mu$m ]{}to 90 $\mu$m. The demonstrated heavily overcoupled microring resonators have a maximum FSR of 5.7 nm and a large 3dB resonance bandwidth that both agree well with the design and simulation. In contrast to previous work who focused on reaching high Qs, this work aims at filtering applications where strong coupling between the ring and bus-waveguide is desired which results in a larger bandwidth of the cavity resonance and a reduced Q correspondingly. Furthermore, we demonstrate the versatility of our fabrication process, etching down to 300nm trenches in LNOI, critical for advanced photonic components. We further report 3 pm/V electro-optic tuning of a 70 $\mu$m-radius microring resonator—to the authors’ knowledge, this is the largest to date in $Z$-cut LNOI. We expect the microring resonators in this work to pave the way towards on-chip filtering in LNOI with ring networks, as well as field enhancement applications such as switching and nonlinear photon generation. {width="0.8\linewidth"} Design and fabrication ====================== Microring resonators with radii 30–90 [$\mu$m ]{}were designed to obtain an FSR from 1.5 to 5.7 nm and were simulated using the commercially available software, Lumerical. Rings of varying radii were fabricated to analyze the FSR and performance for the TE and TM modes. The small bending loss needed for good operation of a 30 [$\mu$m ]{}microring resonator required high-index contrast single mode waveguides at 1550 nm. A mode solver was used to determine the dimensions required to ensure a sufficiently small TM polarization bend radius. The design of the waveguide includes the following parameters: rib height, top width, sidewall angle, refractive indices of the waveguide and claddings, and film thickness. The cross-section of a $Z$-cut rib waveguide cladded with SiO$_{2}$ is shown in Fig. \[fig1\](b). The small gap of 300 nm was chosen to heavily overcouple the rings and obtain wide bandwidth resonances, rather than extremely narrow resonances that require very precise wavelength tuning to access. A simulation of the $Q$-factor as a function of coupling region gap for the TM mode at 1550 nm of a 30 [$\mu$m ]{}microring resonator is shown in Fig. \[fig1\](a), and indicates that the 3dB bandwidth of the microring resonances (FWHM) is $\mathrm{FWHM}=\lambda_\mathrm{res}/Q=\sim$155pm, where $\lambda_\mathrm{res}$ is the wavelength of the resonance and $Q$ is the $Q$-factor. The simulation was performed using Lumerical Mode; the measured losses, as per the Fabry-Perot measurements presented in Fig. \[fig2\](a), were taken into account in the simulation model. The photonic components were fabricated by the process developed and described in our previous work [@Krasnokutska:18]. The process starts with 500 nm thick LN film, which is fabricated using the smart-cut technique on 2 [$\mu$m ]{}of SiO$_2$ layer and supported by a 500 [$\mu$m ]{}LN substrate. The next fabrication steps rely on electron beam lithography and lift-off of the e-beam evaporated metal layer to obtain a hard mask defining the photonic components. The scanning electron microscopy image (SEM) of a waveguide to a ring coupling region just after the metal lift-off process, is shown in Fig. \[fig1\](e). The components were then dry etched in a reactive ion etcher Fig. \[fig1\](d). Following etching, the waveguides were cladded with 3 [$\mu$m ]{}thick plasma-enhanced chemical vapor deposition (PECVD) SiO$_{2}$. The presented structures were etched deeper than in our previous work to achieve the necessary index contrast, reducing the waveguide bending radius. The rib waveguide cross-section, obtained via focused ion beam (FIB) slicing and scanning electron microscopy (SEM), shows a sidewall angle of 75$^{\circ}$ and an etch depth of 350 nm Fig. \[fig1\](c). Finally, the waveguide facets were diced using optical grade dicing to facilitate butt-coupling. The length of the chip, after all processing steps were completed, is 6 mm. Experimental results ==================== In order to confirm that the photonic components are not limited by the propagation loss, loss measurements were performed prior to the characterization of the microrings, using the Fabry-Perot loss measurement technique [@Regener1985]. Laser light at 1550 nm wavelength is coupled into and out of the polished facets of the waveguide using polarization maintaining (PM) lensed fibers with a mode field diameter of 2 [$\mu$m ]{}. A typical optical transmission spectrum for TM (the TE and TM modes have a similar response) is shown on the Fig. \[fig2\] (a). Linear inverse tapers 200 [$\mu$m ]{}long down to 200 nm width, at the waveguide ends, are used to improve the mode matching between the lensed fibre and the waveguide [@6895281], and improve the signal to noise ratio of the Fabry-Perot measurements. As the waveguide narrows, the mode field diameter at the input and output of the waveguide significantly increases, allowing improved mode matching with the mode of the lensed fiber. The total input and output coupling and propagation loss is 8 dB for a 6 mm long chip, compared to the 15 dB loss achieved with the straight waveguide without tapering section. The estimated propagation loss is less than 0.5 dB/cm for both the TE and TM modes, which is in agreement with the results obtained in our previous work [@Krasnokutska:18] {width="0.8\linewidth"} The fabricated microring resonators were characterized by sweeping the wavelength of the laser between 1530 to 1610 nm and recording their spectral responses with a commercially available high-speed InGaAs photodiode. The laser light was injected into and out of a 6 mm bus-waveguide via PM lensed fibers. To decrease the chance of interference between multiple oscillations inside of the photonic component, the inverse tapering section was not implemented for the microrings—this led to a drop in the mode matching efficiency. We observe that both TE and TM (Fig. \[fig2\](b)) modes reaches the largest FSR for the ring with the smallest radius 30 [$\mu$m ]{}; however, the TE and TM modes show different results in terms of the achievable $Q$-factor for this geometry. A $Q$-factor of $\sim9000$ was achieved for the TM mode whilst for the TE mode the $Q$-factor is significantly smaller $\sim1200$. As the radius of the ring increases, the $Q$-factor for TM mode remains almost unchanged (Fig. \[fig2\](b)); meanwhile, for the TE mode, it significantly increases (Fig. \[fig2\](b)) and the highest $Q$-factor has been achieved for the ring with radius of 90 [$\mu$m ]{}Fig. \[fig2\](d). This dissimilarity can be attributed to the difference in the bending loss between both modes. It was deduced by using our numerical model (Fig. \[fig1\](a)) and the value of intrinsic quality factor that TM mode bending loss for microring resonator of 30 [$\mu$m ]{}of radius is around 1.5 dB/cm, while for TE mode it estimated to be around 12 dB/cm for the ring with the same radius. By comparing theoretical and experimental results, the effective index for the TE mode is 1.85 and for the TM mode is 1.72 and the TM mode was confined to have an index contrast of $\sim0.272$, whilst the TE mode is lower $\sim0.247$. As the TM mode has a higher index contrast, a smaller bend radius is achieved, enabling smaller microring resonators to be realized. The TE mode bending loss decreased with increasing microring resonator radius, leading to an improvement in the $Q$-factor. The group indices for the TE and TM modes respectively, ${n_\mathit{g}^\mathrm{TE}}$ and ${n_\mathit{g}^\mathrm{TM}}$, are deduced from the fully-vectorial mode solver using the Sellmeier equations for lithium niobate: ${n_\mathit{g}^\mathrm{TM}}=2.33$ and ${n_\mathit{g}^\mathrm{TE}}=2.38$. The FSR can be calculated using $\mathrm{FSR}=\lambda^2/({n_\mathit{g}}L)$, where $L$ is the circumference of the ring ($L=2\pi R$), $R$ is the radius of the ring. The simulation curve is plotted with the measured FSR for different microring resonator dimensions in Fig. \[fig3\](a) and Fig. \[fig3\](d). The simulated $E$-field distributions of the fundamental waveguide modes at a wavelength of 1550 nm (found using an in-house mode solver) are included to the figures as insets: Fig. \[fig3\](b) for the TE mode, and Fig. \[fig3\](e) for the TM mode. Also included as insets, Fig. \[fig3\](c) and Fig. \[fig3\](f), show the measured power distribution at a wavelength of 1550 nm in a $3\times3$ [$\mu$m ]{}window; each cell defined by the white grid lines represents a single pixel (a single power measurement). The measured power distribution is performed by sweeping the fiber over the output facet of the waveguide, resulting in a convolution between the fiber mode and the waveguide mode, smearing and enlarging the appearance of the waveguide mode. {width="1\linewidth"} Electro-optic resonant wavelength tuning ======================================== An electrode consisting of Cr (20 nm) and Al (500 nm) is deposited directly on the upper cladding of the waveguide. The separation between the electrode and the waveguide is designed to be 3 $\mu$m, which is estimated to be close enough that the electric field extending from the electrode can effectively influence the LNOI waveguide, but far enough that the optical loss is not increased. Figure \[fig4\](a) shows the simulation result of the static electric potential performed using a finite element solver, with the voltage applied across the top and bottom electrodes. The bottom electrode, serving as a ground plane, is made from Cr (10 nm), Au (100nm) and Cr (10 nm). To demonstrate the electro-optic tuning of the device we apply a DC voltage from 0V down to -55V to the top electrode of the ring resonator with a radius of 70 $\mu$m. The resonance shifts with the applied voltage as shown in Fig \[fig4\](b) corresponding to a EO tunability of 3 pm/V. {width="1\linewidth"} Discussion ========== The Fabry-Perot transmission measurements were conducted on straight waveguides with inverse tapers at both ends and indicate low propagation loss for this platform. The overall insertion loss of the waveguides is dominated by mode-mismatch between waveguide and optical fiber, despite the significant improvement of provided the inverse tapers. Given that the straight waveguides measured have identical dimensions to the waveguides used in the ring resonators and were fabricated on the same chip, the propagation loss in the rings are concluded to be equally low loss. The demonstrated ring resonators are designed to be strongly overcoupled, increasing their 3 dB resonance bandwidth (and, conversely, reducing their $Q$-factor). A 300 nm gap in the bus waveguide to microring coupling region provides strong overcoupling. The potential of the nanofabrication process used in this work [@Krasnokutska:18] could be further extended to photonic components including grating couplers and compact directional couplers. The $Q$-factor measurements show that it is possible to achieve small and high performance microring resonators for the TM mode—critical for electro-optic and nonlinear applications. Meanwhile, the TE mode bending losses significantly limit the $Q$-factor of the smaller radius microring resonators; however, increasing the ring radius leads to a substantial increase in $Q$-factor. It demonstrated that the TM mode can achieve a smaller bend losses than the TE mode, as the index contrast of the TE fundamental mode is less than that of the TM fundamental mode, as verified by both our in-house mode solver, and by the $Q$-factor simulations conducted in Lumerical Mode for the 30 [$\mu$m ]{}ring. It was found that the theoretically predicted results for the microring resonators demonstrated in this paper are in a good agreement with the experimental results (Fig. \[fig3\]). The deviation for $n_{g}$ is less than $2\%$ leading to precise agreement between the designed and measured FSRs for different ring geometries. Using 350 nm deep ribs, a small TM bend radius was achieved to enable 30 [$\mu$m ]{}TM microring resonators with an FSR of 5.7 nm. This result is competitive with other high-index contrast leading platforms, such as SiN and AlN, For comparison, we report in Table \[tab:EO comparison\] a summary of experimental results on resonant wavelength tuning. It can be seen the tunable ring resonators have been realized in a multitude of photonic platforms. Silicon has reported very high EO tuning with large FSR [@Li:11]. More recent work has shown good performance in hybrid Si on LN, although it requires extra fabrication steps [@Chen:13; @Chen:14]. While using AlN has so far resulted in limited tunability [@Jung:14], LNOI photonics presents a promising approach to tunable ring resonators [@Wang:18; @Guarino:2007bv; @Siew:cx]. The results presented in this work combine good EO tunability with simple fabrication process of $Z$-cut LNOI single mode waveguides, which are readily compatible with other single mode photonic components and and will enable future low-loss and tunable filtering in LNOI. Material Radius of a ring ([$\mu$m ]{}) Q-factor FSR (nm) EO tuning (pm/V) ------------------------------------------------- -------------------------------- ---------- ---------- ------------------ SOI PN junction [@Li:11] 7.5 8000 12.6 26 SOI on LN with integrated electrodes [@Chen:13] 15 11500 7.15 12.5 SOI on LN [@Chen:14] 15 14000 7.15 3.3 AlN [@Jung:14] 60 500000 n.a. 0.18 $X$-cut LNOI [@Wang:18] n.a. 50000 n.a. 7 $Z$-cut LNOI [@Guarino:2007bv] 100 4000 1.66 1.05 $Z$-cut LNOI [@Siew:cx] 50 2800 3.2 2.15 $Z$-cut LNOI (this work) 70 7500 2.5 3 Conclusion ========== We have analyzed in detail the performance of large FSR microring resonators in $Z$-cut LNOI, fabricating rings of varying radii and reporting their characterization for TE and TM polarizations. The demonstrated advanced fabrication enables minimal separation (300nm) between monolithically defined adjacent features, whilst maintaining smooth waveguide sidewalls. We have verified that the optical characteristics of the fabricated microring resonators correspond well with the design and simulation. We have further demonstrated 3pm/V EO tuning of a 70 $\mu$m radius microring. These results will precede more complex photonic devices in LNOI, ranging from precise filtering with multistage microring resonators to electro-optically tunable devices.\ **Funding**\ Australian Research Council Centre for Quantum Computation and Communication Technology CE170100012; Australian Research Council Discovery Early Career Researcher Award, Project No. DE140101700; RMIT University Vice-Chancellors Senior Research Fellowship. **Acknowledgments**\ We thank Jochen Schröder for discussions. This work was performed in part at the Melbourne Centre for Nanofabrication in the Victorian Node of the Australian National Fabrication Facility (ANFF) and the Nanolab at Swinburne University of Technology. The authors acknowledge the facilities, and the scientific and technical assistance, of the Australian Microscopy & Microanalysis Research Facility at RMIT University. [35]{}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 [****,  ()](\doibase 10.1002/lpor.201100017),  [****,  ()](\doibase 10.1038/nature01939) [****,  ()](\doibase 10.1364/OE.15.011934) [****,  ()](\doibase 10.1364/OE.21.011869) [****,  ()](\doibase 10.1364/OE.19.011415) [****,  ()](\doibase 10.1364/OE.20.015079) [****,  ()](\doibase 10.1364/OE.16.004309) [****,  ()](\doibase 10.1063/1.3696082), [****,  ()](\doibase 10.1364/OL.31.002571) [****,  ()](\doibase 10.1364/OE.20.012261) [****,  ()](\doibase 10.1063/1.4722941),  [****,  ()](\doibase 10.1364/OL.25.000554) [****,  ()](\doibase 10.1364/OL.27.000803) [****, ()](\doibase 10.1364/OE.21.000556) [****,  ()](\doibase 10.1109/JSTQE.2005.862945) [****,  ()](\doibase 10.1364/OL.38.002810) [****,  ()](\doibase 10.1364/OE.24.009501) [****,  ()](\doibase 10.1002/lpor.200810073) [****,  ()](\doibase 10.1063/1.93615),  [****,  ()](http://jlt.osa.org/abstract.cfm?URI=jlt-34-9-2206) @noop [****,  ()]{} [****,  ()](\doibase 10.1109/LPT.2015.2508103) [****,  ()](\doibase 10.1364/OPTICA.4.001536) [****,  ()](\doibase 10.1364/OE.26.000897) @noop [ ()]{},  [****,  ()](\doibase 10.1364/OPTICA.4.001251) [****,  ()](\doibase 10.1364/OE.19.020435) [****,  ()](\doibase 10.1364/OE.21.027003) [****,  ()](\doibase 10.1364/OPTICA.1.000112) [****,  ()](\doibase 10.1364/OL.39.000084) [****,  ()](\doibase 10.1364/OE.26.001547) @noop [****,  ()]{} @noop [****,  ()]{} [****, ()](\doibase 10.1007/BF00691779) [****,  ()](\doibase 10.1109/LPT.2014.2357177) [^1]: These authors contributed equally to this work [^2]: These authors contributed equally to this work [^3]: alberto.peruzzo@rmit.edu.au

--- author: - | \ Institut de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology (BIST)\ Campus UAB, 08193 Bellaterra (Barcelona), Spain\ E-mail: - | O. Blanch$^{a}$, E. de Oña Wilhelmi$^{b}$, D. Galindo$^{c}$, J. Herrera$^{d}$, M. Ribó$^{c}$, J. Rico$^{a}$, A. Stamerra$^{e}$ (for the MAGIC Collaboration), F. Aharonian$^{f,g}$, V. Bosch-Ramon$^{c}$ and R. Zanin$^{f}$\ $^{a}$ IFAE-BIST, Campus UAB, 08193 Bellaterra (Barcelona), Spain\ $^{b}$ CSIC/IEEC, E-08193 Barcelona, Spain\ $^{c}$ Departament de Fśica Quàntica i Astrofísica, Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, IEEC-UB, Barcelona, Spain\ $^{d}$ IAC and Universidad de La Laguna, E-38200/E-28206 La Laguna, Tenerife, Spain\ $^{e}$ INAF - National Institute for Astrophysics, I-00136 Rome, Italy\ $^{f}$ Max-Planck-Institut fur Kernphysik, 69029 Heidelberg, Germany\ $^{g}$ Dublin Institute for Advanced Studies, Dublin 2, Ireland\ title: 'Gamma rays from microquasars Cygnus X-1 and Cygnus X-3' --- Introduction ============ CygnusX-1 is an X-ray binary comprised by a (19.2$\pm$1.9) M$_{\odot}$ O9.7Iab supergiant star and a (14.8$\pm$1.0) M$_{\odot}$ BH [@Orosz2011], classified as a microquasar after the detection of a one-sided relativistic radio-jet [@Stirling2001]. The jet seems to create a 5 pc ring-like structure detected in the radio that extends up to $10^{19}$ cm from the BH [@Gallo2005]. The system follows an almost circular orbit of $\sim 5.6$ d period [@Brocksopp1999a]. Flux modulation with the orbital period is detected in X-ray and radio [@Wen1999; @Brocksopp1999b; @Szostek2007], produced by the absorption/scattering of the radiation by the stellar wind. CygnusX-1 displays the two principal X-ray states of BH transients, the soft state (SS) and the hard state (HS). Both are described by the sum of a blackbody-like emission from the accretion disk that peaks at $\sim 1$ keV (dominant in the SS) and a power-law tail with a cutoff at hundred keV, expected to be originated by inverse Compton (IC) scattering on disk photons by thermal electrons in the so-called *corona* (dominant in the HS). During HS the source displays persistent jets from which synchrotron radio emission is detected, whilst in the SS, these jets are disrupted. CygnusX-1 showed a $4\sigma$-hint above 100 MeV during HS reported by [@Malyshev], using 3.8 yr of *Fermi*-LAT data. Evidences of flaring activity were also reported by *AGILE* ($> 100$ MeV, [@AGILE2010Sabatini; @AGILE2010Bulgarelli; @AGILE2013]) and by MAGIC ($> 100$ GeV, [@Albert2007]). The microquasar CygnusX-3 hosts a Wolf-Rayet (WR) star, although it follows a short 4.8 hr-orbit. The compactness of the system produces an unusually high absorption, which complicates the identification of the compact object (1.4 M$_{\odot}$ neutron star (NS) [@Stark2003] or $< 10$ M$_{\odot}$ BH [@Hanson2000]). Despite this high absorption, its X-ray spectrum shows the two aforementioned states. CygnusX-3 is the strongest radio source among the X-ray binaries, whose flux can vary several orders of magnitude during its frequent radio outbursts. These major flares happen only during SS (see [@Szostek2008]). CygnusX-3 was detected above 100 MeV, during SS by AGILE [@Tavani2009] and *Fermi*-LAT [@Fermi2009]. Its spectrum was described as a power law with photon indices 1.8$\pm$0.2 and $2.70\pm0.25$, respectively. Here, we present the results for GeV and TeV searches on CygnusX-1 using 7.5yr of *Fermi*-LAT data and $\sim 100$ hr of MAGIC data. We also show the latest results of CygnusX-3 obtained with MAGIC during the August-September 2016 flare. Observations and Analysis ========================= *Fermi*-LAT is the principal scientific instrument on the Fermi Gamma-ray Space Telescope spacecraft that studies the gamma-ray sky within an energy range of $\sim 20$ MeV to $\sim 500$ GeV (see [@PerformanceFermi]). To study CygnusX-1 in the high-energy (HE; $>60$ MeV) regime, we used 7.5 years of `Pass8` *Fermi*-LAT data (from MJD 54682–57420). The analysis was performed using *Fermipy*[^1], a package of python tools to automatize the analysis with the FERMI SCIENCE TOOLS (v10r0p5 package). We selected photon-like events between 60 MeV and 500 GeV, within a 30$^{\circ}$ radius centered at the position of CygnusX-1. Find more details in [@Zanin2016]. MAGIC is a stereoscopic system of two 17 m diameter Cherenkov Telescopes located in La Palma (Spain). Until 2009, MAGIC consisted in just one telescope [@Aliu2009]. After autumn 2009, MAGICII started operation [@Alecksic2012] and between 2011-2012, both telescopes underwent a major upgrade [@Alecksic2016]. MAGIC observed CygnusX-1 for $\sim 100$ hours between 2007 and 2014 mostly during its HS (see [@FernandezBarral2017]). This analysis was carried out with standard MAGIC software (MARS, [@Zanin2013]). Upper limits (ULs) at 95% confidence level (CL) were computed with the full likelihood analysis developed by [@AleksicLikelihood], assuming 30% systematic uncertainty. Between August and September 2016, CygnusX-3 experienced strong flaring activity in radio and HE regimes during its SS [@RadioATel; @FermiATel]. MAGIC observed the source $\sim 70$ hours between MJD 57623 to 57653, under different moonlight conditions (moon analysis performed following [@MoonPerformance]). ULs at 95% CL were computed following Rolke method [@Rolke2005]. Results ======= CygnusX-1 --------- *Fermi*-LAT skymap, between 60 MeV and 500 GeV, showed a point-like source at the position of CygnusX-1 with a TS=53. Moreover, detection only happens during HS (Figure \[FermiSkypmaps\]) with TS=49 above 60 MeV (division between HS and SS done following [@Gringberg2013]). Therefore, CygnusX-1 is only detected while displaying persistent radio-jets, as claimed by [@Malyshev] and confirmed afterwards by [@Zdziarski2016]. Making use of the HS sample, we searched for orbital modulation (assuming ephemeris $T_{0}=52872.788$ HJD, [@Gies2008]). Orbital phases ($\phi$) were split into two bins, one centered at $\phi=0$, the superior conjunction of the compact object (0.75 $< \phi <$ 0.25) and other at the inferior conjunction (0.25 &lt; $\phi$ &lt; 0.75). Detection only occurred during superior conjunction (TS=31). CygnusX-1 spectrum, from 60 MeV up to $\sim 20$ GeV, is well defined by a power law with photon index $\Gamma=2.3\pm0.1$ and normalization factor of $N_{0}=(5.8\pm0.9)\times 10^{-13}$ MeV$^{-1}$ cm$^{-2}$ s$^{-1}$, at an energy pivot of 1.3 GeV. Daily basis analysis was also performed, but no short-term flux variability was observed. The results between 0.1-20 GeV can be found in Figure \[CygX1LC\]. ![TS maps above 1 GeV centered in CygnusX-1, using HS (*left*) and SS subsamples (*right*).[]{data-label="FermiSkypmaps"}](./FermiSkypmaps.pdf){width="0.9\linewidth"} ![Multi-wavelength light curve for CygnusX-1. *From top to bottom:* Daily MAGIC ULs ($> 200$ GeV), HE gamma rays from the *Fermi*-LAT analysis (flux points are computed when $TS>9$), hard X-rays from *Swift*-BAT (15-50 keV, [@Krimm2013]), soft X-rays from MAXI (2–20 keV, [@Matsuoka2009]) and *RXTE*-ASM (3–5 keV range), and radio from AMI (15 GHz) and RATAN-600 (4.6 GHz). In the HE pad, dashed lines correspond to *AGILE* transient events. The horizontal green line in *Swift*-BAT pad shows the limit at 0.09 cts cm$^{-2}$ s$^{-1}$ given by [@Gringberg2013] to differentiate between X-ray states. HS and SS periods are highlighted with grey and blue bands, respectively.[]{data-label="CygX1LC"}](./CygX1LC_pre.pdf){width="0.7\linewidth"} With MAGIC, we searched for steady emission at energies above 200 GeV, making use of the total data set of $\sim 100$ hr. No significant excess was found, which led to an integral UL of $2.6\times 10^{-12}$ photons cm$^{-2}$ s$^{-1}$, assuming a power-law function with photon index $\Gamma=3.2$ (following former MAGIC results, [@Albert2007]). We also looked for gamma-ray emission at each X-ray state separately. In the HS, the source was observed for $\sim83$ hours between 2007-2011, which yielded no significant excess. Differential ULs are included in the spectral energy distribution (SED) shown in Figure \[CygX1SED\]. Orbital phase-folded and daily analysis were also carried out, with no evidence of emission. Integral ULs in a night-by-night basis are depicted in Figure \[CygX1LC\]. During SS, this microquasar was observed for $\sim 14$ hours in 2014. We searched for steady, orbital and short-term variability modulation, resulting in no detection. ![SED of CygnusX-1. Soft X-rays from *BeppoSAX* are shown in green stars [@DiSalvo2001], while hard X-rays are taken from *INTEGRAL*-ISGRI (red diamonds,[@Rodriguez2015]) and *INTEGRAL*-PICsIT (brown diamonds, [@Zdziarski2012]). In the HE and VHE band, results presented in this proceeding obtained with *Fermi*-LAT (violet points) and MAGIC (black ULs) are depicted. Sensitivity curves for CTA-North for 50 hours (https://www.cta-observatory.org/science/cta- performance/) and scaled to 200 hours of observations are shown in light blue and dark blue, respectively. No statistical errors are drawn, apart from the *Fermi*-LAT butterfly.[]{data-label="CygX1SED"}](./CygX1SED_preliminary.pdf){width="0.7\linewidth"} CygnusX-3 --------- We searched for steady emission with the MAGIC telescopes, making use of the available $\sim 70$ hours. No excess was found at energies above 300 GeV (accounting for the energy threshold of the sample with the highest moonlight) nor 100 GeV (using $\sim 52$ hours of dark data, i.e. under absence of Moon). Differential ULs, assuming a power-law function with photon index $\Gamma=2.6$, are presented in Figure \[CygX3SED\]. In this figure, *Fermi*-LAT spectrum from [@Fermi2009] is taken, nevertheless *Fermi*-LAT data for the August-September 2016 flare is currently being studied. No orbital (assuming ephemeris $T_{0}=2440949.892\pm 0.001$ JD, [@Singh2002]) or daily modulation was detected either. ![SED of CygnusX-3. Blue butterfly corresponds to *Fermi*-LAT spectrum during 2009 flare [@Fermi2009]. MAGIC ULs for the August-September flare are represented in light orange ($\sim 52$ hours, dark data) and dark orange ($\sim 70$ hours, dark+moon data). Sensitivity curves for CTA-North for 50 hours (dot-dashed line) and 200 hours (dashed lines) observations are shown.[]{data-label="CygX3SED"}](./CygX3SED_ICRC.pdf){width="0.6\linewidth"} Discussion and conclusions ========================== HE and VHE gamma-ray emission were proposed in the literature from both leptonic and hadronic mechanisms (see e.g. [@BoschRamon2006; @Romero2003]). Among these mechanism, the most efficient process seems to be a leptonic one, the IC. The target photons depend on the distance of the production site with respect to the compact object: close to it, thermal photons from the disk or synchrotron photons would dominate [@Romero2002; @BoschRamon2006]; at a binary scales ($\sim R_{orb}$, the size of the system), IC would take place on stellar photons; and finally, gamma-ray emission could also be produced in the interaction between the jet and the medium (as seen in radio for CygnusX-1, [@Gallo2005]). In the first two scenarios, gamma rays may suffer high absorption due to pair creation. CygnusX-1 --------- At the base of the jet, GeV photons would be absorbed by $\sim 1$ keV X-rays. Given the detection achieved with *Fermi*-LAT, and following [@Aharonian] approach, we estimated the smallest region size for HE gamma-ray production at $2\times 10^{9}$ cm. The radius of the corona is $\sim 20-50~R_{g}\sim 5-10 \times 10^{7}$ cm [@Poutanen1998], which allows us to conclude that the observed GeV emission is not originated in the corona, but most likely inside the jets. This scenario is reinforced by the fact that *Fermi*-LAT detection only happens during HS. If the hint of orbital modulation here reported is finally confirmed, GeV emission must arrive from inside the jets and not from their interaction with the environment. Assuming so, we can set an UL on the largest distance of the production site at $< 10^{13}$ cm (few times $R_{orb}$ for this source). On the other hand, this flux variability is only expected if the radiative process that leads to GeV emission is anisotropic IC on stellar photons [@Khangulyan2014]. Given that the density of stellar photons is dominant over other photon fields at distances $>10^{11}$ cm, we place the GeV emitter at $10^{11}$–$10^{13}$ cm from the BH. On the other side, the MAGIC non-detection above 200 GeV allows us to discard jet-medium interaction as possible region for VHE emission above MAGIC sensitivity level, since these regions are not affected by photon-photon absorption. At binary scales this non-detection is less conclusive because of the pair production. Although VHE radiation is predicted in the models (see e.g. [@Pepe2015; @Khangulyan2008]), several factors can prevent detection: low flux below MAGIC sensitivity even under negligible absorption [@Zdziarski2016], no efficient acceleration on the jets or strong magnetic field. Nevertheless, transient events by relaxation of attenuation at some distance from the BH or extended pair cascade [@Zdziarski2009; @BoschRamon2008] cannot be discarded. Transient emission related to discrete radio-emitting-blobs between HS and SS could also happen, as observed in the HE regime for Cygnus X-3. Hint of transient event was indeed reported previously by MAGIC [@Albert2007]. More sensitive instruments, like the future CTA (see Figure \[CygX1SED\]), could provide interesting information on CygnusX-1. CygnusX-3 --------- Despite observing the source during strong radio and HE outbursts, no significant excess was found by MAGIC. One has to consider the extremely high absorption due to the WR, which may affect VHE gamma-ray emission. At energies above 300 GeV, the maximum absorption is produced by near-infrared (NIR) photons ($E_{target}\sim 1.7 $ eV). Following [@Aharonian2005], absorption can be estimated as $\tau\sim \sigma_{\gamma \gamma}\cdot n_{NIR} \cdot R$, where $\sigma_{\gamma \gamma}\sim 1\times 10^{-25}$ cm$^{2}$ is the cross-section of the process, $n_{NIR}\sim L_{NIR}/(4 \pi R^{2} c E_{target})$ is the density of NIR photons and $R$ the size of the emitting region. Assuming the $L_{NIR}$ to be the bolometric luminosity, $L_{NIR}=10^{38}$ erg s$^{-1}$, the absorption is not negligible until a radius $R\sim 10^{13}$ cm, i.e. outside the binary scale ($R_{orb,CygX3}\sim2.5\times 10^{11}$ cm). Given the MAGIC non-detection, acceleration up to VHE could still happen inside the jets at a distance $\lesssim 10^{13}$ cm, maybe related to the HE emission site (produced at $>10^{11}$ cm to avoid absorption by X-rays). On the other hand, MAGIC observed the source simultaneously with the strongest radio flare (at 9.5 Jy on MJD 57651), being the MAGIC significance for this day compatible with background. This could reinforce the idea that VHE gamma rays, if produced, are originated inside the binary scale and not at the radio-emitting regions of the jets far from the compact object. Note, however, that the amount of time observed during strong radio flares is very limited. Figure \[CygX3SED\] shows the CygnusX-3 SED with the results at VHE during the 2016 flare, along with *Fermi*-LAT spectrum taken from the 2009 flare [@Fermi2009]. As mentioned above, dedicated *Fermi*-LAT analysis for the August-Sept 2016 flare is currently being performed. Our constraining ULs are also put in context with the CTA-North sensitivity curve for 50 hours of observations[^2] and the scaled one for 200 hours. [99]{} Orosz J. A., et al., 2011, ApJ, 742, 84 Stirling A. M., et al., 2001, MNRAS, 327, 1273 Gallo E., et al., 2005, Nature, 436, 819 Brocksopp C., et al., 1999b, A&A, 343, 861 Wen L., et al., 1999, ApJ, 525, 968 Brocksopp C., et al., 1999a, MNRAS, 309, 1063 Szostek A., et al., 2007, MNRAS, 375, 793 Done C., Gierli[ń]{}ski M., Kubota A., 2007, A&ARv, 15, 1 Malyshev D., et al., 2013, MNRAS, 434, 2380 Sabatini S., et al., 2010, ApJ, 712, L10 Bulgarelli A., et al., 2010, The Astronomer’s Telegram, 2512 Sabatini S., et al., 2013, ApJ, 766, 83 Albert J., et al., 2007, ApJ, 665, L51 Stark, M. J. et al. 2003, ApJ, 587, L101 Hanson, M. M., et al. 2000, ApJ, 541, 308 Szostek, A., et al. 2008, MNRAS, 388, 1001 Tavani, M., et al. 2009, Nature, 462, 620 Fermi LAT Collaboration, et al. 2009, Science, 326, 1512 Atwood, W. B., et al., The Astrophysical Journal 697, 1071–1102 (2009) Zanin R., et al., 2016, A&A, 596, A55 Aliu E., et al., 2009, Astroparticle Physics, 30, 293 Aleksić J., et al., 2012b, Astroparticle Physics, 35, 435 Aleksić J., et al., 2016b, Astroparticle Physics, 72, 76 MAGIC Collaboration et al., 2017, submitted to MNRAS Zanin R., et al., 2013, in Proc. of the 33st ICRC, Rio de Janeiro, Brasil. Aleksić J., et al., 2012a, J. Cosmology Astropart. Phys., 10, 032 Trushkin, S. A., et al. 2016a, The Astronomer’s Telegram, 9416 Cheung, C. C. et al. 2016, The Astronomer’s Telegram, 9502 MAGIC Collaboration, et al., 2017, ArXiv e-prints \[1704.00906\] Rolke, W. A., et al. 2005, Nuclear Instruments and Methods in Physics Research A, 551, 493 Grinberg V., et al., 2013, A&A, 554, A88 Zdziarski A. A., et al., 2016, ArXiv \[1607.05059\] Gies D. R., et al., 2008, ApJ, 678, 1237 DiSalvoT., et al., 2001, ApJ, 547, 1024 Rodriguez J., et al., 2015, ApJ, 807, 17 Zdziarski A. A., et al., 2012, MNRAS, 423, 663 Krimm H. A., et al., 2013, ApJS, 209, 14 Matsuoka M., et al., 2009, PASJ, 61, 999 Singh, N. S., et al. 2002, A&A, 392, 161 Bosch-Ramon V., et al., 2006, A&A, 447, 263 Romero G. E., et al., 2003, A&A, 410, L1 Romero G. E., et al., 2002, A&A, 393, L61 Aharonian, F. A., et al., apss 115, 201–225 October (1985) Poutanen J. and Coppi P. S., Physica Scripta Volume T 77, p. 57 (1998) Khangulyan D., et al., The Astrophysical Journal 783, p. 100 (2014). Pepe C., Vila G. S., Romero G. E., 2015, A&A, 584, A95 Khangulyan D., et al., 2008, MNRAS, 383, 467 Zdziarski A. A., et al., 2009, MNRAS, 394, L41 Bosch-Ramon V., et al., 2008, A&A, 489, L21 Aharonian, F., et al. 2005, Science, 309, 746 [^1]: http://fermipy.readthedocs.io/en/latest/ [^2]: Taken from (https://www.cta-observatory.org/science/cta- performance/

--- abstract: 'Despite greatly improved observational methods, the presence of magnetic fields at cosmological scales and their role in the process of large-scale structure formation still remains unclear. In this paper we want to address the question how the presence of a hypothetical primordial magnetic field on large scales influences the cosmic structure formation in numerical simulations. As a tool for carrying out such simulations, we present our new numerical code [`AMIGA`]{}. It combines an $N$-body code with an Eulerian grid-based solver for the full set of MHD equations in order to conduct simulations of dark matter, baryons and magnetic fields in a self-consistent way in a fully cosmological setting. Our numerical scheme includes effective methodes to ensure proper capturing of shocks and highly supersonic flows and a divergence-free magnetic field. The high accuracy of the code is demonstrated by a number of numerical tests. We then present a series of cosmological MHD simulations and confirm that, in order to have a significant effect on the distribution of matter on large scales, the primordial magnetic field strength would have to be significantly higher than the current observational and theoretical constraints.' bibliography: - 'archive.bib' date: 'accepted by MNRAS 2009 December 1.' title: 'Investigating the influence of magnetic fields upon structure formation with AMIGA – a C code for cosmological magnetohydrodynamics' --- \[firstpage\] cosmology: theory – magnetohydrodynamics – methods: numerical Introduction {#sec:introduction} ============ Today, we arrived at an era where cosmology has finally reached the stage of a precision science. The cosmological parameters have been determined to a typical precision of very few percents, resulting in the standard $\Lambda$CDM model of cosmology [@Komatsu09], and in this context the process of the cosmic structure formation can be studied in great detail. The non-linear nature of the gravitational dynamics and gas physics make the problem of structure formation virtually intractable analytically, and therefore the field relies on numerical simulations, which have been the driving force behind much of the theoretical progress. While the first codes were just able to follow the evolution of dark matter with the $N$-body method [e.g. @Klypin83; @Efstathiou85; @Davis85; @Barnes86; @Villumsen89; @Couchman91; @Suisalu95; @Kravtsov97; @Knebe01], the tremendous increase in computational power in the last years made it possible to include more and more different components into the simulations [e.g. @Couchman95; @Teyssier02; @OShea04; @Springel05; @Li08; @Xu08; @Collins09]. The inclusion of baryon physics, star formation, AGN feedback, radiative transfer and magnetic fields in modern cosmological codes opens the way to study many different aspects of the structure formation process. Magnetic fields play an important role in astrophysical phenomena on many different scales. Since most of the visible matter in the universe is ionized and magnetic fields are found on every scale where they can be observed, it is natural that they could also play a cosmological role. Unfortunately, magnetic fields on scales larger than individual galaxies are much more difficult to observe – any measurement of magnetic fields must rely on the presence of radiation and a magnetized medium. So far, the largest-scale observable magnetic fields are inside the atmospheres of galaxy clusters [@Carilli02; @Govoni04], reaching strengths of the order of $\mu$G in the core regions. Detection methods include studies of radio synchrotron and inverse Compton X-ray emission from clusters [@Harris79; @Rephaeli87] and surveys of Faraday rotation measures of polarized radio sources passing through the cluster atmosphere [@Clarke01]. However, it could be possible that there are magnetic fields on even larger scales than the largest observable objects. In fact, all the “empty space” in the universe could be magnetized [@Kronberg99]. A truly cosmological magnetic field would not be associated with collapsing or gravitationally bound structures, and would be coherent on scales greater than the largest known structures ($\sim 100$ Mpc) or even the Hubble radius, permeating the whole universe. Although a new generation of highly sensitive radio telescopes like LOFAR and SKA is underway, the detection of such a cosmological field will probably stay out of reach for the next decades. Currently, it is only possible to estimate an upper limit for the strength of such a large-scale field [@Vallee90; @Kronberg94], which should not be higher than $\sim$ 1 nG. In order to learn more about the nature and effects of magnetic fields, we have to rely on theoretical models and numerical simulations. Even if we may not be able to directly prove the existence of a large-scale magnetic field, the subject has important cosmological implications that must be considered. A large-scale magnetic field can have a significant impact on the dynamics of cosmic baryon flows, the thermal and ionization history of the universe, and the onset of structure formation [@Sethi08]. Different theories exist on the origin of a cosmological magnetic field. One class of theories suggests that the creation of a universal, “primordial” magnetic field happened already during a very early stage of the evolution of the universe. Unfortunately, at present such theories are highly parameter-dependent and rather inconclusive [@Subramanian08], and they do not yet allow to derive the field strength of such a primordial field; it is currently only possible to estimate some upper limit. Again, numerical simulations seem to be a very good alternative to learn more about this subject. Cosmological simulations including magnetic fields have already been conducted during the last decade. It has been shown by numerical simulations that indeed a large-scale primordial field of order $\sim$ 1 nG is needed to explain the presence of the observed magnetic fields in galaxy clusters [@Dolag99]. There have also been simulations of magnetic fields in filaments [@Brueggen05], cosmological simulations studying cosmic-ray electrons [@Miniati01], and the influence of magnetic pressure on the growth of baryonic structures [@Gazzola07]. However, aside from @Dolag99, these earlier codes only included magnetic fields passively, neglecting any possible back-reaction effects on the baryons; or, as in @Gazzola07, the magnetic component was modelled simply by adding an additional, isotropic pressure term in the hydrodynamic equations to account for the magnetic pressure. In order to track magnetic fields, baryons and dark matter simultaneously in a self-consistent way inside a cosmological framework, it is necessary to numerically solve the full set of equations of cosmological magnetohydrodynamics (MHD). Codes capable of this task have started to be developed only very recently. They include grid codes [@Fromang06; @Li08; @Collins09] as well as an SPH code [@Dolag09], none of which were publicly available at the time of writing. In this paper, we present the new cosmological MHD code [`AMIGA`]{}, aimed to close this gap.[^1] The [`AMIGA`]{} code originally started as a pure $N$-body code to study dark matter structure formation. It is the successor of the [`MLAPM`]{} code, a very powerful and memory-efficient AMR code published by @Knebe01. Here, we present a new numerical solver for cosmological MHD, now implemented into the [`AMIGA`]{} code. It greatly improves the possibilities of the code, allowing to model dark matter, baryons and magnetic fields simultaneously in a fully cosmological setting. The code utilizes the transformation to *supercomoving* coordinates, which greatly simplifies the numerical solution of cosmological MHD equations. There are implemented techniques to properly resolve strong shockwaves and supersonic flows in the baryon component, and to ensure the important condition of a divergence-free magnetic field down to machine precision. We also present a series of test problems, in order to verify the high accuracy of the code. After a technical description of the underlying principles and numerical methods, we use this new powerful tool to investigate and quantify the influence of a primordial magnetic field on the cosmic structure formation on large scales. Recent numerical efforts in cosmological MHD have concentrated on the modelling of magnetic fields inside individual galaxy clusters, since they are directly observable. For example, @Dubois08 focused specifically on the magnetic field inside one simulated cluster and found a relation between the field strength in the cluster core and cooling processes of the intracluster gas. We want to go a different path and study the influence of a hypothetical universal magnetic field, filling the whole universe, on large-scale structure formation. We currently only have some constraints on the maximum value of such a field, and no working general theory describing its origin or its development until the present time. At this juncture, it seems reasonable to choose a more pragmatic strategy. We want to address an important question for future cosmological simulations: suppose a cosmological primordial field exists – could it have a dynamically significant influence on the other constituents, dark matter and baryons? Does it need to be considered when performing simulations of the large-scale structure? At what strengths of a primordial field do its effects become relevant, and how do these field strengths compare to the current constraints? A series of numerical simulations of the large-scale structure formation, conducted with the new cosmological MHD code [`AMIGA`]{} and including primordial magnetic fields of different strength, is presented here to address these questions. The outline of this paper is as follows. Section \[sec:amiga\] is dedicated to our new cosmological MHD code [`AMIGA`]{}. We present the supercomoving framework, in which we formulate the equations of ideal MHD (\[sec:equations\]), the numerical scheme implemented in [`AMIGA`]{} (\[sec:scheme\]), and then we carry out different numerical tests to ensure that the code is functioning accurately (\[sec:codetesting\]). Section \[sec:cosmomhd\] presents our simulations of structure formation with a primordial large-scale magnetic field. We first use the MHD pancake formation as a toy model to estimate the effect of the fields (\[mhdpancake\]), and then present our 3D cosmological simulations with magnetic fields and analyse the obtained numerical data (\[cosmoMHDsimulations\]). Our conclusions are summarized in section \[sec:summary\]. The derivation of the supercomoving MHD equations is given in the Appendix. AMIGA {#sec:amiga} ===== [`AMIGA`]{} is a cosmological grid code containing the $N$-body solver with adaptive mesh refinement from its predecessor, the [`MLAPM`]{} code [@Knebe01], which is used for the dark matter and gravity equations, and a newly developed MHD solver to track the baryon physics and magnetic fields on a regular grid. Supercomoving ideal MHD equations {#sec:equations} --------------------------------- Our simulations contain dark matter particles, treated by an $N$-body code, and a baryon component that behaves like an ideal, superconducting plasma, together with a magnetic field. To treat all these components simultaneously in a self-consistent way, the equations of ideal magnetohydrodynamics (MHD) have to be solved together with the dark matter particle equations in a fully cosmological setting. For this, the equations of MHD are usually transformed to the comoving frame, defined by $$\begin{aligned} \boldsymbol x= \frac{\boldsymbol r}{a}\end{aligned}$$ In this frame, in the absence of additional forces, the mass points are at rest and the local density remains constant. However, if applied to the equations of MHD, the transformation renders them into equations with lots of additional factors explicitly depending on the cosmological expansion factor $a(t)$; these equations no longer have the form of hyperbolic conservation laws. Nevertheless, most other cosmological MHD codes use this formulation [@Li08; @Collins09]. A different transformation to so-called *supercomoving* coordinates has been proposed by @Martel98 to cast the equations into a more convenient form.[^2] It is defined in a very similar way, but additionally, the physical time $t$ is replaced by a new function $t_x$ depending on the expansion: $$\begin{aligned} \label{definition} \boldsymbol x= \frac{\boldsymbol r}{a}\;; \;\;\; \textrm{d}t_x=\frac{\textrm{d}t}{a^2}\end{aligned}$$ All time derivatives are now formulated in respect to that new function, and the equations are transformed accordingly. Here, we apply this transformation to the full set of MHD equations. Additionally, the physical quantities therein get substituted by a set of new ‘supercomoving’ quantities: $$\begin{aligned} \label{transformation} \rho_x&=\rho a^3 \;\;\; &T_x&=a^2T\\ \notag \phi_x&=a^2(\phi+\frac{1}{2}a \ddot a x^2) &S_x&=a^{-(3\gamma-8)}S\\ \notag p_x&=a^5p &\boldsymbol B_x &= a^{5/2} \boldsymbol B\\ \notag \varepsilon_x&=a^2\varepsilon &\mathcal H_x&=a\dot a\end{aligned}$$ where $\rho$ is the baryon density, $\phi$ the total gravitational potential, $p$ the thermal baryonic pressure, $\varepsilon$ the thermal baryonic energy, $T$ the temperature, $S$ the modified entropy (definition see section \[dualenergy\]), $\mathcal H$ the supercomoving Hubble constant and $\boldsymbol B$ the magnetic field strength. With the supercomoving framework defined in that way, the substitution causes most of the $a(t)$ depending terms to cancel out and results in the following equations (the $x$ subscripts are dropped from here on): $$\begin{aligned} \label{dmx_eq} &\frac{\textrm{d} \boldsymbol x_{DM}}{\textrm{d}t}=\boldsymbol v_{DM} \displaybreak[0] \\ \label{dmv_eq} &\frac{\textrm{d} \boldsymbol v_{DM}}{\textrm{d}t}=-\boldsymbol\nabla\phi \displaybreak[0] \\ \label{poisson_eq} &\Delta\phi=4\pi G(\rho_{tot}-\bar\rho_{tot})\cdot a(t)\displaybreak[0] \\ \label{density_eq} &\frac{\partial \rho}{\partial t}+\boldsymbol\nabla\cdot(\rho \boldsymbol v)=0\displaybreak[0] \\ \label{momentum_eq} &\frac{\partial \rho \boldsymbol v}{\partial t}+\boldsymbol\nabla\cdot\left [\rho \boldsymbol v \boldsymbol v + \left(p+\frac{B^2}{2\mu}\right)I-\frac{1}{\mu} \boldsymbol B \boldsymbol B\right]=-\rho\, \boldsymbol\nabla\phi\displaybreak[0] \\ \label{energy_eq} &\frac{\partial \rho E}{\partial t}+\boldsymbol\nabla\cdot\left[\boldsymbol v \left(\rho E+p+\frac{B^2}{2\mu}\right)-\frac{1}{\mu}\boldsymbol B(\boldsymbol v\cdot\boldsymbol B)\right]\displaybreak[0] \\ \notag &\;\;=-\rho \boldsymbol v \cdot(\boldsymbol\nabla\phi)+\mathcal H\frac{B^2}{2\mu}\displaybreak[0] \\ \label{induction_eq} &\frac{\partial \boldsymbol B}{\partial t}+\boldsymbol\nabla\times(-\boldsymbol v \times\boldsymbol B)=\frac{1}{2}\mathcal H\boldsymbol B\displaybreak[0] \\ \label{divB_eq} &\boldsymbol\nabla\cdot\boldsymbol B=0\end{aligned}$$ The derivation is presented in the appendix. Equations (\[dmx\_eq\]) and (\[dmv\_eq\]) are the equations of motion for the collisionless dark matter (DM) particles, where $\boldsymbol x_{DM}$ is the position and $\boldsymbol v_{DM}$ the velocity, respectively. (\[poisson\_eq\]) is Poisson’s equation for the total gravitational potential $\phi$, where $\rho_{tot}$ is the total density of combined gas and dark matter, and $\bar\rho_{tot}$ is the average total density of the simulated box, given by $$\begin{aligned} \label{average_dens} \bar\rho_{tot}=\Omega_0\,\rho_{crit}=\Omega_0\frac{3 H_0^2}{8\pi G}\end{aligned}$$ Next there are the supercomoving ideal MHD equations in conservative form, where equation (\[density\_eq\]) is the conservation law for gas density $\rho$, equation (\[momentum\_eq\]) for the gas flow momentum $\rho \boldsymbol v$, and (\[energy\_eq\]) for the total energy density $\rho E$ of the gas. $\boldsymbol B$ is the supercomoving magnetic field, whose evolution is given by the law of induction (\[induction\_eq\]), subject to the divergence-free condition (\[divB\_eq\]). The thermal pressure $p$ is obtained via an ideal equation of state, $$\begin{aligned} \label{eq_of_state} p=(\gamma -1)\rho \varepsilon\end{aligned}$$ where the adiabatic index equals $\gamma=5/3$ for a non-relativistic, monoatomic ideal gas, while the internal energy density $\rho\varepsilon$ of the gas follows from the total, kinetic and magnetic energy densities: $$\begin{aligned} \label{edens} \rho E=\frac{1}{2}\rho v^2+\frac{B^2}{2}+\rho\varepsilon\end{aligned}$$ Note that formally the equations (\[dmx\_eq\]) to (\[induction\_eq\]) closely resemble their non-comoving counterparts. The only differences are in Poisson’s equation for the gravitational potential and the two additonal magnetic Hubble terms at the right side of equations (\[energy\_eq\]) and (\[induction\_eq\]). These are now the only places where cosmology explicitly enters, namely in the form of the supercomoving $a(t)$ function, which has to be determined depending on the adopted cosmological model. These properties of the supercomoving MHD equations make them easier to implement than comoving MHD, while still containing the same physics. In particular they make it very easy to employ numerical schemes originally designed for non-cosmological purposes. The code uses the following internal units: The distance unit is the comoving boxsize $B_0$, so that $x,y,z\in[0,1]$ always; the density unit is the average density (\[average\_dens\]), so internally $\delta=\rho-1$; the unit for supercomoving time is the Hubble time $1/H_0$; and the magnetic field unit is defined by setting the magnetic constant to unity: $\mu=1$, so it disappears from all equations. The expansion factor $a(t)$ is evaluated by numerically integrating $$\begin{aligned} \label{codetimeline} \frac{\textrm{d}a}{\textrm{d}t}=a^2 \left[ \Omega_\Lambda (a^2-1) + \frac{\Omega_m}{a} - \Omega_m + 1 \right]^{1/2}$$ internally in the code (this relation results from the Friedmann equation). Note that here, $t$ is the supercomoving time (hence the additional $a^2$) and $H_0\equiv 1$ due to the internal units. Numerical scheme {#sec:scheme} ---------------- For an elaborate description of the AMR solver for equations (\[dmx\_eq\])-(\[poisson\_eq\]) we refer the reader to the `MLAPM` paper by @Knebe01. Below, we present the new solver for the MHD equations (\[density\_eq\])-(\[divB\_eq\]). ### MHD solver The MHD solver of [`AMIGA`]{} serves to solve the cosmological MHD equations (\[density\_eq\]) – (\[divB\_eq\]). It consists of a second-order unsplit Godunov-type central scheme and a constrained transport scheme to ensure a divergence-free magnetic field down to machine precision. It is essentially an expanded, cosmological version of the solver used by the <span style="font-variant:small-caps;">Nirvana</span> code [@Ziegler04; @Ziegler05], which in turn adopts the KNP solver for hyperbolic conservation laws [@Kurganov01].[^3] In the following section, we present the numerical algorithm, including the KNP solver, as implemented in [`AMIGA`]{}; for more on the theory behind the scheme, we refer the reader to these articles. The three hydrodynamic conservation laws (\[density\_eq\]), (\[momentum\_eq\]) and (\[energy\_eq\]) can be written in general vector form: $$\begin{aligned} \label{general_conserv} \frac{\partial \boldsymbol{u}}{\partial t}+ \frac{\partial \boldsymbol f^x}{\partial x}+ \frac{\partial \boldsymbol f^y}{\partial y}+ \frac{\partial \boldsymbol f^z}{\partial z}=\boldsymbol S_u\end{aligned}$$ where $\boldsymbol{u}$ is a vector containing the hydrodynamic variables, $$\begin{aligned} \boldsymbol{u}=\left( \begin{array}{c} \rho \\ \rho v_x \\ \rho v_y \\ \rho v_z \\ \rho E \end{array} \right)\end{aligned}$$ $\boldsymbol{f}^x, \boldsymbol{f}^y, \boldsymbol{f}^z$ are the flux functions $$\begin{aligned} \label{def_fluxfunc} \boldsymbol{f}^x=\left( \begin{array}{c} \rho v_x\\ \rho v_x^2+p+B^2/2-B_x^2 \\ \rho v_x v_y - B_x B_y \\ \rho v_x v_z - B_x B_z \\ \rho v_x( E + p + B^2/2)-B_x(\boldsymbol vÊ\cdot \boldsymbol B) \end{array} \right)\\ \notag \boldsymbol{f}^y=\left( \begin{array}{c} \rho v_y\\ \rho v_x v_y-B_x B_y \\ \rho v_y^2 +p+B^2/2- B_y^2 \\ \rho v_y v_z - B_y B_z \\ \rho v_y( E + p + B^2/2)-B_y(\boldsymbol vÊ\cdot \boldsymbol B) \end{array} \right)\\ \notag \boldsymbol{f}^z=\left( \begin{array}{c} \rho v_z\\ \rho v_x v_z-B_x B_z \\ \rho v_y v_z - B_y B_z \\ \rho v_z^2+p+B^2/2 - B_z^2 \\ \rho v_z( E + p + B^2/2)-B_z(\boldsymbol vÊ\cdot \boldsymbol B) \end{array} \right)\end{aligned}$$ and $\boldsymbol{S}_u$ are the source terms $$\begin{aligned} \label{u_source} \boldsymbol{S}_u=\left( \begin{array}{c} 0\\ \rho \,\partial_x \phi \\ \rho \,\partial_y \phi \\ \rho \,\partial_z \phi \\ \rho \boldsymbol v \cdot (\boldsymbol\nabla\phi)+HB^2/2 \end{array} \right)\end{aligned}$$ [`AMIGA`]{} was developed from a particle-mesh code and inherited its grid structure. We use the cells of this grid to locally store discrete values of the MHD quantities $\boldsymbol u, \boldsymbol B$. The hydrodynamical quantities $\boldsymbol u$ are stored as cell-averaged values $\boldsymbol u_{i,j,k}$ at the centres of the grid cells $i,j,k$ whereas the magnetic field $\boldsymbol B$ is instead arranged in a “staggered grid”, i.e. it is stored on the *cell faces* with a staggered collocation of the components $B_x$, $B_y$, $B_z$ (see Figure \[fig:grid\] a). Thus, every component of the $\boldsymbol B$ field is stored at another interface of the cell. In three dimensions, the KNP solver requires the reconstruction of all these quantities to all six faces of every cell (see Figure \[fig:grid\] b). We will denote the six interfaces with the letters $W,E,S,N,T,B$. For example, we obtain a hydrodynamical variable $u$ at the interfaces lying in $x$ direction (E and W interfaces) by $$\begin{aligned} u_{i,j,k}^E=u_{i,j,k}+\frac{1}{2}(\delta_x u)_{ijk}\\ u_{i,j,k}^W=u_{i,j,k}-\frac{1}{2}(\delta_x u)_{ijk}\end{aligned}$$ where $(\delta_x u)$ is a TVD slope limiter. For cosmological MHD simulations, we use the slope limiter of @vanLeer77: $$\begin{aligned} (\delta_x u)_{ijk}=\frac{2 \max \{(u_{i+1,j,k}-u_{i,j,k})\cdot(u_{i,j,k}-u_{i-1,j,k}),0\}}{u_{i+1,j,k}-u_{i-1,j,k}}\end{aligned}$$ For the $y$ direction (N and S interfaces) and the $z$ direction (T and B interfaces) the formulae are analogue. Since there are magnetic terms present in the hydrodynamic flux functions, we also reconstruct all magnetic field components at these interfaces. The only difference is that, due to the staggered grid collocation of $\boldsymbol B$, we have to average over pairs of opposing cell interfaces on the way. For the interfaces lying in $x$ direction this means $$\begin{aligned} B_{x\,i,j,k}^E=&B_{x\,i+\frac{1}{2},j,k}\\ \notag B_{y\,i,j,k}^E=&\frac{1}{2}\Big(B_{y\,i,j+\frac{1}{2},k}+B_{y\,i,j-\frac{1}{2},k} +(\delta_x B_y)_{i,j+\frac{1}{2},k}+(\delta_x B_y)_{i,j-\frac{1}{2},k}\Big)\\ \notag B_{z\,i,j,k}^E=&\frac{1}{2}\Big(B_{z\,i,j,k+\frac{1}{2}}+B_{z\,i,j,k-\frac{1}{2}} -(\delta_x B_z)_{i,j,k+\frac{1}{2}}-(\delta_x B_z)_{i,j,k-\frac{1}{2}}\Big)\\ \notag B_{x\,i,j,k}^W=&B_{x\,i-\frac{1}{2},j,k}\\ \notag B_{y\,i,j,k}^W=&\frac{1}{2}\Big(B_{y\,i,j+\frac{1}{2},k}+B_{y\,i,j-\frac{1}{2},k} +(\delta_x B_y)_{i,j+\frac{1}{2},k}+(\delta_x B_y)_{i,j-\frac{1}{2},k}\Big)\\ \notag B_{z\,i,j,k}^W=&\frac{1}{2}\Big(B_{z\,i,j,k+\frac{1}{2}}+B_{z\,i,j,k-\frac{1}{2}} -(\delta_x B_z)_{i,j,k+\frac{1}{2}}-(\delta_x B_z)_{i,j,k-\frac{1}{2}}\Big)\\ \notag\end{aligned}$$ Note that the $B_x$ component does not get reconstructed, since it is already stored at the needed interface. Again, for the other two directions there are analogue expressions. Now, we calculate the flux functions $\boldsymbol{\boldsymbol{f}}$ at E,W,N,S,T,B locations by putting the interface values of $\boldsymbol{u}$ and $\boldsymbol B$ into the definition (\[def\_fluxfunc\]). Once we have them, the numerical fluxes in and out of each cell at each interface are calculated utilizing the KNP flux formula: $$\begin{aligned} \label{knpfluxformula} \boldsymbol{F}^x_{i+\frac{1}{2},j,k}=&\frac{1}{a^+_{i+\frac{1}{2},j,k}-a^-_{i+\frac{1}{2},j,k}} \Big[ a^+_{i+\frac{1}{2},j,k}\boldsymbol{f}^x(\boldsymbol{u}^E_{i,j,k},\boldsymbol{B}^E_{i,j,k}) -\\ \notag &- a^-_{i+\frac{1}{2},j,k}\boldsymbol{f}^x(\boldsymbol{u}^W_{i+1,j,k},\boldsymbol{B}^W_{i+1,j,k}) + a^+_{i+\frac{1}{2},j,k} a^-_{i+\frac{1}{2},j,k} \boldsymbol{u}^W_{i+1,j,k}-\boldsymbol{u}^E_{i,j,k}) \Big]\\ \notag \boldsymbol{F}^y_{i,j+\frac{1}{2},k}=&\frac{1}{b^+_{i,j+\frac{1}{2},k}-b^-_{i,j+\frac{1}{2},k}} \Big[ b^+_{i,j+\frac{1}{2},k}\boldsymbol{f}^x(\boldsymbol{u}^N_{i,j,k},\boldsymbol{B}^N_{i,j,k}) -\\ \notag &- b^-_{i,j+\frac{1}{2},k}\boldsymbol{f}^x(\boldsymbol{u}^S_{i,j+1,k},\boldsymbol{B}^S_{i,j+1,k}) + b^+_{i,j+\frac{1}{2},k} b^-_{i,j+\frac{1}{2},k} \boldsymbol{u}^S_{i,j+1,k}-\boldsymbol{u}^N_{i,j,k}) \Big]\\ \notag \boldsymbol{F}^z_{i,j,k+\frac{1}{2}}=&\frac{1}{c^+_{i,j,k+\frac{1}{2}}-c^-_{i,j,k+\frac{1}{2}}} \Big[ c^+_{i,j,k+\frac{1}{2}}\boldsymbol{f}^x(\boldsymbol{u}^T_{i,j,k},\boldsymbol{B}^T_{i,j,k}) -\\ \notag &- c^-_{i,j,k+\frac{1}{2}}\boldsymbol{f}^x(\boldsymbol{u}^B_{i,j,k+1},\boldsymbol{B}^B_{i,j,k+1}) + c^+_{i,j,k+\frac{1}{2}} c^-_{i,j,k+\frac{1}{2}} \boldsymbol{u}^B_{i,j,k+1}-\boldsymbol{u}^T_{i,j,k+1}) \Big]\end{aligned}$$ In these flux formulae, $a^\pm$ denotes the maximum (+) and minimum ($-$) local speed of the hydro density flow at the cell surface in $x$-direction (wavespeed estimate of @Davis88): $$\begin{aligned} a^+_{i+\frac{1}{2},j,k}=\max\{ (v_x+c_f)^W_{i+1,j,k},(v_x+c_f)^E_{i,j,k},0\}\\ \notag a^-_{i+\frac{1}{2},j,k}=\min\{ (v_x-c_f)^W_{i+1,j,k},(v_x-c_f)^E_{i,j,k},0\}\end{aligned}$$ and $b^\pm$, $c^\pm$ the same for $y$ and $z$, respectively. The expression $$\begin{aligned} c_f=\sqrt{c_s^2+c_A^2}\end{aligned}$$ is an upper limit for the possible characteristic wave speed in the medium (fast magnetosonic speed), where $$\begin{aligned} c_s=\sqrt{\frac{\gamma p}{\rho}}\end{aligned}$$ is the sound speed in the medium and $$\begin{aligned} c_A=\sqrt{\frac{B^2}{\rho}}\end{aligned}$$ is the Alfvén speed. All these quantities get calculated on-the-fly using the reconstructed MHD variables and the equations (\[eq\_of\_state\]) and (\[edens\]). By adding the fluxes through all six cell interfaces, we now have the total flux in and out of the cell: $$\begin{aligned} \label{total_flux} \frac{\textrm{d}}{\textrm{d}t}\boldsymbol{u}_{ijk}=&-\frac{\boldsymbol{F}^x_{i+\frac{1}{2},j,k}-\boldsymbol{F}^x_{i-\frac{1}{2},j,k}}{\Delta x}-\frac{\boldsymbol{F}^y_{i,j+\frac{1}{2},k}-\boldsymbol{F}^y_{i-\frac{1}{2},k}}{\Delta y} \\ \notag &-\frac{\boldsymbol{F}^z_{i,j,k+\frac{1}{2}}-\boldsymbol{F}^z_{i,j,k-\frac{1}{2}}}{\Delta z}\end{aligned}$$ Note that no time discretization has been specified yet. We will later use (\[total\_flux\]) to update $\boldsymbol u_{i,j,k}$ applying a second-order Runge-Kutta scheme for the time integration (see section \[subsec:timeintegration\]). ### Constrained transport (CT) In order to track the time evolution of the magnetic field $\boldsymbol B$ as well, we want to solve the induction equation (\[induction\_eq\]) in a similar way. But when introducing magnetic fields to such grid algorithms, one is immediately faced with the problem that the solution for $\boldsymbol B$ must comply *at all times* to the additional condition $\boldsymbol\nabla\cdot\boldsymbol B=0$ down to the highest possible precision. Otherwise magnetic “sources” (monopoles) would be introduced that would lead to unphysical results (like forces parallel to the field direction). Physically, $\boldsymbol\nabla\cdot\boldsymbol B$ is a conserved quantity. But this is not the case for numerical calculations – a nonzero divergence will inevitably build up due to numerical errors, even if $\boldsymbol B$ was divergence-free at the beginning of the simulation. Even worse, numerical nonzero $\boldsymbol\nabla\cdot\boldsymbol B$ usually grows exponentially [@Brackbill80], and the code will crash. There are a handful of techniques to remedy the situation (see @Toth00 for a review and comparison study). @Brackbill80 introduced the “divergence cleaning” (or “Hodge Projection”) approach, which solves an extra Poisson’s equation to recover $\boldsymbol\nabla\cdot\boldsymbol B=0$ at each time step. But it was found later that the divergence cleaning can introduce substantial amounts of additional spurious structure [@Balsara04]. The second method [@Powell99; @Dedner02] extends the MHD equations to produce an additional divergence wave, which then advects the divergence out of the domain. This generally works; however, in cosmological simulations we always work with periodic boundary conditions. Thus, a wave cannot leave the domain, and this method is not applicable. In [`AMIGA`]{}, we use the arguably most elegant solution, the *constrained transport* (CT) method by @Evans88. In this method, the components of $\boldsymbol B$ are arranged in a way that ensures $\boldsymbol\nabla\cdot\boldsymbol B=0$ by definition. This is the reason why we introduced the staggered grid. Another issue with incorporating the induction law in the chosen finite-volume scheme is that the conserved quantity, i.e. the magnetic flux, is defined on a surface rather than on a volume (such as density). This leads to the fact that, as opposed to equations (\[density\_eq\])–(\[energy\_eq\]), the induction equation (\[induction\_eq\]), albeit it is a conservation law, contains the curl operator $\boldsymbol\nabla\times$ instead of the divergence operator $\boldsymbol\nabla\cdot\;$. Here, it is possible to apply a trick that still allows to use the same numerical scheme for the magnetic field as for the hydro variables $\boldsymbol u$. If we write $\boldsymbol E=-\boldsymbol v\times\boldsymbol B$, then the induction equation becomes $$\begin{aligned} \label{induction_eq_sourcefree} \frac{\partial \boldsymbol B}{\partial t} + \boldsymbol\nabla\times\boldsymbol E = \boldsymbol S_B\end{aligned}$$ with the magnetic Hubble source term $\boldsymbol S_B = \frac{1}{2} \mathcal H \boldsymbol B$ (in the non-cosmological case it is zero). Now this equation can be transformed into divergence form using an antisymmetric flux tensor, i.e. $$\begin{aligned} \label{fluxtensor} \frac{\partial \boldsymbol B}{\partial t} + \boldsymbol\nabla\cdot \left( \begin{array}{ccc} 0&E_z&-E_y\\ -E_z&0&-E_x \\ E_y&-E_x&0 \end{array} \right)= \boldsymbol S_B\end{aligned}$$ which is formally analogous to (\[general\_conserv\]); but instead of flux functions (\[def\_fluxfunc\]), we use the components of the antisymmetric flux tensor. It is essentially just a “resorting” of the vector components to make the curl appear formally as a divergence. With this, it is possible to construct numerical fluxes $\boldsymbol{E}$ using the same KNP flux formula as before: $$\begin{aligned} \label{knpfluxmhd} \boldsymbol{E}^x_{i+\frac{1}{2},j,k}=&\frac{1}{a^+_{i+\frac{1}{2},j,k}-a^-_{i+\frac{1}{2},j,k}} \Bigg[ a^+_{i+\frac{1}{2},j,k}\begin{pmatrix}0\\-E_z\\E_y\end{pmatrix}^E_{i,j,k} -\\ \notag &- a^-_{i+\frac{1}{2},j,k}\begin{pmatrix}0\\-E_z\\E_y\end{pmatrix}^W_{i+1,j,k} + a^+_{i+\frac{1}{2},j,k} a^-_{i+\frac{1}{2},j,k}( \boldsymbol B^W_{i+1,j,k}-\boldsymbol B^E_{i,j,k}) \Bigg] \displaybreak[0] \\ \notag \boldsymbol{E}^y_{i,j+\frac{1}{2},k}=&\frac{1}{b^+_{i,j+\frac{1}{2},k}-b^-_{i,j+\frac{1}{2},k}} \Bigg[ b^+_{i,j+\frac{1}{2},k}\begin{pmatrix}E_z\\0\\-E_x\end{pmatrix}^N_{i,j,k} -\\ \notag &- b^-_{i,j+\frac{1}{2},k}\begin{pmatrix}E_z\\0\\-E_x\end{pmatrix}^S_{i,j+1,k} + b^+_{i,j+\frac{1}{2},k} b^-_{i,j+\frac{1}{2},k}( \boldsymbol B^S_{i,j+1,k}-\boldsymbol B^N_{i,j,k}) \Bigg] \displaybreak[0] \\ \notag \boldsymbol{E}^z_{i,j,k+\frac{1}{2}}=&\frac{1}{c^+_{i,j,k+\frac{1}{2}}-c^-_{i,j,k+\frac{1}{2}}} \Bigg[ c^+_{i,j,k+\frac{1}{2}}\begin{pmatrix}-E_y\\E_x\\0\end{pmatrix}^T_{i,j,k} -\\ \notag &- c^-_{i,j,k+\frac{1}{2}}\begin{pmatrix}-E_y\\E_x\\0\end{pmatrix}^B_{i,j,k+1} + c^+_{i,j,k+\frac{1}{2}} c^-_{i,j,k+\frac{1}{2}}( \boldsymbol B^B_{i,j,k+1}-\boldsymbol B^T_{i,j,k}) \Bigg]\end{aligned}$$ Now, to ensure $\boldsymbol\nabla\cdot\boldsymbol B=0$, the constrained transport method enters. The idea is to discretize the magnetic field and the magnetic fluxes in such a way that the $\boldsymbol\nabla\cdot\boldsymbol B=0$ condition follows by definition of the scheme and is therefore conserved down to machine precision. From the face-centered fluxes (\[knpfluxmhd\]) we calculate edge-centered fluxes. An easy way to calculate $\boldsymbol E$ on a cell edge is averaging over the four interfaces touching this edge (see Figure \[fig:ctcell\]): $$\begin{aligned} \label{CT_ziegler} E_{x\,i,j-\frac{1}{2},k-\frac{1}{2}}= \frac{1}{4}(-\boldsymbol E^y_{z\,i,j-\frac{1}{2},k} -\boldsymbol E^y_{z\,i,j-\frac{1}{2},k-1}\\ \notag +\boldsymbol E^z_{y\,i,j,k-\frac{1}{2}} +\boldsymbol E^z_{y\,i,j-1,k-\frac{1}{2}}) \\ \notag E_{y\,i-\frac{1}{2},j,k-\frac{1}{2}}= \frac{1}{4}(-\boldsymbol E^x_{z\,i-\frac{1}{2},j,k} -\boldsymbol E^x_{z\,i-\frac{1}{2},j,k-1}\\ \notag +\boldsymbol E^z_{x\,i,j,k-\frac{1}{2}} +\boldsymbol E^z_{x\,i-1,j,k-\frac{1}{2}})\\ \notag E_{z\,i-\frac{1}{2},j-\frac{1}{2},k}= \frac{1}{4}(-\boldsymbol E^x_{y\,i-\frac{1}{2},j,k} -\boldsymbol E^x_{y,i-\frac{1}{2},j,k-1}\\ \notag +\boldsymbol E^y_{x\,i,j-\frac{1}{2},k} +\boldsymbol E^y_{x\,i-1,j-\frac{1}{2},k})\\ \notag\end{aligned}$$ The other nine edges bordering cell $i,j,k$ are obtained in the same way with the according indices. It has been pointed out recently by @Gardiner05 [@Gardiner08] that this is actually not the best way to construct edge-centered fluxes and that for certain cases, a reconstruction algorithm with proper upwinding gives better results than the simple averaging. However, for the tests and simulations presented here, this has no relevant effects. If it becomes necessary in the future to improve the algorithm, another scheme like the one by @Gardiner05 [@Gardiner08] may be implemented by simply adjusting equation (\[CT\_ziegler\]) accordingly. Alternatively, a numerical dissipation term can be introduced in the induction equation to smear out any possible numerical noise. Using the edge-centered fluxes, we get the temporal change of the staggered magnetic field components, in analogy to the hydro flux (\[total\_flux\]): $$\begin{aligned} \label{total_fluxmhd} \frac{d}{dt}B_{x\,i-\frac{1}{2},j,k}=-\frac{E_{z\,i-\frac{1}{2},j+\frac{1}{2},k}-E_{z\,i-\frac{1}{2},j-\frac{1}{2},k}}{\Delta y}\\ \notag +\frac{E_{y\,i-\frac{1}{2},j,k+\frac{1}{2}}-E_{y\,i-\frac{1}{2},j,k-\frac{1}{2}}}{\Delta z}\\ \notag \frac{d}{dt}B_{y\,i,j-\frac{1}{2},k}=-\frac{E_{z\,i+\frac{1}{2},j-\frac{1}{2},k}-E_{z\,i-\frac{1}{2},j-\frac{1}{2},k}}{\Delta x}\\ \notag +\frac{E_{x\,i,j-\frac{1}{2},k+\frac{1}{2}}-E_{x\,i,j-\frac{1}{2},k-\frac{1}{2}}}{\Delta z}\\ \notag \frac{d}{dt}B_{z\,i+\frac{1}{2},j,k-\frac{1}{2}}=-\frac{E_{y\,i+\frac{1}{2},j,k-\frac{1}{2}}-E_{y\,i-\frac{1}{2},j,k-\frac{1}{2}}}{\Delta x}\\ \notag +\frac{E_{x\,i,j+\frac{1}{2},k-\frac{1}{2}}-E_{x\,i,j-\frac{1}{2},k-\frac{1}{2}}}{\Delta y}\end{aligned}$$ By writing out $\boldsymbol \nabla\cdot\boldsymbol B$ with these discretizations one can immediately see that $\textrm{d}(\boldsymbol\nabla\cdot\boldsymbol B)/\textrm{d}t = 0$ by definition. Therefore, with compatible initial conditions, $\boldsymbol\nabla\cdot\boldsymbol B=0$ is conserved at all times. ### Time integration {#subsec:timeintegration} In order to calculate the temporal changes of the MHD variables, we discretize the equations (\[total\_flux\]) and (\[total\_fluxmhd\]) in time with a standard second-order accurate two-step Runge-Kutta method. At a given time $t$, we have $\boldsymbol u^t$ and $\boldsymbol B^t$ stored as described before. First, we estimate the appropriate timestep $\Delta t$ by using the usual timestep criteria. The time step $\Delta t$ should not exceed the actual age of the universe, $$\begin{aligned} \Delta t \le \frac{1}{\mathcal H}\;\;\; ,\end{aligned}$$ the fastest dark matter particle in the box should not travel farther than some fraction $\epsilon_1$ of one grid cell during one timestep, $$\begin{aligned} \Delta t \le \frac{\epsilon_1 \; \Delta x}{v_{DM,max}}\;\;\; ,\end{aligned}$$ and the same must hold for the fastest baryon flow speed encountered in the medium, $$\begin{aligned} \label{cflcondition} \Delta t \le \frac{\epsilon_2 \; \Delta x}{v_{max}}\;\;\; ,\end{aligned}$$ where $\epsilon_2$, the so-called CFL number, should always be $\le 0.5$ (CFL criterion). Now, denoting the right-hand side of equation (\[total\_flux\]) as $\boldsymbol F(\boldsymbol u, \boldsymbol B)$ and the right-hand side of equation (\[total\_fluxmhd\]) as $\boldsymbol E(\boldsymbol u, \boldsymbol B)$, we perform a predictor timestep $$\begin{aligned} \boldsymbol u^{t+\Delta t*}&=\boldsymbol u^t+\Delta t \cdot\boldsymbol F(\boldsymbol u^t, \boldsymbol B^t)\\ \notag \boldsymbol B^{t+\Delta t*}&=\boldsymbol B^t+\Delta t \cdot\boldsymbol E(\boldsymbol u^t, \boldsymbol B^t)\end{aligned}$$ These predictor values are used to calculate timestep-centered values $$\begin{aligned} \label{u_tmp} \boldsymbol u^{t+\frac{\Delta t}{2}*}&=\frac{1}{2}(\boldsymbol u^t+\boldsymbol u^{t+\Delta t*})\\ \notag \boldsymbol B^{t+\frac{\Delta t}{2}*}&=\frac{1}{2}(\boldsymbol B^t+\boldsymbol B^{t+\Delta t*})\end{aligned}$$ and in a final step, $\boldsymbol u$ and $\boldsymbol B$ get stepped forward in time using these timestep-centered values: $$\begin{aligned} \boldsymbol u^{t+\Delta t}=&\boldsymbol u^{t+\frac{\Delta t}{2}*}+\frac{\Delta t}{2}\cdot\boldsymbol F(\boldsymbol u^{t+\Delta t*}, \boldsymbol B^{t+\Delta t*})+\Delta t \cdot \boldsymbol S_u^{t+\frac{\Delta t}{2}*}\\ \notag \boldsymbol B^{t+\Delta t}=&\boldsymbol B^{t+\frac{\Delta t}{2}*}+\frac{\Delta t}{2}\cdot\boldsymbol E(\boldsymbol u^{t+\Delta t*}, \boldsymbol B^{t+\Delta t*})+\Delta t \cdot \boldsymbol S_B^{t+\frac{\Delta t}{2}*}\\ \notag\end{aligned}$$ where $\boldsymbol S_u$ are the timestep-averaged hydro source terms (\[u\_source\]) and $\boldsymbol S_B=\mathcal H\boldsymbol B/2$ is the magnetic Hubble term from equation (\[induction\_eq\]). It is important to point out that we must use a different time integration scheme here than the one in the $N$-body part. While leapfrog-based integrators like the one used by the $N$-body solver are well suited for Hamiltonian-type equations of motion and in particular the $N$-body problem, they are unstable for hyperbolic conservation laws like the MHD equations. However, the time integration schemes in the $N$-body solver and the MHD solver are connected only through the gravitational potential $\phi$. For both time integrators, the *timestep-averaged* gravitational potential $\phi^{t+\frac{\Delta t}{2}}$ is needed, so we can compute that from the time-averaged total density $\rho_{tot}^{t+\frac{\Delta t}{2}}$ (the sum of baryon and dark matter density). Figure \[fig:flowchart\] shows a flowchart of a full [`AMIGA`]{} timestep, illustrating how the $N$-body-solver and the MHD algorithm are interconnected. The code is completely modular, i.e. it is possible to run a pure $N$-body simulation, a pure hydrodynamic simulation, an MHD simulation or a combination of everything. For non-cosmological runs like the test cases presented before, it is possible to set $a=1, \dot a=0$, and the supercomoving MHD equations reduce to the ordinary MHD equations in proper physical coordinates. The gravity solver and the periodic boundary conditions can also be modified or disabled. ### Supersonic flows and the dual energy formalism {#dualenergy} When including gas physics in cosmological simulations, due to the extreme gravitational forces the gas flows can be accelerated to highly supersonic speeds, reaching Mach numbers of 100 and more. While the shocks and discontinuities that are created by such flows can be captured very well by the KNP solver, they are also followed by highly supersonic bulk flows of cold gas. A serious numerical problem occurs when trying to describe such flows with ideal MHD equations. At different places in the code, the thermal energy density $\rho\varepsilon$ and pressure $p$ have to be calculated. Normally, this happens through equations (\[eq\_of\_state\]) and (\[edens\]). The problem lies in the fact that in such cold, highly supersonic bulk flows, the value of $\rho \varepsilon$ will become several orders of magnitude smaller than the total energy density $\rho E$. Expression (\[edens\]) then contains a small difference of large numbers, leading to wrong results due to limited floating point precision. The thermal pressure and therefore the gas temperature cannot be tracked accurately anymore. To remedy this situation, @Ryu93 proposed to introduce the modified entropy as an additional equation from which the thermal energy could be calculated. Alternatively, @Bryan95 use an equation for the thermal energy itself (which is a bit problematic because it is not a conservation law). We follow the @Ryu93 method and define the modified entropy as $$\begin{aligned} \label{entropy_def} S=\frac{p}{\rho^{\gamma-1}}\;\;\;.\end{aligned}$$ In supercomoving coordinates, the supercomoving modified entropy follows the conservation law $$\begin{aligned} \label{entropy_eq} \frac{\partial S}{\partial t}+\boldsymbol\nabla\cdot(S\boldsymbol v)=- \mathcal H S (3\gamma -5)\end{aligned}$$ (see appendix for the derivation), where the right-hand side equals zero for $\gamma = 5/3$. For the whole simulation, we solve this equation alongside equations (\[density\_eq\]) through (\[energy\_eq\]) with the KNP solver. Now, whenever the thermal energy cannot be calculated accurately from the usual set of equations (\[density\_eq\]), (\[momentum\_eq\]) and (\[energy\_eq\]) – the “E system” – , we use the set of equations (\[density\_eq\]), (\[momentum\_eq\]) and (\[entropy\_eq\]) – the “S system” – instead. In this system, the pressure and thermal energy are calculated as follows: $$\begin{aligned} \label{S_edens} p=S\rho^{\gamma-1}\; ; \;\; \rho\varepsilon=\frac{S\rho^{\gamma-1}}{\gamma-1}\end{aligned}$$ After each timestep, the two systems have to be resynchronized: if the S system was used, the total energy has to be updated to be consistent with the new internal energy; if the E system was used, the entropy has to be updated according to equation (\[entropy\_def\]). The crucial step here is how to determine when to use the entropy for the calculation. A possible choice is to do this whenever the ratio of $\rho\varepsilon$ to $\rho E$ gets smaller than some threshold parameter (e.g. in @Collins09): $$\begin{aligned} \label{eta1} \frac{\rho E-\rho v^2/2-B^2/2}{\rho E}<\eta_1\end{aligned}$$ This works very well for most cases. However, in cosmological simulations sometimes another situation occurs when *all* energy components are near zero numerically, for example in the low-density regions between the shocks in the double pancake test (see section \[subsec:doublepancake\]). The condition (\[eta1\]) is false, nevertheless the pressure can not be tracked accurately. In order to deal with this issue, we propose a new approach: Instead of using the S system only in cases where (\[eta1\]) is true, we reverse the original condition and use the S system always, given that the entropy is conserved (that is, outside of shocks). Whether we are in a shock or not gets estimated by an additional criterion, checking for steep pressure gradients: $$\begin{aligned} \label{eta2} \frac{|\boldsymbol\nabla p|}{p}<\eta_2\end{aligned}$$ Whenever either (\[eta1\]) or (\[eta2\]) is true in a grid cell, we calculate the thermal energy using the S system. This new method gives accurate thermal quantities not only in strong shocks, but also in very cold low-density regions. Of course, since technically the S system abandons strict energy conservation in favor of accurately tracking the temperature, we must make sure that the use of the S system does not have a dynamical effect on the other hydrodynamic variables by choosing the parameters low enough. For the cosmological MHD simulations presented here, we used $\eta_1=0.001$ and $\eta_2=0.3$. The cell-averaged value of the magnetic energy density $B^2/2$ is required here for compatibility with the other energy terms. It is calculated by averaging over the opposing pairs of face-centered $\boldsymbol B$ components that enclose the cell: $$\begin{aligned} \label{cellcenteredB2} \left (\frac{B^2}{2}\right)_{i,j,k}&=\frac{1}{8}\Big[ (B_{x\;i+\frac{1}{2},j,k}+B_{x\;i-\frac{1}{2},j,k})^2 \\ \notag &+(B_{y\;i,j+\frac{1}{2},k}+B_{y\;i,j-\frac{1}{2},k})^2 +(B_{z\;i,j,k+\frac{1}{2}}+B_{z\;i,j,k-\frac{1}{2}})^2 \Big]\end{aligned}$$ Code testing {#sec:codetesting} ------------ To verify that the code is working correctly we applied it to a set of standard test problems. The $N$-body solver of [`AMIGA`]{} comes from its predecessor [`MLAPM`]{} [@Knebe01]. It has been thoroughly tested therein and successfully used for cosmological dark matter simulations (e.g. @Gill04a [@Gill04b; @Gill05; @Warnick06; @Warnick08]). Therefore we can concentrate here on testing the newly implemented MHD solver and its interplay with the gravity solver. The hydrodynamic part of the solver is applied on a 1D test, the Sod shock tube [@Sod78], and a 3D test, the Sedov-Taylor blast wave [@Sedov59]. To verify the MHD solver and the CT scheme we use the Brio-Wu problem [@BrioWu88] and the Orszag-Tang vortex [@OrszagTang79]. Then, combining MHD with the gravity solver and the cosmological expansion, we present the double pancake test of @Bryan95, which also serves as a stringent test on the dual energy algorithm. The computational domain for all tests is $x,y,z\in [0,1]$, conforming with the internal code units. All numerical runs up to $N=256$ cells of box length have been performed on the full three-dimensional $N^3$ box, even for 1D test problems, to test the code under more realistic circumstances. For higher resolutions we used a reduced 1D box to save computing time. It turned out that both recover the exact same result. Furthermore, in the case of pure hydrodynamic tests with no magnetic field, full MHD runs with the initial $\boldsymbol B$ set to zero recover the exact same result as purely hydrodynamic runs. ### Sod Shock tube For the Sod shock tube test, the simulation box is divided in two halves with constant initial states separated by a barrier between them that is removed at $t=0$. This generates a strong shock wave moving to the lower density region, a sound wave (rarefaction) in the opposite direction and a contact discontinuity. This simultaneous presence of different phases makes the shock tube an excellent method to check how well a code handles strong shocks. We chose the same initial conditions as in the original paper of @Sod78. The left and right initial states at $t=0$ are: $$\begin{aligned} &\rho_L=1\; ; \;\;&\rho_R=0.125\\ &p_L=1\; ; \;\;&p_R=0.1\end{aligned}$$ The initial velocity is zero everywhere, the polytropic index is $\gamma=1.4$. The boundary conditions are non-periodic and the system is evolved until $t_{end}=0.2$. By then the main shock will be located at $x=0.85$. Figure \[fig:shocktube\] shows the results perpendicular to the shock plane. In the code comparison suite of @Tasker08, this test is applied to other astrophysical codes. In direct comparison, [`AMIGA`]{} handles the problem very well. The main shock is between three and four cells wide, an accuracy comparable to PPM grid codes. All features of the analytical solution are recovered very accurately without oscillations or other numerical artefacts, except for a slight overshoot in the internal energy at the contact discontinuity. This is a common feature in grid codes and quickly disappears with higher resolution. The analytical reference solution for this problem was generated with an exact Riemann solver algorithm based on @Toro99, using a resolution of $N=10^4$. ### Sedov Blast wave The Sedov blast wave test is performed by injecting a large amount of thermal energy in a small, point-like region with uniform cold gas around it. This causes a strong explosion with a spherical shock front propagating outwards. The test is particularly useful to check if spherical symmetry is preserved by the code. It is important that on a cubic grid, shock fronts that are aligned parallel to the grid are resolved the same way as those moving at an oblique angle, because otherwise we would introduce an artificial anisotropy. Also, the shock front is very narrow and thus numerically challenging to resolve. As a reference we use the known self-similar analytical solution [@Sedov59]. For this test, the gas is at rest with $\rho=1$ and $v=0$ everywhere. We inject the energy $E_0=1$ in a spherical region of radius 3.5 cells in the centre of the simulation box. Then, the initial pressure equals $$\begin{aligned} p = \;\; \begin{cases} \;\;\dfrac{3(\gamma -1)E_0}{4\pi r^3} & \text{if } r < 3.5\; \Delta x \\[1em] \;\;10^{-5} & \text{else} \end{cases}\end{aligned}$$ We use $\gamma=7/5$ and evolve the blast wave until $t=0.0508$. The shock is then located at $r=0.314$. According to @Tasker08, the $r=3.5$ cells sphere is a good approximation of a point-like energy source, if we use a uniform grid with $N=256$ or higher; so we use exactly this resolution. The results are compared with the analytical solution in figure \[fig:sedov\]. The code conserves spherical symmetry and recovers the analytical solution well. The shock front is smoothed over a width of approx. four grid cells, so there is not much broadening due to a shock propagation on different angles with respect to the grid. The anisotropic scatter is not larger than one grid cell. The peak amplitude is somewhat lower than the analytical solution, but still very well compared with other codes [@Ricker00; @Tasker08]. The lowering is partly due to the fact that we use a finite spherical region instead of a really point-like source, which is just impossible with a grid code. Some codes also suffer from other problems: the shock position is sometimes underestimated by as much as 4%, for example <span style="font-variant:small-caps;">Enzo</span> (<span style="font-variant:small-caps;">Zeus</span>) in the @Tasker08 code comparison, since the initial energy lies in a region made of cubical cells and is therefore not exactly spherically symmetric. It can result in a deformed shockwave lagging behind the analytical solution, and the position will be wrong. However, [`AMIGA`]{} does not suffer from such problems, even if no technique is applied to make the start region more spherical (e.g. Gaussian smoothing or some other weighted distribution), and always recovers the correct shock front position. For both hydrodynamic tests in general, we find that the numerical accuracy of [`AMIGA`]{}’s hydrodynamic shock capturing is on par with the most popular astrophysical grid codes used today. ### Brio-Wu problem Now we want to test whether the MHD equations are implemented correctly into the solver. We use the test of @BrioWu88, which is one-dimensional, so the constrained transport reduces to a simple advection (we will move on to a multi-dimensional test afterwards). The Brio-Wu test is very similar to the Sod shock tube, with left and right initial states and a Riemann discontinuity in between; but in addition it features a magnetic field that has components both parallel and perpendicular to the shock plane, interacting with the shock and the different discontinuities. We use it to check the correct implementation of MHD equations and MHD shock capturing in one dimension, before continuing with multidimensional and cosmological tests. The initial conditions for this test are: $$\begin{aligned} \rho_L&=1\; ; &\rho_R&=0.125\\ p_L&=1\; ; \;\;&p_R&=0.1\\ \boldsymbol B_L&=\begin{pmatrix} 0.75\\ 1\\ 0 \end{pmatrix} \;\;\;\; &\boldsymbol B_R&= \begin{pmatrix} 0.75\\ -1\\ 0 \end{pmatrix}\end{aligned}$$ This leads to the propagation of all seven MHD waves (2 shocks, 2 Alfvén waves, 2 slow magnetosonic waves and a contact discontinuity) to travel through the box. For these initial conditions, two of the waves will have almost the same speed and interfere with each other, causing the overshoots typical for this test. Again, the velocity is zero everywhere, but this time we use $\gamma=2$. The system is evolved until $t=0.1$. Since there is no analytical solution known for this problem, we use a high-resolution run with $N=1024$ as a reference. For comparison, this test can also be found e.g. in @Ryu95. The numerical results obtained with [`AMIGA`]{} are shown in figure \[fig:briowu\]. In the pre-shock region, there is an additional overshoot in the x-direction velocity that disappears only at higher resolutions; otherwise, the results are quite accurate and show a good convergence with higher resolution. We acknowledge that the MHD equations are implemented correctly and the shock capturing is accurately handled by the code in the full MHD case. ### Orszag-Tang vortex The next task is to check the multidimensional MHD behaviour: the correct implementation of the constrained transport algorithm and the conservation of $\boldsymbol\nabla\cdot\boldsymbol B=0$. One of the most popular benchmark tests for that purpose is the Orszag-Tang vortex [@OrszagTang79]. This 2D test features an initially smooth flow that quickly develops MHD shocks and shock-shock interactions, and eventually breaks down into supersonic MHD turbulence. The initial conditions for this test are: $$\begin{aligned} &\rho=\frac{25}{36\pi} \; ; &p &= \frac{5}{12 \pi} \\ \notag &\boldsymbol v= \begin{pmatrix} -\sin (2\pi y)\\ \sin (2 \pi x)\\ 0 \end{pmatrix} \; ; &\boldsymbol{B}&= \frac{1}{\sqrt{4\pi}} \begin{pmatrix} -\sin (2\pi y)\\ \sin (4 \pi x)\\ 0 \end{pmatrix} \end{aligned}$$ We use periodic boundary conditions and $\gamma=5/3$, leading to $c_s=\sqrt{\gamma p/\rho}=1$ everywhere. Note that here, as for any MHD test, the initial magnetic field must be chosen so that it is divergence-free. The system is then evolved until $t=0.5\;$. The numerical results are shown in figure \[fig:orszagtang\]: maps of the temperature and magnetic field energy density in the computational plane, and the gas pressure along a cut at $y=0.4277$. [`AMIGA`]{} recovers the characteristic, complex shape of the solution in great detail, including the thin thread-like structure in the middle of the box. Magnetic field components are tracked correctly, and most importantly, $\boldsymbol\nabla\cdot\boldsymbol B$ equals machine zero at all times and positions. For comparison, the same test performed on other MHD codes can be found e.g. in @Ryu98, @Fromang06 (grid codes) and @Borve06, @Dolag09 (SPH codes). ### Double pancake test {#subsec:doublepancake} The cosmological pancake formation [@Zeldovich70] is a very popular test for cosmological hydrocodes, because it combines all of the essential physics (hydrodynamics, cosmological expansion and gravity) and is a very stringent test due to the strong shocks and non-linearities present after the caustic formation. Also, since it describes the evolution of a periodic, sinusoidal density perturbation with a certain wavelength $\lambda=2\pi / k$, it can be seen as a single-mode analysis of actual cosmological structure formation. The original pancake problem has an analytical solution [@Anninos94], which describes the collapse of a pressureless fluid up to caustic formation, happening at redshift $z_{c}$ (the moment of the first shell crossing). For baryonic collapse, it is valid as long as gas pressure is still negligible and can be used to set up the initial conditions. The Lagrangian positions and velocities are given by $$\begin{aligned} \label{pancake_ic} \rho(x_l)&=\rho_0\left[1-\frac{1+z_c}{1+z}\cos (kx_l)\right]^{-1}\\ \notag v(x_l)&=-H_0 \frac{1+z_c}{\sqrt{1+z}}\frac{\sin(kx_l)}{k}\end{aligned}$$ To set up the test, we transform them to Eulerian coordinates: $$\begin{aligned} x=x_l -\frac{1+z_c}{1+z} \frac{\sin (kx_l)}{k}\end{aligned}$$ This ‘single pancake’ test has been used by many authors to test cosmological hydrocodes, e.g. @Ryu93, @Gheller96, @Ricker00. We skip it here (it will appear again in section \[mhdpancake\]) and directly move on to the ‘double pancake’ test. It has been proposed by @Bryan95 and not been considered by any other group ever since. In this test, a second wave with one fourth of the wavelength is superimposed on the original wave, utilizing the same formula (\[pancake\_ic\]). The parameters of the two waves are $$\begin{aligned} \lambda_1&=64 \;\textrm{Mpc}/h & \lambda_2&=16\;\textrm{Mpc}/h\\ z_{c1}&=1 & z_{c2}&=1.45\end{aligned}$$ and are evolved from $z=30$ to $z=0$. The initial baryon temperature is set to $T_{init}=13\;\textrm{K}$ according to the formula given in @Anninos96. The double pancake is a much more challenging test than the single pancake, not only because it introduces stronger shocks, but also because the superimposed additional wave leads to much finer features which are harder to resolve by the code, especially the temperature peaks. The ratio of thermal to kinetic energy density $\rho\varepsilon/\frac{1}{2}\rho v^2$ in this test covers an extremely wide range between $10^{-9}$ and $10^5$, making it a very stringent test on the correct implementation of the dual energy formalism. The numerical results are shown in Figure \[fig:double\_pancake\]. While the low-resolution run with $N=64$ fails to recover all features of the solution (the structure of the central density peak, the peak separation in the temperature), they are present in higher resolutions. The high-resolution run with $N=1024$ impressively recovers the solution of @Bryan95, and due to the dual energy method, the sharp side peaks in the temperature are resolved extremely well. Also, in the extremely cold regions outside of the peaks the temperature is tracked correctly without any oscillations or other artefacts. We could not reproduce this result without our dual energy implementation or with other cosmological codes publically available. We acknowledge that the gravitational solver and the cosmological expansion (through supercomoving coordinates) are implemented correctly and that our variation of the dual energy formalism effectively improves tracking of the temperature without having a dynamical effect on the density or velocity of the gas. We took a closer look at how well the solution of this test converges. For this, we ran the exact same test as described above, with different resolutions from $N=8$ to $N=512$. As long as the behaviour is linear, that is, well before caustic formation, we can define the relative $\Delta_1$ error norm of a quantity $q$ as: $$\begin{aligned} \label{Delta_1} \Delta_1q=\frac{1}{N}\sum_i \frac{|q_i-q_{ref}|}{|q_{ref}|}\end{aligned}$$ The left side of figure \[fig:pancake\_error\] shows this error for the density and velocity as a function of resolution at $z=20$. We took the analytical solution (see above) as reference. Before the calculation of the error, the velocity was shifted by a constant, so that it does not approach zero. We find a constant convergence rate around $N^{-1.1}$ for the whole resolution range. For the final solution at $z=0$, we are far in the non-linear regime, and the solution features strong discontinuities and even singularities in the density. If we are to make a similar study here, we have to redefine what we take as the error. The analytical solution is not valid at this point, so we compare against a high-resolution run ($N=1024$), which is binned down accordingly, and use the relative self-convergence error: $$\begin{aligned} \Delta q=\frac{1}{N} \sum_i \frac{|q_i-q_{1024}|}{\max(|q_i|,|q_{1024}|)}\end{aligned}$$ The denominator is chosen this way to force all terms to be between 0 and 1. It leads to more meaningful results, because the differences can span over several orders of magnitude due to the strong discontinuities present. This error is plotted again for density and velocity at $z=0$ (the right side of figure \[fig:pancake\_error\]). Because of the non-linearity of the system, the solution converges much slower at first and reaches $N^{-1}$ only at high resolutions. There is a minimum of resolution required to get the shape of the solution right (around $N=128$), for lower resolutions there are features missing. This is especially the case for the velocity distribution with its pronounced minima and maxima, producing the kink in the convergence rate between $N=64$ and $N=128$, and after that the convergence improves. When comparing this performance, it turns out that even in @Bryan95, where the same test is run with a third-order accurate piece-wise parabolic (PPM) code (while our scheme is second-order piece-wise linear), the density distribution converges as $N^{-1.5}$ in the linear regime; and for $z=0$, it does not get better than $N^{-1}$ either. The PPM code of @Ricker00 reaches only $N^{-0.6}$ at $z=7$ for the single pancake. In this context, we can safely state that our code performs adequately well. Being a second-order scheme, the MHD solver requires less computational steps and is faster than higher order methods, while attaining comparable accuracy in the nonlinear regime of structure formation. Cosmological MHD simulations {#sec:cosmomhd} ============================ Having tested the functionality of the [`AMIGA`]{} code, we now move on and combine the $N$-body solver, MHD and gravity to perform full cosmological MHD simulations. The aim of this section is to quantify the impact that the introduction of initial large-scale magnetic fields has on simulations of the evolution of dark matter and baryons in a cosmological context. MHD pancake {#mhdpancake} ----------- Before running simulations with realistic cosmological initial conditions, we use the Zel’dovich pancake collapse model in the sense of a single-mode analysis to get an idea of what effects to expect from the presence of a magnetic field. We use a wave in $x$-direction with $z_c=1$ and $\lambda=64\textrm{ Mpc}/h$, but with a few modifications. First, we also want to study the dark matter component. So we include dark matter particles and baryons simultaneously in the simulation, using a baryon fraction $f_b=0.165$ (like later in the full simulations). Both follow the same density and velocity distribution initially. Then, we apply a perpendicular, constant initial magnetic field $\boldsymbol B_{init}$ pointing in $y$-direction. This whole system is then evolved in a flat EdS universe ($\Omega_0=1,\;\Omega_\Lambda=0$) until $z=0$. We repeat the same simulation for a wide range of different initial magnetic fields from $B_{init}=10^{-11}$ G, where the magnetic terms are neglectably small and dynamically unimportant, up to $B_{init}=2\cdot 10^{-6}$ G, where the magnetic field accounts for several percent of the total energy density of the gas. Figure \[fig:MHDpancake\] shows the density and temperature of the baryons, the distribution of the dark matter particles and the magnetic field strength at $z=0$ for different runs. Initial fields up to up to 0.05 $\mu$G do not have any significant effect on the density profiles (and the other quantities) and the field strength just follows the density profile of the baryons. Higher fields, however, induce large changes at the shock and post-shock regions, slowing down the baryon collapse and smearing out the baryon density profile. High field strengths effectively prevent the build-up of sharp, high-temperature baryon peaks. Although the situation is not directly transferrable to a full 3D simulation, this study gives us a hint on the general behaviour of density peak formation under the influence of a magnetic field. The dark matter distribution is generally much less affected than the baryon component – it does not interact directly with the magnetic field, but only indirectly through the gravitational force of the baryons. Since the dark matter particles are collisionless, they do not clump all together in the centre, but pass through each other (this happens exactly at $z=z_c$) and form side peaks. The shape of the central dark matter peak gets somewhat distorted if the baryons clump differently due to the field, but the height and position of the side peaks that form after $z_c$ lie outside the shocked regions and are practically not affected. For a more quantitative view on these effects, we took the non-magnetic numerical pancake solution at $z=0$ as a reference and calculated the deviation created by an initial magnetic field as the relative $\Delta_1$ “error” norm (\[Delta\_1\]) of baryon and DM density distributions over the interval of interest $x \in [0.43,0.57]$, a quantity very sensitive to numerical deviations (Figure \[fig:MHDpancake2\]). Expectedly, the deviation rises with higher initial fields, and the dark matter is less affected than the baryons. It should be noted that there is a certain level of numerical noise: If one changes a code parameter, like the CFL number, initial timestep, or dual energy parameters (within reasonable values of course), the relative deviation will be typically around $\Delta_1 \rho \approx 10^{-4}$. Smaller deviations can therefore be considered statistically insignificant. We can see that, in order to have a significant effect of, say, $\Delta_1 \rho = 1\%$, the energy density of the initial magnetic field at $z=30$ has to be at least around $10^{-7}$G. Cosmological MHD simulations {#cosmoMHDsimulations} ---------------------------- ### The initial magnetic field In order to conduct simulations of the cosmic structure formation that take into account the primordial magnetic field, one must choose appropriate initial conditions. Yet, the possible strength[^4], shape or origin of a primordial field remains unclear at the moment. Theories on this subject suggest that a cosmological large-scale magnetic field was already present before recombination ($z\sim 1000$). Such a primordial magnetic field could have been produced during inflation [@Turner88; @Gasperini06], much like the primordial density fluctuations that led to structure formation, or during subsequent phase transitions [@Gopal05]. Unfortunately, the predictions of such models involving string theory and particle physics are presently highly parameter dependent and rather inconclusive. Certain models can lead to fields as large as 1 nG, while others predict fields that are many orders of magnitude smaller [@Subramanian08]. While at the present there is no working theory to estimate the possible strength of a primordial magnetic field, it is feasible to give some constraints. If a magnetic field coherent at cosmological scales was present in the early universe, it should have left its imprint on the linear polarization of the CMB by Faraday rotation. Based on that, @Kahniashvili09 derived an upper limit for a primordial magnetic field based on the CMB polarization power spectrum from the WMAP 5-year data. They find that at a scale of 100 Mpc, the field amplitude must have been smaller than 0.7 nG. At smaller scales of 1 Mpc, the upper limit may be as high as 30 nG, depending on the assumed power spectrum. Another way to constrain the primordial magnetic field stems from Big Bang Nucleosynthesis (BBN). The presence of a magnetic field during BBN would have changed the nuclear reaction rates, thus resulting in an altered abundance of lighter elements like ${}^{3}$He, ${}^{4}$He, ${}^{7}$Li. To be compatible with the current agreement between BBN theory and element abundancy observations, a primordial field must be smaller than some critical value. First constraints derived in that manner were pretty high, up to 1 $\mu$G [@Kernan96]; later, @Grasso01 deduced a more realistic value of 1 nG for Mpc-scale fields with the help of some additional assumptions. For the simulations presented in the following section, we assume a primordial field already present before the starting time of the simulation. We further assume it to be constant and homogeneous in the whole simulation box, since the focus lies on how structure formation is affected by magnetic fields on scales larger than individual structures. It could be argued that a homogeneous primordial field pointing in one direction contradicts the assumption of an isotropic universe by creating a direction of preference; but the anisotropy created by such initial conditions has no impact on a statistical analysis of the baryon evolution, because the angle between the field vector and the baryon flow, a crucial quantity for the magnetic force on the baryons, is still randomly distributed, and the magnetic pressure does not depend on the field direction at all. It is also worth noticing that ideal MHD predicts the magnetic field lines to follow the baryon distribution. In fact, gravitationally collapsing baryonic structures would completely reshape the magnetic field distribution up to the point that any information on the original shape of the primordial field would be lost [@Dolag02], so the initial field shape should not significantly influence the results. ### Overview and initial conditions For this study, we carried out a set of cosmological 3D simulations with varying primordial field strengths $B_{init}$. The simulations model a universe containing baryons and dark matter particles in a three-dimensional 64 Mpc$/h$ box with periodic boundary conditions, using a baryon fraction of $f_b=0.165$. The initial conditions used for all of the simulation runs were created from an initial CDM power spectrum corresponding to the WMAP 5 parameters [@Komatsu09]: $\Omega_0=0.273$ and $\Omega_\Lambda=0.726$ with the <span style="font-variant:small-caps;">PMCODE</span> IC package [@Klypin97]. The initial density distribution of the baryons follows the dark matter. The initial field is set to a constant magnetic field in $y$-direction: $\boldsymbol B_{init} = (0,B_{init},0$). Apart from the different initial magnetic field strengths, the initial conditions are identical for all the runs. The simulations were run from the chosen starting redshift $z_{init}=30$ until $z=0$ with the full MHD version of the [`AMIGA`]{} code on a regular $256^3$ grid utilizing OpenMP statements for parallelization. ------------ ------------------------------------- ----------------------------- Simulation Initial comoving magnetic Initial *physical* magnetic field strength $B_{init}$ at $z=30$ field strength at $z=30$ B0 0 G 0 G B1 5.79 $\cdot 10^{-10}$ G 5.56 $\cdot 10^{-7}$ G B2 5.79 $\cdot 10^{-9}$ G 5.56 $\cdot 10^{-6}$ G B3 5.79 $\cdot 10^{-8}$ G 5.56 $\cdot 10^{-5}$ G ------------ ------------------------------------- ----------------------------- : Identifiers and initial magnetic field strengths for the MHD simulations used in this paper \[tab:overview\] The initial magnetic field values are summarized in Table 1. While the lowest initial field (B1) is compatible with current theoretical and observational estimates for a large-scale field, the highest initial field (B3) is significantly higher than all current upper limits. The energy density of the B1 field at $z=30$ is equivalent to the kinetic energy density of a gas with the average gas density at that redshift moving at 0.4 km/s. This is clearly too small to be dynamically important at any stage of the simulation. The B3 field, on the other hand, is $10^2$ times stronger and has $10^4$ times more energy density, so we should expect an effect due to its presence. ### Magnetic field influence on the large-scale distribution of baryons Figure \[fig:MHD256maps\] shows maps of the baryon density and magnetic energy density at the final redshift $z=0$ of our simulations, with the weakest and strongest initial magnetic field that was simulated. Even for the very high magnetic field of the B3 run, the baryon distribution does not change significantly; a closer look reveals that the distribution becomes slightly smeared out, featuring less fine structure. The weaker initial fields do not have an influence at all. The magnetic field shows the expected behaviour: in ideal MHD, it is frozen into the gas motion and largely follows the gas distribution. In the following, we want to analyse quantitatively whether the presence of magnetic fields in these simulations alters the result for the other constituents. The smearing of structural features by magnetic fields can be quantified by looking at the power spectrum at $z=0$. In figure \[fig:relativePk\] we plot the power spectrum $P_{bar}(k)$ of the baryons relative to the power spectrum of the non-magnetic run. It can be seen that the additional magnetic pressure and the altering of baryon flows by magnetic tension leads to a characteristic suppression of structure formation at finer scales. This is, however, a very small effect that becomes important only at smaller scales and high magnetic fields. The high initial field in the B3 run lowers the power spectrum by 10 % on a scale of 1 Mpc$/h$, the weaker field in the B2 model only has a (very slight) effect below 1 Mpc$/h$, and the more realistic field of the B1 model has no effect whatsoever. On scales above a few Mpc$/h$, no influence can be seen even with the strongest field in our simulation runs. On scales well below 1 Mpc$/h$, additional processes inside the individual collapsing structures become important, especially in their core regions. There, the magnetic field is influenced by cooling flows, turbulence and field tangling, which is not resolved by the large-scale runs conducted here. They can be better addressed by finer simulations of individual objects like the study of @Dubois08 on a single magnetized galaxy cluster. Figure \[fig:contourplot\] shows the number of cells with a certain value of baryon density $\rho$ and temperature $T$ within the simulated box, again for the strongest initial field and without a field. While the latter shows the characteristic shape known from other cosmological codes (see e.g. @Ryu93 for similar figures), a strong magnetic field leads to an additional maximum at the bottom, where regions with very cool gas are located. We can compare this directly to the result for a single Zel’dovich wave in Figure \[fig:MHDpancake\], where a similar effect occurs: a broadening and smearing of the density profile and the formation of a cool region behind the shock front. These results are in good agreement with @Gazzola07, where the same smoothing of the mass distribution with shallower density profiles and “washed out” finer density clumps can be seen, although they add the magnetic field simply as an additional isotropic pressure term instead of a proper MHD treatment. In any case, for magnetic fields of order $\sim1$ nG and below, no effect whatsoever can be seen on the scales resolved by our simulation. To summarize, on scales of $\sim1$ Mpc and above, only primordial fields significantly higher than of order $\sim$ nG have a noticeable effect on the baryon dynamics and gas distribution, which can be safely stated to be outside the upper theoretical and observational limits. Summary and Conclusions {#sec:summary} ======================= In this paper, we present the new numerical $C$ code [`AMIGA`]{} designed to perform cosmological magnetohydrodynamic simulations. It contains the powerful and memory-efficient AMR $N$-body code from its predecessor [`MLAPM`]{} @Knebe01, as well as a newly developed Eulerian grid-based MHD solver based on @Ziegler04 and @Ziegler05. The new code allows to simulate dark matter, baryon physics and magnetic fields in a self-consistent way inside a full cosmological framework. To facilitate the numerical solution of cosmological MHD equations, the code is working with *supercomoving* coordinates, a transformation that greatly simplifies the equations, while preserving the fully cosmological setting. There are implemented techniques to properly resolve strong shockwaves and supersonic flows in the baryon component, and to ensure the important condition of a divergence-free magnetic field down to machine precision. By conducting a series of test problems we acknowledge the high accuracy of this new code. As a first application of the new code, we present simulations of the cosmic structure formation with primordial magnetic fields. Such large-scale magnetic fields, possibly of cosmological origin, can be expected from different theoretical models and observational evidence of magnetic fields inside galaxy clusters. We want to address the question whether they could be a relevant factor for the large-scale dynamics in cosmological simulations. The simulations carried out with [`AMIGA`]{} model a $\Lambda$CDM universe with the WMAP-5 cosmology in a comoving 64 Mpc$/h$ computational volume, and its evolution from redshift $z_{init}=30$ to $z=0$. The applied primordial field strengths range from about 0.5 nG (a likely value from current constraints) to about 50 nG, which is significantly higher than current theoretical and observational constraints for magnetic fields on such large scales. The analysis of the simulations reveals that only in this last case, a large-scale magnetic field has a statistically significant influence on the baryon dynamics. Then, the magnetic pressure and tension leads to a suppression of baryonic small-scale structure and smears out density peaks, visible in the baryonic power spectrum. However, even the highest simulated initial field has no noticeable effect on scales above a few Mpc$/h$. We can therefore conclude that, since current theoretical and observational constraints predict a large-scale field not much stronger than $\sim$ 1 nG, at least outside of the core regions of gravitationally collapsed structures it cannot have a significance for the baryonic component during large-scale structure formation, neither on the power spectrum nor on the actual distribution. Even though our simulations do not have the required resolution to study the (internal) properties of individual objects, we nevertheless like to close with a brief discussion of our findings in that direction. We observed (though not explicitly presented here) that the mass function of collapsed structures remains unaffected even for magnetic fields as large as the ones in model B3. Furthermore, the shape of (dark matter) haloes also appeared unaltered when increasing the strength of the primordial magnetic field. And for the baryon fraction – for which @Gazzola07 have shown a dependence on the magnetic field strength – our own results are unfortunately affected by resolution effects: while stronger magnetic fields lead to a depletion of baryons in smaller mass objects (cf. Figure 8 in @Gazzola07), the same is caused by a lack of resolution in cosmological codes (@Crain07 [@Rudd08]); hence, we observe this effect but attribute it to our resolution. Further studies and more refined simulations in this direction are necessary to clarify this subject in greater detail. Acknowledgements {#acknowledgements .unnumbered} ================ TD acknowledges support through the AstroSim network of the European Science Foundation (Exchange Grant 2496). AK acknowledges funding through the Emmy Noether programme of the DFG (KN 755/1). AK is further supported by the Ministerio de Ciencia e Innovación (MICINN) in Spain through the Ramon y Cajal programme. We further acknowledge the LEA Astro-PF collaboration and the AstroSim network (Science Meeting 2387) for the financial support of the workshop “The local universe: from dwarf galaxies to galaxy clusters” held in Jabłonna near Warsaw in June/July 2009, during which a part of this work has been conducted. The simulations presented herein have been performed on the <span style="font-variant:small-caps;">Babel</span> cluster at the Astrophysical Institute Potsdam (AIP). The authors wish to thank Jochen Klar from AIP for providing the analytical reference solutions for the hydrodynamic test cases. \[lastpage\] Appendix: Derivation of the supercomoving MHD equations {#appendix-derivation-of-the-supercomoving-mhd-equations .unnumbered} ======================================================= As the starting point for the supercomoving transformation we take the full set of equations describing dark matter, baryons and magnetic fields in the ordinary non-comoving frame: $$\begin{aligned} \label{proper_a} &\frac{\textrm{d} \boldsymbol r_{dm}}{\textrm{d}t}=\boldsymbol v_{dm} \\ \label{proper_b} &\frac{\textrm{d} \boldsymbol v_{dm}}{\textrm{d}t}=-\boldsymbol\nabla\phi\\ \label{proper_c} &\Delta\phi=4\pi G\rho_{tot}\\ \label{proper_d} &\frac{\partial \rho}{\partial t}+\boldsymbol\nabla\cdot(\rho \boldsymbol v)=0 \\ \label{proper_e} &\frac{\partial (\rho \boldsymbol v)}{\partial t}+\boldsymbol\nabla\cdot\left [\rho \boldsymbol v \boldsymbol v + \left(p+\frac{B^2}{2\mu}\right)I-\frac{1}{\mu} \boldsymbol B \boldsymbol B\right]=-\rho\, \boldsymbol\nabla\phi\\ \label{proper_f} &\frac{\partial (\rho E)}{\partial t}+\boldsymbol\nabla\cdot\left[\boldsymbol v \left(\rho E+p+\frac{B^2}{2\mu}\right)-\frac{1}{\mu}\boldsymbol B(\boldsymbol v\cdot\boldsymbol B)\right]=-\rho \boldsymbol v \cdot(\boldsymbol\nabla\phi)\\ \label{proper_g} &\frac{\partial \boldsymbol B}{\partial t}+\boldsymbol\nabla\times(-\boldsymbol v \times\boldsymbol B)=0\\ \label{proper_h} &\boldsymbol\nabla\cdot\boldsymbol B=0 \\ \label{proper_i} &\frac{\partial S}{\partial t}+\boldsymbol\nabla\cdot(S \boldsymbol v)=0\end{aligned}$$ Here, equations (\[proper\_a\]) and (\[proper\_b\]) are the equations of motion for the dark matter ($dm$) particles; equation (\[proper\_c\]) is Poisson’s equation, where $\rho_{tot}$ is the total (dm + baryons) density; and the equations (\[proper\_d\]) – (\[proper\_h\]) are the equations of ideal MHD. For the dual energy formalism, we also need to transform the equation (\[proper\_i\]) describing the modified entropy. Below, we will apply the supercomoving transformation to each of these equations individually and construct a new set of supercomoving equations, using the definitions (\[definition\]) and (\[transformation\]). The new supercomoving quantities and derivatives will be denoted by a subscript $x$. Throughout this appendix, an overdot denotes the temporal derivative with respect to the proper, non-supercomoving time $t$. Dark matter particle equations of motion ---------------------------------------- We define the supercomoving velocity $\boldsymbol v_{x,dm}$ as the derivative of $\boldsymbol x_{dm}$ with respect to the supercomoving time $\textrm{d}t_x=\textrm{d}t/a^2$: $$\begin{aligned} \frac{\textrm{d}\boldsymbol x_{dm}}{\textrm{d}t_x}=\boldsymbol v_{x,dm}\end{aligned}$$ From this follows the relation between physical and supercomoving velocity: $$\begin{aligned} \boldsymbol v_{x,dm} = a \boldsymbol v_{dm} - \dot a \boldsymbol r_{dm} \;\;\;\;\;\Longleftrightarrow \;\;\;\;\; \boldsymbol v_{dm} = \frac{1}{a}\boldsymbol v_{x,dm} + \dot a \boldsymbol x_{dm}\end{aligned}$$ It follows also from the definition that the spatial derivatives change to $$\begin{aligned} \boldsymbol\nabla_x=a\boldsymbol\nabla\;\;\;;\;\;\;\Delta_x = a^2 \Delta\;\;\;.\end{aligned}$$ The goal is to obtain an equation of motion analogous to equation \[proper\_b\] for the supercomoving dark matter velocities $\boldsymbol v_{x,dm}\;$. Let us consider the supercomoving acceleration: $$\begin{aligned} \frac{\textrm{d}\boldsymbol v_{dm}}{\textrm{d}t_x}=\frac{\textrm{d}}{\textrm{d}t_x}\left( \frac{\textrm{d}\boldsymbol x_{dm}}{\textrm{d}t_x} \right)\end{aligned}$$ Now we replace the supercomoving time and position with the physical time and position, and rewrite the result to obtain a relation with the supercomoving gravitational force: $$\begin{aligned} \frac{\textrm{d}\boldsymbol v_{x,dm}}{\textrm{d}t_x}&=a^2\frac{\textrm{d}}{\textrm{d}t}(\boldsymbol v_{x,dm}) \\ &=a^2\frac{\textrm{d}}{\textrm{d}t}(a \boldsymbol v_{dm} - \dot a \boldsymbol r_{dm})\displaybreak[0] \\ &=a^2 \left(a\frac{\textrm{d}\boldsymbol v_{dm}}{\textrm{d}t}+\dot a \boldsymbol v_{dm} -\ddot a \boldsymbol r_{dm}- \dot a \boldsymbol v_{dm}\right )\displaybreak[0]\\ &=a^2 \left(a\frac{\textrm{d}\boldsymbol v_{dm}}{\textrm{d}t}-\ddot a \boldsymbol r_{dm} \right )\displaybreak[0]\\ &=a^2\left(a\frac{\textrm{d}\boldsymbol v_{dm}}{\textrm{d}t}-a\ddot a \boldsymbol x_{dm}\right)\displaybreak[0] \\ &=a^2\left[ -\boldsymbol \nabla_x \phi + a \ddot a \left (-\boldsymbol \nabla_x \left(\frac{1}{2} x_{dm}^2\right)\right) \right] \displaybreak[0] \\ &=-\boldsymbol\nabla_x\left[ a^2 \left( \phi + \frac{1}{2}a \ddot a x^2_{dm}\right) \right] \\ &=-\boldsymbol\nabla_x \phi_x\end{aligned}$$ Here, we used equation (\[proper\_b\]) and the definition of the comoving potential $\phi_x\,$. Now we see that the supercomoving equations of motion are formally identical to their proper physical counterparts, although the quantities are defined differently: $$\begin{aligned} \frac{\textrm{d}\boldsymbol x_{dm}}{\textrm{d}t_x}&=\boldsymbol v_{x,dm} \\ \frac{\textrm{d}\boldsymbol v_{x,dm}}{\textrm{d}t_x}&=-\boldsymbol\nabla_x \phi_x\end{aligned}$$ This is the main advantage of supercomoving coordinates over the comoving coordinates, which explicitly include additional factors depending on $a$. We will see that the other equations behave in a similar way. Poisson’s equation ------------------ Poisson’s equation determines the potential $\phi$ of the system, and as such it is the only equation where cosmology enters explicitly. Although equation (\[proper\_c\]) describes the gravitational potential in an ordinary physical setting, in a cosmological framework we also have to consider the cosmological constant $\Lambda$. This is realized by adding a $\Lambda$ term that has the dimension of a density, and then using this “effective density” in Poisson’s equation. Let us consider the second Friedmann equation, which relates the average total mass density $\bar\rho_{tot}$ and the cosmological constant to the accerelation of the cosmic expansion: $$\begin{aligned} \frac{\ddot a}{a}= -\frac{4 \pi G}{3}\left( \bar \rho_{tot} + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3}\end{aligned}$$ Dark matter is by definition pressureless $(p=0)$, and the pressure of the small baryonic component can be neglected. Then we can write $$\begin{aligned} \label{effectivefriedmann2} \frac{\ddot a}{a}= -\frac{4 \pi G}{3}\left( \bar \rho_{tot} - \rho_\Lambda \right )\end{aligned}$$ with $\rho_\Lambda=-\Lambda c^2/4\pi G$. Then, if the cosmological constant is not zero, Poisson’s equation effectively becomes $$\begin{aligned} \label{poissonwithlambda} \Delta\phi=4\pi G (\rho_{tot}-\rho_\Lambda)\;\;\;.\end{aligned}$$ We formulate the left-hand side in terms of the supercomoving gravitational potential: $$\begin{aligned} \Delta \phi &= \frac{1}{a^2}\Delta_x\left( \frac{\phi_x}{a^2} - \frac{1}{2} a \ddot a x^2 \right)\\ &= \frac{1}{a^4}\Delta_x\phi_x -\frac{\ddot a}{2 a} \Delta_x x^2\\ &= \frac{1}{a^4}\Delta_x\phi_x -3\frac{\ddot a}{a} \end{aligned}$$ Using again the second Friedmann equation \[effectivefriedmann2\], we get $$\begin{aligned} \Delta \phi &= \frac{1}{a^4}\Delta_x\phi_x -3\left[\frac{-4\pi G}{3} (\bar\rho_{tot}-\rho_\Lambda) \right] \;\;\;.\end{aligned}$$ Equating this with the right-hand side of equation \[poissonwithlambda\] yields $$\begin{aligned} \frac{1}{a^4}\Delta_x\phi_x -4\pi G (\bar\rho_{tot}-\rho_\Lambda)&=-4\pi G(\rho_{tot}-\rho_\Lambda) \;\;\;. $$ The $\Lambda$ term cancels, and we arrive at the supercomoving Poisson’s equation: $$\begin{aligned} \notag \Delta_x \phi_x &= 4\pi G a^4 (\bar\rho_{tot}-\rho_{tot})\\ &= 4\pi G a (\bar\rho_{x,tot}-\rho_{x,tot})\;\;\;.\end{aligned}$$ The supercomoving version of Poisson’s equation looks slightly different than the non-cosmological one: the density contrast enters instead of the total density, because the supercomoving potential is responsible for peculiar motions due to density fluctuations, while the total density governs the overall expansion. Baryon mass density ------------------- Now we will transform the conservation law for the baryon density $\rho$, $$\begin{aligned} \frac{\partial \rho}{\partial t}+\boldsymbol \nabla (\rho \boldsymbol v)=0\;\;\;.\end{aligned}$$ First, we replace proper time, density and velocity with their comoving counterparts. We begin by replacing the partial time derivative $\partial/\partial t$ at constant position $\boldsymbol r$ with the one at constant comoving position $\boldsymbol x$: $$\begin{aligned} \frac{\partial}{\partial t}\bigg|_r \rho&=\frac{\partial}{\partial t}\bigg|_x \rho-\frac{\dot a}{a}\boldsymbol x\cdot \boldsymbol \nabla_x \rho\\ &=\frac{1}{a^2}\frac{\partial}{\partial t_x} \left(\frac{1}{a^3}\rho_x\right)-\frac{\dot a}{a^4}\boldsymbol x\cdot\boldsymbol \nabla_x \rho_x \displaybreak[0]\\ &=\frac{-3\dot a}{a^4}\rho_x+\frac{1}{a^5}\frac{\partial\rho_x}{\partial t_x} - \frac{\dot a}{a^4} \boldsymbol x \cdot\boldsymbol \nabla_x\rho_x\end{aligned}$$ Evaluating the flux term, using $\boldsymbol v=\boldsymbol v_x/a + \dot a \boldsymbol x$: $$\begin{aligned} \boldsymbol \nabla (\rho \boldsymbol v) &= \frac{1}{a} \boldsymbol \nabla_x \left( \frac{1}{a^4} \rho_x \boldsymbol v_x + \frac{\dot a}{a^3}\rho_x \boldsymbol x \right) \\ & = \frac{1}{a^5} \boldsymbol \nabla_x (\rho_x \boldsymbol v_x ) + \frac{\dot a}{a^4} \boldsymbol \nabla_x(\rho_x \boldsymbol x) \displaybreak[0]\\ & = \frac{1}{a^5} \boldsymbol \nabla_x (\rho_x \boldsymbol v_x ) + \frac{\dot a}{a^4} \boldsymbol x (\boldsymbol \nabla_x \rho_x) + \frac{3\dot a}{a^4} \rho_x\end{aligned}$$ Putting these two expressions together, four of the six terms cancel, leaving only $$\begin{aligned} \frac{\partial \rho_x}{\partial t_x} + \boldsymbol \nabla_x(\rho_x \boldsymbol v_x)=0\;\;\;.\end{aligned}$$ Baryon momentum density ----------------------- The conservation law for the baryon momentum can be written as: $$\begin{aligned} \label{momentumwithA} \frac{\partial( \rho \boldsymbol v)}{\partial t}+\boldsymbol\nabla\cdot(\rho \boldsymbol v \boldsymbol v +A)=-\rho\, \boldsymbol\nabla\phi\\end{aligned}$$ where we used the abbreviation $A=\left(p+\frac{B^2}{2\mu}\right)I-\frac{1}{\mu} \boldsymbol B \boldsymbol B$. From the definitions of the supercomoving pressure $p_x$ and magnetic field $\boldsymbol B_x$ one immediately sees that $$\begin{aligned} A_x = \left(p_x+\frac{B_x^2}{2\mu}\right)I-\frac{1}{\mu} \boldsymbol B_x \boldsymbol B_x= a^5 A\;\;\;.\end{aligned}$$ We decompose the first term of the momentum equation into two parts: $$\begin{aligned} \frac{\partial( \rho \boldsymbol v)}{\partial t}\bigg|_r=\rho \frac{\partial \boldsymbol v}{\partial t}\bigg|_r+\boldsymbol v \frac{\partial \rho}{\partial t}\bigg|_r\end{aligned}$$ The first part equals $$\begin{aligned} \rho \frac{\partial \boldsymbol v}{\partial t}\bigg|_r &= \frac{1}{a^3}\rho_x\left( \rho \frac{\partial \boldsymbol v}{\partial t}\bigg|_x - \frac{\dot a}{a} \boldsymbol x ( \boldsymbol \nabla_x\boldsymbol u) \right)\\\end{aligned}$$ Using the abbreviation $K=\frac{\dot a}{a} \boldsymbol x ( \boldsymbol \nabla_x\boldsymbol u) $ , it evaluates to $$\begin{aligned} \rho \frac{\partial \boldsymbol v}{\partial t}\bigg|_r &= \frac{1}{a^3}\rho_x\left( \rho \frac{\partial \boldsymbol v}{\partial t}\bigg|_x - K\right)\\ &= \frac{1}{a^3}\rho_x\left[ \frac{\partial }{\partial t}\bigg|_x \left( \frac{1}{a} \boldsymbol v_x + \dot a \boldsymbol x\right) - K\right ] \\ &=\frac{1}{a^3}\rho_x\left( \frac{-\dot a}{a^2} \boldsymbol v_x +\frac{1}{a^3}\frac{\partial \boldsymbol v_x}{\partial t_x} + \ddot a \boldsymbol x - K \right)\end{aligned}$$ while we write the second part as (equation \[proper\_b\]): $$\begin{aligned} \boldsymbol v \frac{\partial \rho}{\partial t}\bigg|_r = - \boldsymbol v \boldsymbol \nabla \cdot (\rho \boldsymbol v)\end{aligned}$$ The second term on the left-hand side of equation (\[momentumwithA\]) transforms as follows: $$\begin{aligned} \boldsymbol \nabla (\rho \boldsymbol v\boldsymbol v+A) &= \boldsymbol v \boldsymbol \nabla \cdot (\rho \boldsymbol v) + \rho \boldsymbol v \cdot \boldsymbol \nabla \boldsymbol v + \boldsymbol \nabla A \\ &=\boldsymbol v \boldsymbol \nabla \cdot (\rho \boldsymbol v) + \frac{1}{a^3}\rho_x \left[ \left( \frac{1}{a} \boldsymbol v_x + \dot a \boldsymbol x\right ) \cdot \frac{1}{a} \boldsymbol \nabla_x \left( \frac{1}{a} \boldsymbol v_x + \dot a \boldsymbol x\right ) + K \right] + \frac{1}{a^6} \boldsymbol \nabla_x A_x \\ &=\boldsymbol v \boldsymbol \nabla \cdot (\rho \boldsymbol v) + \frac{1}{a^3}\rho_x \left[ \frac{1}{a^3} (\boldsymbol v_x \cdot \boldsymbol \nabla_x) \boldsymbol v_x + \frac{\dot a}{a^2} (\boldsymbol v_x \cdot \boldsymbol \nabla_x) \boldsymbol x + K \right] + \frac{1}{a^6} \boldsymbol \nabla_x A_x \\ &=\boldsymbol v \boldsymbol \nabla \cdot (\rho \boldsymbol v) + \frac{1}{a^3}\rho_x \left[ \frac{1}{a^3} (\boldsymbol v_x \cdot \boldsymbol \nabla_x) \boldsymbol v_x + \frac{\dot a}{a^2}\boldsymbol v_x + K \right] + \frac{1}{a^6} \boldsymbol \nabla_x A_x\end{aligned}$$ Combining all terms from the left-hand side of equation (\[momentumwithA\]) gives $$\begin{aligned} \frac{\partial( \rho \boldsymbol v)}{\partial t}\bigg|_r+\boldsymbol \nabla (\rho \boldsymbol v \boldsymbol v + A)&=\rho \frac{\partial \boldsymbol v}{\partial t}\bigg|_r+\boldsymbol v \frac{\partial \rho}{\partial t}\bigg|_r+\boldsymbol \nabla (\rho \boldsymbol v\boldsymbol v + A) \\ & = \frac{1}{a^3} \rho_x \left[ \frac{1}{a^3} \frac{\partial \boldsymbol v_x}{\partial t_x} + \frac{1}{a^3} (\boldsymbol v_x \cdot \boldsymbol \nabla_x) \boldsymbol v_x + \ddot a \boldsymbol x \right ] + \frac{1}{a^6} \boldsymbol \nabla_x A_x\end{aligned}$$ When comparing this to the right-hand side of equation (\[momentumwithA\]), $$\begin{aligned} -\rho \boldsymbol \nabla \phi &= \frac{-1}{a^4}\rho_x\boldsymbol \nabla_x \left( \frac{\phi_x}{a^2} - \frac{1}{2}a \ddot a x^2 \right) \\ &= \frac{-1}{a^6}\rho_x(\boldsymbol \nabla_x \phi_x) + \frac{1}{a^3}\rho_x \boldsymbol \nabla_x \left( \frac{1}{2} \ddot a x^2 \right)\\ &= \frac{-1}{a^6}\rho_x(\boldsymbol \nabla_x \phi_x) + \frac{1}{a^3}\rho_x (\ddot a \boldsymbol x) \;\;\;,\end{aligned}$$ we notice that the $(\ddot a \boldsymbol x)$ term cancels, leaving $$\begin{aligned} \notag \rho_x \frac{\partial \boldsymbol v_x}{\partial t_x} + &\rho_x ( \boldsymbol v_x \cdot \boldsymbol \nabla_x) \boldsymbol v_x + \boldsymbol \nabla_x A_x = -\rho_x \boldsymbol \nabla_x \phi_x \\ \notag \Longleftrightarrow\;\;\;\;\; \frac{\partial (\rho_x \boldsymbol v_x)}{\partial t_x} &+ \boldsymbol \nabla_x ( \rho_x \boldsymbol v_x \boldsymbol v_x + A_x) = -\rho_x \boldsymbol \nabla_x \phi_x \\ \Longleftrightarrow\;\;\;\;\; \frac{\partial (\rho_x \boldsymbol v_x)}{\partial t_x}& + \boldsymbol \nabla_x \left[ \rho_x \boldsymbol v_x \boldsymbol v_x + \left(p_x+\frac{B_x^2}{2\mu}\right) I-\frac{1}{\mu} \boldsymbol B_x \boldsymbol B_x\right] = -\rho_x \boldsymbol \nabla_x \phi_x\;\;\;.\end{aligned}$$ Induction equation ------------------ Before transforming the total energy density equation in the next subsection, the supercomoving induction equation has to be derived as it will be needed for it. We start from the equation $$\begin{aligned} \frac{\partial \boldsymbol B}{\partial t}\bigg|_r + \boldsymbol \nabla \times (- \boldsymbol v \times \boldsymbol B) = 0\end{aligned}$$ subject to the condition $$\begin{aligned} \boldsymbol \nabla \cdot \boldsymbol B = 0\;\;\;.\end{aligned}$$ The divergence-free condition turns out to be useful as it causes several terms to vanish. First we substitute the temporal and spatial derivatives: $$\begin{aligned} &\frac{\partial \boldsymbol B}{\partial t}\bigg|_r + \boldsymbol \nabla \times (- \boldsymbol v \times \boldsymbol B) = 0 \\ \Longleftrightarrow\;\;\;\;\;&\frac{\partial \boldsymbol B}{\partial t}\bigg|_x - \frac{\dot a}{a} \boldsymbol x (\boldsymbol \nabla_x \cdot \boldsymbol B) + \frac{1}{a}\boldsymbol \nabla_x \times \left[- \left( \frac{1}{a} \boldsymbol v_x + \dot a \boldsymbol x \right) \times \boldsymbol B\right] =0\\ \Longleftrightarrow\;\;\;\;\;&\frac{1}{a^2}\frac{\partial \boldsymbol B}{\partial t_x} - \frac{1}{a^2} \boldsymbol \nabla_x \times (\boldsymbol v \times \boldsymbol B) - \frac{\dot a }{a} \boldsymbol \nabla_x \times (\boldsymbol x \times \boldsymbol B) =0\end{aligned}$$ Again, the divergence-free condition allows us to simplify: $$\begin{aligned} \boldsymbol \nabla_x \times (\boldsymbol x \times \boldsymbol B) = (\boldsymbol B \cdot \boldsymbol \nabla) \boldsymbol x - \boldsymbol B (\boldsymbol \nabla \cdot\boldsymbol x) = \boldsymbol B - 3 \boldsymbol B = -2 \boldsymbol B\end{aligned}$$ and therefore $$\begin{aligned} &\frac{1}{a^2}\frac{\partial \boldsymbol B}{\partial t_x} - \frac{1}{a^2} \boldsymbol \nabla_x \times (\boldsymbol v \times \boldsymbol B) + 2 \frac{\dot a}{a} \boldsymbol B=0 \\ \Longleftrightarrow\;\;\;\;\;&\frac{\partial \boldsymbol B}{\partial t_x} - \boldsymbol \nabla_x \times (\boldsymbol v \times \boldsymbol B) + 2 \dot a a \boldsymbol B=0\end{aligned}$$ Now we substitute the supercomoving magnetic field $\boldsymbol B_x = a^{5/2} \boldsymbol B$: $$\begin{aligned} &a^{-5/2}\frac{\partial \boldsymbol B_x}{\partial t_x} + \boldsymbol B_x \frac{\partial}{\partial t_x}\left(a^{-5/2}\right) - a^{-5/2} \boldsymbol \nabla_x \times (\boldsymbol v_x \times \boldsymbol B_x) +2 \dot a a^{-3/2} \boldsymbol B_x = 0 \\ \Longleftrightarrow\;\;\;\;\;& a^{-5/2}\frac{\partial \boldsymbol B_x}{\partial t_x} - a^{-5/2} \boldsymbol \nabla_x \times (\boldsymbol v_x \times \boldsymbol B_x) - \frac{1}{2}\dot a a^{-3/2} \boldsymbol B_x = 0\end{aligned}$$ We define the supercomoving Hubble constant $$\begin{aligned} \mathcal{H}:= \frac{1}{a} \frac{\textrm{d}a}{\textrm{d}t_x} = \dot a a\end{aligned}$$ With this notation, we have: $$\begin{aligned} \frac{\partial \boldsymbol B_x}{\partial t_x} + \boldsymbol \nabla_x \times (-\boldsymbol v_x \times \boldsymbol B_x) =\frac{1}{2} \mathcal{H}\boldsymbol B_x \end{aligned}$$ We defined the frame of reference such that it is comoving with the magnetic energy density, and not with the magnetic field strength. This is the reason why a magnetic Hubble drag term must appear at the right-hand side of the supercomoving induction equation and it is *not* formally identical to the non-comoving induction equation. However, the Hubble term only ensures that the magnetic field scales properly with $a$; it does not have any physical meaning. Total energy density -------------------- Instead of directly transforming the total energy equation (\[proper\_f\]), we derive the supercomoving energy conservation law by putting together all the quantities we have so far. The easiest way is to first derive the energy conservation for the hydrodynamic case and then add the magnetic energy density and flux to the result. In the hydrodynamic case, $\rho E = \frac{1}{2}\rho v^2 + \rho\varepsilon$. One immediately notices from the definitions of the supercomoving variables that the supercomoving total energy is $$\begin{aligned} \rho_x E_x = \frac{1}{2}\rho_x v_x^2 + \rho_x\varepsilon_x\end{aligned}$$ We start by calculating the temporal change of the kinetic energy $\frac{1}{2}\rho_x v_x^2$ with the help of the already derived equations: $$\begin{aligned} \frac{\partial}{\partial t_x}\left(\frac{1}{2}\rho_x v_x^2\right)&= \rho \boldsymbol v_x \frac{\partial \boldsymbol v_x}{\partial t_x} + \frac{1}{2}v_x^2\frac{\partial \rho_x}{\partial t_x} \\ & = \rho_x \boldsymbol v_x \left[ (\boldsymbol v_x \cdot \boldsymbol \nabla_x) \boldsymbol v_x - \boldsymbol \nabla_x \phi_x - \frac{1}{\rho_x} \boldsymbol \nabla_x p_x\right] - \frac{1}{2}v_x^2 \boldsymbol \nabla_x (\rho_x \boldsymbol v_x) \displaybreak[0]\\ & = \rho_x \boldsymbol v_x \left[ -(\boldsymbol v_x \cdot \boldsymbol \nabla_x) \boldsymbol v_x \right] - \frac{1}{2}v_x^2\boldsymbol \nabla_x\cdot(\rho_x \boldsymbol v_x) - \ \boldsymbol v_x \cdot \boldsymbol \nabla_x p_x - \rho_x \boldsymbol v_x \cdot (\boldsymbol \nabla_x \phi_x) \displaybreak[0]\\ & = -\boldsymbol \nabla_x \cdot \left (\boldsymbol v_x \frac{1}{2}\rho_x v_x^2\right) - \boldsymbol v_x \cdot \boldsymbol \nabla_x p_x - \rho_x \boldsymbol v_x \cdot (\boldsymbol \nabla_x \phi_x)\end{aligned}$$ With $\boldsymbol v_x \cdot \boldsymbol p_x = \boldsymbol \nabla_x \cdot (\rho_x \boldsymbol v_x) - p_x \boldsymbol \nabla_x \cdot \boldsymbol v_x$ , we can write: $$\begin{aligned} \label{supercomovingkineticenergy} \frac{\partial}{\partial t_x}\left(\frac{1}{2}\rho_x v_x^2\right) + \boldsymbol \nabla_x \cdot \left[ \left(\frac{1}{2}\rho_x v_x^2 + p_x\right) \boldsymbol v_x \right] = p_x \boldsymbol \nabla_x \cdot \boldsymbol v_x - \rho_x \boldsymbol v_x \cdot (\boldsymbol \nabla_x \phi_x)\end{aligned}$$ Next, we need an equation for the thermal energy $\varepsilon_x$. In proper coordinates, such an equation exists (e.g. @Bryan95). In the case of a monoatomic ideal gas ($\gamma=5/3$), which will be assumed from here on, it reads: $$\begin{aligned} \frac{\partial \varepsilon}{\partial t} + \boldsymbol v \cdot \boldsymbol \nabla \varepsilon = -\frac{1}{\rho}p \boldsymbol \nabla \cdot \boldsymbol v\end{aligned}$$ By plugging in the definitions of the supercomoving variables it easily proves that the same equation holds in supercomoving coordinates: $$\begin{aligned} \frac{\partial \varepsilon_x}{\partial t_x} + \boldsymbol v_x \cdot \boldsymbol \nabla_x \varepsilon_x = -\frac{1}{\rho_x}p_x \boldsymbol \nabla_x \cdot \boldsymbol v_x\end{aligned}$$ We can rewrite that as $$\begin{aligned} p \boldsymbol \nabla_x \cdot \boldsymbol v_x &= - \rho_x \frac{\partial \varepsilon_x}{\partial t_x} - \rho \boldsymbol v_x \cdot \boldsymbol \nabla_x \varepsilon_x \\ &= -\frac{\partial (\rho_x \varepsilon_x)}{\partial t_x} - \boldsymbol \nabla_x [ \boldsymbol v_x (\rho_x \varepsilon_x) ]\end{aligned}$$ and plug it into equation (\[supercomovingkineticenergy\]), yielding $$\begin{aligned} \label{supercomovinghydroenergy} \frac{\partial }{\partial t_x}\left( \rho_x E_x\right) + \boldsymbol \nabla_x \cdot \left[ \left(\rho_x E_x+ p_x\right) \boldsymbol v_x \right] = - \rho_x \boldsymbol v_x \cdot (\boldsymbol \nabla_x \phi_x)\end{aligned}$$ This is the supercomoving total energy equation for the hydrodynamic case, again formally equivalent to the corresponding non-comoving equation. Now we can consider the magnetic energy $B_x^2/2\mu$. Its temporal derivative is easily obtained from the supercomoving induction equation: $$\begin{aligned} \frac{\partial}{\partial t_x} \left( \frac{B_x^2}{2\mu}\right)& = \frac{1}{\mu} \boldsymbol B_x \cdot \frac{\partial \boldsymbol B_x}{\partial t_x} \\ & =\frac{1}{\mu}\boldsymbol B_x \cdot \left [ \boldsymbol \nabla_x \times (\boldsymbol v_x \times \boldsymbol B_x ) + \frac{1}{2} \mathcal H \boldsymbol B_x \right ]\displaybreak[0] \\ & =\boldsymbol B_x \cdot \left [ \frac{1}{\mu} \boldsymbol \nabla_x \times (\boldsymbol v_x \times \boldsymbol B_x ) \right] + \mathcal H \frac{B_x^2}{2 \mu}\displaybreak[0] \\ & = \boldsymbol \nabla_x \cdot \left [ \frac{1}{\mu} (\boldsymbol v_x \times \boldsymbol B_x) \times \boldsymbol B_x \right] + \mathcal H \frac{B_x^2}{2 \mu} \\ & =\boldsymbol \nabla_x \cdot \left [ \frac{1}{\mu} \boldsymbol B_x (\boldsymbol v_x \cdot \boldsymbol B_x) - \boldsymbol v_x \frac{B_x^2}{2\mu} \right] + \mathcal H \frac{B_x^2}{2 \mu}\end{aligned}$$ Adding this equation to (\[supercomovinghydroenergy\]), we get the supercomoving total energy equation for the full MHD case: $$\begin{aligned} \frac{\partial }{\partial t_x}\left( \rho_x E_x\right) + \boldsymbol \nabla_x \cdot \left[ \left(\rho_x E_x+ p_x\right) \boldsymbol v_x - \frac{1}{\mu} \boldsymbol B_x (\boldsymbol v_x \cdot \boldsymbol B_x) \right] = - \rho_x \boldsymbol v_x \cdot (\boldsymbol \nabla_x \phi_x)+ \mathcal H \frac{B_x^2}{2 \mu}\end{aligned}$$ where now $$\begin{aligned} \rho_x E_x = \frac{1}{2}\rho_x v_x^2 + \rho_x\varepsilon_x + \frac{B_x^2}{2\mu}\;\;\;.\end{aligned}$$ Modified entropy ---------------- This additional equation is needed to use the “S system” in the dual energy formalism. Transforming the first term: $$\begin{aligned} \frac{\partial S}{\partial t}\bigg|_r \rho&=\frac{\partial S}{\partial t}\bigg|_x \rho-\frac{\dot a}{a}\boldsymbol x\cdot \boldsymbol \nabla_x S\\ &=\frac{1}{a^2}\frac{\partial}{\partial t_x} \left( a^{3\gamma -8} S_x\right)-\frac{\dot a}{a}\boldsymbol x\cdot\boldsymbol \nabla_x \left( a^{3\gamma - 8}S_x\right) \displaybreak[0]\\ &=a^{3\gamma-10} \left[\frac{\partial S_x}{\partial t_x} + \mathcal H (3\gamma-8) S_x - \mathcal H \boldsymbol x \cdot \boldsymbol \nabla_x S_x \right]\end{aligned}$$ and the second term: $$\begin{aligned} \boldsymbol \nabla \cdot (S \boldsymbol v) &= \frac{1}{a} \boldsymbol \nabla_x \cdot \left[ a^{3\gamma -8} S_x \left( \frac{1}{a} \boldsymbol v_x + \dot a \boldsymbol x \right) \right ] \\ &= a^{3\gamma-10} \boldsymbol \nabla_x \cdot \left[ S_x \boldsymbol v_x + \mathcal H \boldsymbol x S_x \right] \\ &= a^{3\gamma-10} \left[ \boldsymbol \nabla_x \cdot \left( S_x \boldsymbol v_x \right) + \mathcal H \boldsymbol x \cdot \boldsymbol \nabla_x S_x + 3 \mathcal H S_x \right]\end{aligned}$$ Putting both together yields $$\begin{aligned} \notag &\frac{\partial S_x}{\partial t_x} + \mathcal H(3\gamma-8) S_x + \boldsymbol \nabla_x \cdot (S_x \boldsymbol v_x) + 3\mathcal HS_x =0 \\ \Longleftrightarrow\;\;\;\;\; &\frac{\partial S_x}{\partial t_x} + \boldsymbol \nabla_x \cdot (S_x \boldsymbol v_x) = -\mathcal H (3\gamma-5)\;\;\;\;\;\;.\end{aligned}$$ [^1]: [`AMIGA`]{} can be downloaded from the following web site: `http://popia.ft.uam.es/AMIGA`. [^2]: We like to note that codes such as, for instance, `RAMSES` [@Teyssier02] and `ART` [@Kravtsov97; @Kravtsov02] also implement supercomoving coordinates. However, neither of the code description papers has yet shown the full set of the corrresponding equations and their derivation as presented here and in the Appendix, respectively. [^3]: We like to note that the KNP flux is equivalent to the HLL flux formula introduced by @Harten83 as two-speed approximate Riemann solver. [^4]: When talking about primordial field strengths at earlier times (higher redshifts), we mean the *comoving* magnetic field strength $B_{comoving}=a^{-2}B_{proper}$, that is, the strength such a field would have when extrapolated to the present-day scale factor. It is convenient because it allows for the magnetic field strengths from different epochs to be compared directly. The relation $B(z)=B_0/a^2$ follows from magnetic flux conservation.

--- abstract: 'In the core of the vortex of a superconductor, energy levels appear inside the gap. We discuss here through a random matrix approach how these levels are broadened by impurities. It is first shown that the level statistics is governed by an ensemble consisting of a symplectic random potential added to a non-random matrix. A generalization of previous work on the unitary ensemble in the presence of an external source (which relied on the Itzykson-Zuber integral) is discussed for this symplectic case through the formalism introduced by Harish-Chandra and Duistermaat-Heckman. This leads to explicit formulae for the density of states and for the correlation functions, which describe the cross-over from the clean to the dirty limits.' address: | $^{1}$ Laboratoire de Physique Th[é]{}orique, Ecole Normale Sup[é]{}rieure, 24 rue Lhomond 75231, Paris Cedex 05, France[[^1] ]{}\ $^{2}$ Department of Pure and Applied Sciences, University of Tokyo\ [Meguro-ku, Komaba, Tokyo 153, Japan]{}\ [$^{c)}$ Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA]{}\ [$^{d)}$ Landau Institute for Theoretical Physics, 117334 Moscow, Russia]{}\ author: - 'E. Br[é]{}zin$^{1}$, S. Hikami$^{2}$ and A. I. Larkin$^{c),d)}$' title: LEVEL STATISTICS INSIDE THE VORTEX OF A SUPERCONDUCTOR AND SYMPLECTIC RANDOM MATRIX THEORY IN AN EXTERNAL SOURCE --- \#1[[$\backslash$\#1]{}]{} INTRODUCTION ============ The energy levels inside a vortex of a superconductor have been characterized long ago [@CDM], but recent studies have dealt with the broadening of these levels by impurities. The density of states has been computed both in the clean limit and for the dirty case. In this article, we discuss the crossover between these two limits. The impurity potential is handled through a random matrix theory. However the matrix elements between different energy levels due to the impurities are strongly correlated, and therefore one is far from a usual Wigner-Dyson random matrix theory. It is shown below that the crossover is characterized in this case by a random symplectic matrix coupled to an external non-random source matrix. For handling this problem we develop a technique which generalizes earlier work on the unitary ensemble in which one considered the cross-over from a deterministic hamiltonian to a fully random hermitian hamiltonian [@BH1; @BHZ; @BH2; @BH3]. In the hermitian case, the formalism relied on an integral over the unitary group due to Harish-Chandra [@Harish-Chandra] and rederived in the context of random matrix theory by Itzykson-Zuber [@Itzykson]. All the correlation functions for the energy levels were then found explicitely. For the vortex problem considered here the perturbation due to impurities is a matrix with a symplectic structure, which we treat as a random potential added to the unperturbed diagonal matrix consisting of the regularly spaced energy levels. It is shown below that for this problem a similar integral over the symplectic group, considered by Harish-Chandra [@Harish-Chandra], and more recently generalized by Duistermaat and Heckman [@Duistermaat] leads also to explicit formulae. The crossover from the clean spectrum to the dirty limit follows from this formalism. We will show how this general behavior for the crossover is relevant to the actual problem of the excitation energy spectrum in a superconducting vortex. SYMPLECTIC STRUCTURE OF THE PERTURBATION FOR THE ENERGY LEVELS INSIDE A VORTEX ============================================================================== The energy eigenvalues and eigenfunctions of a quasi-particle inside the vortex of a superconductor, were obtained long ago [@CDM]. The two component wave functions of excitations $\hat \psi = $ $( \psi_1,\psi_2 )$ in a superconductor satifies the Bogolubov-de Gennes equation, \[2.1\] \[ \_z (H\_0 + V\_[imp]{}) + \_x [Re]{}(r) + \_y [Im]{} (r)\] = E where $H_0 = p^2/2m - E_F$, and $\Delta(r)$ is a gap order parameter. The impurity potential $V_{imp}$ is a sum of short range scattering sites $r_i$: \[2.2\] V\_[imp]{}(r) = \_i V\_i(r - r\_i) In the absence of impurity, the Schrodinger equation (\[2.1\]) leads to a spectrum of equidistant states $E_n^0$ \[2.3\] E\_n\^0 = - \_0(n - [1]{}) where $n$ is an integer defining the quantized angular momentum of the vortex, and $\hbar \omega_0$ = $\Delta^2/E_F$. We assume that the spacing $\hbar \omega_0$ between these levels is much smaller than the gap $\Delta$, and consequently there are many excitation levels in the vortex core. Using the explicit wave functions for the unperturbed eigenstates, one finds that the matrix elements of the interaction due to the $i$-th impurity are \[2.7\] A\_[nm]{}\^i = [V\_i e\^[- 2 K(r\_i)]{}]{} e\^[i ( m - n)\_i]{}\[ J\_n(k\_F r\_i) J\_m(k\_F r\_i) - J\_[n - 1]{} (k\_F r\_i) J\_[m - 1]{}(k\_F r\_i)\] where $(r_i,\phi_i)$ is the position of the i-th impurity in polar coordinates. The indices $n$ and $m$ run from $-N$ to $+ N$ where $N \sim \Delta/\omega_0 \sim E_F/\Delta$ is assumed to be a large number. The scattering time $\tau$, given by $1/\tau = 2 n_i V_i^2 m$, where $n_i={N_i}/{\xi^2}$ is the density of impurities in the vortex ; $N_i$ the total of impurities in the cross-section of a vortex of radius $\xi$.\ Three different situations may occur: a) a dirty limit $\Delta << 1/\tau << E_F$, b) a clean limit $\omega_0 < 1/\tau << \Delta$, c) a superclean case $1/\tau << \omega_0 \sim \Delta^2/E_F$. However, in [@KL1; @LO] it has been shown that the level statistics depends not only on the parameter $\tau$ but also on the number of impurities $N_i$ . This is clear in the superclean case. Then the density of states has narrow peaks centered around the unperturbed energy levels (\[2.3\]). The form of these peaks is Gaussian when the number of impurities is large ; however as shown in [@KL2; @LO] these peaks have a different form when there is only a single impurity. In the clean case $\omega_0 < 1/\tau < \Delta$, the level statistics also depends upon the number of impurity $N_i$. If $N_i \rightarrow \infty$ and if the random potential is a white noise, it was shown in [@SKF] that one can apply standard random matrix theory. However if the number of impurities $N_i$ is smaller than a certain number $N_{ic}$ ($N_{ic} > (E_F/\Delta)^{1/2}$), Koulakov and Larkin [@KL1] have found that the spectrum is of a different type. In the present article, we have defined the dirty and the clean limits by $N_i > N_{ic}$, and $ N_i < N_{ic}$ respectively. (These definitions differ slightly from the usual ones, quoted hereabove.) When there is only one impurity in the vortex core, with a radius $\xi$, the impurity is located at a position $r_i \sim \xi$. We are thus in a limit in which $k_F r_i >> 1$, since the correlation length is $\xi \sim \hbar^2 k_F/m\Delta$, and we have thus $k_F \xi \sim k_F^2/\Delta \sim E_F/\Delta >> 1$. Therefore, in the clean limit, this condition is always satisfied. In the dirty limit, the number of impurities increase and the typical value of the quantity $k_F r_i$ is reduced. In that limit the Bessel functions in (\[2.7\]) may be replaced by their asymptotic expansions and the matrix element $A_{nm}^i$, produced by the scattering site $i$ between two states $n$ and $m$, is given for $k_Fr_i\gg 1$ by \[2.11\] A\_[nm]{}\^i e\^[i (m - n)\_i]{} ( 2 k\_F r\_i - [n + m]{} ) where ${\tilde C} = {V_i e^{- 2K(r_i)}\over{\lambda_F \xi}} ({2\over{\pi k_F r_i}})$. The corresponding secular equation for the clean case was solved by Koulakov and Larkin[@KL1] who have found that, after averaging over the location of the impurity, the density of states $\rho(E)$ was given by \[2.12\] (E) = [2]{} \^2 ([E]{}) (normalized to one after integration on the interval $0 < E < \omega_0$).\ In the dirty limit, $k_F r_i$ is not as large and it is necessary to take into account the sub-asymptotic behavior of the Bessel functions, and to sum over the impurities. Since the matrix elements $A_{nm}^i$ have an oscillatory behavior, it is natural to regard them in the dirty limit as the elements of a random matrix. However it is necessary to keep some of the structure exhibited by the clean limit (\[2.11\]) in mind. In the dirty limit or in the moderate clean case, a random matrix theory has already been considered for the superconductor-normal interface, and for a quantum dot with a superconductor boundary. [@AZ; @AST]. The excitation levels of the superconductor vortex have also been analysed in this way [@SKF]. The random matrix ensemble, for such cases, has been suggested to be invariant by the symplectic group $Sp(N)$. If the full Hamiltonian is treated as a random matrix in a symplectic ensemble $Sp(n)$ , which is valid in the extreme dirty limit, the density of states has been conjectured [@AZ; @SKF] to be \[2.13\] (E) = 1 - [([ E ]{})([E]{})]{} This differs markedly from the clean result (\[2.12\]).\ It is therefore challenging to find the crossover behavior between these two limits, and it might even be important for the transport problems associated with the excitations of quasi-particles inside the vortex core [@KL2]. In order to achieve our goal, instead of treating the full Hamiltonian as random, we consider here only the scattering by impurities as a random matrix but not the full Ha miltonian.\ In order to motivate the use of the symplectic structure in this problem it is instructive to examine the form of the matrix $A_{nm}^i$. For a given impurity $i$, let $A$ be the matrix whose elements are $<n|A|m>=A_{nm}^i $. The state index $|m>$ varies over $2 N$ levels. When $N = 2$, for example, the states $|m>$ are $|-1>,|0>,|1>,|2>$. Since $E_n^0 = - \omega_0( n - {1\over{2}})$, we have $E_0 = -E_1 = {1\over{2}}\omega_0$ and $E_2 = - E_{-1} = -{3\over{2}}\omega_0$. Labelling the lines and rows of the matrix $A$ in the order $|0>,|2>,|1>,|-1>$, in order to split it into even-even, even-odd, odd-odd $2\times 2$ blocks, we write \[2.14\] A = () Since $<0|A|2> = A_{02}^i = A_{20}^i= <2|A|0>$, and $<2|A|1> = <0|A|-1>$, etc. from (\[2.7\]) (we have used the $J_{-n}(x) = (-1)^n J_{n}(x)$), it is easy to see that the matrix $A$ has, for arbitrary $N$, the structure \[2.15\] A = () where $a$ is an $N\times N$ hermitian matrix $a^{\dag} = a$, and $b$ is an $N\times N$ symmetric complex matrix $b^{T} = b$ ($b^{T}$ is the transpose of $b$). The number of degrees of freedom in the random matrix $A$ is thus $2 N^2 + N$, ( $N^2$ for the hermitian matrix $a$ and $N^2 + N$ for the complex symmetric matrix $b$). The number of generators of the symplectic $Sp(N)$ group is indeed also $N(2 N + 1)$. The symplectic structure of $A$ is exhibited by the algebraic relation \[2.17\] A\^[T]{} J + J A = 0 where $J$ is \[2.18\] J = ( ). This structure implies that the eigenvalues are real and pairwise opposite, giving a chiral structure to the eigenvalues. The Lie algebra $X$ is diagonalized by the symplectic group $G \in Sp(N)$. For example, for $N = 2$, we have \[2.19\] X = G\^ () G If we mutiply the matrix $A$ by $i =\sqrt{-1}$ one recovers an element of the Lie algebra of the group $Sp(N)$. If we consider now that the position $(\phi_i,r_i)$ of the impurity is random, it is natural to take $<n|A|m>$ in this $Sp(N)$ invariant random ensemble. For instance when there is only one impurity (i = 1), the matrix $A$ has a periodic structure, with alternating signs for each 2 by 2 block. For example, in the case of N = 4, we have for the full Hamiltonian $H = E^0 + A$, ($E^0$ is a diagonal matrix) \[2.20\] A = () where $ s = \tilde C \sin(2 k_F a)$ and $c = \tilde C \cos( 2 k_F a)$. ( In the single impurity case the phase $\phi$ can be set equal to zero by a choice of gauge condition ). This even-odd structure has been studied earlier in [@LO; @KL1]. The periodic structure becomes clearer if we write the matrix $A = (A_{nm})$, (\[2.7\]) for N=2 with the approximation (\[2.11\]), \[2.21\] A = () in which rows and lines are in the order $|2>,|1>,|0>,|-1>$. The determinant of this matrix is a kind of Toeplitz determinant (constant along parallel to the anti-diagonal). We have omitted the diagonal part $E_n^0$. It is clear that the eigenvalues are periodic, depending on the odd-even parity of $n$. When we consider several impurities, a sum over impurities $\sum_i A_{nm}^i$ should be taken. In the clean limit, we replace the quantities $s$ and $c$ by $s = \sum_i {\tilde C} \sin (2 k_F r_i)$ and $c = \sum_i {\tilde C} \cos (2 k_F r_i)$. The random average over the positions of these impurities gives the density of state (\[2.13\]) [@KL1]. ITZYKSON-ZUBER INTEGRAL AND ITS EXTENSION ========================================= In the previous section, we have discussed how we are led to study the crossover from the clean case to the dirty case. We now divide the matrix $\sum_{i=1}^{N_i}A_{nm}^i$ into two parts. One part, denoted by $H_0$, corresponds to the clean case, namely the matrix has a periodic structure as in (\[2.20\]), like a Toeplitz matrix, but with a sum over $i$. The other part is the remaining difference between the matrix $\sum_{i=1}^{N_i}A_{nm}^i$ and $H_0$, which is considered as a random matrix $V$. Thus we have \[3.1\] H = H\_0 + V The matrices $H_0$ and $V$ are both symplectic, represented by the symplectic group Lie algebra $Sp(N)$, which has the form (\[2.15\]). In the analogous, but simpler, problem of a unitary invariant random perturbation, ($H_0$ and $V$ were then both complex Hermitian matrices), a technique to study the crossover from the non-stochastic $H_0$ to a pure random matrix $V$ has been developped earlier [@BH1; @BHZ; @BH2]. Here we have to consider the symplectic case. As in the unitary case, we have to integrate over the Lie group which diagonalizes the matrix $H$. In the unitary case, the first step, the integration over the unitary group, was introduced in the study of two coupled random matrices by Itzykson and Zuber. [@Itzykson]. We now summarize the formulae of integration over Lie Groups, which are useful for the subsequent discussion. We assume that the probability distribution of $V$ is a Gaussian, and that $H_0$ is a fixed non-random matrix. \[It is possible to generalize it to some non-Gaussian distributions for $V$ ; for instance in the appendix B, we have considered the case of a general hypergeometric function whose argument is a matrix.\] The Itzykson-Zuber integral [@Itzykson] is an integral over the unitary group $U(n)$, \[3.2\] \_[U(n)]{} e\^[[tr]{} ( u a u\^ b )]{} du = C\_N[[det ]{}( e\^[a\_i b\_j]{})]{} where $\Delta(a) = \prod_{i<j} (a_i - a_j)$, and similarly $\Delta(b)$, are the Vandermonde determinants of the eigenvalues of the Hermitian matrices $a$ and $b$. The constant is found to be $C_N = \prod_{j=1}^N (j - 1)!$. By expressing ${\rm det}(e^{a_ib_j})$ as an alternating sum over the symmetric group $S_n$, one may write \[3.3\] \_[U(n)]{} e\^[&lt;[Ad]{}(u)a|b&gt;]{} du = C\_N[(w) e\^[&lt;w a|b&gt;]{} ]{} where ${\rm Ad}(u)\cdot a = u a u^{-1}$ is the adjoint action of $u\in U(n)$ on the matrix $a$. In this formula the matrix $a$ is diagonal, $a = {\rm diag}(a_1,...,a_n)$, $w$ is a permutation and $\epsilon(w)$ is its signature. We have used the notation $<a|b> = {\rm tr} (a b)$. This formula is a special case of a more general formula due to Harish-Chandra [@Harish-Chandra]. Let $G$ be a compact connected Lie group. Then Harish-Chandra’s result reads [@Harish-Chandra; @AltItzykson; @GrossRichards1] \[HC\] \_G e\^[&lt;[Ad]{}(g)a|b&gt;]{} dg = [\_[wW]{} ([det]{} w) e\^[&lt;wa|b&gt;]{} ]{} where $a, b$, $h$ belong to a Lie algebra $h$, and for any $H\in h$, \[3.5\] (H) = \_[\_[+]{}]{} (H) $\Delta_{+}$ is the collection of the positive roots, $w$ is the finite reflection group, called the Weyl (or Coxeter) group and $h$ is the Cartan subalgebra. The Harish-Chandra formula has been interpreted more recently as an integration over an orbit, with a symplectic structure. This structure implies that the saddle point method is exact, provided one sums over all the critical points [@Duistermaat; @Witten]. This is also related to the localization theorems [@Atiyah]. As an illustration of the semi-classical nature of this formula, we consider again the integral over the unitary goup $U(N)$, \[3.6\] I = g e\^[[tr]{}(a g b g\^)]{} Representing an element $g$ of $U(N)$ as \[3.7\] g = g\_0 e\^[i X]{} where $X$ is an element of the Lie algebra, we expand the argument of the exponential. up to second order in $X$, and impose the stationarity condition, \[3.8\] [tr]{} \[a g\_0 i X b g\_0\^\] + [tr]{} \[a g\_0 b ( - i X) g\_0\^\] = 0 for any $X$, i.e. \[3.9\] \[ b, g\_0\^ a g\_0\] = 0 If $a$ and $b$ are diagonal, it implies that $g_0$ should be a unitary permutation matrix, $g_0 = p$. Then, the integral $I$ becomes \[3.10\] I = \_p e\^[[tr]{}(a p b p\^[-1]{})]{} dX Performing the Gaussian integral over $X$ one recovers the Itzykson-Zuber formula. (This is not a derivation of course, but a way of verifying that for this integral the one-loop approximation is exact provided one sums over all the saddle-points). A similar saddle point technique may be applied to the symplectic case. We take $g \in Sp(N)$ in (\[3.6\]) and for $a$ and $b$ diagonal matrices with chiral eigenvalues, $a = {\rm diag}(a_1, ..., a_N, -a_1, ..., -a_N)$ and similarly for $b$. Then the same calculation yields \[3.11\] I = C [[det]{}\[ 2 [ ]{}(2 a\_i b\_j)\]]{} and this result will be repeatedly used below. DENSITY OF STATE ================ Using the method developped for the unitary case [@BH1; @BHZ; @BH2; @BH3; @Kazakov], we consider now the density of states with an external source matrix. Indeed for a random Gaussian $V$, the resulting probability for $H$ is a Gaussian with an external matrix source $A$ linearly coupled to $V$. We assume that the matrix $A$ belongs to the Lie algebra of the symplectic group and, without a loss of generality, we may take it as a diagonal matrix, $A = {\rm diag}(a_1,...,a_N, -a_1,...,-a_N)$. We take this $A$ as the unperturbed $H_0$. The probability distribution of the random matrix $V$ is assumed to be Gaussian. The density of states $\rho(\lambda)$ is given by \[4.1\] () = [1]{}&lt; \_[=1]{}\^N (- \_) &gt; The probability distribution $P(M)$ of the random matrix $M$ is $$\begin{aligned} \label{4.2} P(M) &=& \exp [ - {\rm tr} M^2 - {\rm tr} M A ]\nonumber\\ &=& \exp [ - \sum \lambda_i^2 - {\rm tr}( g \Lambda g^{\dag} A )]\end{aligned}$$ where $\Lambda = {\rm diag}(\lambda_1,...,\lambda_N,-\lambda_1,...,-\lambda_N)$ and $g$ is an element of the symplectic group. (Note that we use here a normalization which differs from our previous treatment [@BH1; @BH2; @BH3] of the unitary case). With the present normalization the edge of the density of states becomes of order $\sqrt{N}$ ; in the large N limit, the support of the density of states lies in an interval of order $[-\sqrt{2N},\sqrt{2N}]$. The density of states is obtained as the Fourier transform of the evolution operator \[4.4\] () = \_[-]{}\^ [dt]{} e\^[- i t ]{} U\_A(t) and using the symplectic measure $\Delta^2(\lambda)$ with $\Delta(\lambda) = \prod (\lambda_i^2 - \lambda_j^2) \prod \lambda_k$ [@HM], and the previous Harish-Chandra integral formula (\[3.11\]), we obtain $$\begin{aligned} \label{4.3} U_A(t) &=& {1\over{N}}\sum_{\alpha=1}^N \int_{0}^{\infty} \prod_{i=1}^N{\rm d\lambda_i} {\prod_{1\le i<j\le N}(\lambda_i^2 - \lambda_j^2) \prod_{1\le k \le N} \lambda_k \over{\prod_{1\le i<j\le N}(a_i^2 - a_j^2)\prod_{1\le k \le N}a_k}} e^{- \sum \lambda_i^2 + i t \lambda_{\alpha}} \nonumber\\ &\times & {\rm det}[ {\rm sh} ( 2 \lambda_i a_j)]\end{aligned}$$ (up to a factor 2 in (\[3.11\]) absorbed in the coefficient $C$). By the reflexion symmetry $\lambda_i\rightarrow - \lambda_i$, and by the exchange symmetry between $\lambda_i$ and $\lambda_j$, we can extend the integrations over the $\lambda_i$’s from $-\infty$ to $\infty$ . The change of sign for $i t \lambda_i$ can be absorbed since we can change $t \rightarrow - t$ by parity. Then the expression for $U_A(t)$ simplifies to \[4.5\] U\_A(t) = [1]{} \_[=1]{}\^N \_[-]{}\^ \_[i=1]{}\^N [d]{}\_i [\_[1i&lt;jN]{}(\_i\^2 - \_j\^2) \_[1k N]{} \_k ]{} e\^[- \_i\^2 + i t \_ + 2 \_[i]{}\^N a\_i \_i]{} Before proceeding to the explicit integrations over the $\lambda_i$’s in $U_A(t)$, let us investigate the expression for the density of states $\rho(\lambda)$ , in the absence of any external source, which one may then calculate by the usual method with orthogonal polynomials . The N-point level distribution is given \[4.6\] \_N( \_1,...,\_N) = C \_[1i&lt;jN]{}(\_i\^2 - \_j\^2)\^2 \_[1k N]{} \_k\^2 Using orthogonal polynomials method [@Mehta], this distribution may be written as \[4.7\] \_N( \_1,...,\_N) = [det]{} \[ K\_N(\_i,\_j)\] with the kernel $K_N(\lambda_i,\lambda_j)$ given by a sum of Laguerre polynomials, and in particular the density of states $\rho(\lambda)= K_N(\lambda,\lambda)$ is equal to \[4.8\] () = [1]{} \_[n=0]{}\^[N-1]{} [n!]{} L\_n\^[([1]{})]{}(\^2) L\_n\^[([1]{})]{}(\^2)\^2 e\^[-\^2]{}. $L_n^{(1/2)}(x)$ are associated (generalized) Laguerre polynomials, orthogonal with the normalization \[4.9\] \_[0]{}\^ e\^[-x]{} x\^[[1]{}]{} L\_n\^[([1]{})]{}(x) L\_m\^[([1]{})]{}(x) dx = [([3]{} + n)]{} \_[n,m]{} ; leading to $ L_0^{(1/2)} (x) = 1, L_1^{(1/2)}(x) = 3/2 - x$, $ L_n^{(1/2)}(x) = \sum_{r=0}^n (-1)^r [\Gamma(n + 3/2)/ \Gamma(n - r + 1)\Gamma( r + 3/2)]x^r/r!$. The asmptotic behavior of these Laguerre polynomials in the limit $N\rightarrow \infty$ is [@Erdelyi] \[4.10\] L\_N\^[([1]{})]{}(x) = [1]{}e\^[[1]{}x]{} (2 ) + O([1]{}). With the Christoffel-Darboux identity, we have $$\begin{aligned} \label{4.11} & &\sum_{n=0}^{N-1} {n!\over{\Gamma({3\over{2}} + n)}} L_n^{({1\over{2}})}(x) L_n^{({1\over{2}})}(y)\nonumber\\ &=& - {N!\over{\Gamma({1\over{2}} + N)}} {L_{N}^{({1\over{2}})}(x) L_{N-1}^{({1\over{2}})}(y) - L_{N-1}^{({1\over{2}})}(x) L_{N}^{({1\over{2}})}(y)\over{x - y}}\end{aligned}$$ and using the large $N$ aymptotic behavior (\[4.10\]), we obtain $$\begin{aligned} \label{4.12} && x e^{-x}[\sum_{n=0}^{N-1}{n!\over{\Gamma({3\over{2}} + n)}} L_n^{({1\over{2}})}(x) L_n^{({1\over{2}})}(x)]\nonumber\\ &=& x e^{-x}[L_{N}^{({1\over{2}})}(x) {d\over{d x}}L_{N-1}^{({1\over{2}})}(x) - L_{N-1}^{({1\over{2}})}(x) {d\over{d x}}L_{N}^{({1\over{2}})}(x)]\nonumber\\ &\simeq&{1\over{\pi}}[ {\sqrt{N}\over{\sqrt{x}}}\sin (\sqrt{{x\over{N}}}) - {1\over{4\sqrt{N x} }} \sin(4\sqrt{N x})] \nonumber\\ &\simeq& {1\over{\pi}}[1 - {1\over{4\sqrt{N x} }} \sin(4\sqrt{N x})]\end{aligned}$$ Thus, putting $x = \lambda^2$, we get \[4.13\] () = [1]{}\[ 1 - [(4 )]{}\]. This result is identical to (\[2.13\]), when we substitute $\omega_0 = \frac{\pi}{4 N}$ and $E = \lambda/\sqrt{N}$. We now return to the integral (\[4.5\]) over the $\lambda_i$’s in the presence of the external source. We first replace $t \rightarrow 2 t$ and following [@BH1; @BHZ], for fixed $\alpha$, we denote by $b_i$ the sum $b_i = a_i + i t \delta_{i,\alpha}$ . One then uses the integral \[4.14\] \_[-]{}\^ [d]{}\_i \_[i&lt;j]{}(\_i\^2 - \_j\^2) \_[i=1]{}\^N \_i e\^[- (\_i - b\_i)\^2 ]{} = \_[i&lt;j]{} (b\_i\^2 - b\_j\^2)\_[i=1]{}\^N b\_i ; (after the translation $\lambda_i \rightarrow\lambda_i+b_i$ the result follows easily from antisymmetry under permutation of the $b_i$’s, parity in $b_i$, and counting the degree of the resulting polynomial in those $b$’s). Using the normalization $U_A(0) = 1$, and writing the sum over $\alpha$ as a contour integral around the $a_j^2$’s in the complex $u$-plane, we obtain \[4.15\] U\_A(t) = [1]{} \_[j=1]{}\^N [(( + i t)\^2 - a\_j\^2)]{} [1]{}(1 + [i t]{}) e\^[- t\^2 + 2 i t ]{}. Fourier transforming $U_A(t)$ we obtain the density of states $\rho(\lambda)$ \[4.16\] () = \_[-]{}\^ U\_A(t) e\^[- 2 i t ]{} [dt]{} Let us verify the consistency with the orthogonal polynomial result (\[4.8\]) when the external source vanishes. Then U\_0(t) = [1]{} ( 1 + [i t]{}) e\^[- t\^2 + 2 i t ]{} and $$\begin{aligned} \label{4.18} \rho(\lambda) &=& \int_{-\infty}^{\infty} {dt\over{2 \pi}} U_0(t) e^{- 2 i t\lambda}\nonumber\\ &=& {i (-1)^N\over{N}} \int_{-\infty}^{\infty}{dx\over{2 \pi}} \oint{du\over{2 \pi i}} ({x\over{\sqrt{u}}})^{2N + 1} {1\over{u + x^2}} e^{-x^2 - 2 i x \lambda - u + 2 \lambda \sqrt{u}}\end{aligned}$$ (we have shifted $t$ to $ t = x + i \sqrt{u}$). The integration contour is a small loop around the origin $u = 0$. Expanding $1/(u + x^2) = x^{-2}( 1 - {u\over{x^2}} + \cdots )$, and noting that \[4.19\] e\^[- u + 2 ]{} = [ 2 ([3]{})]{} (-1)\^n L\_n\^[([1]{})]{} (\^2) and \[4.20\] \_[-]{}\^ [dx]{} x\^[2n + 1]{} e\^[- x\^2 - 2 i x ]{} = - [i n!]{}L\_n\^[([1]{})]{}(\^2) e\^[-\^2]{} we v erify the consistency of the integral representation (\[4.18\]) with the orthogonal polynomial result (\[4.8\]). The integral representation (\[4.20\]), gives an easy way to recover the large $n$ asymptotic behavior (\[4.11\]) of the Laguerre polynomials through the steepest descent method. Note that the chiral invariance leads as expected to an even density of states. In the presence of the external source, the density of states becomes () = [i (-1)\^[N - 1]{}]{} \_[-]{}\^[dx]{} ([x]{}) \_[=1]{}\^N ([x\^2 + a\_\^2 ]{}) [1]{} e\^[-x\^2 - 2 i x - u + 2 ]{}\ (We shall analyze this density of states as the limit of $\lambda \rightarrow \mu $ of the kernel $K_N(\lambda,\mu )$ in the next section). We note here that if we make the change of variable, $u = v^2$, we have () = [i (-1)\^[N -1]{}]{} \_[-]{}\^ [dx]{} \_[=1]{}\^N ([x\^2 + a\_\^2 ]{}) [x]{} e\^[-x\^2 - 2 i x - v\^2 + 2 v]{}\ where the contour encloses all the $\pm a_{\gamma}$. For example when $N= 1$, we obtain $$\begin{aligned} \rho(\lambda) &=& {\lambda\over{2 \sqrt{\pi}}}{{\rm sinh}( 2 \lambda a_1 ) \over{a_1}}e^{- a_1^2 - \lambda^2}\nonumber\\ &=& {\lambda\over{4 \sqrt{\pi} a_1}}[ e^{-(\lambda - a_1)^2} - e^{-(\lambda + a_1)^2}]\end{aligned}$$ Near the origin $\lambda = 0$, the density of states $\rho(\lambda)$ becomes $\rho(\lambda) \sim {\lambda^2\over{\pi}} {\rm exp}( - a_1^2 - \lambda^2)$. Thus the density of states is an even function and it vanishes near the origin as $\lambda^2$. CORRELATON FUNCTION =================== The $n$-level correlation functions are expressed as the determinant of the kernel $K_N(\lambda,\mu )$, exactly as for Hermitian random matrices in a source. Indeed the formula of (\[4.14\]) is quite similar to (3.8) of [@BH3]. Therefore, all the derivations of the Hermitian matrix model in [@BH1; @BH2; @BH3], may be repeated almost identically for the $Sp(N)$ case. For instance, the two-level correlation function $\rho^{(2)}(\lambda,\mu )$ is obtained as the double Fourier transform of $U_A(t_1,t_2)$, \[5.1\] \^[(2)]{}(,) = e\^[- i t\_1 - i t\_2 ]{} U\_A(t\_1,t\_2) in which $U_A(t_1,t_2)$ is defined as \[5.2\] U\_A(t\_1,t\_2) = &lt; [1]{} \_[=1]{}\^N e\^[i t\_1\_]{} [1]{} \_[=1]{} e\^[i t\_2 \_]{} &gt; Using the integral formulae (\[3.10\]) and (\[4.14\]), we find $$\begin{aligned} \label{5.3} U_A(t_1,t_2) &=& {1\over{N^2}}\sum_{\alpha\neq \beta} {(a_{\alpha} + i t_1)^2 - ( a_{\beta} + i t_2)^2\over{ (a_{\alpha}^2 - a_{\beta}^2)}} \prod_{\gamma \neq (\alpha,\beta)} {(a_{\alpha} + i t_1)^2 - a_{\gamma}^2\over{ a_{\alpha}^2 - a_{\gamma}^2 }} \nonumber\\ &\times &\prod_{\gamma \neq (\alpha,\beta)} {(a_{\beta} + i t_1)^2 - a_{\gamma}^2\over{ a_{\beta}^2 - a_{\gamma}^2 }} (1 + {it\over{a_{\alpha}}})(1 + {i t_2\over{a_{\beta}}}) e^{- t_1^2 - t_2^2 + 2 i \sqrt{u} t_1 + 2 i \sqrt{v} t_2} \nonumber\\ \end{aligned}$$ (wa have subtracted the term $\alpha = \beta$ in $U_A(t_1,t_2)$). The summation over $\alpha$ and $\beta$ is expressed by contour integration over two complex variables $u$ and $v$, $$\begin{aligned} \label{5.4} U_A(t_1,t_2) &=& {1\over{N^2}} \oint {du dv\over{(2 \pi i)^2}} \prod_{\gamma=1}^N {(\sqrt{u} + i t_1)^2 - a_{\gamma}^2\over{ (u - a_{\gamma}^2)}} \prod_{\gamma = 1}^N {(\sqrt{v} + i t_2)^2 - a_{\gamma}^2\over{ v - a_{\gamma}^2 }}\nonumber\\ &\times & {1\over{(\sqrt{u} + i t_1)^2 - v}} {1\over{(\sqrt{v} + i t_2)^2 - u}} (1 + {i t_1\over{\sqrt{u}}})(1 + {i t_2\over{\sqrt{v}}}) \nonumber\\ &\times & {(u - v) [(\sqrt{u} + i t_1)^2 - (\sqrt{v} + i t_2)^2] \over{[(\sqrt{u} + i t_1)^2 - u][(\sqrt{v} + i t_2)^2 - v]}} e^{- t_1^2 - t_2^2 + 2 i \sqrt{u} t_1 + 2 i \sqrt{v} t_2} \nonumber\\ \end{aligned}$$ We now shift $t_1\rightarrow t_1 + i \sqrt{u}$ and $ t_2\rightarrow t_2 + i \sqrt{v}$, note that = [\[(it\_1)\^2 - u\]\[(it\_2)\^2 - v\]]{} - 1 and divide $U_A(t_1,t_2)$ into two terms. The first one, which comes from (-1) gives the product of the two density of states $\rho(\lambda)\rho(\mu )$, and the second one is a product of two integrals over $t_1,u$ and $t_2,v$ respectively. Therefore, we end up with \^[(2)]{}(,) = K\_N(,) K\_N(,) - K\_N(,)K\_N(,) with $$\begin{aligned} \label{5.7} K_N(\lambda,\mu ) &=& {1\over{N}} \oint {du\over{2 \pi i}}\int_{-\infty}^{\infty} {dt\over{2\pi}} e^{- t^2 - 2 i t \mu - u + 2 \sqrt{u} \lambda} \prod_{\gamma=1}^N {(i t)^2 - a_{\gamma}^2\over{u - a_{\gamma}^2}}{1\over{ (it)^2 - u}} \nonumber\\ &\times & ({it\over{\sqrt{u}}}) ; \end{aligned}$$ (the contour in the $u$-plane encircles all the $a_{\gamma}^2$) . From this expression, we recover the previous result (\[4.15\],\[4.16\]) for the density of states as $\rho(\lambda)= K_N(\lambda,\lambda)$. The n-point correlation can be analyzed as in [@BH3] and leads here also to a determinant form, (\_1,,\_n) = [det]{}\[K\_N(\_i,\_j)\] for an arbitrary external source. We now discuss Dyson’s short-distance universality, within our model, i.e. when we vary the external source $A$, through the explicit expression of the kernel $K_N(\lambda,\mu )$. We return first to the sourceless case,$a_{\gamma} = 0$, and consider the large N limit of $K_N(\lambda,\mu )$ for $\lambda,\mu $ are order of one. We scale $t \rightarrow \sqrt{N} t$, and $u \rightarrow N u$. The kernel is then \[5.9\] K\_N(,) = - [1]{} e\^[- N (f(t) + g(u))]{} ([it]{})[1]{} where $f(t) = t^2 + 2i\mu /\sqrt{N} - 2 \ln t$ and $g(u) = u - 2 \lambda \sqrt{u}/\sqrt{N} + \ln u$. In the large N-limit we have the saddle points, $t_c^{(1)} = 1 - i\mu /2\sqrt{N}$, $t_c^{(2)} = - 1 - i\mu /2\sqrt{N}$,$u_c^{(1)} = -1 + i\lambda/2\sqrt{N}$ and $u_c^{(2)} = -1 - i \mu /2\sqrt{N}$. Adding these saddle-points contributions, we obtain \[5.10\] K\_N(,) = [1]{}\[[(2(- )) ]{} - [(2(+ )) ]{}\] When $\lambda\rightarrow \mu $, we recover the expression (\[4.13\]) of the density of states. In the presence of the external source, we find a similar expression to Eq.(\[5.9\]), with \[5.11\] f(t) = t\^2 + [2it ]{} - [1]{}(t\^2 + [a\_\^2 ]{}) \[5.12\] g(u) = u - [2 ]{} + [1]{} ( u - [a\_\^2]{}) Using the definition , $f^{\prime}(t_c)=0,g^{\prime}(u_c) = 0$, of the saddle-points we find \[5.13\] K\_N(,) = [1]{}\[[(2 t\_[c0]{}(- )) ]{} - [(2 t\_[c0]{}(+ )) ]{}\] where $t_{c0}$ is a solution of \[5.14\] [1]{} \_[= 1]{}\^N [1]{} = 1 Here we have assumed that the order of $a_{\gamma}$ is $a_{\gamma}^2 \leq O(N)$. The derivation of (\[5.13\]) is the same as (\[5.10\]). The difference is just a normalization, due to the change of the saddle-point $t_{c0}$. Note that the support of the density of states is inside the interval of $-\sqrt{N}$ and $\sqrt{N}$. Therefore, universality holds provided that the eigenvalues of the external source matrix are located in an interval of the same order of magnitude ; ( it has to be of same order as the support of the density of states for the zero external source case). In this universal regime for the correlation function, the energies $\lambda$ and $\mu $ are assumed of order one, namely $\lambda/\sqrt{N} \rightarrow 0$ in the large N limit. Since $\lambda$ can be order of $\sqrt{N}$, the universal behavior of (\[5.13\]) appears only near the origin, as exepcted, and is differnt from the non-universal bulk behaviour. The universality found here, is similar to that found in other chiral random matrix models studied in [@BHZ]; there it was the density of states which took a universal form near the origin, and became independent of the external source or of the non-Gaussian distributions (as seen by a rescaling of the energy). When the $a_{\gamma}^2$ spread over an interval larger than $N$, the universal form of (\[5.13\]) does not hold any more, since the saddle-point method of (\[5.14\]) breaks down. In our formulation, the amplitude of the random matrix is fixed. We now change the strength of the external source. For this purpose, we introduce a parameter $C$, which determines the strength of the external source $H_0$, respective to the random potential $V$. Let us consider the example of the external source, \[5.15\] a\_ = C [tan]{} \[[(2- 1)]{}\] where $\gamma = 1, 2,..., N$, and $C$ is a parameter. When $\gamma/( 2N + 1) << 1$, we have \[5.16\] a\_ = [C ]{} ( - [1]{}) and the eigenvalues of the external source are equally distributed, and thus reproduce the excitation spectrum inside the clean superconductor vortex, if we identify $\hbar \omega_0 = C\pi/(2N + 1)$ in (\[2.3\]). Since we are interested in the behavior near the origin for the energies $\lambda$ and $\mu $ in $K_N(\lambda, \mu )$, this linear approximation of the ${\rm tan}(x) \simeq x$ is valid. More generally, using the formula, \[5.17\] \_[m = 1]{}\^N ( x\^2 + [tan]{}\^2 \[[( 2 m - 1) ]{}\]) = [1]{} \[ ( 1 + x)\^[2 N + 1]{} + ( 1 - x)\^[2 N + 1]{}\] we may replace the product involving the $a_\gamma$ in (\[5.7\]), and obtain $$\begin{aligned} \label{5.18} K_N(\lambda,\mu ) &=& - {1\over{N}} \oint {du\over{2 \pi i}} \int {dt\over{2 \pi}} \left[ {( 1 + { t\over{C}})^{2N + 1} + ( 1 - { t\over{C}})^{2N + 1}\over{ ( 1 + {i \sqrt{ u} \over{C}})^{2N + 1} + ( 1 - {i \sqrt{u} \over{C}})^{2N + 1}}}\right ] \nonumber\\ &\times & {1\over{t^2 + u}} ({i t\over{\sqrt{u}}}) e^{- t^2 - 2 i t \mu - u + 2 \sqrt{u } \lambda} \end{aligned}$$ This expression is exact for finite $N$ when the $a_{\gamma}$ are given by (\[5.15\]). It is easily seen that when $C\rightarrow 0$, we recover the previous external source free result (\[5.9\]). For non-zero $C$ we have two different cases according to the size of $C$,\ i) if $C$ is of order $\sqrt{N}$, we recover the universal behavior of (\[5.13\]). (We simply scale $t\rightarrow \sqrt{N}t$, and $u \rightarrow Nu$, and use the saddle point method). We recover then the result (\[5.13\]), with t\_[c0]{} = [1]{}( - [C ]{} ) .\ ii) in the second case $C \sim O(N)$, we change $u = v^2$ in (\[5.18\]) : $$\begin{aligned} \label{5.20} K_N(\lambda,\mu ) &=& - {1\over{N}} \oint {dv\over{2 \pi i}} \int {dt\over{2 \pi}} \left[ {( 1 + { t\over{C}})^{2N + 1} + ( 1 - { t\over{C}})^{2N + 1}\over{ ( 1 + {i v \over{C}})^{2N + 1} + ( 1 - {i v \over{C}})^{2N + 1}}}\right ] \nonumber\\ &\times & {i t\over{t^2 + v}} e^{- t^2 - 2 i t \mu - v^2 + 2 v \lambda} \end{aligned}$$ In the large N-limit (\[5.20\]) reduces to $$\begin{aligned} \label{5.21} K_N(\lambda,\mu ) &=& - {1\over{N}} \oint {dv\over{2 \pi i}}\int {dt\over{2 \pi}} {{\rm cosh} ( {2 N t\over{C}} )\over{{\rm cos} ( {2 N v\over{C}} )}}{i t\over{t^2 + v^2}} \nonumber\\ &\times & e^{- t^2 - 2 i t \mu - v^2 + 2 v \lambda} \end{aligned}$$ If we rescale $t \rightarrow \sqrt{N} t$ and $v \rightarrow \sqrt{N} v$ poles appears in the $v$-plane at $ v = C ( m - {1\over{2}})\pi/ 2 N^{3/2}$ for which ${\rm cos}( 2 N^{3/2} v/C) = 0$ ; $m$ is an integer. Within the residues of these poles, we have the Gaussian factors e\^[- N (v - )\^2]{} = e\^[-([C( m - [1]{})]{} - )\^2]{} There are thus two different regions for $\lambda$. We first consider $\lambda \sim O(\sqrt{N})$, i.e. the bulk case. Then $\lambda/\sqrt{N} >> \sqrt{N}/C$, and the saddle point of $t$ is $t_0 = - i \lambda/\sqrt{N}$. We set $\lambda = \mu $, and for this this saddle point, we have the density of states $\rho(\lambda) = K_N(\lambda,\lambda)$ $$\begin{aligned} \label{5.23} \rho(\lambda) &=& - {1\over{2\sqrt{\pi N}}} \sum_m (-1)^m [{\lambda{\rm cos} ( {2 N\lambda\over{C}}) \over{- \lambda^2 + ({C (m - {1\over{2}})\pi\over{2 N}})^2}}] \nonumber\\ &\times & e^{-({C( m - {1\over{2}})\pi\over{2 N}}- \lambda )^2} \end{aligned}$$ Note that the zeros of the denominator are cancelled by zeros of the numerator ${\rm cos}( 2 N \lambda/ C)$. When $C/N >> 1$, the exponential factor of (\[5.23\]) damps the result and we may set $\lambda = C( m - 1/2)\pi/(2 N)$, then = - (- 1)\^m [N]{} Thus we find () = \_m e\^[-([C( m - [1]{})]{}- )\^2]{} which is a sum of Gaussians : the density of states is just sum of Gaussian peaks around the levels $\lambda = C( m - 1/2)\pi/2 N$. This result is valid provided $\lambda$ is not too close to the origin. When $\lambda$ is near the origin, namely $\lambda \sim O(1)$, we have $\lambda/\sqrt{N} \sim \sqrt{N}/C$ since $C$ is order $N$. In this case, we have to take into account the term ${\rm cosh}( 2N t/C)$ in the saddle-point equation. The saddle-point becomes $t_0 = - i \lambda/\sqrt{N} \pm \sqrt{N}/C$. Using this value in (\[5.21\]), we find () = [Re]{} (-1)\^n \[ [+ i [N]{}]{}\] e\^[-\[[C]{}( n - [1]{})- \]\^2]{} Near $\lambda = 0$, the density of states behaves as $\rho(\lambda) \sim \lambda^2$. For example, when $C = N$, we have $\rho(\lambda ) \sim \pi \lambda^2 {\rm exp} ( - \pi^2/16)/(1 + \pi^2/16)$, and $\rho(\lambda)$ has peaks at $\lambda = \pm \pi/4, \pm 3 \pi/4,...$. Thus, when $C \sim N$, which means that the random potential is weak compared to the external source, the density of states takes a form which is different from the universal form (\[5.13\]). CROSSOVER FROM THE CLEAN LIMIT TO THE DIRTY LIMIT ================================================= In the clean case, when $k_F r_i \sim k_F \xi >> 1$, we approximate $A_{nm}^i$ by (\[2.19\]). Then, the periodic spectrum is obtained \[6.1\] E\_n = - \_0 ( n - [1]{} + z (- 1)\^n ) When the number of impurities inside the vortex $N_i \simeq 1$, we are in the superclean case, and $\tilde z$ in (\[6.1\]) is a function of the position of this impurity. If $1 < N_i < N_{ic}$, we are in a clean case, and we need to average over the different values of $\tilde z$. In the previous section, we have discussed the case where the deterministic term $E_n = - \hbar \omega ( n - {1\over{2}})$ is coupled to the random matrix $A$. Here we consider for the external source the matrix whose eigenvalues are (\[6.1\]). We may then use the formula \[6.2\] \_[r = 1]{}\^N \[x\^2 + [tan]{}\^2 ( [r ]{})\] = [1]{} \[ (1 + x )\^[2(N + 1)]{} - ( 1 - x)\^[2(N + 1)]{}\] , which is similar to (\[5.17\]). ( We assume that $N$ is odd here). We divide this product into two parts, r-odd and r-even . The r-even part is obtained immediately from (\[6.2\]) as \[6.3\] \_[r = even]{}\^[N - 1]{} \[x\^2 + [tan ]{}\^2 ( [r ]{} )\] = [1]{} \[ (1 + x)\^[N + 1]{} - (1 - x)\^[N + 1]{}\] Dividing (\[6.2\]) by (\[6.3\]), we obtain the expression for the r-odd part, \[6.4\] \_[r = odd]{}\^N \[x\^2 + [tan]{}\^2 ( [r ]{})\] = [1]{}\[ (1 + x)\^[N + 1]{} + ( 1 - x)\^[N + 1]{} \] We now consider the eigenvalues (\[6.1\]) as an external source, \[6.5\] a\_ = \[C /2(N + 1)\](- [1]{} + z ( - 1)\^), and introduce $z_0 = C \pi/4(N + 1)$, $z = - C \pi \tilde z/2(N + 1)$. Using the expressions (\[6.3\]) and (\[6.4\]), we write $$\begin{aligned} \label{6.6} &&({x - iz_0 - iz\over{C}}) \prod_{r = even}^{N - 1} [({x - iz_0 - iz\over{C}})^2 + {\rm tan}^2({r \pi\over{2 ( N + 1)}})]\nonumber\\ &\times & \prod_{r = odd}^N [ ({x - z_0 + z\over{C}})^2 + {\rm tan}^2 ({r \pi\over{2( N + 1)}})]\nonumber\\ &=& {1\over{4( N + 1)}} [ ( 1 + {x - iz_0 - iz\over{C}})^{N + 1} - ( 1 - {x - iz_0 - iz\over{C}})^{N + 1}] \nonumber\\ &\times & [( 1 + {x - iz_0 + iz\over{C}})^{N + 1} + ( 1 + {x - iz_0 + iz\over{C}})^{N + 1}]\end{aligned}$$ In the large N limit, when $r\pi/(2(N + 1)) << 1$, the left hand side of (\[6.5\]) vanishes at $ x = - i(\pm r - 1/2 + z(-1)^r) C \pi /[2(N + 1)]$ . Noting that $$\begin{aligned} \label{6.7} \prod_{\gamma} (t^2 + a_{\gamma}^2) &=& \prod_{\gamma} [ t - i(z_0 + z) + i {C\gamma \pi\over{2(N + 1)}}] [ t - i(z_0 + z) - i {C\gamma \pi\over{2(N + 1)}}]\nonumber\\ &=& \prod_{\gamma} [ ( t - i(z_0 + z))^2 + C^2 {\rm tan}^2 ({\gamma \pi \over{2(N + 1)}})]\end{aligned}$$ we have from (\[5.7\]) (and the change of variable $\sqrt{u} = v$), $$\begin{aligned} \label{6.8} K_N(\lambda,\mu ) &=& - {1\over{N}} \oint{dv\over{2 \pi i}}\int {dt\over{2 \pi}} e^{-t^2 - 2 i t \mu - v^2 + 2 v \lambda} {it\over{t^2 + v^2}} \nonumber\\ &\times & {[( 1 + {i\over{C}}( - it - z_0 - z))^{N + 1} - ( 1 - {i\over{C}}( - i t - z_0 - z))^{N + 1}] \over{[( 1 + {i\over{C}}( v - z_0 - z))^{N + 1} - ( 1 - {i\over{C}}( v - z_0 - z))^{N + 1}] }}\nonumber\\ &\times &{ [( 1 + {i\over{C}}( - i t - z_0 + z))^{N + 1} + ( 1 - {i\over{C}}( - i t - z_0 + z))^{N + 1}]\over{ [( 1 + {i\over{C}}( v - z_0 + z))^{N + 1} + ( 1 - {i\over{C}}( v - z_0 + z))^{N + 1}]}} \nonumber\\\end{aligned}$$ As in the previous section, we have to distinguish two different cases, i) $C \sim O(N)$ and ii) $C \sim O(\sqrt{N})$. There are also two regions $\lambda \sim O(\sqrt{N})$ and $\lambda \sim O(1)$. When $C \sim O(N)$, we exponentiate the factor in the bracket of (\[6.7\]), $$\begin{aligned} \label{6.9} &&{{\rm sin}[ {N\over{C}}( - i t - z_0 - z) ]{\rm cos} [{N\over{C}} ( - i t - z_0 + z)]\over{ {\rm sin}[ {N\over{C}}( v - z_0 - z)] {\rm cos} [ {N\over{C}} (v - z_0 + z)]}}\nonumber\\ &=& {{\rm sin} [ {2 N\over{C}} ( - i t - z_0)] - {\rm sin} ({2 N \over{C}} z)\over{ {\rm sin} [ {2 N\over{C}} ( v - z_0)] - {\rm sin} ({2 N \over{C}} z)}}\end{aligned}$$ Since $z_0 = C \pi/[4(N + 1)]$, we have $$\begin{aligned} \label{6.10} K_N(\lambda,\mu ) &=& - {1\over{N}} \oint {dv\over{2 \pi i}}\int {dt\over{2 \pi}} {{\rm cosh} ( {2 N t\over{C}} ) + {\rm sin}({2N\over{C}} z) \over{{\rm cos} ( {2 N v\over{C}} ) + {\rm sin}({2N\over{C}} z) }}{i t\over{t^2 + v^2}} \nonumber\\ &\times & e^{- t^2 - 2 i t \mu - v^2 + 2 v \lambda} \end{aligned}$$ When $z=0$, we obtain the same expression as in the previous section (\[5.21\]). Poles in the integral over $v$ are present at $ v = z_0 + z + C n\pi/ N $ and $v = z_0 - z + (2 n + 1) (C \pi/2 N)$ from (\[6.8\]). The residues $R$ for the first poles are \[6.11\] R = (- 1)\^n i t [[sin]{}\[ [N]{}( - i t - z\_0 - z)\] [cos]{} \[ [N]{} ( - i t - z\_0 + z)\]]{}\ When $\lambda = \mu \sim O(\sqrt{N})$, the saddle point becomes $t_0 = - i \lambda/\sqrt{N}$ and we set this value in $R$. The second pole gives a similar expression. Furthermore, if $C/N >> 1$, we can approximate $\lambda \sim z_0 + z + C n\pi/ N$. Then, we find that the residue $R$ in (\[6.10\]) becomes $R = N/2C$. Therefore, we have the density of states for $C/N >>1$, and $\lambda \sim O(\sqrt{N})$, \[6.12\] () = \_n \[ e\^[-( z\_0 + z + [C n ]{} - )\^2]{} + e\^[- ( z\_0 - z + [C ( 2 n + 1)]{} - )\^2]{}\] which is a sum of Gaussian distributions. In the clean case we have several impurities and one may average this density of states over $z$ . As discussed in [@KL1], the measure for this $z$-average is \[6.13\] \_0(z) = A[cos]{}\^2\[ [ 2 N z ]{}\] where $A = 4N/C \pi$. The matrix elements are represented by a quaternion, and the measure is isomorphic to the uniform measure over the four-dimensional sphere $s^{3}$, $Sp(1) \sim S^3$. In spherical coordinate this leads to (\[6.13\]). When $C/N >> 1$, the integration of (\[6.12\]) over $z$ with the measure (\[6.14\]) yields $$\begin{aligned} \label{6.14} <\rho(z)> &=& \int_{-\infty}^{\infty} e^{-(z_0 + z + {C n \pi\over{N}} - \lambda)^2} A {\rm cos}^2 ({2 N z\over{C}}) dz \nonumber\\ &=& A {\sqrt{\pi}\over{2}} [ 1 - e^{- {4 N^2\over{C^2}}} {\rm cos} ( {4 N\over{C}} \lambda )] \end{aligned}$$ (in which we have used $N/C << 1$). When $C/N \rightarrow \infty$, the exponential factor ${\rm exp}( - 4 N^2/C^2)$ becomes one and the density of states becomes ${\rm sin}^2( 2 N \lambda/C)$. This is indeed the result for the clean case found in [@KL1]. When $C/N$ is finite, at the energy $\lambda = n \pi C/2N = n \hbar \omega_0$, the density of state becomes non-vanishing, except at the origin. At the origin $\lambda = 0$, we have a zero due to the factor $ i t = \lambda $ in (\[6.11\]). It may be interesting to note this deviation from zero has been observed in the numerical work of [@KL1]. When $C$ becomes order of $\sqrt{N}$, we may apply the saddle point method in (\[6.8\]). We take the saddle point equation (\[5.14\]) for $a_{\gamma} = {\pi C\over{2 N}} (n - {1\over{2}} + ( -1)^{\gamma} \tilde z)$. However, the second and third term $( -1/2 + (-1)^{\gamma} \tilde z)$ can be neglected since they remain of order one, compared to $n$ which can be order of $N$. Thus we may approximate $a-{\gamma}$ by $a_{\gamma} = \pi^2 C^2 n^2/4 N^3$, which is independent of $z$. Therefore the average over $z$ does not change the result for the density of states, and we have a universal result for the kernel and for the density of states (\[5.13\]). It is possible to take the average (\[6.10\]) by the formula, \[6.15\] \_[-]{}\^ dz [a + [sin]{} z]{} [cos]{}\^2 z = Then, changing the contour integration in the $v$-plane to the integration near the saddle-points for $v$ in the large N limit, we have $$\begin{aligned} \label{6.16} K_N(\lambda,\mu ) &=& - {CA\pi\over{4 N^2}} \int{dt\over{2 \pi}}\int {dv\over{2 \pi}}({t\over{t^2 + v^2}}) [ ({\rm cosh}({2 N t\over{C}}) - {\rm cos}({2 N v\over{C}}) ) \nonumber\\ &\times & {\rm sin} ( {2 N v\over{C}})] e^{- t^2 - 2 i t \mu - v^2 + 2 v \lambda} \end{aligned}$$ The term ${\rm sin}( 2N v/C )$ comes from the sum of integration paths in opposite directions. We assume here that $\lambda$ and $\mu $ are of order one, and $O(\lambda) \sim O({N\over{C}})$. After the shift $t \rightarrow \sqrt{N} t$, the saddle-point is at $t_0 = - i \mu /\sqrt{N} \pm \sqrt{N}/C$ and $v = \lambda/\sqrt{N} \pm i \sqrt{N}/C$ for the terms in (\[6.16\]). Inserting the values of these saddle-points, we have $$\begin{aligned} \label{6.17} \rho(\lambda) &=& {N\over{C}}[ 1 - {C\over{4 N \lambda}} {\rm sin} ({4 N \lambda\over{C}})] + {C\over{2 N}}{\rm sin}^2 ({2 N \lambda\over{C}}) \nonumber\\ &-& {C \lambda^2\over{4 N ( \lambda^2 + {N^2\over{C^2}})}} [ 1 + e^{- {4 N^2\over{C^2}}}{\rm cos} ({4 N \lambda\over{C}})] \nonumber\\ &+& {\lambda\over{4 (\lambda^2 + {N^2\over{C^2}})}} e^{- {4 N^2\over{C^2}}}{\rm sin} ( {4 N \lambda\over{C}}) \end{aligned}$$ The above expression reduces to that of the dirty case when $C/N \rightarrow 0$, since the first term is then dominant. Thus (\[6.17\]) gives the correction to the dirty limit, and it is valid for $C \sim N$. In the large $C$, $C >> N$, it is easy to recover the clean case from the expression (\[6.16\]). The saddle-point is given by $t = - i\mu $ and $v = \lambda$. Putting these values in (\[6.16\]), and taking $\mu \rightarrow \lambda$, we obtain immediately () = [N]{} [sin]{}\^2 ([2 N ]{}) Thus, (\[6.16\]) gives the result both for the dirty case and for the clean case when one varies the parameter $C$. DISCUSSION ========== We have discussed a random matrix theory for the energy levels inside a superconductor vortex and investigated the crossover from the clean case to the dirty case. The technique involves a transposition of earlier results for the Hermitian case to the symplectic group $Sp(N)$. An extension of the Itzykson-Zuber integral is presented (in an appendix). In the symplectic case we have to consider an external source matrix, which describes the spectrum of energies in the absence of impurities, whose eigenvalues are of the form $a = (a_1,...,a_N,-a_1, ...,-a_N)$. In the symplectic case the measure $\Delta (a)= \prod (a_i^2 - a_j^2) \prod a_i$ replaces the Vandermonde determinant of the unitary case. Indeed, in the superconductor vortex, the eigenvalues appear in opposite pairs, i.e. with an exact particle-hole symmetry, as seen in the Andreev reflexion. We have then discussed generalized integral formulae related to the $Sp(N)$ group. In an appendix we have investigated hypergeometric functions of matrix argument both for the unitary and symplectic groups. This result may be useful for non-Gaussian random distributions. This random matrix theory is phenomenological. One phenomenological parameter, which can be viewed as the intensity of the disorder, is sufficient to describe the distribution functions of the random matrices. We have fixed this disorder parameter to one, which is why it does not appear explicitly in (\[4.2\]). Instead for the external source matrix $A$, we have introduced a parameter $C$ in (\[5.16\]). We find the a universal formula for the density of states and the correlation functions for the different regimes of this parameter: a clean case $C > N$, a dirty case $C < N$, and a crossover region $C \sim N$. Nevertherless the identification of this parameter $C$ with microscopic parameters, such as the strength of the impurity potential $V_i$ and the number of impurities $N_i$ in (\[2.7\]), requires a microscopic study which is outside the scope of this work. This work was supported by the CREST of JST. S. H. thanks a Grant-in-Aid for Scientific Research by the Ministry of Education, Science and Culture. A. L. thanks a Grant NSF DMR-9812340. S. H. and A. L. thanks ICTP in Trieste where this work was started. [**Appendix A: [Harish-Chandra formula for the unitary, orthogonal and symplectic group]{}**]{} Let $G$ be a compact connected Lie group. The Harish-Chandra integral formula is [@Harish-Chandra; @AltItzykson; @GrossRichards1] \[A1\] \_G e\^[&lt;[Ad]{}(g)a|b&gt;]{} dg = [\_[wW]{} ([det]{} w) e\^[&lt;wa|b&gt;]{} ]{} where $a, b\in h$; $h$ is a Lie algebra, and for any $H\in h$, \[A2\] (H) = \_[\_[+]{}]{} (H) $\Delta_{+}$ is the collection of positive roots, $W$ is the finite reflection group, called the Weyl (or Coxeter) group and $h$ is the Cartan subalgebra of the Lie algebra of the group. In the general classification theory, the irreducible finite refelection groups are categorized as belonging to various types, $A_n,B_n,C_n,D_n,....$, which are associated with certain compact Lie groups. In (\[HC\]), ${\rm det} (w)$ is simply $\pm 1$, since each $w\in W$ is an orthogonal transformation. Let us illustrate this result in the simplest case of the unitary group. The Lie algebra of $SU(n)$ , $u = su(n)$, consists $n\times n$ complex skew-Hermitian traceless matrices. The complexification of $u$ is $g = sl(n,C)$, the Lie algebra of all $n\times n$ complex matrices with zero trace. The Cartan subalgebra $h$ of $g$ consists of diagonal $n\times n$ complex matrices $H = diag(h_1,...,h_n)$ such that $h_1 + .... +h_n = 0$. We define the linear functional $e_j$ on $h$ by $e_j(H) = h_j$, and the $n\times n$ matrix $E_{jk}$ which consists of 1 in the (j,k)th position and 0 elsewhere. The linear functional $\alpha = e_j - e_k,j\neq k$ is a root of $g$ with respect to $h$, i.e. $\Delta = [e_j - e_k: 1 \leq j\neq k\leq n]$. One then verifies that (\[A1\]) reduces the Itzykson-Zuber formula (\[HC\]) in the unitary case. The orthogonal group $O(N)$ consists of real $N\times N$ matrices $u$ such that $u u^{t} = 1$. The Harish-Chandra formula applies to compact connected groups $G$ and we thus restrict ourselves to the special orthogonal subgroup $SO(N)$, of orthogonal matrices with determinant one. The Lie algebra $g = so(N)$ of $ G = SO(N)$ consists of all $N\times N$ real skew-symmetric traceless matrices. We need to consider separately the even case $N = 2n$ and the odd one $N = 2n + 1$. For $SO(2n)$, the Cartan subalgebra $h$ is the complex Lie algebra of all $2n \times 2n$ complex block-diagonal matrices of the form [@KNAPP] \[A3\] H = () This matrix $H$ is written as the direct sum of $v$’s defined as \[A4\] v = () \[A5\] H = h\_1 v h\_2 v h\_n v Let $[e_1,...,e_n]$ be the standard basis for $R^n$. A root system for $SO(2n)$ is $\Delta = [\pm e_j \pm e_k: 1\leq j \le k \leq n]$, which is a root system of type $D_n$. For $\alpha = e_j \pm e_k\in \Delta_{+}$ and the Cartan subalgebra $H$, we have $\alpha(H) = h_j \pm h_k$ and \[A6\] V(H) = \_[1j &lt; k n]{} (h\_j\^2 - h\_k\^2). A fundamental Weyl chamber is defined as $S = [(h_1,...,h_n)\in R^n:h_1 >\cdots > h_{n-1} > |h_n|]$. The group $G(n)$ of permutations $w$ of the set $[-n,...,-1,1,...,n]$ restricted to $w(-j)$ = - $w(j)$, acts on the set of $[h_{-n},...,h_{-1},h_1,...,h_n]$ as \[A7\] w(h\_1,...,h\_n) = (h\_[w(1)]{},...,h\_[w(n)]{}) where we denote $h_{-j} = - h_j, j = 1,...,n$. The Weyl group $W$ consists of $W$ = \[permutations and even number of sign changes of $[e_1, ...,e_n]$\]; thus $|W| = 2^{n-1}n!$. For $H$, we have \[A8\] wH = h\_[w(1)]{}vh\_[w(2)]{} vh\_[w(n)]{} v Therefore the Harish-Chandra formula (\[A1\]) gives for $a = a_1 v \oplus a_2 v\oplus \cdots \oplus a_n v$ and $b = b_1 v\oplus b_2 v\oplus \cdots \oplus b_n v$ \[A9\] \_[SO(2n)]{} e\^[[tr]{} ( g a g\^[-1]{} b)]{} dg = C\_[SG(n)]{} [\_[wSG(n)]{} ([det]{} w) ( 2 \_[j=1]{}\^n w(a\_j) b\_j)]{} with $C_{SG(n)} = ( 2n - 1)! \prod_{j=1}^{2n -1} (2 j - 1)!$. For $SO(2n + 1)$, the matrix $H$ is \[A10\] H = h\_1 v h\_2 vh\_[n]{} v0 and the root system for $SO(2n + 1)$ is \[A11\] = \[ e\_j e\_k: 1j k n \]Thus for $\alpha\in \Delta_{+}$ and $H$, we have $$\begin{aligned} \label{A12} \alpha(H) =& & h_j\pm h_k \hskip 24mm if \hskip 5mm\alpha = e_j \pm e_k\cr & & h_j \hskip 35mm if \hskip 5mm\alpha = e_j\end{aligned}$$ and \[A13\] V(H) = \_[1j &lt; k n ]{}( h\_j\^2 - h\_k\^2) \_[j=1]{}\^n h\_j The weyl group of $SO(2n + 1)$ is $W$ =\[ permutations and sign changes of $[e_1, ...,e_n]$\]. $|W| = 2^n n!$. The action of $G(n)$ on the Lie algebra $h$ is \[A14\] wH = h\_[w(1)]{} vh\_[w(2)]{} vh\_[w(n)]{} v0 The Harish-Chandra formula (\[HC\]) becomes for $a = a_1 v\oplus \cdots \oplus a_n v\oplus 0$, $b = b_1 v\oplus \cdots \oplus b_n v \oplus 0$, \[A15\] \_[SO(2n + 1)]{} e\^[[tr]{}(g a g\^[-1]{} b)]{} dg = C\_[G(n)]{} [\_[wG(n)]{} ([det]{} w) (2 \_[j=1]{}\^n w(a\_j) b\_j) ]{} where $C_{G(n)} = \prod_{j=1}^n(2j - 1)!\prod_{j=2n}^{4n - 1} j! $. For the case of the symplectic group $Sp(N)$, we have \[A16\] H = () and $e_j(H) = h_j$. A root system for $Sp(N)$ is \[A17\] = \[e\_j e\_k: 1j k n \]The Weyl group for the Sp(N) algebra is $W$ = \[permutations and sign changes of $[e_1,...,e_n]$\]. $|W| = 2^{n}n!$. [**Appendix B: [Generalization of the Itzykson-Zuber formula]{}**]{} The Itzykson-Zuber formula for Hermitian matrices $a$ and $b$ is again \[B1\] \_[U(n)]{} e\^[[tr]{} (g a g\^[-1]{} b)]{} dg = [[det]{}(e\^[a\_ib\_j]{})]{} where $a_i$ and $b_i$ are eigenvalues of $a$ and $b$, respectively. We wish now to consider generalizations of this type of integrals for which \[B2\] \_[U(n)]{} \_n( g a g\^[-1]{} b ) dg = [[det]{} ( f(a\_i b\_j))]{} where $\psi_n$ is a function of a matrix argument, and $f$ is a real function. The Itzykson-Zuber formula of (\[B1\]) corresponds to $f(x y) = \exp[ x y]$ and $\psi_n = \exp [ {\rm tr}(g a g^{-1} b)]$. We may take for $f(x y)$ an hypergeometric function $_p{\cal F}_q(\alpha_1,...,\alpha_p;\beta_1,...,\beta_q;xy)$ . Note that $f(xy) = \exp[ x y ] = {_0{\cal F}_0} ( x y )$. Then , the corresponding function $\psi_n (t)$ is also an hypergeometric function of the matrix argument $_pF_q(\alpha_1 + n - 1, ..., \alpha_p + n - 1; \beta_1 + n - 1,..., \beta_q + n - 1; t)$. For example,in the $p=1,q=0$ case, we have $$\begin{aligned} \label{B3} \int_{U(n)} {\rm det} ( 1 - a g b g^{-1} )^{-\alpha -n + 1} dg &=& _1F_0(\alpha; a,b)\nonumber\\ &=& {{\rm det}(_1{\cal F}_0(\alpha; a_i b_j))\over{\Delta(a)\Delta(b)}} \end{aligned}$$ where $_1{\cal F}_0(\alpha;a_ib_j) = (1 - a_ib_j)^{-\alpha}$. This is easily checked directly for the $n = 2$, $g = U(2)$ case. We represent $g$ as \[B3a\] g = () with the measure $J = \cos \phi \sin \phi d \phi \prod_{i=1}^{3} d\theta_i$. More generally, the formula of the integration over the $U(N)$ group may be written as $$\begin{aligned} \label{B4} &&_pF_q(\alpha_1,...,\alpha_p;\beta_1,...,\beta_q; a, b)\nonumber\\ &=& \int_{U(n)} {_pF_q}(\alpha_1,...,\alpha_p;\beta_1,...,\beta_q; a g b g^{-1}) d g \nonumber\\ &=& C {{\rm det}(_p{\cal F}_q(\alpha_1 - n + 1,..., \alpha_p -n + 1; \beta_1 - n + 1,...,\beta_q - n + 1; a_i b_j)) \over{\Delta(a)\Delta(b)}}\nonumber\\ \end{aligned}$$ This derivation of (\[B4\]) proceeds by induction[@Gross]. First, we introduce the zonal polynomials $Z_m(A)$ [@James] which are homogeneous polynomials of degree m, which are symmetric functions of the $n$ eigenvalues of the matrix $A$. From their definition they have the simple property that \[B7\] \_[G]{} Z\_m( g A g\^[-1]{} B) dg = [Z\_m(A) Z\_m(B)]{} in which the integral runs over the elements of a compact Lie group $G$ with a Haar measure normalized to one. The coefficients of these polynomials are group-dependent, but they can be constructed from this property inductively. The polynomials are thus expressed as decompositions of products of $ Tr(A^m_1)Tr(A^m_2)\cdots Tr(A^m_n)$, with $|m| = m_1 + \cdots + m_n$, and the $m_j$ are the partitions of m characterizing a Young tableau. It then follows that \[B5\] ([tr]{} A)\^[k]{} = \_[|m| = k]{} Z\_m(A) in which the sum runs over all the Young tableaux with k boxes. We may now take for a generating function the hypergeometric function with matrix argument defined as \[B6\] \_pF\_q(\_1,...,\_p;\_1,...,\_q;t) = \_[k=0]{}\^ [1]{} \_[|m| = k]{} [\[\_1\]\_m \[\_p\]\_m]{} Z\_m(t). Then, one has $$\begin{aligned} \label{B8} _pF_q(\alpha_1,...,\alpha_p;\beta_1,...,\beta_q;a,b) &=& \int_{G}{_p}F_q(\alpha_1,...,\alpha_p;\beta_1,...,\beta_q; gag^{-1}b) dg \nonumber\\ &=& \sum_{k=0}^{\infty} {1\over{k!}} \sum_{|m| = k} {[\alpha_1]_m\cdots [\alpha_p]_m\over{[\beta_1]_m\cdots [\beta_q]_m}} {Z_m(a)Z_m(b)\over{Z_m(1)}}\nonumber\\ \end{aligned}$$ where \[B9\] \[\]\_m = \_[j=1]{}\^n ( - j + 1)\_[m\_j]{} \[B10\] ()\_k = (+ 1) (+ k - 1) In the case of the unitary group $G=U(n)$ the zonal polynomial $Z_m(A)$ is a Schur function, namely it is given by the Weyl formula for the characters of the representations of G \[BSchur\] Z\_m(A) = . The Euler integral gives \[B11\] Z\_m(A) = [\_n()]{} [\[\]\_m]{} \_[0&lt;r&lt;1]{} Z\_m(r A) [det]{}(r)\^[- n]{} [det]{}(1 - r)\^[- - n]{}d r where the integration is over Hermitian matrices $r$ whose eigenvalues are between 0 and 1. The Gamma function $\Gamma_n(\alpha)$ is defined by \[B11a\] \_n() = \^[n(n-1)/2]{} \_[i=1]{}\^n (- i + 1) Using these notations, we have established the recurrence formula, $$\begin{aligned} \label{B12} &&_{p+1}F_{q+1}(\alpha_1,...,\alpha_{p+1};\beta_1,...,\beta_{q+1};a,b) = {\Gamma_n(\beta_{q+1})\over{\Gamma_n(\alpha_{p+1}) \Gamma_n(\beta_{q+1}-\alpha_{p+1})}}\nonumber\\ &&\times \int_{0<r<1} {\rm det}(r)^{\alpha_{p+1}- n} {\rm det}(1 - r)^{\beta_{q+1} - \alpha_{p+1} - n} {_p}F_q(\alpha_1,..,\alpha_p;\beta_1,..,\beta_q;ra,b) dr \nonumber\\ \end{aligned}$$\[B13\] which proves (\[B4\]) inductively. For the case of the confluent and Gaussian hypergeometric functions, $_1F_1$ and $_2F_1$, these formulae reduce to \[B14\] \_1F\_1(;;t) = [\_n()]{} \_[0&lt;r&lt;1]{} dr e\^[[tr]{}(r t)]{} [det]{}(r)\^[- n]{} [det]{}( 1 - r)\^[- - n]{} $$\begin{aligned} \label{B15} _2F_1(\alpha,\beta;\gamma;t) &=& {\Gamma_n(\gamma)\over{\Gamma_n(\beta)\Gamma_n(\gamma - \beta)}} \int_{0<r<1} dr {\rm det}(r)^{\beta - n} {\rm det}( 1 - r)^{\gamma - \beta - n}\nonumber\\ &&\times {\rm det}(1 - r t)^{-\alpha} \end{aligned}$$ For the symplectic group $Sp(n)$, we have similar formulae with hypergeometric functions. In the case $p=0,q=0$, and $p=1,q=0$, which is similar to (\[B1\]) and (\[B3\]), $$\begin{aligned} \label{B15a} {_0}F_0(a,b) &=& \int_{Sp(n)} e^{{\rm tr}(g a g^{-1} b)} dg\nonumber\\ &=& {{\rm det} [ 2 {\sinh}(2 a_i b_j) ]\over{\Delta(a) \Delta(b)}}\end{aligned}$$ $$\begin{aligned} \label{B16} {_1}F_{0}(\alpha;a,b) &=& \int_{Sp(n)} {\rm det} ( 1 - a g b g^{-1})^{-\alpha - 2 n + 1} {\rm d}g \nonumber\\ &=& {{\rm det}[({1\over{1 -a_ib_j}})^{2 \alpha} - ({1\over{1 + a_ib_j}})^{2 \alpha}]\over{\Delta(a)\Delta(b)}} \end{aligned}$$ where $\Delta(a) = \prod (a_i^2 - a_j^2) \prod a_i$. These formula are easily checked for the simplest case, $n = 1$, $Sp(1)$, which is isomorphic to $SU(2)$, with \[B17\] g = () It may be useful to notice that these results for $Sp(1)$ may also be derived immediately from the $U(2)$ case by setting $a = a_1 = - a_2$, and $b = b_1 = - b_2$ in (\[B1\]) and (\[B3\]). Indeed the Lie group $g$ of $Sp(1)$ in (\[B17\]) is derived from (\[B3a\]) with the condition $\theta_3 = - \theta_2$. The denominator $\Delta(a)$ for $Sp(n)$ is also derived from the $U(n)$ case with the condition on the eigenvalues $ a = {\rm diag}(a_1, \cdots, a_n, - a_1, \cdots, - a_n)$. This equivalence between $U(n)$ and $Sp(n)$ holds for $n \neq 1$. Since the eigenvalues appear here in pairs $(a_i,- a_i)$, we have to replace $n$ by $2 n$ in the hypergeometric relations as shown in (\[B16\]). For a general hypergeometric function, we have for the $Sp(n)$ case, $$\begin{aligned} \label{B18} &&_pF_q(\alpha_1,...,\alpha_p;\beta_1,...,\beta_q; a, b)\nonumber\\ &=& \int_{Sp(n)} {_pF_q}(\alpha_1,...,\alpha_p;\beta_1,...,\beta_q; a g b g^{-1}) d g \nonumber\\ &=& {C\over{\Delta(a)\Delta(b)}} {\rm det}[(_p{\cal F}_q(\alpha_1 - 2n + 1,..., \alpha_p -2n + 1; \beta_1 - 2n + 1,...,\nonumber\\ && \beta_q -2 n + 1; a_i b_j))^2 \nonumber\\ &-& (_p{\cal F}_q(\alpha_1 - 2n + 1,..., \alpha_p -2n + 1; \beta_1 - 2n + 1,...,\nonumber\\ &&\beta_q - 2n + 1; - a_i b_j))^2] \nonumber\\ \end{aligned}$$ C. Caroli, P. G. De Gennes and J. Matricon, Phys. Lett. [**9**]{}, (1964) 307. E. Br[é]{}zin and S. Hikami, Nucl. Phys. [**B479**]{} (1996) 697. E. Br[é]{}zin, S. Hikami and A. Zee, Nucl. Phys. [**464**]{} (1996) 411. E. Br[é]{}zin and S. Hikami, Phys. Rev. [**E 55**]{} (1997) 4067. E. Br[é]{}zin and S. Hikami, Phys. Rev. [**E 56**]{} 264 (1997). Harish-Chandra, Proc. Nat. Acad. Sci. [**42**]{} 252 (1956). C. Itzykson and J. -B. Zuber, J. Math. Phys. [**21**]{} (1980) 411. J. J. Duistermaat and G. H. Heckman, Invent. Math. [**69**]{} (1982) 259. V. A. Kazakov, Nucl. Phys. [**B 354**]{} (1991) 697. M.L. Mehta, Random Matrices, [*[Academic Press]{}*]{} (1991) A. A. Koulakov and A. I. Larkin, Cond-mat/9810125. A. I. Larkin and Yu. N. Ovchinnikov, Phys. Rev. [**B 57**]{} (1998) 5457 M. A. Skvortsov, V. E. Kravtsov and M. V. Feigel’man, Cond-mat/9805296. A. A. Koulakov and A. I. Larkin, Cond-mat/9802002. A. Altland and M. R. Zirnbauer, Phys. Rev. Lett. [**76**]{} (1996) 3420. A. Altland, B. D. Simon and D. Taras-Semchuk, cond-mat/9807371. D. Altshuler and C. Itzykson, Ann. Inst. Heri Poincar[é]{}, [**54**]{} (1991) 1. K. I. Gross and D. S. P. Richards, J. Approx. Theory [**59**]{} (1989) 224, J. Approx. Theory [**82**]{}(1995) 60. E. Witten, J. Geom. Phys. [**9**]{} (1992) 303. M. F. Atiyah and R. Bott, Topology [**23**]{} (1984) 1. S. Hikami and T. Maskawa, Prog. Theor. Phys. [**67**]{} (1982) 1038. A. W. Knapp, [*Representation theory of semisimple groups*]{}, Princeton University Press, Princeton, New Jersey 1986. Erdelyi et al. Higher transcendental functions, Vol.2, P.199, (1953) McGraw-Hill, New York. A. T. James, Ann. Math. Statist. [**35**]{} (1964) 475. K. I. Gross and D. S. P. Richards, Trans. American Math. Soc. [**301**]{} (1987) 781. [^1]: Unit[é]{} propre du centre national de la Recherche Scientifique, Associ[é]{}e [à]{} l’Ecole Normale Sup[é]{}rieure et [à]{} l’Universit[é]{} de Paris-Sud

--- abstract: 'This paper proposes a novel approach to regularize the *ill-posed* blind image deconvolution (blind image deblurring) problem using deep generative networks. We employ two separate deep generative models — one trained to produce sharp images while the other trained to generate blur kernels from lower-dimensional parameters. To deblur, we propose an alternating gradient descent scheme operating in the latent lower-dimensional space of each of the pretrained generative models. Our experiments show excellent deblurring results even under large blurs and heavy noise. To improve the performance on rich image datasets not *well learned* by the generative networks, we present a modification of the proposed scheme that governs the deblurring process under both generative and classical priors.' bibliography: - 'egbib.bib' title: Blind Image Deconvolution using Pretrained Generative Priors --- Introduction and Related Work {#sec:introduction} ============================= Blind image deblurring aims to recover a true image $i$ and a blur kernel $k$ from blurry and possibly noisy observation $y$. For a uniform and spatially invariant blur, it can be mathematically formulated as $$\label{eq:bid} y = i \otimes k + n,$$ where $\otimes $ is a convolution operator and $n$ is an additive Gaussian noise. In its full generality, the inverse problem is severely ill-posed as many different instances of $i$, and $k$ fit the observation $y$ [@campisi2016blind; @kundur1996blind]. To resolve between multiple instances, priors are introduced on images and/or blur kernels in the image deblurring algorithms. Priors assume an *a priori* model on the true image/blur kernel or both. These natural structures expect images or blur kernels to be sparse in some transform domain; see, for example, [@chan1998total; @fergus2006removing; @levin2009understanding; @hu2010single; @zhang2011sparse; @cai2009blind]. Some of the other penalty functions to improve the conditioning of the blind image deblurring problem are low-rank [@ren2016image], and total variation based priors [@pan2014motion]. A recently introduced dark channel prior [@pan2016blind] also shows promising results; it assumes a sparse structure on the dark channel of the image, and exploits this structure in an optimization program [@xu2011image] to solve the blind image deblurring problem. Other works include extreme channel priors [@yan2017image], outlier robust deblurring [@dong2017blind], learned data fitting [@pan2017learning], and discriminative prior based blind image deblurring approaches [@li2018learning]. Although generic and applicable to multiple applications, these engineered models are not very effective as many unrealistic images also fit the prior model [@hand2017global]. \ \ Recently deep learning based blind image deblurring approaches have shown impressive results due to their power of learning from large training data [@hradivs2015convolutional; @nah2016deep; @schuler2016learning; @xu2017learning; @nimisha2017blur; @kupyn2017deblurgan]. Generally, these deep learning based approaches invert the forward acquisition model of blind image deblurring via end-to-end training of deep neural networks in a supervised manner. The main drawback of this end-to-end deep learning approach is that it does not explicitly take into account the knowledge of forward map , but rather learns implicitly from training data. Consequently, the deblurring is more sensitive to changes in the blur kernels, images, or noise distributions in the test set that are not representative of the training data, and often requires expensive retraining of the network for a competitive performance [@lucas2018using]. Meanwhile, neural network based implicit generative models such as generative adversarial networks (GANs) [@goodfellow2014generative] and variational autoencoders (VAEs) [@kingma2013auto] have found much success in modeling complex data distributions especially that of images. Recently, GANs and VAEs have been used for blind image deblurring but only in an end-to end manner [@xu2017learning; @nimisha2017blur; @kupyn2017deblurgan] , which is completely different from our approach as will be discussed in detail. These methods show competitive performance, but since these generative model based approaches are end-to-end they suffer from the same draw backs as other deep learning based debluring approaches. On the other hand, pretrained generative models have recently been employed as regularizers to solve inverse problems in imaging including compressed sensing [@bora2017compressed; @shah2018solving], image inpainting [@yeh2017semantic], Fourier ptychography [@shamshad2018deep], and phase retrieval [@shamshad2018robust; @hand2018phase]. However the applicability of these pretrained generative models in blind image deblurring is relatively unexplored. Recently [@gandelsman2018double] employ a combination of multiple untrained deep generative models and show their effectiveness on various image layer decomposition tasks including image water mark removal, image dehazing, image segmentation, and transparency separation in images and videos. Different from their approach, we show the effectiveness of our blind image deblurring method by leveraging trained generative models for images and blurs. In this work, we use the expressive power of pretrained GANs and VAEs to tackle the challenging problem of blind image deblurring. Our experiments in Figure \[fig:intro-results\] confirm that integrating deep generative priors in the image deblurring problem enables a far more effective regularization yielding sharper and visually appealing deblurred images. Specifically, our main contributions are - To the best of our knowledge, this is the first instance of utilizing pretrained generative models for tackling challenging problem of blind image deblurring. - We show that simple gradient descent approach assisted with generative priors is able to recover true image and blur kernel, to with in the range of respective generative models, from blurry image. - We investigate a modification of the loss function to allow the recovered image some leverage/slack to deviate from the range of the image generator. This modification effectively addresses the performance limitation due to the range of the generator. - Our experiments demonstrate that our approach produce superior results when compared with traditional image priors and unlike deep learning based approaches does not require expensive retraining for different noise levels. Problem Formulation and Proposed Solution {#sec:Problem-Formulation} ========================================= We assume the image $i \in \R^n$ and blur kernel $k \in \R^n$ in are members of some structured classes $\mathcal{I}$ of images, and $\mathcal{K}$ of blurs, respectively. For example, $\setI$ may be a set of celebrity faces and $\setK$ comprises of motion blurs. A representative sample set from both classes $\setI$ and $\setK$ is employed to train a generative model for each class. We denote $G_{\mathcal{I}}: \mathbb{R}^l \rightarrow \mathbb{R}^n$ and $G_{\mathcal{K}}: \mathbb{R}^m \rightarrow \mathbb{R}^n$ as the generators for class $\setI$, and $\setK$, respectively. Given low-dimensional inputs $z_i \in \mathbb{R}^l$, and $z_k \in \mathbb{R}^m$, the pretrained generators $G_{\setI}$ and $G_{\setK}$ generate new samples $G_{\setI}(z_i)$, and $G_{\setK}(z_k)$ that are representative of the classes $\setI$ and $\setK$, respectively. Once trained, the weights of the generators are fixed. To recover the sharp image and blur kernel $(i,k)$ from the blurred image $y$ in , we propose minimizing the following objective function $$\begin{aligned} \label{eq:Optimization-Ambient} (\hat{i},\hat{k}) := \underset{\substack{i \in\text{Range}(G_{\setI}) \\k \in\text{Range}(G_{\setK})}}{\text{argmin}} \ \|y - i \otimes k \|^2, \end{aligned}$$ where $\|\cdot\|$ is the $\ell_2$-distance, Range($G_\setI$) and Range($G_\setK$) is the set of all the images and blurs that can be generated by $G_\setI$ and $G_\setK$, respectively. In words, we want to find an image $i$ and a blur kernel $k$ in the range of their respective generators, that best explain the forward model . Ideally, the range of a pretrained generator comprises of only the samples drawn from the probability distribution of the training image or blur class. Constraining the solution $(\hat{i},\hat{k})$ to lie only in generator ranges forces the solution to be the members of classes $\setI$ and $\setK$. The minimization program in can be equivalently formulated in the lower dimensional, latent representation space as follows: $$\begin{aligned} \label{eq:Optimization-latent} (\hat{z}_i, \hat{z}_k) = \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}} \ \| y - G_{\mathcal{I}}(z_i) \otimes G_\setK(z_k) \|^2.\end{aligned}$$ This optimization program can be thought of as tweaking the latent representation vectors $z_i$ and $z_k$, (input to the generators $G_{\setI}$, and $G_{\setK}$, respectively) until these generators generate an image $i$ and blur kernel $k$ whose convolution comes as close to $y$ as possible. Incorporating the fact that latent representation vectors $z_i$, and $z_k$ are assumed to be coming from standard Gaussian distributions, we further augment the measurement loss in with $\ell_2$ penalty terms on the latent representations. The resultant optimization program is then $$\label{eq:regularized-program} \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}}\ \| y - G_\setI(z_i) \otimes G_\setK(z_k) \|^2+ \gamma\| z_i \|^2 + \lambda\| z_k \|^2,$$ where $\gamma$ and $\lambda$ are free scalar parameters. For brevity, we denote the objective function above by $\setL(z_i,z_k)$. Importantly, the weights of the generators are always fixed as they enter into this algorithm as pretrained models. To minimize this non-convex objective, we begin by initializing $z_i$ and $z_k$ by sampling from standard Gaussian distribution, and resort to an alternating gradient descent algorithm by taking a gradient step in one of these while fixing the other to find a minima $(\hat{z}_i, \hat{z}_k)$. To avoid being stuck in a not good enough local minima, we restart the algorithm with a new random initialization (Random Restarts) when the measurement loss in does not reduce sufficiently after reasonably many iterations. We dubbed proposed deblurring algorithm as *Deep Deblur* and denote blurry image deblurred via *Deep Deblur* as $\hat{\textit{{i}}}_\text{DD}$.The estimated deblurred image and the blur kernel are acquired by a forward pass of the solutions $\hat{z}_i$ and $\hat{z}_k$ through the generators $G_{\mathcal{I}}$ and $G_{\mathcal{K}}$. Mathematically, $(\hat{i},\hat{k}) = ( G_\mathcal{I}(\hat{z}_i), G_\mathcal{K}(\hat{z}_k))$. Problem Formulation {#sec:Problem-Formulation} =================== We assume the image $i \in \R^n$ and blur kernel $k \in \R^n$ in are members of some structured classes $\mathcal{I}$ of images, and $\mathcal{K}$ of blurs, respectively. For example, $\setI$ may be a set of celebrity faces and $\setK$ comprises of motion blurs. A representative sample set from both classes $\setI$ and $\setK$ is employed to train a generative model for each class. We denote by the mappings $G_{\mathcal{I}}: \mathbb{R}^l \rightarrow \mathbb{R}^n$ and $G_{\mathcal{K}}: \mathbb{R}^m \rightarrow \mathbb{R}^n$, the generators for class $\setI$, and $\setK$, respectively. Given low-dimensional inputs $z_i \in \mathbb{R}^l$, and $z_k \in \mathbb{R}^m$, the pretrained generators $G_{\setI}$, and $G_{\setK}$ generate new samples $G_{\setI}(z_i)$, and $G_{\setK}(z_k)$ that are representative of the classes $\setI$, and $\setK$, respectively. Once trained, the weights of the generators are fixed. To recover the sharp image, and blur kernel $(i,k)$ from the blurred image $y$ in , we propose minimizing the following objective function $$\begin{aligned} \label{eq:Optimization-Ambient} (\hat{i},\hat{k}) := \underset{\substack{i \in\text{Range}(G_{\setI}) \\k \in\text{Range}(G_{\setK})}}{\text{argmin}} \ \|y - i \otimes k \|^2, \end{aligned}$$ where Range($G_\setI$) and Range($G_\setK$) is the set of all the images and blurs that can be generated by $G_\setI$ and $G_\setK$, respectively. In words, we want to find an image $i$ and a blur kernel $k$ in the range of their respective generators, that best explain the forward model . Ideally, the range of a pretrained generator comprises of only the samples drawn from the distribution of the image or blur class. Constraining the solution $(\hat{i},\hat{k})$ to lie only in generator ranges, therefore, implicitly reduces the solution ambiguities inherent to the ill-posed blind deconvolution problem, and forces the solution to be the members of classes $\setI$, and $\setK$. The minimization program in can be equivalently formulated in the lower dimensional, latent representation space as follows $$\begin{aligned} \label{eq:Optimization-latent} (\hat{z}_i, \hat{z}_k) = \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}} \ \| y - G_{\mathcal{I}}(z_i) \otimes G_\setK(z_k) \|^2.\end{aligned}$$ This optimization program can be thought of as tweaking the latent representation vectors $z_i$, and $z_k$, (input to the generators $G_{\setI}$, and $G_{\setK}$, respectively) until these generators generate an image $i$ and blur kernel $k$ whose convolution comes as close to $y$ as possible. The optimization program in is obviously non-convex owing to the bilinear convolution operator, and non-linear deep generative models. We resort to an alternating gradient descent algorithm to find a local minima $(\hat{z}_i, \hat{z}_k)$. Importantly, the weights of the generators are always fixed as they enter into this algorithm as pretrained models. At a given iteration, we fix $z_i$ and take a descent step in $z_k$, and vice verse. The gradient step in each variable involves a forward and backward pass through the generator networks. Section \[proposedapproach\] talks about the back propagation, and gives explicit gradient forms for descent in each $z_i$ and $z_k$ for this particular algorithm. The estimated deblurred image and the blur kernel are acquired by a forward pass of the solutions $\hat{z}_i$ and $\hat{z}_k$ through the generators $G_{\mathcal{I}}$ and $G_{\mathcal{K}}$. Mathematically, $(\hat{i},\hat{k}) = ( G_\mathcal{I}(\hat{z}_i), G_\mathcal{K}(\hat{z}_k))$. Image Deblurring Algorithm {#sec:proposedapp} ========================== Our approach requires pretrained generative models $G_\setI$ and $G_\setK$ for classes $\setI$ and $\setK$, respectively. We use both GANs and VAEs as generative models on the clean images and blur kernels. {width="75.00000%"} Naive Deblurring {#backprop} ---------------- To deblur an image $y$, a simplest possible strategy is to find an image closest to $y$ in the range of the given generator $G_{\setI}$ of clean images. Mathematically, this amounts to solving the following optimization program $$\begin{aligned} \label{eq:naive-deblur} \underset{z_i \in \R^l}{\text{argmin}} \ \| y - G_\setI(z_i) \|, \quad \quad \hat{i} = G_\setI(\hat{z}_i), \end{aligned}$$ where we emphasize again that in the optimization program above, the weights of the generator $G_\setI$ are fixed (pretrained). Although non-convex, a local minima $\hat{z}_i$ can be achieved via gradient descent implemented using the back propagation algorithm. The recovered image $\hat{i}$ is obtained by a forward pass of $\hat{z}_i$ through the generative model $G_{\setI}$. Expectedly, this approach fails to produce reasonable recovery results; see Figure \[fig:Back-Propagation\]. The reason being that this back projection approach completely ignores the knowledge of the forward blur model in . We now address this shortcoming by including the forward model and blur kernel in the objective (\[eq:naive-deblur\]). Deconvolution using Deep Generative Priors {#proposedapproach} ------------------------------------------ We discovered in the previous section that simply finding a clean image close to the blurred one in the range of the image generator $G_{\setI}$ is not good enough. A more natural and effective strategy is to instead find a pair consisting of a clean image and a blur kernel in the range of $G_{\setI}$, and $G_{\setK}$, respectively, whose convolution comes as close to the blurred image $y$ as possible. As outlined in Section \[sec:Problem-Formulation\], this amounts to minimizing the measurement loss $$\label{eq:measurement-loss} \| y - G_\setI(z_i) \otimes G_\setK(z_k) \|^2,$$ over both $z_i$, and $z_k$, where $\otimes$ is the convolution operator. Incorporating the fact that latent representation vectors $z_i$, and $z_k$ are assumed to be coming from standard Gaussian distributions in both the adversarial learning and variational inference framework, outlined in Section \[generative\], we further augment the measurement loss in with $\ell_2$ penalty terms on the latent representations. The resultant optimization program is then $$\label{eq:regularized-program} \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}}\ \| y - G_\setI(z_i) \otimes G_\setK(z_k) \|^2+ \gamma\| z_i \|^2 + \lambda\| z_k \|^2,$$ where $\lambda$, and $\gamma$ are free scalar parameters. For brevity, we denote the objective function above by $\setL(z_i,z_k)$. To minimize this non-convex objective, we begin by initializing $z_i$, and $z_k$ as standard Gaussian vectors, and then take a gradient step in one of these while fixing the other. To avoid being stuck in a not good enough local minima, we may restart the algorithm with a new random initialization (Random Restarts) when the measurement loss in does not reduce sufficiently after reasonably many iterations. Algorithm \[alg:generative-prior-deblurring\] formally introduces the proposed alternating gradient descent scheme. Henceforth, we will denote the image deblurred using Algorithm \[alg:generative-prior-deblurring\] by $\hat{i}_\text{DD}$. Beyond the Range of Generator {#sec:proposed-method2} ----------------------------- As described earlier, the optimization program implicitly constrains the deblurred image to lie in the range of the generator $G_{\setI}$. This may leads to some artifacts in the deblurred images when the generator range does not completely span the set $\setI$. This inability of the generator to completely learn the image distribution is often evident in case of more rich and complex natural images. In such cases, it makes more sense to not strictly constrain the recovered image to come from the range of the generator, and rather also explore images a bit outside the range. To accomplish this, we propose minimizing the measurement loss of images inside the range exactly as in together with the measurement loss $\| y - i \otimes G_\setK(z_k) \|^2$ of images not necessarily within the range. The in-range image $G_{\setI}(z_i)$, and the out-range image $i$ are then tied together by minimizing an additional penalty term, $\text{Range Error(i)} := \| i - G_\setI(z_i) \|^2$. The idea is to strictly minimize the range error when pretrained generator has effectively learned the image distribution, and afford some slack when it is not the case. The amount of slack can be controlled by tweaking the weights attached with each loss term in the final objective. Finally, to guide the search of a best deblurred image beyond the range of the generator, one of the conventional image priors such as total variation measure $\|\cdot\|_{\text{tv}}$ is also introduced. This leads to the following optimization program $$\begin{aligned} \label{eq:opt-admm} \underset{i, z_i, z_k}{\text{argmin}} \ & \| y - i \otimes G_\setK(z_k) \|^2 + \tau \| i - G_\setI(z_i) \|^2 + \zeta\| y - G_\setI(z_i) \otimes G_\setK(z_k) \|^2 + \rho\|i\|_{\text{tv}}.\end{aligned}$$ All of the variables are randomly initialized, and the objective is minimized using gradient step in each of the unknowns, while fixing the others. The computations of gradients is very similar to the steps outlined in Section \[proposedapproach\]. We take the solution $\hat{i}$, and $G(z_k)$ as the deblurred image, and the recovered blur kernel. The iterative scheme is formally given in Algorithm \[alg:generative+classical-prior-deblurring\]. For future references, we will denote the recovered image using Algorithm \[alg:generative+classical-prior-deblurring\] by $\hat{i}_\text{DDS}$. ![[]{data-label="fig:proposed_approach"}](./figures/block_diag.png){width="90.00000%"} A more natural and effective strategy is to instead find a pair consisting of a clean image and a blur kernel in the range of $G_{\setI}$, and $G_{\setK}$, respectively, whose convolution comes as close to the blurred image $y$ as possible; see Figure \[fig:proposed\_approach\]. As outlined in Section \[sec:Problem-Formulation\], this amounts to minimizing the measurement loss $$\label{eq:measurement-loss} \| y - G_\setI(z_i) \otimes G_\setK(z_k) \|^2,$$ over both $z_i$ and $z_k$. Incorporating the fact that latent representation vectors $z_i$, and $z_k$ are assumed to be coming from standard Gaussian distributions in both the adversarial learning and variational inference framework, we further augment the measurement loss in with $\ell_2$ penalty terms on the latent representations. The resultant optimization program is then $$\label{eq:regularized-program} \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}}\ \| y - G_\setI(z_i) \otimes G_\setK(z_k) \|^2+ \gamma\| z_i \|^2 + \lambda\| z_k \|^2,$$ where $\gamma$ and $\lambda$ are free scalar parameters. For brevity, we denote the objective function above by $\setL(z_i,z_k)$. Importantly, the weights of the generators are always fixed as they enter into this algorithm as pretrained models. To minimize this non-convex objective, we begin by initializing $z_i$ and $z_k$ as standard Gaussian vectors, and then take a gradient step in one of these while fixing the other. To avoid being stuck in a not good enough local minima, we may restart the algorithm with a new random initialization (Random Restarts) when the measurement loss in does not reduce sufficiently after reasonably many iterations. Algorithm \[alg:generative-prior-deblurring\] formally introduces the proposed alternating gradient descent scheme. Henceforth, we will denote the image deblurred using Algorithm \[alg:generative-prior-deblurring\] by $\hat{i}_1$. \[alg:AltGradDescent\] **Input:** $y$, $G_\setI$ and $G_\setK$\ **Output:** Estimates $\hat{i}_1$ and $\hat{k}$\ **Initialize:**\ $z_i^{(0)} \sim \setN(0,I), ~~ z_k^{(0)} \sim \setN(0,I)$ (\[eq:regularized-program\])\ (\[eq:regularized-program\])\ $\hat{i}_1 \leftarrow G_\setI(z^{(T)}_i), \hat{k} \leftarrow G_\setK(z^{(T)}_k)$ \[alg:generative-prior-deblurring\] Beyond the Range of Generator {#sec:proposed-method2} ----------------------------- \[alg:admm\] **Input:** $y$, $G_\setI$ and $G_\setK$\ **Output:** Estimates $\hat{i}_2$ and $\hat{k}$\ **Initialize:**\ $z_i^{(0)} \sim \setN(0,I), z_k^{(0)} \sim \setN(0,I), i^{(0)} \sim \setN(0.5,10^{-2}I)$ (\[eq:opt-admm\])\ (\[eq:opt-admm\])\ (\[eq:opt-admm\])\ $\hat{i}_2 \leftarrow {i^{(T)}}, \hat{k} \leftarrow G_\setK(z_k^{(T)})$ \[alg:generative+classical-prior-deblurring\] \[algorithm:admm\] Beyond the Range of Generator {#sec:proposed-method2} ----------------------------- As described earlier, the optimization program implicitly constrains the deblurred image to lie in the range of the generator $G_{\setI}$. This may lead to some artifacts in the deblurred images when the generator range does not completely span the set $\setI$. In such case, it makes more sense to not strictly constrain the recovered image to come from the range of the generator, and rather also explore images a bit outside the range. To accomplish this, we propose minimizing the measurement loss of images inside the range exactly as in together with the measurement loss $\| y - i \otimes G_\setK(z_k) \|^2$ of images not necessarily within the range. The in-range image $G_{\setI}(z_i)$ and the out-range image $i$ are then tied together by minimizing an additional penalty term, $\text{Range Error(i)} := \| i - G_\setI(z_i) \|^2$. The idea is to strictly minimize the range error when pretrained generator has effectively learned the image distribution, and afford some slack otherwise. Finally, to guide the search of a best deblurred image beyond the range of the generator, one of the conventional image priors such as total variation measure $\|\cdot\|_{\text{tv}}$ is also introduced. This leads to the following optimization program $$\begin{aligned} \label{eq:opt-admm} \underset{i, z_i, z_k}{\text{argmin}} \ & \| y - i \otimes G_\setK(z_k) \|^2 + \tau \| i - G_\setI(z_i) \|^2 +\zeta\| y - G_\setI(z_i) \otimes G_\setK(z_k) \|^2 + \rho\|i\|_{\text{tv}}\end{aligned}$$ All of the variables are randomly initialized, and the objective is minimized using gradient step in each of the unknowns, while fixing the others. We take the solution $\hat{i}$, and $G(\hat{z}_k)$ as the deblurred image, and the recovered blur kernel. We dubbed this approach as *Deep Deblur with Slack* (DDS) and the image deblurred using this approach is referred to as $\hat{\textit{i}}_\text{DDS}$. Experimental Results {#experiments} ==================== In this section, we provide a comprehensive set of experiments to evaluate the performance of *Deep Deblur* and *Deep Deblur with Slack* against iterative and deep learning based baseline methods. We also evaluate performance under increasing noise and large blurs. In all experiments, we use noisy blurred images, generated by convolving images $i$, and blurs $k$ from their respective test sets and adding 1$\%$ [^1] Gaussian noise (unless stated otherwise). The choice of free parameters in both algorithms for each dataset are provided in the supplementary material. Implementation Details ---------------------- **Datasets**: We choose three image datasets. First dataset, SVHN, consists of house number images from Google street view. A total of 531K images, each of dimension $32\times32 \times 3$, are available out of which 30K are held out as test set. Second dataset, Shoes [@yu2014fine] consists of 50K RGB examples of shoes, resized to $64 \times 64 \times 3$. We leave $1000$ images for testing and use the rest as training set. Third dataset, CelebA, consists of relatively more complex images of celebrity faces. A total of 200K, each center cropped to dimension $64 \times 64 \times 3$, are available out of which 22K are held out as a test set. A motion blur dataset is generated consisting of small to very large blurs of lengths varying between 5 and 28; following strategy given in [@boracchi2012modeling]. We generate 80K blurs out of which 20K is held out as a test set. **Generative Models**: We choose VAE as the generative model for SVHN images and motion blurs. For Shoes and CelebA, the generative model $G_{\setI}$ is the default deep convolutional generative adversarial network (DCGAN) [@salimans2016improved]. Further details on architectures of generative models are provided in the supplementary material. **Baseline Methods**: Among the conventional algorithms using engineered priors, we choose dark prior (DP) [@pan2016blind], extreme channel prior (EP) [@yan2017image], outlier handling (OH) [@dong2017blind], and learned data fitting (DF) [@pan2017learning] based blind deblurring as baseline algorithms. We optimized the parameters of these methods in each experiment to obtain the best possible baseline results. Among driven approaches for deblurring, we choose [@hradivs2015convolutional] that trains a convolutional neural network (CNN) in an end-to-end manner, and [@kupyn2017deblurgan] that trains a neural network (DeblurGAN) in an adversarial manner. Deblurred images from these baseline methods will be referred to as $i_\text{DP}$, $i_\text{EP}$, ${i_\text{OH}}$, ${i_\text{DF}}$, $i_\text{CNN}$ and $i_\text{DeGAN}$. Deblurring Results under Pretrained Generative Priors {#sec:Exps-PretrainedPriors} ----------------------------------------------------- The central limiting factor in the *Deep Deblur* performance is the ability of the generator to represent the (original, clean) image to be recovered. As pointed out earlier that often the generators are not fully expressive (cannot generate new representative samples) on a rich/complex image class such as face images compared to a compact/simple image class such as numbers. Such a generator mostly cannot *adequately* represent a new image in its range. Since *Deep Deblur* strictly constrains the recovered image to lie in the range of image generator, its performance depends on how well the range of the generator spans the image class. Given an arbitrary test image $i_{\text{test}}$ in the set $\setI$, the closest image $i_{\text{range}}$, in the range of the generator, to $i_{\text{test}}$ is computed by solving the following optimization program $$\begin{aligned} z_{\text{test}} := \underset{z} {\text{argmin}} \|i_{\text{test}}-G_{\setI}(z)\|^2, \quad i_{\text{range}} = G_{\setI}(z_{\text{test}})\end{aligned}$$ We solve the optimization program by running $10,000(6,000)$ gradient descent steps with a step size of $0.001(0.01)$ for CelebA(SVHN). Parameters for Shoes are the same as CelebA. A more expressive generator leads to a better deblurring performance as it can well represent an arbitrary original (clean) image $i_{\text{test}}$ leading to a smaller mismatch $ \text{range error} := \|i_{\text{test}} - i_{\text{range}}\|$ to the corresponding range image $i_{\text{range}}$. ### Impact of Generator Range on Image Deblurring To judge the proposed deblurring algorithms independently of generator range limitations, we present their deblurring performance on range image $i_{\text{range}}$; we do this by generating a blurred image $y = i_{\text{range}} \otimes k + n$ from an image $i_{\text{range}}$ already in the range of the generator; this implicitly removes the range error as now $i_{\text{test}} = i_{\text{range}}$. We call this range image deblurring, where the deblurred image is obtained using *Deep Deblur*, and is denoted by $\hat{i}_{\text{range}}$. For completeness, we also assess the overall performance of the algorithm by deblurring arbitrary blurred images $y = i_{\text{test}} \otimes k +n$, where $i_{\text{test}}$ is not necessarily in the range of the generator. Unlike above, the overall error in this case accounts for the range error as well. We call this arbitrary image deblurring, and specifically the deblurred image is obtained using *Deep Deblur*, and is denoted by $\hat{i}_\text{DD}$. Figure \[fig:range-images\] shows a qualitative comparison between $i_{\text{test}}$, $i_{\text{range}}$, and $\hat{i}_\text{DD}$ on CelebA dataset. It is clear that the recovered image $\hat{i}_\text{DD}$ is a good approximation of the range image, $i_{\text{range}}$, indicating the limitation of the image generative network. \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_19\_x\_hat\_from\_test.png]{} (64,0) [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_19\_x\_hat\_from\_test.png]{} (64,0) [./figures/CelebA/“Original Images”/x\_orig\_19.png]{} (64,0) \ *Deep Deblur with Slack* mitigates the range error by not strictly constraining the recovered image to lie in the range of the image generator, for details, see Section \[sec:proposed-method2\]. As shown in Figure \[fig:celebA-results\], estimate $\hat{i}_\text{DDS}$ of true image $i_\text{test}$ from blurry observations is close to $i_\text{test}$ instead of $i_\text{range}$, thus mitigating the range issue. \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_92\_x\_hat\_from\_test.png]{} (55,0) [./figures/SVHN/“Original Images”/x\_orig\_92.png]{} (55,0) \ \ \ \ ### Deblurring Results on CelebA, Shoes and SVHN **Qualitative results on CelebA**: Figure \[fig:celebA-results\] gives a qualitative comparison between $i$, $i_{\text{range}}$, $\hat{i}_\text{DD}$, $\hat{i}_\text{DDS}$, and baseline approaches on CelebA and Shoes dataset. The deblurred images under engineered priors are qualitatively a lot inferior than the deblurred images $\hat{i}_\text{DD}$, and $\hat{i}_\text{DDS}$ under the proposed generative priors, especially under large blurs. On the other hand, the end-to-end training based approaches CNN, and DeblurGAN perform relatively better, however, the well reputed CNN is still displaying over smoothed images with missing edge details, etc compared to our results $\hat{i}_\text{DDS}$. DeblurGAN, though competitive, is outperformed by the proposed *Deep Deblur with Slack* by more than 1.5dB as shown in Table \[table:psnr-ssim-results\]. The images $\hat{i}_\text{DD}$ are sharp and with well defined facial boundaries and markers owing to the fact they strictly come from the range of the generator, however, in doing so these images might end up changing some image features such as expressions, nose, etc. On a close inspection, it becomes clear that how well $\hat{i}_\text{DD}$ approximates $i_{\text{test}}$ roughly depends (see, images specifically in Figure \[fig:range-images\]) on how close $i_{\text{range}}$ is to $i_{\text{test}}$ exactly. While as $\hat{i}_\text{DDS}$ are allowed some leverage, and are not strictly confined to the range of the generator, they tend to agree more closely with the ground truth. We go on further by utilizing pretrained PGGAN [@karras2017progressive] in *Deep Deblur* by convolving sampled images with large blurs ($30 \times 30$); see Figure \[fig:pg-gan-results\]. It has been observed that pre-trained generators struggle at higher resolutions [@athar2018latent], so we restrict our results at $128 \times 128$ resolution. In Figure \[fig:pg-gan-results\], it can be seen that under expressive generative priors our approach exceeds all other baseline methods recovering fine details from extremely blurry images. **Qualitative Results on SVHN**: Figure \[fig:svhn-results\] gives qualitative comparison between proposed and baseline methods on SVHN dataset. Here the deblurring under classical priors again clearly under performs compared to the proposed image deblurring results $\hat{i}_\text{DD}$. CNN also continues to be inferior, and the DeblurGAN also shows artifacts. We do not include the results $\hat{i}_\text{DDS}$ in these comparison as $\hat{i}_\text{DD}$ already comprehensively outperform the other techniques on this dataset. The convincing results $\hat{i}_\text{DD}$ are a manifestation of the fact that unlike the relatively complex CelebA and Shoes datasets, the simpler image dataset SVHN is effectively spanned by the range of the image generator. \ **Quantitative Results**: Quantitative results for CelebA, Shoes[^2] and SVHN using peak-signal-to-noise ratio (PSNR) and structural-similarity index (SSIM) [@wang2004image], averaged over 80 respective test set images, are given in Table \[table:psnr-ssim-results\]. On CelebA and Shoes, the results clearly show a better performance of *Deep Deblur with Slack*, on average, compared to all baseline methods. On SVHN, the results show that *Deep Deblur* outperforms all competitors. The fact that *Deep Deblur* performs more convincingly on SVHN is explained by observing that the range images $i_{\text{range}}$ in SVHN are quantitatively much better compared to range images of CelebA and Shoes. ### Robustness against Noise and Large Blurs **Robustness against Noise**: Figure \[fig:psnr-ssim-blursize\] gives a quantitative comparison of the deblurring obtained via *Deep Deblur* (the free parameters $\lambda$, $\gamma$ and random restarts in the algorithm are fixed as before), and baseline methods CNN, DeblurGAN (trained on fixed 1% noise level and on varying 1-10% noise levels) in the presence of Gaussian noise. We also include the performance of deblurred range images $\hat{i}_{\text{range}}$, introduced in Section \[sec:Exps-PretrainedPriors\], as a benchmark. Conventional prior based approaches are not included as their performance substantially suffers on noise compared to other approaches. On the vertical axis, we plot the PSNR and on the horizontal axis, we vary the noise strength from 1 to 10%. In general, the quality of deblurred range images (expressible by the generators) $\hat{i}_{\text{range}}$ under generative priors surpasses other algorithms on both CelebA, and SVHN. This in a way manifests that under expressive generative priors, the performance of our approach is far superior. The quality of deblurred images $\hat{i}_\text{DD}$ under generative priors with arbitrary (not necessarily in the range of the generator) input images is the second best on SVHN, however, it under performs on the CelebA dataset; the most convincing explanation of this performance deficit is the range error (not as expressive generator) on the relatively complex/rich images of CelebA. The end-to-end approaches trained on fixed 1% noise level display a rapid deterioration on other noise levels. Comparatively, the ones trained on 1-10% noise level, expectedly, show a more graceful performance. Qualitative results under heavy noise are depicted in Figure \[fig:results-heavynoise\]. Our deblurred image $\hat{i}_\text{DD}$ visually agrees better with $i_{\text{test}}$ than other methods. **Robustness against Large Blurs**: Figure \[fig:results-largeblur\] shows the deblurred images obtained from a very blurry face image. The deblurred image $\hat{i}_\text{DDS}$ using *Deep Deblur with Slack* is able to recover the true face from a completely unrecognizable face. The classical baseline algorithms totally succumb to such large blurs. The quantitative comparison against end-to-end neural network based methods CNN, and DeblurGAN is given in Figure \[fig:psnr-ssim-blursize\]. We plot the blur size against the average PSNR for both Shoes, and CelebA datasets. On both datasets, deblurred images $\hat{i}_\text{DDS}$ convincingly outperforms all other techniques. For comparison, we also add the performance of $\hat{i}_{\text{range}}$. Excellent deblurring under large blurs can also be seen in Figure \[fig:pg-gan-results\] for PGGAN. To summarize, the end-to-end approaches begin to lag a lot behind our proposed algorithms when the blur size increases. This is owing to the firm control induced by the powerful generative priors on the deblurring process in our newly proposed algorithms. Conclusion {#sec:conc} ========== This paper proposes a novel framework for blind image deblurring that uses deep generative networks as priors rather than in a conventional end-to-end manner. We report convincing deblurring results under the generative priors in comparison to the existing methods. A thorough discussion on the possible limitations of this approach on more complex images is presented along with an effective remedy to address these shortcomings. Our main contribution lies in introducing pretrained generative model in solving blind deconvolution. Introducing more expressive generative models with new novel architectures would improve the performance. We leave these exciting directions for future work. Supplementary Stuff =================== Algorithm Parameters -------------------- The choice of free parameters for SVHN, Shoes and CelebA of *Deep Deblur* and *Deep Deblur with Slack* are given in Table \[table:alg1-param\] and \[table:alg2-param\]. Both algorithms were implemented in Tensorflow and the code will be made publicly available. Generative Models ================= The generative model of SVHN images is a trained VAE with the network architecture described in Table \[table:vae-architectures\]. The dimension of the latent space of VAE is fixed at 100, and training is carried out on SVHN with a batch size of 1500, and a learning rate of $10^{-5}$ using the Adam optimizer. After training, the decoder part is extracted as the desired generative model $G_{\setI}$. For CelebA and Shoes dataset, the generative model $G_{\setI}$ is the default deep convolutional generative adversarial network (DCGAN)[@salimans2016improved]. The generative model of motion blur dataset is a trained VAE with the network architecture given in Table \[table:vae-architectures\]. This VAE is trained using Adam optimizer with latent dimension 50, batch size 5, and learning rate $10^{-5}$. After training, the decoder part is extracted as the desired generative model $G_{\setK}$. [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_10\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_10\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_10.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_18\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_18\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_18.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_26\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_26\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_26.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_48\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_48\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_48.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_73\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_73\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_73.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_53\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_53\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_53.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_35\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_35\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_35.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_02\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_02\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_02.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_79\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_79\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_79.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_06\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_06\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_06.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_40\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_40\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_40.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_63\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_63\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_63.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_72\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_72\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_72.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_39\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_39\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_39.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_00\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_00\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_00.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_07\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_07\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_07.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_03\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_03.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_04\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_04.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_07\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_07.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_10\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_10.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_17\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_17.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_31\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_31.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_37\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_37.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_44\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_44.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_57\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_57.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_61\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_61.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_68\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_68.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_76\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_76.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_12\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_12\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_12.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_24\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_24\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_24.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_26\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_26\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_26.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_31\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_31\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_31.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_33\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_33\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_33.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_40\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_40\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_40.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_35\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_35\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_35.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_37\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_37\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_37.png]{} \ \ \ \ [^1]: For an image scaled between 0 and 1, Gaussian noise of $1\%$ translates to Gaussian noise with standard deviation $\sigma = 0.01$ and mean $\mu=0$. [^2]: For qualitative results, see supplementary material.

--- abstract: 'Optical harmonic generation occurs when high intensity light ($>10^{10}$W/m$^{2}$) interacts with a nonlinear material. Electrical control of the nonlinear optical response enables applications such as gate-tunable switches and frequency converters. Graphene displays exceptionally strong-light matter interaction and electrically and broadband tunable third order nonlinear susceptibility. Here we show that the third harmonic generation efficiency in graphene can be tuned by over two orders of magnitude by controlling the Fermi energy and the incident photon energy. This is due to logarithmic resonances in the imaginary part of the nonlinear conductivity arising from multi-photon transitions. Thanks to the linear dispersion of the massless Dirac fermions, ultrabroadband electrical tunability can be achieved, paving the way to electrically-tuneable broadband frequency converters for applications in optical communications and signal processing.' author: - 'G. Soavi$^1$, G. Wang$^1$, H. Rostami$^2$, D. Purdie$^1$, D. De Fazio$^1$, T. Ma$^1$, B. Luo$^1$, J. Wang$^1$, A. K. Ott$^1$, D. Yoon$^1$, S. Bourelle$^1$, J. E. Muench$^1$, I. Goykhman$^1$, S. Dal Conte$^{3,4}$, M. Celebrano$^{4}$, A. Tomadin$^2$, M. Polini$^2$, G. Cerullo$^{3,4}$, A. C. Ferrari$^1$' title: 'Broadband, electrically tuneable, third harmonic generation in graphene' --- The response of a material to interaction with an optical field can be described by its polarization[@Shen1984]: $$\vec{P} = \epsilon_0 [\chi^{(1)} \cdot \vec{E}+\chi^{(2)} : \vec{E}\vec{E}+\chi^{(3)}\vdots \vec{E}\vec{E}\vec{E}+ \cdots ] \label{eq:Pol}$$ where $\vec{E}$ is the incident electric field and $\epsilon_{0}$ is the permittivity of free space. $\chi^{(1)}$(dimensionless) is the linear susceptibility, while the tensors $\chi^{(2)}$ \[m/V\] and $\chi^{(3)}$ \[m$^{2}$/V$^{2}$\] are the second- and third-order nonlinear susceptibilities[@units]. Thanks to the nonlinear terms of $\vec{P}$, new frequencies can be generated inside a material due to harmonic generation[@FranPRL1961] and frequency mixing[@StolAPL1974]. E.g., in Second Harmonic Generation (SHG) an incident electromagnetic wave with angular frequency $\omega_{0}=2\pi\nu$, with $\nu$ the photon frequency, generates via $\chi^{(2)}$ a new electromagnetic wave with frequency $2\omega_{0}$[@FranPRL1961]. The SHG efficiency (SHGE) is defined as the ratio between the SH intensity and the intensity of the incoming light. Analogously, Third Harmonic Generation (THG) is the emission of a photon with energy triple that of the incident one. The THG efficiency (THGE) is defined as the ratio between the TH intensity and the intensity of the incoming light. Second-order nonlinear processes are also known as three-wave-mixing, as they mix two optical fields to produce a third one[@ArmsPhysRev1962]. Third-order nonlinear processes are known as four-wave-mixing (FWM)[@ArmsPhysRev1962], as they mix three fields to produce a fourth one. Nonlinear optical effects are exploited in a variety of applications, including laser technology[@SteinScience1999], material processing[@ChanJLA1998] and telecommunications[@GarmOptExp2013]. E.g., to generate new photon frequencies (532nm from SHG in a Nd:YAG laser at 1.06$\mu$m)[@MillOptLett1997] or broadly tuneable ultrashort pulses (fs-ps) by optical parametric amplifiers (OPAs)[@CeruRSI2003] and optical parametric oscillators (OPOs)[@BoseOptLEtt1996]. High harmonic generation is also used for extreme UV light[@CorkPRL1993] and attosecond pulse generation[@CorkNP2007], while difference frequency generation is used to create photons in the THz range[@FergNM2002]. Second order nonlinear effects can only occur in materials without inversion symmetry, while third order ones occur in any system independent of symmetry[@Boyd2003], and they thus represent the main intrinsic nonlinear response for most materials. THG intensity enhancement was achieved by exploiting magnetic dipole[@ShchNL2014] and excitonic resonances[@ChenPRB1993], surface plasmons in Ag films[@TsanOptLett1996] and photonic-crystal waveguides[@CorcNP2009], by exploiting spatial compression of the optical energy, resulting in an increase of the local optical field. Nonlinear optical effects depend on the characteristics of the impinging light beam(s) (frequency, polarization) and on the properties of the nonlinear material, dictated by its electronic structure. The ability to electrically control the nonlinear optical response of a material by a gate voltage opens up disruptive applications to compact nanophotonic devices with novel functionalities. However, to the best of our knowledge, external electrical control of the THGE has not been reported to date in any material. ![**Samples used for THG experiments.** (a) CVD SLG on Sa for transmission and (b) exfoliated SLG on Si/SiO$_{2}$ for reflection measurements.[]{data-label="fig:samples"}](fig_samples.eps){width="90mm"} Layered materials (LMs) have a strong nonlinear optical response[@SeylNatNano2015; @HendPRL2010; @WangPRL2015; @LiuNatPhys2017; @SaynNC2017; @KleinNanoLett2017; @SunACSNano2010; @BonaNP2010]. Electrically tunable SHG was reported for monolayer WSe$_{2}$ for photon energies close to the A exciton ($\sim$1.66eV)[@SeylNatNano2015]. However, the tunability was limited to a narrow band ($\sim$10meV) in the proximity of the excitonic transition. Electrically tunable SHG was also reported by inversion symmetry breaking in bilayer MoS$_{2}$ close to the C exciton ($\sim$2.75eV)[@KleinNanoLett2017], but SHGE was strongly dependent on the laser detuning with respect to the C exciton transition energy. Thus, in both cases electrical control was limited to narrow energy bands ($\sim$10-100meV) around the excitonic transitions. Graphene, instead, can provide electrically tunable nonlinearities over a much broader bandwidth thanks to the linear dispersion of the Dirac Fermions. In single layer graphene (SLG), SHG is forbidden due to symmetry[@MikhPRB2011; @DeanPRB2010; @AnPRB2014; @AnNanoLett2013]. SHG was reported in the presence of an electric current[@AnNanoLett2013; @AnPRB2014], but weak compared to third-order nonlinear effects. The data in Refs., imply that THG in SLG is at least one order of magnitude stronger than SHG activated by inversion symmetry breaking. Thus, third order nonlinear effects are the most intense terms of the intrinsic nonlinear optical response of SLG. Third order nonlinearities in SLG were studied both theoretically[@ChengNJPhys2014; @ChengPRB2015; @MikhPRB2016; @RostPRB2016; @RostPRB2017] and experimentally[@HendPRL2010; @KumaPRB2013; @AlexCondMat2017; @HongPRX2013]. Ref. reported that SLG has $\chi^{(3)}\sim$10$^{-15}$m$^{2}$/V$^{2}$ ($\sim$10$^{-7}$esu), several orders of magnitude higher than typical metals (*e.g.* $\chi^{(3)}\sim 7.6 \times 10^{-19}$m$^{2}$/V$^{2}$ in Au[@Boyd2003]) and dielectrics (*e.g.* $\chi^{(3)}\sim 2.5 \times 10^{-22}$m$^{2}$/V$^{2}$ in fused silica[@Boyd2003]). Ref. also reported a $1/\omega_{0}^{4}$ proportionality of the third order optical nonlinear response in a narrow band (emitted photons between$\sim$1.47 and 1.63eV), but no resonant behavior nor doping dependence. Ref. reported a factor$\sim$2 enhancement of the third order nonlinear signal in a FWM experiment at the onset of inter-band transitions ($\hbar\omega_{0}$=2$|E_{\rm F}|$, where $E_{\rm F}$ is the Fermi Energy) for SLG on SiN waveguides in a narrow band (emitted photon energies between$\sim$0.79 and 0.8eV). Thus, to date, evidence of tunable third order nonlinear effects in SLG is limited to narrow bands and weak enhancements. Here we show that THGE in SLG can be tuned by almost two orders of magnitude over a broad energy range (emitted photon energies between$\sim$1.2-2.2eV) and over 20 times by electrical gate tuning. These results, in agreement with calculations based on the intrinsic third-harmonic conductivity of massless Dirac fermions, confirm that SLG is a unique nonlinear material since it allows electrical tuning of $\chi^{(3)}$ over an ultra-broad range, only limited by the linearity of the Dirac cone ($\pm 2$eV[@NetoRevModPhys2009]). In order to test both the photon energy dependence and the electrical tunability of THGE we use two sets of samples: SLG placed on a transparent substrate (sapphire, Sa), Fig.\[fig:samples\]a, and back-gated SLG on a reflective substrate (Si/SiO$_2$), Fig.\[fig:samples\]b. To study the THGE photon energy dependence, we measure it over a broad range (incident photon energy$\sim$0.4-0.7eV, with a THG signal at$\sim$1.2-2.1eV only limited by the absorption of the Si based charge-coupled device, CCD, used in our set-up). Transmission measurements allow us to derive the absolute THGE, by taking into account the system losses and by minimizing the chromatic aberrations of the optical components (e.g. in reflection one needs to use beam splitters and these do not have a flat response over the$\sim$0.4-2.2eV range). Thus, we use chemical vapor deposition (CVD) to obtain a large area SLG ($\sim$cm size) and simplify the alignment, given the low optical contrast of SLG on sapphire[@CasiNL2007]. When measuring the THGE electrical tunability we need to follow the THG intensity normalized to its minimum, as function of gate voltage ($V_G$). For each $\omega_0$ we measure 62 spectra, one for each $V_G$. For each spectrum we calculate the total number of counts on the CCD, which is proportional to the total number of THG photons, and divide all the spectra by that with the minimum counts. The key here is a precise control of $E_F$, while any system uncertainties on the absolute THGE are removed by the normalization. Thus, we use an exfoliated SLG back-gated field-effect transistor (FET) on Si+285nm SiO$_{2}$. The two sets of samples are prepared and characterized as described in Methods (Sect.\[subs:samples\]). $E_F$ for the CVD SLG on Sa is$\sim$250meV, and $<$100meV in the exfoliated SLG on Si+285nm SiO$_2$. The defect density is n$_{\rm D}\sim$6$\times$10$^{10}$cm$^{-2}$ for SLG on Sa and n$_{\rm D}\sim$2.4$\times$10$^{10}$cm$^{-2}$ for SLG on Si/SiO$_2$. The different $E_F$ is considered in our theory and addressed experimentally, since $E_F$ is one of the parameters of our study. The small difference in defects suggests that the two samples are comparable. In fact, as we discuss in the following, THGE has a negligible dependence on disorder, impurities and imperfections over a range of values that covers the vast majority of SLG reported in literature. ![**Energy dependence of SLG THGE** (a) THG spectra for $\hbar\omega_{0}\sim0.4$ to$\sim0.7$eV and an average power$\sim$1mW. (b) THGE for SLG on Sa as a function of $\hbar\omega_{0}$ (x bottom axis) and $3\hbar\omega_{0}$ (x top axis). Curves are calculated for $\tau \gg$ 1 ps and increasing $T_e$ for $E_F$=250meV and $I_{\omega_{0}}\sim 2.4\times 10^{12}$Wm$^{-2}$.[]{data-label="fig:THG_energy"}](fig_THG_energy.eps){width="90mm"} ![**Gate tunability of THG** (a) THG spectrum as a function of $E_F$ for exfoliated SLG on Si/SiO$_2$. (b) THG intensity (left y axis, blue dots) and $I_{SD}$ (right y axis, red dots) as a function of $E_F$ (bottom x axis) and corresponding $V_G$ (top x axis) for SLG on Si/SiO$_2$.[]{data-label="fig:THG_gate"}](fig_THG_gate.eps){width="90mm"} {width="180mm"} The THGE measurements are performed in air at room temperature for both transmission and reflection, as detailed in Methods (Sect.\[subs:setup\]). Fig.\[fig:THG\_energy\]a plots representative TH spectra for different incident $\hbar\omega_0$ for SLG on Sa. We assign the measured signal to THG because the energy of the detected photons is equal to three times that of the incident one: $\hbar\omega_{THG}=3\hbar\omega_0$ and its intensity scales with the cube of the incident power ($I_{3\omega_0} \propto I^3_{\omega_0}$)[@Boyd2003]. Fig.\[fig:THG\_energy\]b shows that when $\hbar\omega_0$ decreases from$\sim0.7$ to$\sim 0.4$eV THGE is enhanced by a factor$\sim$75, almost one order of magnitude larger than the$\sim9.5$ expected from the $1/\omega_0^4$ dependence of THGE in SLG at $E_F$=0[@RostPRB2016]. This can be explained by taking into account the dependence of the SLG third order optical conductivity $\sigma^{(3)}_{\ell\alpha_1\alpha_2\alpha_3}$, where $\ell,\alpha_{i=1,2,3}$ are the Cartesian indexes, on $\omega_0$, $E_F$ and electronic temperature ($T_e$). Note that $\chi^{(3)} \equiv i \sigma^{(3)}/(3\epsilon_0\omega_0 d_{eff})$[@ChengNJPhys2014], by considering SLG with a thickness $d_{eff}$. Our modeling is based on $\sigma^{(3)}_{\ell\alpha_1\alpha_2\alpha_3}$, therefore we do not need to use $d_{eff}$. The dependence on $E_F$ was calculated in Refs.[@MikhPRB2016; @RostPRB2016] and gives resonances at $\hbar\omega_0=2|E_F|,|E_F|,2|E_F|/3$ for $T_e$=0K. A finite $T_e$ modifies the height and broadening of these resonances, as derived in Methods (Sect.\[subs:theory\]). A comparison between theoretical curves, for $E_F$=250meV and different $T_e$, and experiments is plotted in Fig.\[fig:THG\_energy\]b. This indicates that $E_F$ plays a key role, in particular when $\hbar\omega_0\leq$2$|E_F|$. The effects of disorder, impurities and imperfections can be phenomenologically introduced by a relaxation rate, $\Gamma=\hbar/\tau$, through the density matrix approach[@ChengPRB2015; @MikhPRB2016]. Our analysis (see Methods Sect.M3.3) shows that the effect of a finite $\tau $ in the $\sim$0.1fs-1ps range on THGE is negligible. Since $\Gamma\sim e\hbar v^2_{\rm F}/(\mu_{e}E_{\rm F})$[@ref:proof] this range of $\tau$, for SLG with $E_F$ between 100 and 600meV, would correspond to mobilities$\sim$1-10$^5$ cm$^2$V$^{-1}$s$^{-1}$, covering the vast majority of experimental SLG in literature. Refs. predicted that gate tunability of THGE should be possible. Fig.\[fig:THG\_gate\]a plots the THG spectra for different V$_G$ and $\hbar\omega_0$=0.59eV. Fig.\[fig:THG\_gate\]b shows the THG intensity over -600meV$\leq E_{\rm F} \leq$+150meV corresponding to-150V$\leq V_G \leq$+150V. $E_F$ is derived from each $V_G$ as discussed in Methods Sect.M1. Fig.\[fig:THG\_gate\]b shows that, as a function of $V_G$, there is a THG intensity enhancement by over a factor of 20 when $\hbar\omega_0<2|E_F|$. Fig.\[fig:THG\_gate\]b also indicates that THGE in SLG follows an opposite trend compared to FWM[@AlexCondMat2017]: it is higher for intra-band ($\hbar\omega_0<2|E_F|$) than inter-band ($\hbar\omega_0>2|E_F|$) transitions. This is reproduced by the calculations in Methods (Sect.\[subs:theory\]). THGE for SLG, considered as a nonlinear interface layer between air and substrate, under normal incidence can be written as (see Methods Sect.M3.1): $$\label{eq:THG_efficiency} \eta^{THG}(\omega_0,E_F,T_e)=\frac{I_{3\omega_0}}{I_{\omega_0}}=f(\omega_0)\frac{I^2_{\omega_0}}{4\epsilon^4_0 c^4}\left|\sigma^{(3)}_{\ell\ell\ell\ell}(\omega_0,E_F,T_e)\right |^2$$ where $\epsilon_0\sim 8.85\times 10^{-12}{\rm C (V m)^{-1}}$ and $c=3\times10^8$m/s are the vacuum permittivity and the speed of light; $f(\omega_0)=n^{-3}_1(\omega_0)n_2(3\omega_0)[n_1(3\omega_0)+n_2(3\omega_0)]^{-2}$ in which $n_{i=1,2}(\omega)$ is the refractive index of air (i=1) and substrate (i=2). For SLG on any substrate $n_1({\omega})\sim1$ and $n_2({\omega})=\sqrt{\epsilon_{2}(\omega)}$, with $\epsilon_{2}(\omega)$ the substrate dielectric function. For Sa, we use $\epsilon_2\sim10$[@SaSpecs]. According to the $C_{6v}$ point group symmetry of SLG on a substrate[@RostNC2017], the relative angle between laser polarization and SLG lattice is not important for the third-order response (see Methods Sect.M3.2). We can thus assume the incident light polarization $\hat{\ell}$ along the zigzag direction without loss of generality. $\sigma^{(3)}_{\ell\ell\ell\ell}$ can then be calculated employing a diagrammatic technique[@RostPRB2016; @RostPRB2017; @RostNC2017], where we evaluate a four-leg Feynman diagram for the TH response function (see Methods Sect.\[subs:theory\]). The light-matter interaction is considered in a scalar potential gauge in order to capture all intra-, inter-band and mixed transitions[@ChengNJPhys2014; @ChengPRB2015; @MikhPRB2016; @RostPRB2016]. Fig.\[fig:THG\_TheoryExp\] compares experiments and theory for THGE for four incident photon energies between 0.41 and 0.69eV and different T$_e$. Both theory and experiments display a plateau-like feature for THGE at low $E_F$ ($2|E_F|<\hbar\omega_0$), which corresponds to inter-band transitions for the incident photon. By further increasing $|E_F|$, we reach the energy region for intra-band transitions ($\hbar\omega_0<2|E_F|$), where we observe a THGE rise up to a maximum for $|E_F|\sim1.25\hbar\omega_0$ (Fig.\[fig:THG\_TheoryExp\]a). This is due to the merging of the two $T_e$=0K resonances at $|E_F|/\hbar\omega_0=$1 and 1.5 as a result of high $T_e$ (see Methods Sect.M3.4). Fig.\[fig:THG\_TheoryExp\] indicates that the best agreement between theory and experiments is reached when $\sim$1500K$\leq T_e\leq$2000K. $T_e$ can be also independently estimated as follows. When a pulse of duration $\Delta t$ and fluence ${\cal F}$, with average absorbed power per unit area P/A, photoexcites SLG, the variation dU of the energy density in a time interval dt is dU=(P/A)dt. The corresponding $T_e$ increase is dT$_e$=dU/c$_v$, where $c_v$ is the electronic heat capacity of the photoexcited SLG, as derived in Methods (Sect.\[subs:Te\]). When the pulse is off, $T_e$ relaxes towards the lattice temperature on a time-scale $\tau$. This reduces $T_e$ by $dT_e=-(T_e/\tau)dt$ in a time interval dt. Thus: $$\frac{dT_{\rm e}}{dt}=\frac{1}{c_{\rm v}}\frac{P}{A}-\frac{T_{\rm e}}{\tau}~.$$ If the pulse duration is: (i) much longer than$\sim$20fs, which is the time-scale for the electron distribution to relax to the Fermi-Dirac profile in both bands[@BridNC2013]; (ii) comparable to the time-scale$\sim150-200fs$ needed to heat the optical phonon modes[@BridNC2013; @LazzPRL2005; @PiscPRL2004], it is safe to assume that $T_e$ reaches a steady-state during the pulse, given by: $$\label{eq:steady_temp} T_e= \frac{\tau}{c_v(\mu_c,\mu_v, T_e)}\frac{P}{A}~.$$ The $T_e$ dependence of $c_v$ in Eq.\[eq:steady\_temp\] is discussed in Methods (Sect.\[subs:Te\]). In our experiments we have: ${\cal F}$=70$\mu$J/cm$^2$, $\Delta$t=300fs, P/A=2.3%$\times {\cal F}/\Delta t$. The resulting $T_e$, as a function of $\hbar\omega_0$, for several $\tau$, are in Fig.\[fig:steady\_state\_temperature\]. $T_e$ increases for more energetic photons and for longer $\tau$. Overall, $T_e$ ranges between$\simeq$1000 and 1500K, in excellent agreement with the estimate from Fig.\[fig:THG\_TheoryExp\]. ![\[fig:steady\_state\_temperature\] **Steady-state T$_e$ in photoexcited SLG**. $T_e$ as a function of $\hbar\omega_0$ for $\tau$=100 (black), 200 (blue), and 300fs (red).](fig_steady_state_temperature.eps){width="80mm"} ![**Multi-photon resonances in SLG.** Resonances corresponding to the three logarithmic peaks in the imaginary part of the SLG nonlinear conductivity that occur at $T_e$=0K for $\hbar \omega_0=2|E_F|/3$, $|E_F|$, $3|E_F|/2$. The red arrows represent the incident $\omega_{0}$ photons and the blue arrows represent the TH photons at $3\omega_0$.[]{data-label="fig:THG_sketch_resonances"}](fig_THG_sketch_resonances.eps){width="80mm"} The observed gate-dependent enhancement of the THGE can be qualitatively understood as follows. The linear optical response of SLG at T$_e$=0K has a “resonance” for $\hbar\omega_0=2|E_F|$, the onset of intra- and inter-band transitions[@MakPRL2008]. Around this energy, a jump occurs in the real part of $\sigma^{(3)}_{\ell\alpha_1\alpha_2\alpha_3}$ due to the relaxation of the Pauli blocking constraint for vertical transitions between massless Dirac bands. From the Kramers-Kronig relations[@KronJOSA1926], this jump corresponds to a logarithmic peak in the imaginary part of $\sigma^{(3)}_{\ell\alpha_1\alpha_2\alpha_3}$. In a similar way, for the SLG third-order nonlinear optical response, logarithmic peaks in the imaginary part of $\sigma^{(3)}_{\ell\alpha_1\alpha_2\alpha_3}$ occur at T$_e$=0K for multi-photon transitions such that $m \hbar \omega = 2|E_F|$ with $m$=1,2,3, which correspond to incident photon energies $\hbar \omega_{0}=2|E_F|$, $|E_F|$, 2/3$|E_F|$[@MikhPRB2016; @RostPRB2016], as sketched in Fig.\[fig:THG\_sketch\_resonances\]. At high T$_e$, due to the broadening of the Fermi-Dirac distribution, these peaks are smeared and merge (see Methods Sect.M3.4). Our work provides experimental evidence of this resonant structure (Fig.\[fig:THG\_TheoryExp\]), in agreement with theory. In summary, we demonstrated that the THG efficiency in SLG can be modulated by over one order of magnitude by controlling its $E_F$ and by almost two orders of magnitude by tuning the incident photon energy in the range$\sim$0.4-0.7eV. The observation of a steep increase of THGE at $|E_F|$=$\hbar\omega_0/2$ for all the investigated photon energies suggests that the effect can be observed over the entire linear bandwidth of the SLG massless Dirac fermions. These results pave the way to novel SLG-based nonlinear photonic devices, in which the gate tuneability of THG may be exploited to implement on-chip schemes for optical communications and signal processing, such as ultra-broadband frequency converters. \[Meth\]Methods {#methmethods .unnumbered} =============== Sample preparation and characterization {#subs:samples} --------------------------------------- SLG on Sa is prepared as follows. SLG is grown by CVD on Cu as for Ref.[@LiS2009]. A Cu foil (99.8% pure) substrate is placed in a furnace. Annealing is performed at 1000 $^{\circ}$C in a 20sccm (standard cubic centimeters per minute) hydrogen atmosphere at $\sim$196mTorr for 30min. Growth is then initiated by introducing 5sccm methane for 30mins. The grown film is characterized by Raman spectroscopy[@FerrPRL2006; @FerrNN2013] with a Horiba Labram HR800 spectrometer equipped with a 100x objective at 514nm, with a power on the sample$\sim$500$\mu$W to avoid any possible heating effects. The D to G intensity ratio is I(D)/I(G)$\ll$0.1, corresponding to a defect density n$_{\rm D}\ll2.4\times10^{10}$cm$^{-2}$[@CancNL2011; @BrunaACSNano2014]. The 2D peak position (Pos) and full width at half maximum (FWHM) are Pos(2D)$\sim$2703cm$^{-1}$ and FWHM(2D)$\sim$36cm$^{-1}$, respectively, while Pos(G)$\sim$1585cm$^{-1}$ and FWHM(G)$\sim$18cm$^{-1}$. The 2D to G intensity and area ratios are I(2D)/I(G)$\sim$3.3 and A(2D)/A(G)$\sim$6.5, respectively. The CVD SLG is then transferred on Sa by polymer-assisted Cu wet etching[@BonaMT2012], using polymethyl methacrylate (PMMA). After transfer Pos(2D)$\sim$2684cm$^{-1}$, FWHM(2D)$\sim$24cm$^{-1}$, Pos(G)$\sim$1584cm$^{-1}$, FWHM(G)$\sim$13cm$^{-1}$, I(2D)/I(G)$\sim$5.3, A(2D)/A(G)$\sim$10. From Refs. we estimate $E_F\sim$250meV. After transfer, I(D)/I(G)$\sim$0.14, which corresponds to $n_{\rm D}\ll6.0\times10^{10}$cm$^{-2}$[@CancNL2011; @BrunaACSNano2014] with a small increase of defect density. The back-gated SLG sample is prepared by micromechanical exfoliation of graphite on Si+285nm SiO$_2$[@NovoPNAS2005]. Suitable single-layer flakes are identified by optical microscopy[@CasiNL2007] and Raman spectroscopy[@FerrPRL2006; @FerrNN2013]. The device is then prepared as follows. We deposit a resist (A4-495) on the exfoliated SLG on Si/SiO$_2$ and we pattern it with electron beam lithography. Then, we develop the resist in a solution of isopropanol (IPA) diluted with distilled water, evaporate and lift-off 5/70nm of Cr/Au. Cr is used to improve adhesion of the Au, while Au is the metal for source-drain contacts. Raman spectroscopy is then performed after processing. We get Pos(2D)$\sim$2678cm$^{-1}$, FWHM(2D)$\sim$25cm$^{-1}$, Pos(G)$\sim$1581cm$^{-1}$, FWHM(G)$\sim$12cm$^{-1}$, I(2D)/I(G)$\sim$4.9, A(2D)/A(G)$\sim$10.3, indicating E$_F<$100meV[@BaskPRB2009; @DasNN2008]. I(D)/I(G)$\ll$0.1, corresponding to n$_D\ll2.4\times10^{10}$cm$^{-2}$[@CancNL2011; @BrunaACSNano2014]. When V$_G$ is applied, $E_{\rm F}$ is derived from V$_G$ as follows[@DasNN2008]: $$\label{eq:Ef} E_{\rm F} = \hbar v_{\rm F} \sqrt{\pi n}$$ where $\hbar$ is the reduced Plank constant, and $n$ is the SLG carrier concentration. This can be written as[@DasNN2008]: $$\label{eq:nFET} n = \frac{C_{\rm BG}}{e} (V_{\rm G}-V_0)$$ where $C_{BG}=\epsilon\epsilon_0/d_{BG}=1.2 \times 10^{-8}$Fcm$^{-2}$ is the back-gate capacitance (d$_{BG}$=285nm is the back-gate thickness and $\epsilon\sim$4 the SiO$_2$ dielectric constant[@DasNN2008]), $e>0$ is the fundamental charge and $V_0$ is the voltage at which the resistance of the SLG back-gated device reaches its maximum (minimum of the current between source and drain). We note that the SLG quantum capacitance (C$_{QC}$) is negligible in this context. In fact the SiO$_2$ layer and SLG can be considered as two capacitors in series and the SLG C$_{QC}$ is $\sim$10$^{-6}$Fcm$^{-2}$[@XiaNN2009]. Thus the total capacitance C$_{tot}=$(1/C$_{BG}$+1/C$_{QC}$)$^{-1}\sim$C$_{BG}$. THGE measurements and calibration {#subs:setup} --------------------------------- ![Setup used for THG experiments in both transmission and reflection. BS, Beam Splitter.[]{data-label="fig:setup"}](fig_setup.eps){width="90mm"} THGE measurements are performed in air at room temperature for both transmission and reflection, as shown in Fig.\[fig:setup\]. For excitation we use the idler beam of an OPO (Coherent) tuneable between$\sim$0.31eV (4$\mu$m) and$\sim$0.73eV (1.7$\mu$m). This is seeded by a mode-locked Ti:Sa laser (Coherent) with 150fs pulse duration, 80MHz repetition rate and 4W average power at 800nm. The OPO idler is focused by a 40X reflective objective (Ag coating, numerical aperture NA=0.5) to avoid chromatic aberrations. The THG signal is collected by the same objective (in reflection mode) or collimated by an 8mm lens (in transmission mode) and delivered to a spectrometer (Horiba iHR550) equipped with a nitrogen cooled Si CCD, Fig.\[fig:setup\]. The idler spot-size is measured with the razor-blade technique[@DomiEL1999] to be$\sim$4.7$\mu$m. This corresponds to an excitation fluence$\sim$70$\mu$J/cm$^{2}$ for the average power (1mW) used in our experiments. The idler pulse duration is checked by autocorrelation measurements based on two-photon absorption on a single channel Si photodetector and is$\sim$300fs. Under these excitation conditions, the THG signal is stable over at least 1 hour. For the electrical dependent THGE measurements we use a Keithley 2612B dual channel source meter to apply $V_G$ between -150 and +150V, a source-drain voltage (10mV), as well as to read the source-drain current ($I_{SD}$). For the photon energy dependence measurements we use 60s acquisition time and 10 accumulations (giving a total of 10 minutes for each spectrum). For the gate dependence measurements we proceeded as follows. We tune $V_{\rm G}$ (62 points between -150 and +150V) and for each $V_G$ we measure the THG signal by using 10s acquisition time and 1 accumulation. We use a shorter accumulation time compared to the photon energy measurements to reduce the total time required for each $V_{\rm G}$ scan. A lower accumulation time implies that less photons are collected by the CCD. We consider the amplitude of THG in counts/s, obtained by dividing the number of counts detected on the CCD by the accumulation time. Thus, in the case of $V_G$ dependent measurements, SLG is kept at a given $V_G$ for 10s before moving to the next point (next value of $V_G$). This corresponds to$\sim$10minutes for each measurement (*i.e.* a full $V_G$ scan between -150 and + 150V). In this way, for each $V_G$ and, consequently, for each $E_F$, we record one THG signal spectrum. To estimate the $\omega_0$ dependent THGE, it is necessary to first characterize the photon energy dependent losses of the optical setup. The pump-power is measured on the sample (by removing it and measuring the power after the objective). The major losses along the optical path are the absorption of Sa, the grating efficiency, and the CCD quantum efficiency. We also need to consider the CCD gain. The Sa transmittance is$\sim$85% in the energy range of our THG experiments[@SaSpecs]. To evaluate the losses of the grating and the absorption of the CCD, we align the Ti:Sa laser, tuneable between$\sim$1.2 and 1.9eV ($\sim$650-1050nm), with the microscope and detect it on the CCD. We then measure the signal on the spectrometer, given a constant number of photons for all wavelengths, and compare this with the spectrometer specifications[@SymphonySpecs]. We get an excellent agreement between the two methods (*i.e.* evaluation of the losses from detection of the fundamental beam and spectrometer specifications). Thus we use the spectrometer specifications to estimate the losses due to grating and CCD efficiencies. We also account for the CCD gain, *i.e.* the number of electrons necessary to have 1 count. The instrument specs[@SymphonySpecs] give a gain$\sim$7. TGHE modeling {#subs:theory} ------------- $\sigma^{(3)}_{\ell\ell\ell\ell}$ is calculated through a diagrammatic technique, with the light-matter interaction taken in the scalar potential gauge in order to capture all intra-, inter-band and mixed transitions[@ChengPRB2015; @MikhPRB2016; @RostPRB2016]. We evaluate the diagram in Fig.\[fig:feyn\] and denote by $\Pi^{(3)}_{\ell}$ the response function. $\hat{n}$ and $\hat{j}_\ell$ are the density and paramagnetic current operators. Then, $\sigma^{(3)}_{\ell\ell\ell\ell}=(i e)^3 \lim_{{\vec q}\to { 0}} \partial^3\Pi^{(3)}_{\ell}/\partial q^3_{\ell}$, where $e>0$ is the fundamental charge[@RostPRB2016]. We use the Dirac Hamiltonian of low-energy carriers in SLG as ${\cal H}_{k} = \hbar v_{\rm F} {\vec \sigma}\cdot{\vec k}$ where ${ \vec \sigma} = (\tau\sigma_x,\sigma_y)$ stands for the Pauli matrices in the sublattice basis. Note that $\tau=\pm$ stands for two valleys in the SLG Brillouin zone. We get $\sigma^{(3)}_{xxxx}(\omega,E_{\rm F},0)=i\sigma^{(3)}_{0} \bar{\sigma}^{(3)}_{xxxx}(\omega,E_{\rm F},0)$ at $T_{\rm e}=0$[@ChengPRB2015; @MikhPRB2016; @RostPRB2016]: $$\begin{aligned} \label{eq:clean} \bar{\sigma}^{(3)}_{xxxx} (\omega,E_{\rm F},0) &=\frac{ 17 G(2|E_{\rm F}|,\hbar\omega_{+}) - 64 G(2|E_{\rm F}|,2\hbar\omega_{+})|}{24 (\hbar\omega_{+})^4} \nonumber\\ &+ \frac{45 G(2|E_{\rm F}|,3\hbar\omega_{+})}{24 (\hbar\omega_{+})^4}\end{aligned}$$ where $G(x,y)= \ln|(x+y)/(x-y)|$, $\sigma^{(3)}_0 ={N_{\rm f} e^4 \hbar v^2_{\rm F}}/({32\pi})$ with $N_{\rm f}=4$ and $\hbar\omega_{+} \equiv \hbar\omega+i0^+$. At finite $T_{\rm e}$, $\sigma^{(3)}_{\ell\ell\ell\ell}$ is evaluated as[@Vignale_book]: $$\sigma^{(3)}_{xxxx}(\omega,E_{\rm F},T_{\rm e}) =\frac{1}{4k_{B}T_{\rm e}} \int^{\infty}_{-\infty}dE~\frac{ \sigma^{(3)}_{xxxx}(\omega,E,0) }{ \cosh^2 \left (\frac{E-\mu}{2k_{B}T_{\rm e}} \right ) }.$$ ![Feynamn diagram for $\Pi^{(3)}_{\ell}$ in the scalar potential gauge. Solid/wavy lines indicate non-interacting Fermionic propagators/external photons. Solid circles and square indicate density and current vertexes[]{data-label="fig:feyn"}](fig_feyn.eps){width="50mm"} ![Schematic of SLG on substrate. The TH radiated waves in the top and bottom medium obey the TH Snell’s law: $n_i(3\omega_0)\sin\theta_i=n_1(\omega_0)\sin\theta$. The red dashed arrows indicate the propagation direction of in-coming and out-going waves.[]{data-label="fig:snell"}](fig_snell.eps){width="80mm"} ### THGE of SLG as an interface layer {#subss:THGElayer} In order to evaluate the THGE for SLG on a substrate we consider SLG as an interface layer between air and substrate, see Fig.\[fig:snell\], and implement electromagnetic boundary conditions for the non-harmonic radiations. We follow Ref., and provide explicit details for THG in SLG. The Maxwell equations in the nonlinear medium in the $m(\ge 2)$-th order of perturbation are given by[@Boyd2003; @Jackson_book]: $$\begin{aligned} &{\vec \nabla}\cdot {\vec B}^{(m)} = 0~,\\ & {\vec \nabla}\cdot {\vec D}^{(m)} = \frac{\rho^{(m)}_{\rm f}}{\epsilon_0} -\frac{1}{\epsilon_0}{\vec \nabla}\cdot {\vec P}^{(m)}~, \\ &{\vec \nabla}\times {\vec E}^{(m)} = i \omega_\Sigma {\vec B}^{(m)} ~, \\ &{\vec \nabla}\times {\vec B}^{(m)} = \mu_0 {\vec J}^{(m)}_{\rm f} -i\frac{\omega_\Sigma}{c^2} {\vec D}^{(m)} -i\omega_\Sigma \mu_0 {\vec P}^{(m)}~.\end{aligned}$$ where ${\vec D}^{(m)}=\epsilon (\omega_\Sigma) {\vec E}^{(m)}$ is the [*conventional*]{} displacement vector. $\rho^{(m)}_{\rm f}$ and ${\vec J}^{(m)}_{\rm f}$ are the $m$-th order Fourier components of free charge and free current. Note that $\omega_\Sigma=\sum^m_{i}\omega_i$ in which $\omega_i$ correspond to the incoming photons frequency into the nonlinear medium. In the case of THG, we have $m=3$, $\omega_{1,2,3}=\omega_0$ and $\omega_\Sigma=\omega_{\rm THG}=3\omega_0$. $\epsilon(\omega)$ is the isotropic and homogenous linear relative dielectric function. Only electric-dipole contributions are included. We consider SLG in the $x$-$y$ plane embedded between air and a substrate. SLG is modeled by a dielectric function $\epsilon_{\rm s}(\omega)$, nonlinear polarization, free surface charge and free surface current: $$\begin{aligned} {\vec P}^{(m)} &= \delta(z) {\vec {\mathcal P}}^{(m)}~, \\ \rho^{(m)}_{\rm f} &=\delta(z)\sigma^{(m)}_{\rm f}~,\\ {\vec J}^{(m)}_{\rm f} &= \delta(z) {\vec K}^{(m)}_{\rm f}~.\end{aligned}$$ Having the Dirac delta, $\delta(z)$, in the above relations implies that SLG only shows up in the electromagnetic boundary conditions. Note that ${\vec {\mathcal P}}^{(m)}$ and $ {\vec K}^{(m)}_{\rm f}$ are in-plane vectors with zero component along the interface normal, $\hat z$. The interface layer is the only source of nonlinearity. We assume $\sigma^{(m)}_{\rm f}=0$ and ${\vec K}^{(m)}_{\rm f}={0}$, consistent with our experiments, where there are no free surface charges and currents that oscillate at frequency $m\omega$ with $m=2,3,\dots$. The boundary conditions for the nonlinear fields at z=0 are obtained as: $$\begin{aligned} \label{eq:BC} &~{\vec B}^{(m)}_{1} - {\vec B}^{(m)}_{2} = \mu_0 ({\vec K}^{(m)}_{\rm f}-i\omega_\Sigma {\vec {\mathcal P}}^{(m)}) \times \hat {z} ~, \nonumber \\ &\left \{ \epsilon_{1}(\omega_\Sigma){\vec E}^{(m)}_{1} - \epsilon_{2}(\omega_\Sigma){\vec E}^{(m)}_{2} \right \} \cdot \hat { z} = \frac{\sigma^{(m)}_{\rm f}- {\vec \nabla}_{\rm 2D}\cdot {\vec {\mathcal P}}^{(m)}}{\epsilon_0} ~,\nonumber\\ &~({\vec E}^{(m)}_{1} - {\vec E}^{(m)}_{2})\times \hat{ z} = { 0} ~.\end{aligned}$$ Where the sub-indexes 1,2 stand for the top(bottom) medium and ${\vec \nabla}_{\rm 2D}= \hat{ x}\partial /\partial x+ \hat{ y}\partial /\partial y$. The dielectric function of the interface layer, $\epsilon_{\rm s}(\omega)$, does not emerge in the above boundary conditions. The wave equation in the top and bottom media, with vanishing nonlinear polarization, follows: $${\vec \nabla} \times {\vec \nabla} \times {\vec E}^{(m)} - \ \frac{\omega^2_\Sigma}{c^2} \epsilon(\omega_\Sigma) {\vec E}^{(m)} = 0~.$$ which has a plane wave solution[@Jackson_book]: $$\begin{aligned} {\vec E}^{(m)} = \hat{ \ell} {\cal E}^{(m)} e^{i({\vec q}_{\Sigma} \cdot {\vec r} -\omega_\Sigma t)} +c.c.\end{aligned}$$ $\hat{ \ell} \cdot {\vec q}_{\Sigma} = 0$ and the dispersion relation in the top and bottom media is given by: $$\begin{aligned} q_{\Sigma} =\left |{\vec q}_{\Sigma} \right |= \frac{\omega_\Sigma}{c} n(\omega_{\Sigma})~.\end{aligned}$$ where $n(\omega_\Sigma) =\sqrt{\epsilon(\omega_\Sigma)} $ is the refractive index of the lossless media. We consider a linearly polarized incident laser with arbitrary incident angle exposed to the interface layer: $$\begin{aligned} {\vec E}_{\rm in} =\left\{ \hat{ x} {\cal E}_{x} + \hat{ y} {\cal E}_{y}+ \hat{ z} {\cal E}_{z} \right\} e^{i({\vec q}\cdot {\vec r} - \omega_0 t)} +c.c.\end{aligned}$$ where $$\begin{aligned} {\vec q}= \frac{\omega_0}{c} n_{1}(\omega_0) [-\cos\theta \hat{ z}+\sin\theta \hat{ x}]~.\end{aligned}$$ The leading nonlinearity of SLG is encoded in a third-order conductivity tensor, $\overset\leftrightarrow {\sigma}^{(3)}$. Using the SLG symmetry, the third-order nonlinear polarization follows: $$\begin{aligned} {\vec {\mathcal P}}^{(3)} =\vec {\widetilde{ \mathcal {P}}}^{(3)} \exp \left\{ i\frac{3\omega_0}{c} \left [n_{1}(\omega_0) x \sin\theta -c t \right ]\right\} +c.c.\end{aligned}$$ where $$\begin{aligned} \label{eq:p3} &\widetilde{\cal P}^{(3)}_x =\frac{i}{3\omega_0}\sigma^{(3)}_{xxxx} \left \{ {\cal E}^3_{x} +{\cal E}_{x} {\cal E}^2_{y}\right \}~, \nonumber \\ &\widetilde{\cal P}^{(3)}_y =\frac{i}{3\omega_0}\sigma^{(3)}_{xxxx} \left \{ {\cal E}^3_{y} + {\cal E}_{y} {\cal E}^2_{x}\right \} ~, \nonumber \\ &\widetilde{\cal P}^{(3)}_z=0~.\end{aligned}$$ The wave-vectors of TH radiated waves in the top and bottom media are then: $$\begin{aligned} \label{eq:q3tb} &{\vec q}_{3\omega_0,1}=\frac{3\omega_0}{c} n_{1}(3\omega_0) [\cos\theta_{1}\hat{ z}+\sin\theta_{1}\hat{ x}]~, \nonumber \\ &{\vec q}_{3\omega_0,2}= \frac{3\omega_0}{c} n_{2}(3\omega_0) [-\cos\theta_{2}\hat{ z}+\sin\theta_{2}\hat{ x}]~.\end{aligned}$$ According to the boundary condition relations of Eq.\[eq:BC\], we find $q_{3\omega_0,1,x}=q_{3\omega_0,2,x}=3q_x$. Therefore, we derive the Snell’s law for THG: $$\begin{aligned} \label{eq:snell_like} n_{2}(3\omega_0)\sin\theta_{2} =n_{1}(3\omega_0)\sin\theta_{1}= n_{1}(\omega_0)\sin\theta~.\end{aligned}$$ Considering frequency dependence of the refractive indexes, the Snell’s law for THG implies that $\sin\theta_1=[ n_1(\omega_0)/n_1(3\omega_0)] \sin\theta$ is not generally equal to $\sin\theta$. This is in contrast with the specular reflection for first harmonic generation[@Jackson_book]. The plane wave nature of the TH radiations implies: $$\begin{aligned} \label{eq:plane_wave} \cos\theta_{1} {\cal E}^{(3)}_{1,z}+\sin\theta_{1} {\cal E}^{(3)}_{1,x}=0~,\nonumber \\ -\cos\theta_{2} {\cal E}^{(3)}_{2,z}+\sin\theta_{2} {\cal E}^{(3)}_{2,x}=0~.\end{aligned}$$ By considering Eqs.\[eq:p3\],\[eq:q3tb\], the boundary condition relations Eqs.\[eq:BC\] become: $$\begin{aligned} \label{eq:bc_1} & n_{1}(3\omega_0) \left [\cos\theta_{1} {\cal E}^{(3)}_{1,x} - \sin\theta_{1} {\cal E}^{(3)}_{1,z} \right ] + \nonumber\\ &n_{2}(3\omega_0) \left [\cos\theta_{2} {\cal E}^{(3)}_{2,x} +\sin\theta_{2} {\cal E}^{(3)}_{2,z} \right ] =i \frac{3\omega_0}{c} \frac{\widetilde{\cal P}_x}{\epsilon_0}~, \\ \label{eq:bc_2}& n_{1}(3\omega_0) \cos\theta_{1} {\cal E}^{(3)}_{1,y} + n_{2}(3\omega_0) \cos\theta_{2} {\cal E}^{(3)}_{2,y} = -i \frac{3\omega_0}{c} \frac{\widetilde{\cal P}_y}{\epsilon_0}~, \\ \label{eq:bc_3}& n_{1}(3\omega_0) \sin\theta_{1} {\cal E}^{(3)}_{1,y} - n_{2}(3\omega_0) \sin\theta_{2} {\cal E}^{(3)}_{2,y} = 0~, \\ \label{eq:bc_4}&{\cal E}^{(3)}_{1,x} = {\cal E}^{(3)}_{2,x}~, \\ \label{eq:bc_5}&{\cal E}^{(3)}_{2,y} = {\cal E}^{(3)}_{2,y} ~, \\ \label{eq:bc_6}& n_{1}(3\omega_0)^2{\cal E}^{(3)}_{1,z} - n_{2} (3\omega_0)^2{\cal E}^{(3)}_{2,z}= - i \frac{3\omega_0}{c} \frac{\widetilde{\cal P}_x}{\epsilon_0} n_{1}(\omega_0)\sin\theta~.\end{aligned}$$ From Eqs.\[eq:bc\_1\]-\[eq:bc\_6\],\[eq:snell\_like\],\[eq:plane\_wave\] we get: $$\begin{aligned} &{\cal E}^{(3)}_{i,x}= S_{i,x} \frac{\sigma^{(3)}_{xxxx} }{c\epsilon_0} \left \{ {\cal E}^3_{x} +{\cal E}_{x} {\cal E}^2_{y}\right \} ~,\\ &{\cal E}^{(3)}_{i,y}= S_{i,y} \frac{\sigma^{(3)}_{xxxx} }{c\epsilon_0} \left \{ {\cal E}^3_{y} +{\cal E}_{y} {\cal E}^2_{x}\right \} ~,\\ &{\cal E}^{(3)}_{i,z}= S_{i,z} \frac{\sigma^{(3)}_{xxxx} }{c\epsilon_0} \left \{ {\cal E}^3_{x} +{\cal E}_{x} {\cal E}^2_{y}\right \} ~.\end{aligned}$$ where $$\begin{aligned} &S_{1,x}=S_{2,x}=- \frac{\cos\theta_{1}\cos\theta_{2}}{n_{1}(3\omega_0)\cos\theta_{2}+n_{2}(3\omega_0)\cos\theta_{1}} ~,\\ &S_{1,y}=S_{2,y}=-\frac{1}{n_{1}(3\omega_0)\cos\theta_{2}+n_{2}(3\omega_0)\cos\theta_{1}} ~,\\ &S_{1,z}= \frac{\cos\theta_{2}\sin\theta_{1}}{n_{1}(3\omega_0)\cos\theta_{2}+n_{2}(3\omega_0)\cos\theta_{1}} ~,\\ &S_{2,z}= -\frac{\cos\theta_{1}\sin\theta_{2}}{n_{1}(3\omega_0)\cos\theta_{2}+n_{2}(3\omega_0)\cos\theta_{1}}~.\end{aligned}$$ For normal incidence we have $\theta=0$. From Eq.\[eq:snell\_like\] we have $\theta_1=\theta_2=0$. Therefore, $S_{i,z}=0$ and $S_{i,x}=S_{i,y}=-1/[n_{1}(3\omega_0) +n_{2}(3\omega_0)]$. The time-average of the incident intensity gives $I_{\omega_0} = 2 n_1(\omega_0) \epsilon_0 c |{\vec E}_{\rm in}|^2 $. The intensity of the transmitted TH signal is $I_{3\omega_0} = 2 n_2(3\omega_0) \epsilon_0 c |{\vec E}^{(3)}|^2$. From this we get Eq.\[eq:THG\_efficiency\] for THGE. ### Symmetry considerations {#subss:symmetry} The rank-4 tensor of $\sigma^{(3)}$ transforms as follows under an arbitrary $\phi$-rotation: $$\begin{aligned} \label{eq:sym} \sigma^{(3)}_{\alpha'\beta'\gamma'\delta'}=\sum_{\alpha\beta\gamma} R_{\alpha'\alpha}(\phi) R_{\beta'\beta}(\phi) R_{\gamma'\gamma}(\phi) R_{\delta'\delta}(\phi) \sigma^{(3)}_{\alpha\beta\gamma\delta}~.\end{aligned}$$ We take the $z$-axis as the rotation-axis, perpendicular SLG. Therefore, the rotation tensor is: $$\label{eq:R} \overset\leftrightarrow R(\phi) = \begin{pmatrix} \cos\phi &\sin\phi \\ -\sin\phi &\cos\phi \end{pmatrix}~.$$ We take $\hat{\bm \ell}= \overset\leftrightarrow R(\phi)\cdot\hat{\bm x}$. By plugging Eq.\[eq:R\] in \[eq:sym\], we get: $$\begin{aligned} \label{eq:rot} \sigma^{(3)}_{\ell\ell\ell\ell} & = [\sin\phi]^4 \sigma^{(3)}_{yyyy} +[\cos\phi]^4 \sigma^{(3)}_{xxxx} \nonumber\\& +\cos\phi [\sin\phi]^3 \left [ \sigma^{(3)}_{xyyy} +\sigma^{(3)}_{yxyy} +\sigma^{(3)}_{yyxy} +\sigma^{(3)}_{yyyx} \right ] \nonumber\\& +[\cos\phi]^3\sin\phi \left [\sigma^{(3)}_{xxxy}+\sigma^{(3)}_{xxyx}+\sigma^{(3)}_{xyxx}+\sigma^{(3)}_{yxxx} \right ] \nonumber\\& + [\cos\phi \sin\phi]^2 \big [\sigma^{(3)}_{xxyy} +\sigma^{(3)}_{xyxy}+\sigma^{(3)}_{xyyx} \nonumber\\& + \sigma^{(3)}_{yxxy}+ \sigma^{(3)}_{yxyx} +\sigma^{(3)}_{yyxx} \big].\end{aligned}$$ Because of the $C_{6v}$ symmetry for SLG on a substrate, there are only 4 independent tensor elements[@Boyd2003]: $$\begin{aligned} &\sigma^{(3)}_{xxxx}=\sigma^{(3)}_{yyyy}=\sigma^{(3)}_{xxyy}+\sigma^{(3)}_{xyyx}+\sigma^{(3)}_{xyxy} \nonumber\\& \sigma^{(3)}_{xxyy}=\sigma^{(3)}_{yyxx}, \nonumber\\& \sigma^{(3)}_{xyyx}=\sigma^{(3)}_{yxxy}, \nonumber\\& \sigma^{(3)}_{xyxy}=\sigma^{(3)}_{yxyx}.\end{aligned}$$ By implementing Eq.43 in Eq.\[eq:rot\], we get $\sigma^{(3)}_{\ell\ell\ell\ell}=\sigma^{(3)}_{xxxx}$. ### Effect of finite relaxation rate {#subss:Gamma} The effect of finite $\tau$ in the TH conductivity can be derived from[@ChengPRB2015]: $$\begin{aligned} \label{eq:clean} \bar{\sigma}^{(3)}_{xxxx} (\omega_0,E_{\rm F},0) & \approx & \frac{ 17 G(2|E_{\rm F}|,\hbar\omega_0+i\Gamma) - 64 G(2|E_{\rm F}|,2\hbar\omega_0+i\Gamma) + 45 G(2|E_{\rm F}|,3\hbar\omega_0+i\Gamma)}{24 (\hbar\omega_0)^4} \nonumber \\ &+&\frac{\Gamma}{6(\hbar\omega_0)^4} \Bigg \{ 17 \left [\frac{1}{2 | E_{\rm F} | +3 \hbar\omega_0 +i \Gamma }+\frac{1}{2 | E_{\rm F} | -3 \hbar\omega_0 -i \Gamma}\right ] \nonumber \\ &-& 8 \left [ \frac{1}{2 | E_{\rm F} | +2 \hbar\omega_0 +i \Gamma}+\frac{1}{2 | E_{\rm F} | -2 \hbar\omega_0 -i \Gamma}\right ] \nonumber\\ &+& 3 \hbar\omega_0 \left [\frac{1}{(2 | E_{\rm F} | +3 \hbar\omega_0 +i \Gamma )^2}-\frac{1}{(2 | E_{\rm F} | -3 \hbar\omega_0 -i \Gamma )^2}\right ] \Bigg \}~.\end{aligned}$$ Note that ($\approx$) is because we assume $\Gamma\ll \hbar\omega_0$[@ChengPRB2015]. Fig.\[fig:THGE\_tau\] shows that a finite $\tau$ has a small effect on THGE for most of the SLG in literature, as well as those in this paper. ![THGE for SLG on Sa as a function of $\omega_0$ for different $\tau=\hbar/\Gamma$ at constant $T_e$=2000K and $E_F$=200meV, for incident intensity$\sim2.4\times 10^{12}{\rm Wm}^{-2}$, corresponding to the value used in our experiments[]{data-label="fig:THGE_tau"}](fig_THGE_tau.eps){width="80mm"} ### $T_{\rm e}$ and $E_{\rm F}$ effects on THGE {#subss:THGE_TeandEf} ![\[fig:THGE\_Te\] $E_{\rm F}$ dependence of THGE for SLG on SiO$_2$ at $\hbar\omega_0=500{\rm meV}$ for different $T_{\rm e}$ between 0K and 1800K. (a) Absolute THGE. (b) THGE normalized to the minimum so that THGE at $E_{\rm F} = 0$ is equal to 1 for all $T_{\rm e}$.](fig_THGE_Te.eps){width="90mm"} ![\[fig:THGE\_xi\] $T_e$ dependence of $\xi^{\rm THG}$.](fig_THGE_xi.eps){width="80mm"} The $T_e$ and $E_F$ dependence of THGE for SLG on SiO$_2$ at $\hbar\omega_0=500$meV is shown in Fig.\[fig:THGE\_Te\], where 3 logarithmic singularities at $2|E_F|=\hbar\omega_0,2\hbar\omega_0, 3\hbar\omega_0$ for $T_e$=0K can be seen. By increasing $T_e$, the first peak at $2|E_F|=\hbar\omega_0$ disappears and the two others merge and form a broad maximum, roughly located at $2|E_{\rm F}|\sim (2+3)\hbar\omega_0/2=2.5\hbar\omega_0$. THGE is almost insensitive to $E_{\rm F}$ for $2|E_{\rm F}|<\hbar\omega_0$. This can be explained using the asymptotic relation of the TH conductivity for $|E_{\rm F}|\ll \hbar\omega_0 $. For $T_{\rm e}=0$: $$\begin{aligned} \label{eq:sigma3xxxx} \sigma^{(3)}_{xxxx} \approx \frac{e^4 \hbar v^2_{\rm F}}{(\hbar\omega_0)^4} \left \{ \frac{1}{96} + \frac{i}{\pi} \left(\frac{2|E_{\rm F}|}{3\hbar\omega_0} \right)^3+\dots \right \}\end{aligned}$$ Eqs.\[eq:sigma3xxxx\],\[eq:THG\_efficiency\] explain the flat part of the curves in Fig.\[fig:THGE\_Te\] in the low-doping regime ($\hbar\omega_0>$2$|E_F|$). In order to quantify the tuneability of THG in SLG by altering $E_F$, we define a parameter: $$\begin{aligned} \xi^{\rm THG} \equiv \frac{\eta^{\rm THG}_{\rm max} }{\eta^{\rm THG}_{\rm min}}~,\end{aligned}$$ where $\eta^{THG}_{min}$ stands for THGE in the nearly undoped regime ($|E_F|\ll \hbar\omega_0$). Fig.\[fig:THGE\_xi\] indicates that $\xi^{THG}$ decreases by increasing $T_e$. Fermi energy, Fermi level, chemical potential and the estimation of $T_e$ in photoexcited SLG {#subs:Te} --------------------------------------------------------------------------------------------- ![\[fig:cv\] Numerical calculation of $c_v$ for (a) $E_F$=10 and (b) 300meV. The blue and red dashed lines are Eqs.\[eq:cv\_undoped\], \[eq:cv\_doped\].](fig_cv_doping.eps){width="90mm"} When a pulsed laser interacts with SLG, for several hundred fs after the pump pulse, the electron and hole distributions in valence and conduction bands are given by the Fermi-Dirac functions $f_{\rm FD}(\varepsilon; \mu_{\lambda}, T_{\rm e})$ with the same $T_{\rm e}$ and two chemical potentials $\mu_{\rm v}$ and $\mu_{\rm c}$. By definition $\mu_{\rm c}=-\mu_{\rm v}$. The term Fermi level ($E_{\rm FL}$) is sometimes used in literature to denote $\mu$. The Fermi energy ($E_{\rm F}$) is defined as the value of $\mu$ at $T_{\rm e}=0K$. $E_{\rm F}$ is thus a function of the electron density only. Only after the recombination of the photoexcited electron-hole pairs, a single Fermi-Dirac distribution is established in both bands and the equilibrium condition $\mu_{\rm v} =\mu_{\rm c}$ holds. The recombination time depends on carrier density and laser fluence, and can be much longer than the time$\lesssim 20fs$ needed for thermalization (see Ref. and references therein). $N_{\rm f}=4$ is the product of spin and valley degeneracy; $\nu(\varepsilon)=N_{\rm f}|\varepsilon|/[2 \pi (\hbar v_{\rm F})^{2}]$ the density of electronic states per unit of area. The electronic heat capacity $c_{\rm v}$ is defined as the derivative of the electronic energy density $U$ with respect to $T_{\rm e}$. This quantity depends on all the variables which affect the electronic energy density, such as $T_{\rm e}$ and the carrier density, or equivalently $\mu_{\rm c}$ and $\mu_{\rm v}$. In a photoexcited system, in general, $c_{\rm v}$ depends on both the electron and hole densities, *i.e.* on both $\mu_{\rm c}$ and $\mu_{\rm v}$. In this case, $c_{\rm v}$ can be written as[@Grosso_book]: $$\begin{aligned} \label{eq:cv_full} c_{\rm v}(\mu_{\rm c}, \mu_{\rm v}, T_{\rm e}) &= \frac{\partial}{\partial T_{\rm e}} \int_{0}^{\infty} d\varepsilon \nu(\varepsilon) \varepsilon f_{\rm FD}(\varepsilon; \mu_{\rm c}, T_{\rm e}) \nonumber\\ &+ \frac{\partial}{\partial T_{\rm e}} \int_{0}^{\infty} d\varepsilon \nu(\varepsilon) \varepsilon f_{\rm FD}(\varepsilon; -\mu_{\rm v}, T_{\rm e})~,\end{aligned}$$ where the first integral is the electron and the second the hole contribution, and the Fermi-Dirac distribution is $$f_{\rm FD}(\varepsilon; \mu, T_{\rm e}) = \frac{1}{e^{(\varepsilon-\mu) / (k_{\rm B} T_{\rm e})} + 1}~. \label{eq:fFD}$$ To take the $T_{\rm e}$ derivative in Eq.\[eq:cv\_full\], the $T_{\rm e}$ dependence of $c_{\rm v}$ has to be specified. The electron and hole densities are given by: $$\begin{aligned} &n_{\rm e}(\mu_{\rm c}, T_{\rm e}) = \int_{0}^{\infty} d\varepsilon \nu(\varepsilon) f_{\rm FD}(\varepsilon; \mu_{\rm c}, T_{\rm e}), \quad \nonumber\\ &n_{\rm h}(-\mu_{\rm v}, T_{\rm e}) = \int_{0}^{\infty} d\varepsilon \nu(\varepsilon) f_{\rm FD}(\varepsilon; -\mu_{\rm v}, T_{\rm e})~. \label{eq:nEnH}\end{aligned}$$ Since the total electron density in both bands is constant, the difference between electron and hole densities is constant: $$\label{eq:cons_electr} n_{\rm e}^{(0)} - n_{\rm h}^{(0)} = n_{\rm e}(\mu_{\rm c}, T_{\rm e}) - n_{\rm h}(-\mu_{\rm v}, T_{\rm e})~,$$ where $n_{\rm e}^{(0)}$ and $n_{\rm h}^{(0)}$ are the electron and hole densities. At equilibrium, when $\mu_{\rm c}=\mu_{\rm v}=\mu$, Eq.\[eq:cons\_electr\] can be solved for $\mu$. A photoexcited density $\delta n_{\rm e}$ changes the densities in both bands as follows: $$\begin{aligned} \label{eq:pe_dens} &n_{\rm e}(\mu_{\rm c}, T_{\rm e}) = n_{\rm e}(\mu, T_{\rm e}) + \delta n_{\rm e}, \quad \nonumber\\ &n_{\rm h}(-\mu_{\rm v}, T_{\rm e}) = n_{\rm h}(-\mu, T_{\rm e}) + \delta n_{\rm e}~.\end{aligned}$$ After finding $\mu$ with Eq.\[eq:cons\_electr\], one can get $\mu_{\rm c}$ and $\mu_{\rm v}$ with Eq.\[eq:pe\_dens\]. This defines the $c_{\rm v}$ dependence of $T_{\rm e}$ in Eq.\[eq:cv\_full\], and allows to calculate the temperature derivative. The result of Eq.\[eq:cv\_full\] is shown in Fig.\[fig:cv\] for $\mu_{\rm c}=\mu_{\rm v}=\mu$. In Ref. the following expression is given for $c_{\rm v}$: $$\label{eq:cv_undoped} c_{\rm v}(T_{\rm e}) = \frac{18 \zeta(3)}{\pi (\hbar v_{\rm F})^{2}} k_{\rm B}^{3} T_{\rm e}^{2}~.$$ In principle, as noted in Ref., Eq.\[eq:cv\_undoped\] is valid at the charge neutrality point $|\mu| \ll k_{\rm B} T$ only. For a degenerate system, $k_{\rm B} T \ll |\mu|$, we have[@Vignale_book]: $$\label{eq:cv_doped} c_{\rm v}(\mu, T_{\rm e}) = \frac{\pi^{2}}{3} \nu(E_{\rm F}) k_{\rm B}^{2} T_{\rm e}~,$$ as derived *e.g.* in Eqs.8.10 of Ref., in Eq.4 of Ref. and in Eq.8 in the Supplementary Information of Ref.. However, the numerical calculation in Fig.\[fig:cv\] shows that the quadratic approximation (Eq.\[eq:cv\_undoped\]) is much better in the regime where $T_{\rm e}\sim 1000K$ and $\mu \sim 100$meV. Fig.\[fig:cv\_pe\] shows that, taking into consideration the difference between $\mu_c$ and $\mu_v$, for typical values of the photoexcited density, contributes$\gtrsim 15\%$ to $c_v$. ![\[fig:cv\_pe\] Numerical calculation of (a) electron energy density and (b) $c_v$ for $E_F$=200meV. The blue, and red lines correspond to photoexcited densities $\delta n_e= 10^{12}~{cm}^{-2}$ and $\delta n_e=3 \times 10^{12}~{cm}^{-2}$, while the black line corresponds to a thermalized system with a single $\mu$](fig_U_and_cv.eps){width="90mm"} The number of photoexcited electron-hole pairs per unit area in the time interval $dt$ is given by the number of absorbed photons in the same time interval per unit area, i.e. $(d n_{\rm e} + d n_{\rm h})/2 = (P / A) / (\hbar \omega_0) d t $. In the steady state, the energy delivered by the pump is transferred into the phonon modes. Hence, we identify the electron-hole recombination time with $\tau$. We then get: $$\begin{aligned} &\frac{1}{2} \left ( \frac{d n_{\rm e}}{dt} + \frac{d n_{\rm h}}{dt} \right ) = \frac{1}{\hbar \omega_0} \frac{P}{A} \nonumber\\ &- \frac{1}{2}\frac{[n_{\rm e}(\mu_{\rm c}, T_{\rm e}) + n_{\rm h}(-\mu_{\rm v}, T_{\rm e})] - (n_{\rm e}^{(0)} + n_{\rm h}^{(0)})}{\tau}~.\end{aligned}$$ In the steady state this becomes: $$\label{eq:e_h_pairs} n_{\rm e}^{(0)} + n_{\rm h}^{(0)} = n_{\rm e}(\mu_{\rm c}, T_{\rm e}) + n_{\rm h}(-\mu_{\rm v}, T_{\rm e}) - \frac{2 \tau}{\hbar \omega_0} \frac{P}{A}.$$ Combining Eqs.\[eq:cons\_electr\],\[eq:e\_h\_pairs\], we find: $$\delta n_{\rm e} = \frac{\tau}{\hbar \omega_0} \frac{P}{A}~.$$ To calculate $E_F$ (e.g. for a $n$-doped sample) one needs to solve Eqs.\[eq:fFD\], \[eq:nEnH\], \[eq:cons\_electr\] with $\mu_{\rm c}=\mu_{\rm v}=E_F$, $T_e=0$, and $n_{h}^{(0)}=0$, finding $E_F=\hbar v_{\rm F} \sqrt{\pi n_{\rm e}}$. This relation can be safely used at $T_{\rm e}=300K$ and electron densities $n_{\rm e}^{(0)} \gtrsim 10^{11}$ because the density of thermally excited holes is negligible. Indeed, experimental measurements of the carrier density of $n$-doped SLG at room temperature routinely neglect the hole population contributions. In a photoexcited SLG, even after the recombination of the photoexcited electron-hole pairs, the $T_e$ dependence of $\mu$ cannot be ignored. In this case, to calculate $\mu$, one needs to solve Eqs.\[eq:fFD\],\[eq:nEnH\], \[eq:cons\_electr\] with $\mu_{\rm c}= \mu_{\rm v}=\mu$ and $n_{\rm h}^{(0)}=0$ as a function of $T_{\rm e}$. This gives $\mu=E_F [1 - \pi^{2} T_{\rm e}^{2} / (6 T_{\rm F}^{2})]$ for $T_{\rm e}\lesssim T_{\rm F}$ and $\mu =E_F T_{\rm F}/ (4 {\rm ln}2 \times T_{\rm e})$ for $T_{\rm e}\gtrsim T_{\rm F}$[@hwang_prb_2009], where $T_{\rm F}=E_{\rm F}/K_{\rm B}$, with K$_{\rm B}$ the Boltzmann constant. For a typical case of $E_F$=200meV and $T_e=$1500K, we have $\mu \sim 0.3-0.5 E_F$. \[Ackn\]Acknowledgements {#acknacknowledgements .unnumbered} ======================== We acknowledge funding from EU Graphene Flagship, ERC Grant Hetero2D, EPSRC Grants EP/K01711X/1, EP/K017144/1, EP/N010345/1, and EP/L016087/1. [100]{} Y. R. Shen, *The principles of nonlinear optics* (John Wiley & Sons, New York, 1984). The second and third order nonlinear optical susceptibilities are often reported in electrostatic units (esu). The relation between esu and standard units (SI) is[@Butc1991]: $$\frac{\chi^{n}(SI)}{\chi^{n}(esu)} = \frac{4 \pi}{(10^{-4}c)^{n-1}}\nonumber$$ where $c=3\times 10^{8}$. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. **7**, 118 (1961). R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, Appl. Phys. Lett. **24**, 308 (1974). J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. **127**, 1918 (1962). G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, Science **286**, 1507 (1999). J. J. Chang, B. E. Warner, E. P. Dragon, and M. W. Martinez, J. Laser Appl. **10**, 285 (1998). E. Garmire, Opt. Express **21**, 30532 (2013). G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, Opt. Lett. **22**, 1834 (1997). G. Cerullo, and S. De Silvestri, Rev. Sci. Instrum. **74**, 1 (2003). W. R. Bosenberg, L. E. Myers, R. L. Byer, A. Drobshoff, and J. I. Alexander, Opt. Lett. **21**, 1336 (1996). P. Corkum, Phys. Rev. Lett. **71**, 1994 (1993). P. B. Corkum, and F. Krausz, Nat. Phys. **3**, 381 (2007). B. Ferguson, and X.-C. Zhang, Nat. Mater. **1**, 26 (2002). R. W. Boyd, *Nonlinear Optics* (Academic Press, London, 2008). M. R. Shcherbakov *et al.*, Nano. Lett. **14**, 6488 (2014). R. Chen, D. L. Lin and B. Mendoza, Phys. Rev. B **48**, 11879 (1993). T. Y. F. Tsang, Opt. Lett. **21**, 245 (1996). B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain and T. F. Krauss, Nat. Photon. **3**, 206 (2009). K. L. Seyler, J. R.Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan,D. G. Mandrus, W. Yao and X. Xu, Nat. Nanotech. **10**, 407 (2015). E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko and S. A. Mikhailov, Phys. Rev. Lett. **105**, 097401 (2010) G. Wang, X. Marie, I. Gerber, T. Amand, D. Lagarde, L. Bouet, M. Vidal, A. Balocchi, and B. Urbaszek, Phys. Rev. Lett. **114**, 097403 (2015). H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, Nat. Phys. **13**, 262 (2017). A. Saynatjoki, L. Karvonen, H. Rostami, A. Autere, S. Mehravar, A. Lombardo, R. A. Norwood, T. Hasan, N. Peyghambarian, H. Lipsanen, K. Kieu, A. C. Ferrari, M. Polini, Z. Sun, Nat Comms **8** DOI: 10.1038/s41467-017-00749-4 (2017) J. Klein, J. Wierzbowski, A. Steinhoff, M. Florian, M. Rösner, F. Heimbach, K. Mueller, F. Jahnke, T. O. Wehling, J. J. Finley, and M. Kaniber, Nano Lett. **17**, 392 (2017). Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, ACS Nano **4**, 803 (2010). F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photon. **4**, 611 (2010). S. A. Mikhailov, Phys. Rev. B **84**, 045432 (2011). J. J. Dean, and H. M. van Driel, Phys. Rev. B **82**, 125411 (2010). Y. Q. An, F. Nelson, J. U. Lee, and A. C. Diebold, Nano Lett. **13**, 2104 (2013). Y. Q. An, J. E. Rowe, D. B. Dougherty, J. U. Lee, and A. C. Diebold, Phys. Rev. B **89**, 115310 (2014). J. L. Cheng, N. Vermeulen and J. E. Sipe, New J. Phys. **16**, 053014 (2014). J. L. Cheng, N. Vermeulen and J. E. Sipe, Phys. Rev. B **91**, 235320 (2015); S. A. Mikhailov, Phys. Rev. B **93**, 085403 (2016). H. Rostami and M. Polini, Phys. Rev. B **93**, 161411 (2016). H. Rostami, M. I. Katsnelson and M. Polini, Phys. Rev. B **95**, 035416 (2017). N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H.-Y. Chiu, A. L. Smirl and H. Zhao, Phys. Rev. B **87**, 121406(R) (2013). K. Alexander, B. Kuyken, D. Van Thourhout, N. A. Savostianova and S. A. Mikhailov, arXiv:1704.08567 \[physics.optics\] (2017). S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone and R. M. Osgood, Jr., Phys. Rev. X **3**, 021014 (2013). A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. **81**, 109 (2009). C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, Nano Lett. **7**, 2711 (2007). By definition, $\mu_e=\sigma/(n e)$ where $n$ is the carrier density and $\sigma=D/(\pi\Gamma)$ is the Drude conductivity of graphene in which $D=e^2E_{\rm F}/\hbar$[@Abedinpour_prb_2011] is the Drude weight. From Eq.5 we get $n\pi=[E_{\rm F}/(\hbar v_{\rm F})]^2$, $\Gamma=e\hbar v^2_{\rm F}/(\mu_e E_{\rm F})$. Note that this relation is not valid for the undoped limit when $E_{\rm F}/\Gamma \to 0$, where inter-band effects are dominant and the Drude modeling is not sufficient. Sa sourced from UQG Optics, http://www.uqgoptics.com. This gives a transmission$\sim$85% between$\sim$400nm and $\sim$5$\mu$m. The dielectric constant of sapphire is 11.5/9.4 when parallel/perpendicular to the Sa C-axis. Here we use$\sim$10. A. Saeynaetjoki, L. Karvonen, H. Rostami, A. Autere, S. Mehravar, A. Lombardo, R. A. Norwood, T. Hasan, N. Peyghambarian, H. Lipsanen, K. Kieu, A. C. Ferrari, M. Polini, and Z. Sun, Nature Comms (2017). D. Brida, A. Tomadin, C. Manzoni, Y.J. Kim, A. Lombardo, S. Milana, R.R. Nair, K.S. Novoselov, A.C. Ferrari, G. Cerullo, and M. Polini, Nature Comms. **4**, 1987 (2013) M. Lazzeri, S. Piscanec, F. Mauri, A.C. Ferrari, and J. Robertson, Phys. Rev. Lett. 95, 236802 (2005) S. Piscanec, M. Lazzeri, F. Mauri, A.C. Ferrari, and J. Robertson, Phys. Rev. Lett. 93, 185503 (2004) K. F. Mak, M. Y. Sfeir,2Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. **101**, 196405 (2008). R. Kronig, J. Opt. Soc. Am. **12**, 547 (1926). X. S. Li, W. W. Cai, J. H. An, S. Kim, J. Nah, D. X. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc *et al.*, Science **324**, 1312 (2009). A. C. Ferrari and D. M. Basko, Nat. Nanotechnol. **8**, 235 (2013). A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth and A. K. Geim, Phys. Rev. Lett. **97**, (2006). L. G. Cancado, A. Jorio, E. H. Ferreira, F. Stavale, C. A. Achete, R. B. Capaz, M. V. Moutinho, A. Lombardo, T. S. Kulmala and A. C. Ferrari, Nano Lett. **11**, 3190 (2011). M. Bruna, A. K. Ott, M. Ijäs, D. Yoon, U. Sassi, and A. C. Ferrari, ACS Nano **8**, 7432 (2014). F. Bonaccorso, A. Lombardo, T. Hasan, Z. Sun, L. Colombo, and A. C. Ferrari, Mater. Today **15**, 564 (2012). A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S. K. Saha, U. V. Waghmare, K. S. Novoselov, H. R. Krishnamurthy, A. K. Geim, A. C. Ferrari *et al.*, Nat. Nanotechnol. **3**, 210 (2008) D. M. Basko, S. Piscanec and A. C. Ferrari, Phys. Rev. B **80**, 165413 (2009). K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Proc. Natl. Acad. Sci. U.S.A **102**, 10451 (2005). J. Xia, F. Chen, J. Li2, and N. Tao, Nat. Nanotechnol. **4**, 505 (2009). V. Dominic, S. MacCormack, R. Waarts, S. Sanders, S. Bicknese, R. Dohle, E. Wolak, P.S. Yeh and E. Zucker, Electron. Lett. **35**, 1158 (1999). Symphony II 1024 x 256 Cryogenic Open-Electrode CCD Detector Quantum Efficiency and grating (300 gr/mm , Blazed 600 nm 510 19 140) relative efficiency from user manual and specifications available from Horiba website. Y. R. Shen, Annu. Rev . Phys. Chem. **40**, 327 (1989). J. D. Jackson, *Classical Electrodynamics* (John Wiley & Sons, New York, 1998). A. Tomadin, D. Brida, G. Cerullo, A.C. Ferrari, and M. Polini, Phys. Rev. B [**88**]{}, 035430 (2013). G. Grosso, and G. Pastori Parravicini, *Solid State Physics* (Academic Press, Oxford, 2000) C.H. Lui, K.F. Mak, J. Shan, and T.F. Heinz, Phys. Rev. Lett. [**105**]{}, 127404 (2010). D. Sun, G. Aivazian, A.M. Jones, J.S. Ross, W. Yao, D. Cobden, and X. Xu, Nature Nanotech. [**7**]{}, 114 (2012). G. F. Giuliani and G. Vignale, *Quantum Theory of the Electron Liquid* (Cambridge University Press, Cambridge, 2005). T. Low, V. Perebeinos, R. Kim, M. Freitag, and P. Avouris, Phys. Rev. B [**86**]{}, 045413 (2012). K.J. Tielrooij, J.C.W. Song, S.A. Jensen, A. Centeno, A. Pesquera, A. Zurutuza Elorza, M. Bonn, L.S. Levitov, and F.H.L. Koppens, Nat. Phys. [**9**]{}, 248 (2013). E.H. Hwang and S. Das Sarma, Phys. Rev. B 79, 165404 (2009). P. N. Butcher, and D. Cotter, *The Elements of Nonlinear Optics* (Cambridge university press, London, 1991). S. H. Abedinpour, G. Vignale, A. Principi, M. Polini, W.-K. Tse, and A. H. MacDonald, Phys. Rev. B **84**, 045429 (2011).

--- abstract: 'We study some properties of coisotropic submanifolds of a manifold with respect to a given multivector field. Using this notion, we generalize the results of Weinstein [@wein] from Poisson bivector field to Nambu-Poisson tensor or more generally to any multivector field. We also introduce the notion of Nambu-Lie groupoid generalizing the concepts of both Poisson-Lie groupoid and Nambu-Lie group. We show that the infinitesimal version of Nambu-Lie groupoid is the notion of weak Lie-Filippov bialgebroid as introduced in [@bas-bas-das-muk]. Next we introduce coisotropic subgroupoids of a Nambu-Lie groupoid and these subgroupoids corresponds to, so called coisotropic subalgebroids of the corresponding weak Lie-Filippov bialgebroid.' address: 'Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, West Bengal, India.' author: - Apurba Das title: - - Multiplicative Nambu structures on Lie groupoids --- Introduction ============ In [@wein], Weinstein introduced the notion of coisotropic submanifold of a Poisson manifold generalizing the notion of Lagrangian submanifold of symplectic manifold. A submanifold $C$ of a Poisson manifold $(P, \pi)$ is called coisotropic, if $\pi^{\sharp}(TC)^0 \subset TC,$ or, equivalently $\pi (\alpha, \beta) = 0$, for all $ \alpha, \beta \in (TC)^0$, where $(TC)^0$ is the conormal bundle of $C$. Moreover, Weinstein proved the following results. 1. A map $\phi : P_1 \rightarrow P_2$ between Poisson manifolds is a Poisson map if and only if its graph is a coisotropic submanifold of $P_1 \times P_2^{-},$ where $P_2^{-}$ stands for the manifold $P_2$ with opposite Poisson structure. 2. If $\phi : P \rightarrow Q$ is a surjective submersion from a Poisson manifold $P$ to some manifold $Q$, then $Q$ has a Poisson structure for which $\phi$ is a Poisson map if and only if $$\{(x,y)| \phi(x) = \phi(y)\} \subset P \times P$$ is a coisotropic submanifold of $P \times P^{-}.$ To define the coisotropic submanifold of a Poisson manifold, one does not require the Poisson tensor to be closed, that is, $[\pi, \pi] = 0,$ where $[~,~]$ denotes the Schouten bracket on multivector fields. Therefore, the notion of coisotropic submanifolds make sense for any bivector field, or more generally, for any multivector field. Explicitly, if $M$ is a smooth manifold and $\Pi \in \mathcal{X}^n(M) = \Gamma \bigwedge^n TM$ be an $n$-vector field on $M$, then a submanifold $C \hookrightarrow M$ is called coisotropic with respect to $\Pi$ if $$\Pi^{\sharp} ({\bigwedge}^{n-1}(TC)^0) \subset TC ~~\Leftrightarrow ~~\Pi (\alpha_1, \ldots, \alpha_n) = 0,~~ \mbox{for all}~~ \alpha_1, \ldots, \alpha_n \in (TC)^0,$$ where $\Pi^\sharp : \bigwedge^{n-1}T^*M \rightarrow TM$ is the bundle map induced by $\Pi$. Nambu-Poisson manifolds are generalization of Poisson manifolds. Recall that a Nambu-Poisson manifold of order $n$ is a manifold $M$ equipped with an $n$-vector field $\Pi$ such that the induced bracket on functions satisfies the Fundamental identity (Definition \[nambu-poisson\]). The $n$-vector field $\Pi$ of a Nambu-Poisson manifold is referred to as the associated Nambu tensor. Coisotropic submanifolds of a Nambu-Poisson manifold $M$ are those submanifolds which are coisotropic with respect to the Nambu tensor $\Pi$. In the present paper, we study some basic properties of coisotropic submanifolds of a manifold with respect to a given multivector field and generalize the results of Weinstein to the case of multivector field. More precisely, we prove the following results (Propositions \[nambu map-coiso\] and \[coinduced-coiso\]). 1. Let $(M, \Pi_M)$ and $(N, \Pi_N)$ be two manifolds with $n$-vector fields and $\phi: M \rightarrow N$ be a smooth map. Then $\phi_* \Pi_M = \Pi_N$ if and only if its graph $$\text{Gr}(\phi) := \{(m, \phi(m))| m \in M\}$$ is a coisotropic submanifold of $M \times N$ with resoect to $\Pi_M \oplus (-1)^{n-1} \Pi_N$. 2. Let $(M, \Pi_M)$ be a manifold with an $n$-vector field and $\phi: M \rightarrow N$ be a surjective submersion. Then $N$ has an (unique) $n$-vector field $\Pi_N$ such that $\phi_* \Pi_M = \Pi_N$ if and only if $$R(\phi) := \{(x,y) \in M \times M|\phi(x)=\phi(y)\}$$ is a coisotropic submanifold of $M \times M$ with respect to $\Pi_M \oplus (-1)^{n-1}{\Pi}_M$. Poisson Lie group is a Lie group equipped with a Poisson structure such that the group multiplication map is a Poisson map. Equivalently, a Lie group equipped with a Poisson structure is a Poisson Lie group if the Poisson bivector field is multiplicative [@lu-wein]. These definitions have no natural extension when one wants to define Poisson groupoid. Nevertheless, the notion of coisotropic submanifolds of Poisson manifolds was used by Weinstein [@wein] to introduce the notion of Poisson groupoid. Recall that a Poisson groupoid is a Lie groupoid $G \rightrightarrows M$ with a Poisson structure on $G$ such that the graph of the groupoid (partial) multiplication map is a coisotropic submanifold of $G \times G \times G^{-}.$ In [@xu], P. Xu gave an equivalent formulation of Poisson groupoid which generalizes the multiplicativity condition for Poisson Lie group. More generally, In [@pont-geng-xu], the authors introduced the notion of multiplicative multivector fields on a Lie groupoid. Given a Lie groupoid $G \rightrightarrows M$, an $n$-vector field $\Pi \in \mathcal{X}^{n}(G)$ is called multiplicative, if the graph of the groupoid multiplication is a coisotropic submanifold of $G \times G \times G$ with respect to $\Pi \oplus \Pi \oplus (-1)^{n-1} \Pi$. In this terminology, a Poisson groupoid is a Lie groupoid equipped with a multiplicative Poisson tensor. In the present paper, we extend this approach to the case of Lie groupoid with a Nambu structure. We introduce the notion of a Nambu-Lie groupoid as a Lie groupoid with a Nambu structure $\Pi$ such that the Nambu tensor $\Pi$ is multiplicative (Definition \[nambu-lie groupoid\]). When $G$ is a Lie group, this definition coincides with the definition of Nambu-Lie group given by Vaisman [@vais]. Using results proved in [@pont-geng-xu] for mutiplicative multivector fields on Lie groupoid, we deduce the following facts which are parallel to the case of Poisson groupoid. Suppose $(G \rightrightarrows M, \Pi)$ is a Nambu-Lie groupoid, then 1. $M \hookrightarrow G$ is a coisotropic submanifold of $G$; 2. the groupoid inversion map $i: G \rightarrow G$ is an anti Nambu-Poisson map; 3. there is a unique Nambu-Poisson structure $\Pi_M$ on $M$ for which the source map is a Nambu-Poisson map (Proposition \[inverse-basenambu\]). It is well known that for a Nambu-Poisson manifold $M$ of order $n$, the space of $1$-forms admits a skew-symmetric $n$-bracket which satisfies the Fundamental identity modulo some restriction ([@vais; @gra-mar; @bas-bas-das-muk]). Moreover, the bracket on forms and the de-Rham differential of the manifold satisfy a compatibility condition similar to that of a Lie bialgebroid. This motivates the authors [@bas-bas-das-muk] to introduce a notion of weak Lie-Filippov bialgebroid of order $n$. If $M$ is a Nambu-Poisson manifold of order $n$, then $(TM, T^*M)$ provides such an example. Roughly speaking, a weak Lie-Filippov bialgebroid of order $n$ ($n > 2$) over $M$ is a Lie algebroid $A \rightarrow M$ together with a skew-symmetric $n$-ary bracket on the space of sections of the dual bundle $A^* \rightarrow M$ and a bundle map $\rho : \bigwedge^{n-1}A^* \rightarrow TM$ satisfying some conditions (cf. Definition \[lie-fill-defn\]). Moreover it is proved in [@bas-bas-das-muk] that, if $(A, A^*)$ is a weak Lie-Filippov bialgebroid of order $n$ over $M$, then there is an induced Nambu-Poisson structure of order $n$ on the base manifold $M$. In the present paper, we prove that weak Lie-Filippov bialgebroids are infinitesimal form of Nambu-Lie groupoids. Explicitly, if $C$ is a coisotropic submanifold of a Nambu-Poisson manifold $(M, \Pi)$, then we show that the $n$-ary bracket on the space of $1$-forms on $M$ restricts to the sections of the conormal bundle $(TC)^0 \rightarrow C$ and the induced bundle map $\Pi^{\sharp} : \bigwedge^{n-1}T^*M \rightarrow TM$ maps $\bigwedge^{n-1}(TC)^0$ to $TC$ (Proposition \[coiso-n-bracket\]). Therefore, if $G \rightrightarrows M$ is a Nambu-Lie groupoid of order $n$ whose Lie algebroid is $AG \rightarrow M,$ then as $M$ is a coisotropic submanifold of $G$, the space of sections of the dual bundle $A^*G \cong (TM)^0 \rightarrow M$ is equipped with a skew-symmetric $n$-ary bracket. Moreover, there is a bundle map $\bigwedge^{n-1} A^*G \rightarrow TM$ so that the pair $(AG, A^\ast G),$ with the above data, satisfies the conditions of a weak Lie-Filippov bialgebroid. Thus we prove the following (cf. Theorem \[nambu-grpd-bialgbd\]). Let $(G \rightrightarrows M , \Pi) $ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$. Then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ over $M$. Finally, we compare the Nambu-Poisson structures on the base manifold $M$ induced from the Nambu-Lie groupoid $G \rightrightarrows M$ and the weak Lie-Filippov bialgebroid $(AG, A^*G)$ (cf. Proposition \[compare-two-NP-structure\]). Next, we introduce the notion of a coisotropic subgroupoid $H \rightrightarrows N$ of a Nambu-Lie groupoid $(G \rightrightarrows M, \Pi).$ To study the infinitesimal form of a coisotropic subgroupoid, we introduce the notion of coisotropic subalgebroid of a weak Lie-Filippov bialgebroid. Then we show that the Lie algebroid of a coisotropic subgroupoid $H \rightrightarrows N$ is a coisotropic subalgebroid of the corresponding weak Lie-Filippov bialgebroid $(AG, A^*G)$ (cf. Proposition \[coiso-subgrpd-coiso-subalgbd\]). [**Organization.**]{} The paper is organized as follows. In section 2, we recall basic definitions and conventions. In section 3, we study some properties of coisotropic submanifolds of a manifold with respect to a given multivector field and in section 4 we introduce Nambu-Lie groupoids and study some of its basic properties. In section 5, we show that the infinitesimal object corresponding to Nambu-Lie groupoid is weak Lie-Filippov bialgebroid and in section 6 we introduce coisotropic subgroupoids of a Nambu-Lie groupoid and study their infinitesimal.\ [**Acknowledgements.**]{} The author would like to thank his Ph.D supervisor Professor Goutam Mukherjee for his guidance and carefully reading the manuscript. Preliminaries ============= In this section, we recall some basic preliminaries from [@duf-zung; @mac-book; @xu] and fix the notations that will be used throughout the paper . Nambu-Poisson manifolds are $n$-ary generalizations of Poisson manifolds introduced by Takhtajan [@takhtajan]. \[nambu-poisson\] Let $M$ be a smooth manifold. A [*Nambu-Poisson structure*]{} of order $n$ on $M$ is a skew-symmetric $n$-multilinear bracket $$\begin{aligned} \{~, \ldots, ~\} : C^\infty (M) \times \stackrel{(n)}{\ldots} \times C^\infty (M) \rightarrow C^\infty (M)\end{aligned}$$ satisfying the following conditions: 1. [*Leibniz rule*]{}:\ $\{f_1, \ldots, f_{n-1}, fg\} = f \{f_1, \ldots, f_{n-1}, g \} + \{f_1, \ldots, f_{n-1}, f \} g; $ 2. [*Fundamental identity*]{}:\ $\{f_1, \ldots ,f_{n-1}, \{g_1,\ldots, g_n\}\} = \sum_{k=1}^n \{g_1,\ldots,g_{k-1},\{f_1,\ldots,f_{n-1},g_k\},\ldots, g_n\}; $ for all $f_i, g_j, f, g \in C^\infty (M).$ A manifold together with a Nambu-Poisson structure of order $n$ is called a [*Nambu-Poisson manifold of order $n$*]{}. Thus the space of smooth functions with this bracket forms a Nambu-Poisson algebra. A Nambu-Poisson manifold of order $2$ is nothing but a Poisson manifold [@vais-book]. Since the bracket above is skew-symmetric and satisfies Leibniz rule, there exists an $n$-vector field $\Pi \in \mathcal{X}^{n}(M)$, such that $$\{f_1, \ldots, f_n\} = \Pi (df_1, \ldots, df_n),$$ for all $ f_1, \ldots, f_n \in C^\infty (M)$. Given any $(n-1)$ functions $ f_1, \ldots, f_{n-1} \in C^\infty (M)$, the [*Hamiltonian vector field*]{} associated to these functions is denoted by $X_{f_1\ldots f_{n-1}}$ and is defined by $$X_{f_1\ldots f_{n-1}} (g) = \{f_1, \ldots ,f_{n-1}, g\}.$$ Note that the Fundamental identity, in terms of Hamiltonian vector fields, is equivalent to the condition $$[X_{f_1\ldots f_{n-1}}, X_{g_1\ldots g_{n-1}}] = \sum_{k=1}^{n-1} X_{g_1\ldots \{f_1, \ldots, f_{n-1},g_k\}\ldots g_{n-1}}.$$ for all $f_1, \ldots, f_{n-1}, g_1, \ldots, g_{n-1} \in C^\infty(M).$ The Fundamental identity can also be rephrased as $$\mathcal{L}_{X_{f_1\ldots f_{n-1}}} \Pi = 0,$$ for all $f_1, \ldots, f_{n-1} \in C^\infty(M),$ which shows that every Hamiltonian vector field preserves the Nambu tensor. A Nambu-Poisson manifold is often denoted by $(M, \{~, \ldots, ~\})$ or simply by $(M, \Pi).$ 1. Let $M$ be an orientable manifold of dimension $n$ and $\nu$ be a volume form on $M$. Define an $n$-bracket $\{~, \ldots, ~\}$ on $C^\infty(M)$ by the following identity $$df_1 \wedge \cdots \wedge df_n = \{f_1, \ldots, f_n\} \nu.$$ Then $\{~, \ldots, ~\}$ defines a Nambu-Poisson structure of order $n$ on $M$. Let $\Pi_\nu \in \Gamma{\bigwedge^n TM}$ denotes the associated Nambu-Poisson tensor. If $\Pi \in \Gamma{\bigwedge^n TM}$ is any Nambu-Poisson structure of order $n$ such that $\Pi \neq 0$ at every point, then there exists a volume form $\nu'$ on $M$ such that $\Pi = \Pi_{\nu'}$. If $M = \mathbb{R}^n$ and $\nu = dx_1 \wedge \cdots \wedge dx_n$ is the standard volume form, then one recovers the Nambu structure on $\mathbb{R}^n$ originally discussed by Y. Nambu [@nambu]. 2. Let $\Pi$ be any $n$-vector field on an oriented manifold $M$ of dimension $n$. Then $\Pi$ defines a Nambu structure of order $n$ on $M$ (see [@ibanez1]). 3. Let $M$ be a manifold of dimension $m$ and $X_1, \ldots, X_n$ be linearly independent vector fields such that $[X_i, X_j] = 0$ for all $i, j = 1, \ldots, n$. Then the $n$-vector field $\Pi = X_1 \wedge \cdots \wedge X_n$ defines a Nambu structure of order $n$. 4. Let $(M, \{~, \ldots, ~\})$ be a Nambu-Poisson manifold of order $n$. Suppose $k \leqslant n-2$ and $F_1, \ldots, F_k \in C^\infty(M)$ be any fixed functions on $M$. Define a $(n-k)$-bracket $\{~, \ldots, ~\}'$ on $C^\infty(M)$ by $$\begin{aligned} \{f_1, \ldots, f_{n-k}\}' = \{F_1, \ldots, F_k, f_1, \ldots, f_{n-k}\}\end{aligned}$$ for $f_1, \ldots, f_{n-k} \in C^\infty(M)$. Then $\{~, \ldots, ~\}'$ defines a Nambu-structure of order $(n-k)$ on $M$. This Nambu structure is called the [*subordinate Nambu structure*]{} of $(M, \{~, \ldots, ~\})$ with subordinate function $F_1, \ldots, F_k$. 5. If $\Pi_i$ is a Nambu structure of order $n_i$ on a manifold $M_i$ $(i=1,2)$, then $\Pi = \Pi_1 \wedge \Pi_2$ is a Nambu structure of order $n_1 + n_2$ on $M_1 \times M_2$ [@duf-zung]. More examples of Nambu structures can be found in [@ibanez2; @vais]. Let $(M, \Pi)$ be a Nambu-Poisson manifold of order $n$. For each $m \in M$, let $\mathcal{D}_mM \subset T_mM$ be the subspace of the tangent space at $m$ generated by all Hamiltonian vector fields at $m$. Since the Lie bracket of two Hamiltonians is again a Hamiltonian, therefore $\mathcal{D}$ defines a [*(singular) integrable distribution*]{} whose leaves are either $n$-dimensional submanifolds endowed with a volume form or just singletons [@duf-zung]. Let $(M, \Pi_M)$ and $(N, \Pi_N)$ be two manifolds with $n$-vector fields. A smooth map $\phi: M \rightarrow N$ is called [*$(\Pi_M, \Pi_N)$-map*]{} if the induced brackets on functions satisfies: $$\begin{aligned} \{\phi^*f_1, \ldots, \phi^*f_n\}_M = \phi^* \{f_1, \ldots, f_n\}_N\end{aligned}$$ for all $f_1, \ldots, f_n \in C^\infty(N),$ or equivalently, $\phi_* \Pi_M = \Pi_N.$ The map $\phi$ is called an [*anti $(\Pi_M, \Pi_N)$-map*]{} if $$\begin{aligned} \{\phi^*f_1, \ldots, \phi^*f_n\}_M = (-1)^{n-1} \phi^* \{f_1, \ldots, f_n\}_N\end{aligned}$$ for all $f_1, \ldots, f_n \in C^\infty(N)$. A $(\Pi_M, \Pi_N)$-map $\phi : (M, \Pi_M) \longrightarrow (N, \Pi_N)$ between Nambu-Poisson manifolds of the same order $n$ is called a [*Nambu-Poisson map*]{} or a $N$-$P$-map. The condition for a $(\Pi_M, \Pi_N)$-map can also be expressed in terms of the induced bundle maps as $$\begin{aligned} \Pi^{\sharp}_{N, {\phi(m)}} = T_m \phi \circ \Pi^{\sharp}_{M, {m}} \circ T^*_m \phi ~ ~~\mbox{~~ for each}~~ m \in M,\end{aligned}$$ where $\Pi^{\sharp}_M :\bigwedge^{n-1}T^*M \rightarrow TM$ is the induced bundle map and is given by $$\begin{aligned} \langle \beta, \Pi^{\sharp}_M (\alpha_1 \wedge \cdots \wedge \alpha_{n-1}) \rangle = \Pi_M (\alpha_1, \ldots, \alpha_{n-1}, \beta)\end{aligned}$$ for all $\alpha_1, \ldots, \alpha_{n-1}, \beta \in T_x^* M,$ $x \in M$. A [*Lie groupoid*]{} over a smooth manifold $M$ is a smooth manifold $G$ together with the following structure maps: 1. two surjective submersions $\alpha, \beta : G \rightarrow M $, called the [*source*]{} map and the [*target*]{} map respectively; 2. a smooth [*partial multiplication*]{} map $$G_{(2)} = \{(g,h) \in G \times G | \beta(g) = \alpha (h) \} \rightarrow G , ~~ (g,h)\mapsto gh ;$$ 3. a smooth [*unit*]{} map $\epsilon : M \rightarrow G$, $x \mapsto \epsilon_x ;$ 4. and a smooth [*inverse*]{} map $i : G \rightarrow G$, $g \mapsto g^{-1}$ with $\alpha (g^{-1}) = \beta (g)$ and $\beta (g^{-1}) = \alpha (g)$ such that, the following conditions are satisfied\ $\hspace*{1.5cm}$(i) $\alpha (gh) = \alpha (g)$ and $\beta(gh) = \beta (h)$;\ $\hspace*{1.5cm}$(ii) $(gh)k = g(hk),$ whenever the multiplications make sense;\ $\hspace*{1.5cm}$(iii) $\alpha(\epsilon_x) = \beta (\epsilon_x) = x$, $\forall x \in M$;\ $\hspace*{1.5cm}$(iv) $\epsilon_{\alpha(g)} g = g$ and $ g \epsilon_{\beta(g)} = g$, $\forall g \in G$;\ $\hspace*{1.5cm}$(v) $g g^{-1} = \epsilon_{\alpha(g)}$ and $g^{-1} g = \epsilon_{\beta(g)}$, $\forall g \in G$. A Lie groupoid $G$ over $M$ is denoted by $G \rightrightarrows M$ when all the structure maps are understood. Note that the smooth structure on $G_{(2)}$ comes from the fact that $$G_{(2)} = (\beta \times \alpha)^{-1}(\Delta_M),$$ where $\beta \times \alpha : G \times G \rightarrow M \times M,~~(g,h) \mapsto (\beta(g), \alpha (h))$ and $ \Delta_M = \{(m,m)| m \in M\} \subset M \times M$ is the diagonal submanifold of $M \times M.$ Then these conditions imply that the inverse map $i : G \rightarrow G,$ $g \mapsto g^{-1}$ is also smooth [@mac-book]. Moreover, $\alpha$-fibers and $\beta$-fibers are submanifolds of $G$ as both $\alpha$ and $\beta$ are surjective submersions. Given a Lie groupoid $G \rightrightarrows M$, define an equivalence relation $'\sim'$ on $M$ by the following: two points $x, y \in M$ are said to be equivalent, written as $x \sim y$, if there exists an element $g \in G$ such that $\alpha(g) = x$, $\beta(g) = y$. The quotient $M/\sim$ is called the [*orbit set*]{} of $G$. Given two Lie groupoid $G_1 \rightrightarrows M_1$ and $G_2 \rightrightarrows M_2$, a [*morphism*]{} between Lie groupoids is a pair $(F, f)$ of smooth maps $F : G_1 \rightarrow G_2$ and $f: M_1 \rightarrow M_2$ which commute with all the structure maps of $G_1$ and $G_2$. In other words, $$\alpha_2 \circ F = f \circ \alpha_1, ~ ~ \beta_2 \circ F = f \circ \beta_1, ~ ~ \text{and} ~ ~ F(g_1 h_1) = F(g_1)F(h_1)$$ for all $(g_1,h_1) \in (G_1)_{(2)}.$ Let $G \rightrightarrows M$ be a Lie groupoid. A [*Lie subgroupoid*]{} of it is a Lie groupoid $ H \rightrightarrows N$ together with injective immersions $i : H \rightarrow G$ and $i_0 : N \rightarrow M$ such that $(i, i_0)$ is a Lie groupoid morphism. Let $G \rightrightarrows M$ be a Lie groupoid. A submanifold $\mathcal{K}$ of $G$ is called a [*bisection*]{} of the Lie groupoid, if $\alpha|_{\mathcal{K}} : \mathcal{K} \rightarrow M$ and $\beta|_{\mathcal{K}} : \mathcal{K} \rightarrow M$ are diffeomorphisms. The existence of local bisections through any point $g \in G$ is always guaranted. The space of bisections $\mathcal{B}(G)$ form an infinite dimensional (Fréchet) Lie group under the multiplication of subsets induced from the partial multiplication of $G$. Note that the left (right) multiplication is defined only on $\alpha$-fibers ($\beta$-fibers), therefore, we can not define a diffeomorphism of $G$ using left (right) multiplication by an element, like a Lie group. However we can do so by using bisection instead of an element. Given a bisection $\mathcal{K} \in \mathcal{B}(G)$, let $l_{\mathcal{K}}$ and $r_{\mathcal{K}}$ be the diffeomorphisms on $G$ defined by $$l_{\mathcal{K}} (h) = gh, \hspace{0.15cm} \text{where} \hspace{0.15cm} g \in \mathcal{K} \hspace{0.15cm} \text{is the unique element such that} \hspace{0.15cm} \beta(g) = \alpha(h)$$ and $$r_{\mathcal{K}} (h) = hg', \hspace{0.15cm} \text{where} \hspace{0.15cm} g' \in \mathcal{K} \hspace{0.15cm} \text{is the unique element such that} \hspace{0.15cm} \alpha(g') = \beta(h).$$ Suppose $\mathcal{K}$ is any (local) bisection of $G$ through $g \in G$. Then the restriction of the map $l_{\mathcal{K}}$ to $\alpha^{-1} (\beta(g))$ is the left translation $l_g$ by $g$: $$l_g : \alpha^{-1}(\beta(g)) \rightarrow \alpha^{-1}(\alpha(g)), ~ h \mapsto gh .$$ Then we have the following result [@xu]. \[left-invariant\] Let $G \rightrightarrows M$ be a Lie groupoid and $P$ be an $n$-vector field on $G$. Suppose for any $g \in G$ with $\beta(g) = u$, $P$ satisfies $P(g) = (l_{\mathcal{G}})_* P(\epsilon_u)$, where $\mathcal{G} \in \mathcal{B}(G)$ is any arbitrary bisection through the point $g$. Then $P$ is left invariant. A [*Lie algebroid*]{} $(A, [~, ~], a)$ over a smooth manifold M is a smooth vector bundle $A$ over M together with a Lie algebra structure $[~, ~]$ on the space $\Gamma{A}$ of the smooth sections of $A$ and a bundle map $a : A \rightarrow T M $ , called the [*anchor*]{}, such that 1. the induced map $ a : \Gamma{A} \rightarrow \mathcal{X}^1(M) $ is a Lie algebra homomorphism, where $\mathcal{X}^1(M)$ is the usual Lie algebra of vector fields on $M$. 2. For any $ X, Y \in \Gamma{A} $ and $f \in C^\infty (M)$, we have $$[X, f Y ] = f [X, Y ] + (a(X)f )Y.$$ We may denote a Lie algebroid simply by $A$, when all the structures are understood. Any Lie algebra is a Lie algebroid over a point with zero anchor. The tangent bundle of any smooth manifold is a Lie algebroid with usual Lie bracket of vector fields and identity as anchor.\ [**Lie algebroid of a Lie groupoid.**]{} Given a Lie groupoid $G \rightrightarrows M$, its Lie algebroid consists of the vector bundle $AG \rightarrow M$ whose fiber at $x \in M$ coincides with the tangent space at the unit element $\epsilon_x$ of the $\alpha$-fiber at $x$. Then the space of sections of $AG$ can be identified with the left invariant vector fields $$\begin{aligned} \mathcal{X}^1_{\text{inv}}(G) = \{ X \in \Gamma (T^\alpha G) = \Gamma (\text{ker} (d\alpha))| X_{gh} = (l_g)_* X_h, \forall (g, h) \in G_{(2)} \}\end{aligned}$$ on $G$. Since the space of left invariant vector fields on $G$ is closed under the Lie bracket, therefore it defines a Lie bracket on $\Gamma AG$. The anchor $a$ of $AG$ is defined to be the differential of the target map $\beta$ restricted to $AG$. Let $AG $ be the Lie algebroid of the Lie groupoid $G \rightrightarrows M$. Given any $X \in \Gamma AG$, let $\overleftarrow{X}$ be the corresponding left invariant vector field on $G$. Then there exists an $\epsilon > 0$ and a $1$-parameter family of transformations $\phi_t$ $(|t| < \epsilon)$, generated by $\overleftarrow{X}$ ([@mac-book]). Suppose each $\phi_t$ is defined on all of $M$, where $M$ is identified with a closed embedded submanifold of $G$ via the unit map. We denote the image of $M$ via $\phi_t$ by exp $tX$. Then exp $tX$ is a bisection of the groupoid (for all $|t| < \epsilon$) and satisfies $1$-parameter group like conditions, namely $$\begin{aligned} \text{exp} (t+s) X = \text{exp} \hspace*{0.1cm} tX \cdot \text{exp} \hspace*{0.1cm}sX, \hspace*{1cm} \text {whenever} \hspace*{0.5cm} |t| , |s| , |t+s| < \epsilon,\end{aligned}$$ where on the right hand side, we used the multiplication of bisections. Coisotropic submanifolds ======================== Let $M$ be a manifold and $\Pi \in \mathcal{X}^n(M)$ be a $n$-vector field on $M$. Let $$\begin{aligned} \Pi^{\sharp} : {\bigwedge}^{n-1}T^*M \rightarrow TM\end{aligned}$$ be the induced bundle map given by $$\begin{aligned} \langle \beta, \Pi^{\sharp} (\alpha_1 \wedge \cdots \wedge \alpha_{n-1}) \rangle = \Pi (\alpha_1, \ldots, \alpha_{n-1}, \beta)\end{aligned}$$ for all $\alpha_1, \ldots, \alpha_{n-1}, \beta \in T_x^* M,$ $x \in M$. We recall the following definition from [@pont-geng-xu]. A submanifold $C \hookrightarrow M$ is said to be [*coisotropic*]{} with respect to $\Pi$, if $$\begin{aligned} \Pi^{\sharp} ({\bigwedge}^{n-1} (TC)^0) \subset TC\end{aligned}$$ where $$(TC)^0_x = \{ \alpha \in T_x^* M | \hspace*{0.1cm} \alpha (v) = 0, \forall v \in T_xC\},~~x \in C,$$ or equivalently, $$\begin{aligned} \Pi_x (\alpha_1, \ldots, \alpha_n) = 0, \forall \alpha_i \in (TC)^0_x, x \in C.\end{aligned}$$ We have the following easy observation for coisotropic submanifolds of a Nambu-Poisson manifold. Let $(M, \Pi)$ be a Nambu-Poisson manifold of order $n$ and $C$ be a closed embedded submanifold of $M$. Let $\mathcal{I} (C) = \{ f \in C^\infty(M)\big| f|_C \equiv 0 \}$ denote the vanishing ideal of $C$. Then the followings are equivalent: 1. $C$ is a coisotropic submanifold; 2. $\mathcal{I} (C)$ is a Nambu-Poisson subalgebra; 3. for every $f_1, \ldots, f_{n-1} \in \mathcal{I} (C)$, the Hamiltonian vector field $X_{f_1...f_{n-1}}$ is tangent to $C$. \(1) $\Rightarrow$ (2) Let $f_1, \ldots, f_n \in \mathcal{I} (C) $. Then for any $x \in C$, $d_xf_i \in (TC)^0_x$, for all $i=1, \ldots, n.$ Now since $C$ is a coisotropic submanifold, we have $$\begin{aligned} \{f_1, \ldots, f_n\}(x) = \Pi_x (d_xf_1, \ldots, d_xf_n) = 0,~ \forall x \in C.\end{aligned}$$ Hence $\{f_1, \ldots, f_n\} \in \mathcal{I} (C)$. Therefore $\mathcal{I} (C)$ is a Nambu-Poisson subalgebra. \(2) $\Rightarrow$ (3) Let $f_1, \ldots, f_{n-1} \in \mathcal{I} (C)$ and $x \in C$. Let $\alpha \in (TC)^0_x.$ Then there exists a function $g$ vanishing on $C$ such that $d_xg = \alpha$. Since $\mathcal{I} (C)$ is a Nambu-Poisson subalgebra, we have $$\begin{aligned} \{f_1, \ldots, f_{n-1}, g \}(x) = 0.\end{aligned}$$ Thus, $$\begin{aligned} X_{f_1\ldots f_{n-1}}\big|_x (\alpha) = X_{f_1\ldots f_{n-1}}\big|_x (d_xg) = \{f_1, \ldots, f_{n-1}, g \}(x) = 0\end{aligned}$$ and consequently, $X_{f_1\ldots f_{n-1}}$ is tangent to $C$. \(3) $\Rightarrow$ (1) Let $x \in C$ and $\alpha_1, \ldots, \alpha_n \in (TC)^0_x.$ Then there exist functions $f_1, \ldots, f_n \in \mathcal{I}(C)$ such that $d_xf_i = \alpha_i,$ $\forall i = 1, \ldots, n.$ Therefore, $$\begin{aligned} \Pi_x (\alpha_1, \ldots, \alpha_n) = \Pi_x (d_xf_1, \ldots, d_xf_n) = X_{f_1...f_{n-1}} \big|_x (d_xf_n) = 0.\end{aligned}$$ Hence $C$ is a coisotropic submanifold of $M$. Let $(M, \Pi_M)$ and $(N, \Pi_N)$ be two Nambu-Poisson manifolds of same order $n$ and $C \hookrightarrow N$ be a coisotropic submanifold of $N$ with respect to $\Pi_N$. If $\phi: M \rightarrow N$ is a Nambu-Poisson map transverse to $C$, then $\phi^{-1}(C)$ is a coisotropic submanifold of $M$ with respect to $\Pi_M$ (the result holds true for manifolds with $n$-vector fields such that $\phi_* \Pi_M = \Pi_N$). Since $\phi$ is transverse to $C$, therefore $\phi^{-1}(C)$ is a submanifold of $M$. Moreover $$\begin{aligned} T(\phi^{-1}(C)) = (T \phi)^{-1} TC.\end{aligned}$$ Therefore $T(\phi^{-1}(C))^0 = (T \phi)^{*} (TC)^0$. Observe that $$\begin{aligned} T\phi (\Pi_M^{\sharp} \big( {\bigwedge}^{n-1} T(\phi^{-1}(C))^0 \big)) =& T\phi ( \Pi_M^{\sharp} \big((T \phi)^{*} {\bigwedge}^{n-1} (TC)^0 \big))\\ =& \Pi_N^{\sharp} ({\bigwedge}^{n-1} (TC)^0)\\ \subseteq & TC.\end{aligned}$$ Thus, $$\begin{aligned} \Pi_M^{\sharp} \big( {\bigwedge}^{n-1} T(\phi^{-1}(C))^0 \big) \subseteq (T\phi)^{-1} TC = T(\phi^{-1}(C))\end{aligned}$$ and hence $\phi^{-1}(C)$ is coisotropic with respect to $\Pi_M.$ \[image-coiso\] Let $\phi: (M, \Pi_M) \rightarrow (N, \Pi_N)$ be a Nambu Poisson map between two Nambu-Poisson manifolds $(M, \Pi_M)$ and $(N, \Pi_N)$ and $C\hookrightarrow M$ be a coisotropic submanifold of $M$. Assume that $\phi(C)$ is a submanifold of $N$. Then $\phi(C)$ is a coisotropic submanifold of $N$ (the result holds true for manifolds with $n$-vector fields such that $\phi_{*}\Pi_M = \Pi_N$). We have $T(\phi(C)) \supseteq T\phi (TC)$ and $(T\phi)^* (T(\phi(C)))^0 \subseteq (TC)^0$. Therefore, $$\begin{aligned} \Pi_N^{\sharp} ({\bigwedge}^{n-1} T(\phi(C))^0) =& T\phi (\Pi_M^{\sharp} ((T\phi)^* {\bigwedge}^{n-1} T(\phi(C))^0)) \hspace*{1cm}(\text{since ~} \phi \text{~ is a N-P map})\\ \subseteq & T\phi (\Pi_M^{\sharp} ({\bigwedge}^{n-1} (TC)^0))\\ \subseteq & T\phi (TC) \hspace*{1cm}(\text{since~} C \hookrightarrow M \text{~ is coisotropic})\\ \subseteq & T(\phi(C))\end{aligned}$$ which shows that $\phi(C)$ is a coisotropic submanifold of $N$. Using the terminology of coisotropic submanifold with respect to any multivector field allows us to extend the results of Weinstein [@wein] from Poisson bivector field to Nambu-Poisson tensor or more generally to any multivector field. \[nambu map-coiso\] Let $(M, \Pi_M)$ and $(N, \Pi_N)$ be two manifolds with $n$-vector fields and $\phi: M \rightarrow N$ be a smooth map. Then $\phi$ is a $(\Pi_M, \Pi_N)$-map, that is $\phi_* \Pi_M = \Pi_N$ if and only if its graph $$\begin{aligned} \text{Gr}(\phi) = \{(m, \phi(m))| m \in M \}\end{aligned}$$ is a coisotropic submanifold of $M \times N$ with respect to $\Pi_M \oplus (-1)^{n-1} {\Pi}_N$. Let $C = \text{Gr}(\phi) \subset M \times N$. Then $C$ is a closed embedded submanifold of $M \times N.$ Note that, a tangent vector to the graph consist of a pair $(v_m, (T\phi)(v_m))$, where $m \in M$, $v_m \in T_mM$. Therefore, $(TC)^0$ consists of a pair of covectors $(-(T\phi)^*\psi, \psi)$, where $\psi \in T_{\phi(m)}^*N.$ Therefore, Gr($\phi)$ is a coisotropic submanifold of $M \times N$ with respect to $\Pi_M \oplus (-1)^{n-1} {\Pi}_N$ if and only if $(\Pi_M^{\sharp} \times (-1)^{n-1} \Pi_N^{\sharp})$ maps $(-(T\phi)^*\psi_1, \psi_1) \wedge \cdots \wedge (-(T\phi)^*\psi_{n-1}, \psi_{n-1}) $ into $TC$, for all $\psi_1, \ldots, \psi_{n-1} \in T^*_{\phi(m)}N$ and $m \in M$. In other words, $$\begin{aligned} (T\phi) \bigg(\Pi_M^{\sharp} (- (T\phi)^*\psi_1, \ldots, - (T\phi)^*\psi_{n-1})\bigg) = (-1)^{n-1} {\Pi}_N^{\sharp} (\psi_1, \ldots, \psi_{n-1})\end{aligned}$$ that is, $$\begin{aligned} (T\phi) \bigg(\Pi_M^{\sharp} ( (T\phi)^*\psi_1, \ldots, (T\phi)^*\psi_{n-1})\bigg) = {\Pi}_N^{\sharp} (\psi_1, \ldots, \psi_{n-1}).\end{aligned}$$ This is equivalent to the condition that $\phi$ is a $(\Pi_M, \Pi_N)$-map. Let $(M, \Pi_M)$ be a Nambu-Poisson manifold of order $n$ and $\phi: M \rightarrow N$ be a smooth surjective map. If there exist a Nambu-Poisson structure $\Pi_N$ (of order $n$) on $N$ which makes $\phi$ into a Nambu-Poisson map, then $\Pi_N$ is called the Nambu-Poisson structure [*coinduced*]{} by the mapping $\phi$. The following is a characterization of coinduced Nambu-Poisson structure. \[coinduced\] Let $(M, \Pi_M)$ be a Nambu-Poisson manifold of order $n$ and $\phi: M \rightarrow N$ be a smooth surjective map from $M$ to some manifold $N$. Then $N$ has a Nambu-Poisson structure coinduced by $\phi$ if and only if for all $f_1, \ldots, f_{n} \in C^\infty(N)$, the function $\{\phi^*f_1, \ldots, \phi^*f_n\}_M$ is constant along the fibers of $\phi$. Let $f_1, \ldots, f_n \in C^\infty(N)$. If the function $\{ \phi^*f_1, \ldots, \phi^*f_n \}_M$ is constant along the $\phi$-fibers, then there exists a function on $N$, which we denote by $\{f_1, \ldots, f_n\}_N$ such that $\{\phi^*f_1, \ldots, \phi^*f_n\}_M = \phi^* \{f_1, \ldots, f_n\}_N $. Clearly this bracket defines a coinduced Nambu-Poisson structure on $N$. Conversely, suppose that there is a Nambu-Poisson bracket $\{~, \ldots, ~\}_N$ on $N$ coinduced by $\phi$. Then for any $y \in N,$ $$\begin{aligned} \{\phi^*f_1, \ldots, \phi^*f_n\}_M(\phi^{-1}\{y\}) =& (\phi^* \{f_1, \ldots, f_n\}_N )(\phi^{-1}y)\\ =& \{f_1, \ldots, f_n\}_N (y)\end{aligned}$$ proving $\{\phi^*f_1, \ldots, \phi^*f_n\}_M$ is constant along the $\phi$-fibers. \[rem-coinduced\] Let $(M, \Pi_M)$ be a manifold with an $n$-vector field and $\phi: M \rightarrow N$ be a smooth map. Then there exists an $n$-vector field $\Pi_N$ on $N$ such that $\phi$ is a $(\Pi_M, \Pi_N)$-map if and only if for all $f_1, \ldots, f_n \in C^\infty(N)$, the function $\{\phi^*f_1, \ldots, \phi^*f_n\}_M$ is constant along the fibers of $\phi$. Let $(M, \Pi_M)$ be a Nambu-Poisson manifold and $\phi : M \rightarrow N$ be a surjective submersion with connected fibers. Let $\text{ker} \hspace*{0.1cm} \phi_*(m)$ is spanned by local Hamiltonian vector fields (that is, $\text{ker} \hspace*{0.1cm} \phi_*(m) \subset \mathcal{D}_mM $), for all $m \in M$. Then $N$ has a Nambu-Poisson structure coinduced by $\phi.$ Since $\phi$ is a submersion, the fibers of $\phi$ are submanifolds of $M$. Then for $y \in N,$ $\phi^{-1}(\{y\}) = C$ is a submanifold of $M$. Let $g_1, \ldots, g_{n-1}$ be locally defined functions on $M$ such that $X_{g_1...g_{n-1}} \in \text{ker} \hspace*{0.1cm} \phi_{*}$. Let $f_1, \ldots, f_n \in C^\infty(N).$ To prove that $\{\phi^*f_1, \ldots, \phi^*f_n\}$ is constant on the fibers, it is enough to prove that $$\begin{aligned} X_{g_1...g_{n-1}} \{\phi^*f_1, \ldots, \phi^*f_n\} = 0.\end{aligned}$$ Note that $$\begin{aligned} X_{g_1...g_{n-1}} \{\phi^*f_1, \ldots, \phi^*f_n\} = \sum_{k=1}^n \{\phi^*f_1, \ldots, X_{g_1...g_{n-1}} (\phi^*f_k), \ldots, \phi^*f_n\}\end{aligned}$$ and the functions $\phi^*f_i$ are constant along the fibers. Hence by the Proposition \[coinduced\], there exists a coinduced Nambu-Poisson structure on $N.$ \[coinduced-coiso\] Let $(M, \Pi_M)$ be a manifold with an $n$-vector field and $\phi: M \rightarrow N$ be a surjective submersion. Then $N$ has an (unique) $n$-vector field $\Pi_N$ such that $\phi$ is a $(\Pi_M, \Pi_N)$-map if and only if $R(\phi) = \{(x,y) \in M \times M|\phi(x)=\phi(y)\}$ is a coisotropic submanifold of $M \times M$ with respect to $\Pi_M \oplus (-1)^{n-1}{\Pi}_M$. Note that $R(\phi) = (\phi \times \phi)^{-1} (\Delta_N)$, where $\Delta_N$ is the diagonal of $N \times N$. Since $\phi$ is surjective submersion $R(\phi)$ is a submanifold of $M \times M$. Moreover, for $(x,y) \in R(\phi)$ $$\begin{aligned} T_{(x,y)} (R(\phi)) = \{(X, Y)\in T_xM \times T_yM | (T\phi)_x (X) = (T\phi)_y (Y)\}.\end{aligned}$$ Therefore, $T(R(\phi))^0$ consists of covectors $(-(T\phi)_x^*\psi, (T\phi)_y^*\psi)$, where $\psi \in T_{\phi(x)}^*N$. Thus, $R(\phi)$ be a coisotropic submanifold of $M \times M$ with respect to $\Pi_M \oplus (-1)^{n-1}{\Pi}_M$ if and only if for all $\psi_1, \ldots, \psi_{n-1} \in T_{\phi(x)}^*N$ and $(x, y) \in R(\phi),$ $\Pi_M^{\sharp} \oplus (-1)^{n-1}{\Pi}_M^{\sharp}$ maps $$(-(T\phi)_x^*\psi_1, (T\phi)_y^*\psi_1) \wedge \cdots \wedge (-(T\phi)_x^*\psi_{n-1}, (T\phi)_y^*\psi_{n-1})$$ into $T(R(\phi)).$ That is $$(T\phi)_x \Pi_M^{\sharp} (-(T\phi)_x^*\psi_1, \ldots, -(T\phi)_x^*\psi_{n-1} )= (-1)^{n-1} (T\phi)_y \Pi_M^{\sharp} ((T\phi)_y^*\psi_1, \ldots, (T\phi)_y^*\psi_{n-1}),$$ or equivalently, $$\begin{aligned} \label{firsteqn} (T\phi)_x \Pi_M^{\sharp} ((T\phi)_x^*\psi_1, \ldots, (T\phi)_x^*\psi_{n-1} ) = (T\phi)_y \Pi_M^{\sharp} ((T\phi)_y^*\psi_1, \ldots, (T\phi)_y^*\psi_{n-1})\end{aligned}$$ holds. Let $f_1, \ldots ,f_n \in C^\infty(N)$ and $x \in M.$ Then $$\begin{aligned} \{\phi^*f_1, \ldots , \phi^*f_n\}_M (x) =& \langle \Pi_M^{\sharp} (d_x (\phi^*f_1) \wedge \cdots \wedge d_x (\phi^*f_{n-1})), d_x (\phi^*f_n) \rangle\\ =& \langle \Pi_M^{\sharp} \big((T\phi)_x^* \psi_1 \wedge \cdots \wedge (T\phi)_x^* \psi_{n-1}\big), (T\phi)_x^* \psi_n \rangle\\ =& \langle (T\phi)_x \Pi_M^{\sharp} \big((T\phi)_x^* \psi_1 \wedge \cdots \wedge (T\phi)_x^* \psi_{n-1}\big), \psi_n \rangle\end{aligned}$$ where $\psi_i = d_{\phi(x)}f_i = d_{\phi(y)}f_i\in T_{\phi(x)}^*N$, for all $1 \leqslant i \leqslant n.$ It follows from the Equation (\[firsteqn\]) that the function $\{\phi^*f_1, \ldots , \phi^*f_n\}_M$ is constant along the $\phi$-fibers if and only if $R(\phi)$ is a coisotropic submanifold of $M \times M$ with respect to $\Pi_M^{\sharp} \oplus (-1)^{n-1}{\Pi}_M^{\sharp}$. Hence the result follows by the Remark \[rem-coinduced\]. The uniqueness follows from the surjectivity of $\phi.$ Nambu-Lie groupoids =================== In this section, we recall the definition of multiplicative multivector fields on Lie groupoid ([@pont-geng-xu]) and define Nambu-Lie groupoid (of order $n$) as a Lie groupoid with a multiplicative $n$-vector field which is also a Nambu-Poisson tensor. Let $G \rightrightarrows M$ be a Lie groupoid and $\Pi \in \mathcal{X}^n(G)$ be an $n$-vector field on $G$. Then $\Pi$ is called [*multiplicative*]{} if the graph of the groupoid multiplication $$\begin{aligned} \{ (g, h, gh) \in G \times G \times G | ~ \beta(g) = \alpha(h)\}\end{aligned}$$ is a coisotropic submanifold of $G \times G \times G$ with respect to $\Pi \oplus \Pi \oplus (-1)^{n-1} \Pi$. Then we have the following characterization of multiplicative multivector fields [@pont-geng-xu]: \[multiplicative\] Let $G \rightrightarrows M$ be a Lie groupoid and $\Pi \in \mathcal{X}^n (G)$ be an $n$-vector field on $G$. Then $\Pi$ is multiplicative if and only if the following conditions are satisfied. 1. $\Pi$ is an affine tensor. In other words $$\begin{aligned} \Pi(gh) = (r_{\mathcal{H}})_* \Pi(g) + (l_{\mathcal{G}})_* \Pi(h) - (r_{\mathcal{H}})_* (l_{\mathcal{G}})_* \Pi(u)\end{aligned}$$ where $u = \beta(g) = \alpha(h)$ and $\mathcal{G}, \mathcal{H}$ are (local) bisections through the points $g, h$ respectively. 2. $M$ is a coisotropic submanifold of $G$ with respect to $\Pi$. 3. For all $g \in G$, $\alpha_* \Pi(g)$ and $\beta_* \Pi(g)$ depend only on the base points $\alpha(g)$ and $\beta(g)$ respectively. 4. For all $f, f' \in C^\infty(M)$, the $(n-2)$-vector field $\iota_{d(\alpha^*f) \wedge d(\beta^*f')} \Pi$ is zero. In other words, $$\begin{aligned} \{~, \ldots, \alpha^*f, \beta^*f' \} = 0.\end{aligned}$$ 5. For all $f_1, \ldots, f_k \in C^\infty(M)$, $\iota_{d(\beta^*f_1) \wedge \cdots \wedge d(\beta^*f_k)} \Pi$ is a left invariant $(n-k)$-vector field on $G$, $1 \leqslant k < n.$ \[lie-gp-multiplicative\] Suppose $G$ be a Lie group considered as a Lie groupoid over a point. Then the conditions [*(3)*]{} - [*(5)*]{} of the Theorem \[multiplicative\] are satisfied automatically. The condition [*(2)*]{} implies that $\Pi (e) = 0$ (where $e$ is the identity element of the group), which together with condition [*(1)*]{} implies that $\Pi$ satisfies the usual multiplicativity condition $$\Pi (gh) = (r_h)_* \Pi(g) + (l_g)_* \Pi(h) .$$ \[nambu-lie groupoid\] A [*Nambu-Lie groupoid of order $n$*]{} is a Lie groupoid $G \rightrightarrows M$ with a multiplicative Nambu tensor $\Pi \in \mathcal{X}^n(G)$ of order $n.$ A Nambu Lie groupoid (of order $n$) will be denoted by $(G \rightrightarrows M, \Pi) $. \[exam-nlg\] 1. Poisson groupoids [@wein] are examples of Nambu-Lie groupoids with $n=2.$ 2. Any Lie groupoid with zero Nambu structure is a Nambu-Lie groupoid. 3. Let $(G, \Pi)$ be a Nambu-Lie group (of order $n$) [@vais]. Thus $G$ is a Lie group equipped with a Nambu structure $\Pi$ of order $n$ on $G$ such that $$\begin{aligned} \Pi(gh) = (r_h)_* \Pi(g) + (l_g)_* \Pi(h)\end{aligned}$$ for all $g, h \in G$. Note that the right hand side of the above equality is equal to $m_* (\Pi(g), \Pi(h)),$ where $m_* : \bigwedge^n T_{(g,h)} (G \times G) \rightarrow \bigwedge^n T_{gh}G$ is the map induced by the multiplication map $m : G \times G \rightarrow G$. Therefore, $$\begin{aligned} \Pi(gh) = m_* (\Pi(g), \Pi(h)).\end{aligned}$$ Thus, the group multiplication map $m : G \times G \rightarrow G$ is a $(\Pi \oplus \Pi, \Pi)$-map. Therefore, by the Proposition \[nambu map-coiso\], the graph of the group multiplication map is a coisotropic submanifold of $G \times G \times G$ with respect to $\Pi \oplus \Pi \oplus (-1)^{n-1} \Pi$. Hence $(G, \Pi)$ is a Nambu-Lie groupoid over a point. Conversely, if $(G, \Pi)$ is a Nambu-Lie groupoid over a point, then the group multiplication map $m : G \times G \rightarrow G$ is a $(\Pi \oplus \Pi, \Pi)$-map. Hence $(G, \Pi)$ is a Nambu-Lie group in the sense of [@vais]. One can also see the equivalence between Nambu-Lie groupoid over a point and Nambu-Lie group by using Remark \[lie-gp-multiplicative\]. For a Poisson groupoid the following facts are well known [@wein]. - The groupoid inversion map is a anti-Poisson map. - The Poisson structure on the total space induces a Poisson structure on the base such that the source map is a Poisson map and the target map is a anti-Poisson map. In the next proposition we generalize the above facts to the Nambu-Poisson setting. \[inverse-basenambu\] Let $(G \rightrightarrows M , \Pi)$ be a Nambu-Lie groupoid. Then 1. The inverse map $i : G \rightarrow G$, $g \mapsto g^{-1}$ is an anti-Nambu Poisson map. 2. There is a unique Nambu-Poisson structure on $M$ which we denote by $\Pi_M$ for which $\alpha$ is a Nambu-Poisson map and $\beta$ is an anti Nambu-Poisson map. \(1) It is proved in [@pont-geng-xu] that given a Lie groupoid $G \rightrightarrows M$ with multiplicative $n$-vector field $\Pi \in \mathcal{X}^n(G)$, the groupoid inversion map $i : G \rightarrow G$ satisfies $$\begin{aligned} i_* \Pi = (-1)^{n-1} \Pi.\end{aligned}$$ Hence the result follows as $\Pi$ is a Nambu tensor. \(2) Let $f_1, \ldots, f_n \in C^\infty(M)$ be any functions on $M$. Then for any $g \in G$, we have $$\begin{aligned} \{\alpha^* f_1, \ldots, \alpha^* f_n \} (g) =& \Pi (g) (d_g (\alpha^* f_1), \ldots, d_g (\alpha^* f_n))\\ =& \Pi(g) (\alpha^* (d_{\alpha(g)} f_1), \ldots, \alpha^* (d_{\alpha(g)} f_n))\\ =& \alpha_* \Pi (g) (d_{\alpha(g)} f_1, \ldots, d_{\alpha(g)} f_n).\end{aligned}$$ Since $\alpha_* \Pi(g)$ depends only on the value of $\alpha(g)$, it follows that the function $\{ \alpha^* f_1, \ldots, \alpha^* f_n \}$ is constant on the $\alpha$-fibers. Therefore, by the Proposition \[coinduced\], there exists a Nambu-Poisson structure $\Pi_M$ with the induced bracket denoted by $\{ ~, \ldots, ~\}_M,$ on $M$ for which $\alpha$ is a Nambu-Poisson map. Since $\beta = \alpha \circ i$ and $i$ is anti Nambu-Poisson, therefore $\beta$ is an anti Nambu-Poisson map. Consider the map $(\alpha,\beta): G \rightarrow M \times M.$ Since we have $\alpha_{*} \Pi = \Pi_M$ and $\beta_{*}\Pi = (-1)^{n-1} \Pi_M$, using property [*(4)*]{} of the Theorem \[multiplicative\] we obtain $$\begin{aligned} (\alpha,\beta)_{*} \Pi = \alpha_* \Pi \oplus \beta_* \Pi = \Pi_M \oplus (-1)^{n-1} \Pi_M.\end{aligned}$$ Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid. If the orbit space $M/\sim$ is a smooth manifold, then $M/\sim$ carries a Nambu-Poisson structure such that the projection $q: M \rightarrow M/\sim $ is a Nambu-Poisson map. Let $\Pi_M$ be the induced Nambu structure on the base $M$. For the projection map $q: M \rightarrow M/\sim $, we have $$\begin{aligned} R(q) =& \{(x,y) \in M \times M| q(x) = q(y)\}\\ =& \{(\alpha(g), \beta(g))| g \in G\}\\ =& (\alpha,\beta)(G).\end{aligned}$$ Consider $G$ as a coisotropic submanifold of $G$ with respect to $\Pi$ and also consider the map $(\alpha, \beta) : G \rightarrow M \times M$. By the above Remark we have $(\alpha, \beta)_* \Pi = \Pi_M \oplus (-1)^{n-1} \Pi_M.$ Therefore, by the Proposition \[image-coiso\], $R(q) = (\alpha,\beta)(G)$ is a coisotropic submanifold of $M \times M$ with respect to $\Pi_M \oplus (-1)^{n-1}\Pi_M.$ Hence the result follows from the Proposition \[coinduced-coiso\]. Infinitesimal form of Nambu-Lie groupoid ======================================== The aim of this section, is to study the infinitesimal form of a Nambu-Lie groupoid. We show that if $(G \rightrightarrows M, \Pi)$ is a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$, then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ introduced in [@bas-bas-das-muk]. Before proceeding further, let us briefly recall from [@bas-bas-das-muk] the notion of a weak Lie-Filippov bialgebroid. Lie bialgebroids are generalization of both Poisson manifolds and Lie bialgebras. Recall that a Lie bialgebroid, introduced by Mackenzie and Xu [@mac-xu] is also the infinitesimal form of a Poisson groupoid. It is defined as a pair $(A, A^\ast)$ of Lie algebroids in duality, where the Lie bracket of $A$ satisfies the following compatibility condition expressed in terms of the differential $d_\ast$ on $\Gamma (\bigwedge^\bullet A)$ $$d_\ast[X, Y] = [d_\ast X, Y] + [X , d_\ast Y],$$ for all $X,~ Y \in \Gamma A.$ We note that if $M$ is a Poisson manifold then the Lie algebroid structures on $TM$ and $T^\ast M$ form a Lie bialgebroid. On the other hand, it is well known [@kosmann; @mac-xu] that if $(A, A^\ast)$ is a Lie bialgebroid over a smooth manifold $M$ then there is a canonical Poisson structure on the base manifold $M$. Thus it is natural to ask the following question which was posed in [@bas-bas-das-muk]:\ [*Does there exist some notion of bialgebroid associated to a Nambu-Poisson manifold of order $n > 2$* ]{}? To answer this question, the authors [@bas-bas-das-muk] introduced the notion of weak Lie-Filippov bialgebroid. It is well known [@gra-mar; @vais] that for a Nambu-Poisson manifold $M$ of order $n \geqslant 2$, the space $\Omega^1(M)$ of $1$-forms admits an $n$-ary bracket, called [*Nambu-form bracket*]{}, such that the bracket satisfies almost all the properties of an $n$-Lie algebra (also known as Filippov algebra of order $n$) bracket [@fil] except that the Fundamental identity is satisfied only in a restricted sense as described below. Let $(M, \{~, \ldots, ~\})$ be a Nambu-Poisson manifold of order $n$ with associated Nambu-Poisson tensor $\Pi$. Then one can define the Nambu form-bracket on the space of $1$-forms $$\begin{aligned} [~, \ldots, ~]: \Omega^1(M) \times \cdots \times \Omega^1(M) \rightarrow \Omega^1(M)\end{aligned}$$ by the following $$\begin{aligned} [\alpha_1, \ldots, \alpha_n] = \sum_{k=1}^n (-1)^{n-k} \mathcal{L}_{\Pi^{\sharp}(\alpha_1 \wedge\cdots \wedge \hat{\alpha}_k \wedge\cdots \wedge \alpha_n)} \alpha_k - (n-1) d (\Pi(\alpha_1, \ldots, \alpha_n))\end{aligned}$$ $$\begin{aligned} \label{nambu-bracket} \hspace*{1cm} = d (\Pi(\alpha_1, \ldots, \alpha_n)) + \sum_{k=1}^n (-1)^{n-k} \iota_{\Pi^{\sharp}(\alpha_1 \wedge\cdots \wedge \hat{\alpha}_k \wedge\cdots \wedge \alpha_n)} d\alpha_k\end{aligned}$$ for $\alpha_i \in \Omega^1(M), i=1,\ldots ,n.$ Here $\hat{\alpha}_k$ in a monomial $\alpha_1 \wedge \cdots \wedge \hat{\alpha}_k \wedge \cdots \wedge \alpha_n$ means that the symbol $\alpha_k$ is missing in the monomial. The above bracket satisfies the following properties ([@vais]). 1. The bracket is skew-symmetric. 2. $[df_1, \ldots, df_n] = d \{f_1, \ldots, f_n \}$. 3. $[\alpha_1,\ldots, \alpha_{n-1}, f \alpha_n] = f [\alpha_1,\ldots, \alpha_{n-1}, \alpha_n] + \Pi ^{\sharp} (\alpha_1 \wedge\cdots \wedge \alpha_{n-1})(f) \alpha_n$. 4. The bracket satisfies the Fundamental identity $$\begin{aligned} [\alpha_1, \ldots, \alpha_{n-1}, [\beta_1, \ldots, \beta_n]] = \sum_{k=1}^n [\beta_1, \ldots , \beta_{k-1},[\alpha_1, \ldots, \alpha_{n-1}, \beta_k], \ldots, \beta_n]\end{aligned}$$ whenever the $1$-forms $\alpha_i \in \Omega^1(M)$ are closed, $1 \leqslant i \leqslant n-1$ and for any $\beta_j$. 5. $[\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \alpha_{n-1}), \Pi^{\sharp}(\beta_1 \wedge \cdots \wedge \beta_{n-1})]\\ = \sum_{k=1}^{n-1} \Pi^{\sharp} (\beta_1 \wedge\cdots \wedge [\alpha_1,\ldots , \alpha_{n-1}, \beta_k] \wedge \cdots \wedge \beta_{n-1})$\ for closed $1$-forms $\alpha_i \in \Omega^1 (M)$ and for any $1$-forms $\beta_j$. The Nambu-form bracket on $\Omega^1(M)$, together with the usual Lie algebroid structure on $TM$ yields an example of a notion called a [*weak Lie-Filippov algebroid pair of order $n$*]{}, $n>2,$ on a smooth vector bundle (cf. Definition $5.5$, [@bas-bas-das-muk]). In order to classify such structures, the authors formulate a notion of [*Nambu-Gerstenhaber algebra of order $n$*]{}. It turns out, weak-Lie-Filippov algebroid pair structures of order $n$, $n>2$, on a smooth vector bundle $A$ over $M$, are in bijective correspondence with Nambu-Gerstenhaber brackets of order $n$ on the graded commutative, associative algebra $\Gamma \bigwedge^\bullet A^*$ of multisections of $A^*$, where $A^*$ is the dual bundle (cf. Definition $5.7$, Theorem $5.8$, [@bas-bas-das-muk]). Moreover, for a Nambu-Poisson manifold $M$ of order $n > 2$, the Nambu-Gerstenhaber bracket on $\Omega^\bullet(M)$, extending the Nambu-form bracket on $\Omega^1(M)$ satisfies certain suitable compatibility condition similar to the compatibility condition of a Lie bialgebroid. This motivates the authors to introduce the notion of a [*weak Lie-Filippov bialgebroid structure*]{} of order $n$ on a smooth vector bundle. \[lie-fill-defn\] A [*weak Lie-Filippov bialgebroid of order $n>2$*]{} over a smooth manifold $M$ consists of a pair $(A, A^*)$, where $A$ is a smooth vector bundle over $M$ with dual bundle $A^*$ satisfying the following properties: 1. $A$ is a Lie algebroid with $d_A$ being the differential of the Lie algebroid cohomology of $A$ with trivial representation; 2. the space of smooth sections $\Gamma A^*$ admits a skew-symmetric $n$-ary bracket $$[~, \ldots ,~]: {\Gamma A^* \times \cdots \times \Gamma A^*} \longrightarrow \Gamma A^*$$ satisfying $$[\alpha_1, \ldots , \alpha_{n-1}, [\beta_1, \ldots , \beta_n]] = \sum_{k=1}^n [\beta_1, \ldots , \beta_{k-1}, [\alpha_1, \ldots , \alpha_{n-1}, \beta_k], \ldots ,\beta_n]$$ for all $d_A$-closed sections $\alpha_i \in \Gamma A^*,~ 1 \leqslant i \leqslant n-1$ and for any sections $\beta_j \in \Gamma A^*,~ 1\leqslant j\leqslant n;$ 3. there exists a vector bundle map $\rho : \bigwedge^{n-1}A^* \longrightarrow TM$, called the [*anchor*]{} of the pair $(A, A^*)$, such that the identity $$[\rho (\alpha_1 \wedge \cdots \wedge \alpha_{n-1}), \rho (\beta_1 \wedge \cdots \wedge \beta_{n-1})] = \sum_{k=1}^{n-1}\rho (\beta_1 \wedge \cdots \wedge [\alpha_1, \ldots , \alpha_{n-1}, \beta_k] \wedge \cdots \wedge \beta_{n-1})$$ holds for all $d_A$-closed sections $\alpha_i \in \Gamma A^*,~ 1\leqslant i \leqslant n-1$ and for any sections $\beta_j \in \Gamma A^*,~ 1\leqslant j\leqslant n-1;$ 4. for all sections $\alpha_i \in \Gamma A^*,~ 1\leqslant i \leqslant n$ and any $f \in C^\infty(M)$, $$[\alpha_1, \ldots , \alpha_{n-1}, f\alpha_n] = f [\alpha_1, \ldots , \alpha_{n-1}, \alpha_n] + \rho (\alpha_1 \wedge \cdots \wedge \alpha_{n-1})(f)\alpha_n$$ holds; 5. the following compatibility condition holds: $$d_A[\alpha_1, \ldots , \alpha_n] = \sum_{k=1}^n [\alpha_1, \ldots , d_A\alpha_k, \ldots , \alpha_n],$$ for any $\alpha_i \in \Gamma A^\ast,$ $1 \leqslant i \leqslant n$, where the bracket $[~, \ldots , ~]$ on the right hand side is the graded extension of the bracket on $\Gamma A^\ast$. A weak Lie-Filippov bialgebroid (of order $n$) over $M$ is denoted by $(A, A^*)$ when all the structures are understood. A Lie bialgebroid is a Lie-Filippov bialgebroid of order $2$ such that the conditions (2) and (3) of the above definition has no restriction on $\alpha$. In [@bas-bas-das-muk], the authors have shown that for a Nambu-Poisson manifold $M$ of order $n > 2$, the pair $(TM, T^*M)$ is a weak Lie-Filippov bialgebroid of order $n$ (cf. Corollary $6.3$, [@bas-bas-das-muk]). It is also proved that if $(G, \Pi)$ is a Nambu-Lie group [@vais] of order $n$ with its Lie algebra $\mathfrak{g},$ then $(\mathfrak{g}, \mathfrak{g}^*)$ forms a (weak) Lie-Filippov bialgebroid of order $n$ over a Point. It is known that the base of a Lie bialgebroid carries a natural Poisson structure. In [@bas-bas-das-muk] it has been extended to the Nambu-Poisson set up. \[wlfb-base-nambu\]([@bas-bas-das-muk]) Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid (of order n) over $M$. Then the bracket $$\begin{aligned} \{f_1, \ldots, f_n\}_{(A, A^*)} := \rho (d_A f_1 \wedge \cdots \wedge d_A f_{n-1}) f_n\end{aligned}$$ defines a Nambu-Poisson structure of order n on $M$. It is known that, given a coisotropic submanifold $C$ of a Poisson manifold $M$, the conormal bundle $(TC)^0 \rightarrow C$ is a Lie subalgebroid of the cotangent Lie algebroid $T^*M$ [@wein]. If $M$ is a Nambu-Poisson manifold of order $n$ ($n \geqslant 3$), the cotangent bundle $T^*M$ is not a Filippov algebroid. However we have the following useful result. \[coiso-n-bracket\] Let $C$ be a closed embedded coisotropic submanifold of a Nambu-Poisson manifold $(M, \Pi)$ of order $n$. Then 1. the bundle map $\Pi^{\sharp} : \bigwedge^{n-1}T^*M \rightarrow TM$ maps $\bigwedge^{n-1} (TC)^0$ to $TC$; 2. the Nambu-form bracket on the space of $1$-forms $\Omega^1(M)$ can be restricted to the sections of the conormal bundle $ (TC)^0 \rightarrow C$. The assertion $(1)$ follows from the definition of coisotropic submanifold. To prove $(2)$, let $\alpha_1, \ldots ,\alpha_n \in \Gamma (TC)^0.$ We extend them to $1$-forms on $M$, which we denote by the same notation. Let $X \in \mathcal{X}^1(M)$ be such that $X\big|_C$ is tangent to $C$. From the definition of Nambu-form bracket on $1$-forms, we have $$\begin{aligned} \langle [\alpha_1, \ldots, \alpha_n], X \rangle = \sum_{k=1}^n (-1)^{n-k} \langle \mathcal{L}_{\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \widehat{\alpha}_k \wedge \cdots \wedge \alpha_n)} \alpha_k, X \rangle - (n-1) \langle d (\Pi(\alpha_1,\ldots, \alpha_n)), X \rangle. \end{aligned}$$ Observe that $$\begin{aligned} \langle \mathcal{L}_{\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \widehat{\alpha}_k \wedge \cdots \wedge \alpha_n)} \alpha_k, X \rangle =& \mathcal{L}_{ \Pi^\sharp (\alpha_1 \wedge \cdots \wedge \widehat{\alpha}_k \wedge \cdots \wedge \alpha_n )} \langle \alpha_k, X \rangle \\ &- \langle \alpha_k, [\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \widehat{\alpha}_k \wedge \cdots \wedge \alpha_n), X]\rangle .\end{aligned}$$ This is zero on $C$, because, - $\langle \alpha_k, X\rangle$ is zero on $C$; - $\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \widehat{\alpha_k} \wedge \cdots \wedge \alpha_n)$ and $X$ are both tangent to $C$ and hence their Lie bracket is also tangent to C. Thus its pairing with $\alpha_k$ vanish on $C$. Note that $\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \alpha_{n-1})\big|_C$ is tangent to $C$ and $\alpha_n \big|_C \in (TC)^0.$ As a consequence, the function $$\begin{aligned} \Pi (\alpha_1,\ldots, \alpha_n) = \langle \alpha_n, \Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \alpha_{n-1})\rangle\end{aligned}$$ is zero on $C.$ Therefore, the differential $d (\Pi(\alpha_1, \ldots, \alpha_n))$ restricted to $C$ is in $(TC)^0$, which in turn implies that the second term of the right hand side also vanish on $C$. Hence $$\begin{aligned} [\alpha_1, \ldots, \alpha_n]\big|_C \in (TC)^0.\end{aligned}$$ One can check that the restriction to $C$ does not depend on the chosen extension. Hence it defines a bracket on the sections of the conormal bundle $(TC)^0 \rightarrow C$. 1. Let $m_0 \in M$ such that $\Pi (m_0) = 0$. Then $\{m_0\}$ is a coisotropic submanifold of $M$. In this case, the conormal structure becomes $T_{m_0}^*M$, which is a Filippov algebra. 2. The Nambu structure of a Nambu-Lie group G vanishes at the identity element and therefore the dual $\mathfrak{g}^*$ of the Lie algebra $\mathfrak{g}$ of G has a Filippov algebra structure [@vais]. \[nlg-bracket-anchor\] Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$. By the Proposition \[coiso-n-bracket\], we see that the space of sections of the conormal bundle $ A^*G = (TM)^0 \rightarrow M$ admits a skew-symmetric $n$-bracket $[~,\ldots,~]$ and there exists a bundle map $$\rho := \Pi^{\sharp}\big|_{{\bigwedge}^{n-1} (TM)^0} : {\bigwedge}^{n-1} A^*G = {\bigwedge}^{n-1} (TM)^0 \rightarrow TM ,$$ as $M$ is a coisotropic submanifold of $G$. Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$. Let $f \in C^\infty(M).$ Then by part ($5$) of the Theorem \[multiplicative\], $\iota_{d(\beta^*f)} \Pi$ is a left invariant $(n-1)$-vector field on $G$. Therefore, there exists an $(n-1)$-multisection $\delta^0_\Pi (f)\in \Gamma{\bigwedge^{n-1}AG}$ of the Lie algebroid $AG$ such that $$\begin{aligned} \iota_{d(\beta^*f)} \Pi = \overleftarrow{\delta^0_\Pi (f)} .\end{aligned}$$ Then we have the following result. Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid of order $n$ and $AG \rightarrow M$ be its Lie algebroid. Then for any $ X \in \Gamma AG$, $$\begin{aligned} \mathcal{L}_{\overleftarrow{X}} \Pi := [\overleftarrow{X}, \Pi]\end{aligned}$$ is a left invariant $n$ vector field on $G$, where $\overleftarrow{X}$ is the left invariant vector field on $G$ corresponding to $X$. Moreover $\mathcal{L}_{\overleftarrow{X}} \Pi$ corresponds to the $n$-multisection $- \delta^1_\Pi (X) \in \Gamma{\bigwedge^n AG}$, that is, $$\begin{aligned} \mathcal{L}_{\overleftarrow{X}} \Pi = -\overleftarrow{\delta^1_\Pi (X)}\end{aligned}$$ where $\delta^1_\Pi (X) \in \Gamma{\bigwedge^n AG}$ is given by $$\begin{aligned} \delta^1_\Pi (X) (\alpha_1, \ldots, \alpha_n) = \sum_{k=1}^{n} (-1)^{n-k} \Pi^{\sharp} (\alpha_1 \wedge \cdots \wedge \hat{\alpha}_k \wedge \cdots \wedge \alpha_n) (X(\alpha_k)) - X([\alpha_1,\ldots, \alpha_n])\end{aligned}$$ for $\alpha_1, \ldots, \alpha_n \in \Gamma A^*G = \Gamma (TM)^0.$ Let $\mathcal{X}_t = \text{exp} \hspace*{0.05cm} tX$ be the one-parameter family of bisections generated by $X \in \Gamma AG.$ Let $g \in G$ with $\beta(g) = u$. Let $u_t = (\text{exp} \hspace*{0.05cm} tX) (u)$ be the integral curve of $\overleftarrow{X}$ starting from $u$. If $\mathcal{G}$ is any (local) bisection through $g$, then from the multiplicativity condition of $\Pi$ (cf. Theorem \[multiplicative\]), we have $$\begin{aligned} \Pi ( g u_t) = (r_{\mathcal{X}_t})_* \Pi(g) + (l_{\mathcal{G}})_* \Pi(u_t)- (r_{\mathcal{X}_t})_* (l_{\mathcal{G}})_* \Pi(u).\end{aligned}$$ Therefore, $$\begin{aligned} (r_{\mathcal{X}^{-1}_t})_* \Pi ( g u_t) - \Pi(g) = (r_{\mathcal{X}^{-1}_t})_* (l_{ \mathcal{G}})_* \Pi(u_t) - (l_{\mathcal{G}})_* \Pi (u).\end{aligned}$$ Taking derivative at $t = 0$, one obtains $$\begin{aligned} (\mathcal{L}_{\overleftarrow{X}} \Pi)(g) = (l_{\mathcal{G}})_* ((\mathcal{L}_{\overleftarrow{X}} \Pi)(u)).\end{aligned}$$ Therefore, $\mathcal{L}_{\overleftarrow{X}} \Pi $ is left invariant by the Proposition \[left-invariant\] and hence it corresponds to some $n$-multisection of $AG$. To show that $\mathcal{L}_{\overleftarrow{X}} \Pi$ corresponds to $- \delta^1_\Pi (X) \in \Gamma {\bigwedge^n AG}$, we have to check that $\mathcal{L}_{\overleftarrow{X}} \Pi$ and $- \overleftarrow{\delta^1_\Pi (X)}$ coincide on the unit space $M$ (both being left invariant). Since both of them are tangent to $\alpha$-fibers, it is enough to show that they coincide on the conormal bundle $(TM)^0$. Let $\alpha_1, \ldots, \alpha_n$ be any sections of $(TM)^0$ and $\tilde{\alpha}_1, \ldots, \tilde{\alpha}_n$ be their respective extensions to one forms on $G$. Observe that $$\begin{aligned} & (\mathcal{L}_{\overleftarrow{X}} \Pi)\big|_M (\alpha_1, \ldots, \alpha_n)\\ =& \big[\langle \overleftarrow{X}, d (\Pi(\tilde{\alpha}_1,\ldots,\tilde{\alpha}_n)) \rangle - \sum_{k=1}^{n} \Pi(\tilde{\alpha}_1 ,\ldots, \mathcal{L}_{\overleftarrow{X}} \tilde{\alpha}_k,\ldots, \tilde{\alpha}_n)\big]\big|_M\\ =& \big[\langle \overleftarrow{X}, [\tilde{\alpha}_1,\ldots,\tilde{\alpha}_n] \rangle - \sum_{k=1}^n (-1)^{n-k} \langle \overleftarrow{X}, \iota_{\Pi^{\sharp} (\tilde{\alpha}_1 \wedge\cdots \wedge \hat{\tilde{\alpha}}_k \wedge\cdots \wedge \tilde{\alpha}_n)} d\tilde{\alpha}_k \rangle\\ & - \sum_{k=1}^{n} (-1)^{n-k} \langle \Pi^{\sharp} (\tilde{\alpha}_1 \wedge\cdots \wedge \hat{\tilde{\alpha}}_k \wedge\cdots \wedge \tilde{\alpha}_n), \mathcal{L}_{\overleftarrow{X}} \tilde{\alpha}_k \rangle \big]\big|_M\\ & \hspace*{5cm} (\text{from the Equation}~ (\ref{nambu-bracket}))\\ =& \big[\langle \overleftarrow{X}, [\tilde{\alpha}_1,\ldots,\tilde{\alpha}_n] \rangle - \sum_{k=1}^n (-1)^{n-k} \langle \Pi^{\sharp} (\tilde{\alpha}_1 \wedge\cdots \wedge \hat{\tilde{\alpha}}_k \wedge\cdots \wedge \tilde{\alpha}_n), d \iota_{\overleftarrow{X}} \tilde {\alpha}_k \rangle \big]\big|_M\\ & \hspace*{5cm} (\text{using Cartan formula})\\ =& \langle X, [\alpha_1,\ldots,\alpha_n] \rangle - \sum_{k=1}^n (-1)^{n-k} \Pi^{\sharp} ({\alpha}_1 \wedge\cdots \wedge \hat{\alpha}_k \wedge\cdots \wedge {\alpha_n}) (X (\alpha_k))\\ = & -\delta^1_\Pi (X) (\alpha_1,\ldots, \alpha_n).\end{aligned}$$ To make our notation simple, let us denote $\delta^0_\Pi, \delta^1_\Pi$ by the same symbol $\delta_\Pi$. We extend $\delta_\Pi$ to the graded algebra $\Gamma{\bigwedge^{\bullet}A}$ of multisections of $AG$ by the following rule $$\begin{aligned} \delta_\Pi (P \wedge Q) = \delta_\Pi (P) \wedge Q + (-1)^{|P| (n-1)} P \wedge \delta_\Pi (Q)\end{aligned}$$ for $P \in \Gamma{\bigwedge^{|P|}A}, Q \in \Gamma{\bigwedge^{|Q|}A}$. Then the operator $$\begin{aligned} \delta_\Pi : \Gamma{{\bigwedge}^kAG} \rightarrow \Gamma{{\bigwedge}^{k+n -1} AG}\end{aligned}$$ satisfies $$\begin{aligned} \delta_\Pi ([P,Q]) = [ \delta_\Pi (P), Q] + (-1)^{(|P|-1)(n-1)} [P, \delta_\Pi (Q)].\end{aligned}$$ Note that the operator $\delta_\Pi$ need not satisfy condition $\delta_\Pi \circ \delta_\Pi = 0 .$ We known that, Lie bialgebroids are infinitesimal form of Poisson groupoids. More precisely, given a Poisson groupoid $G \rightrightarrows M$ with Lie algebroid $AG$, it is known that its dual bundle $A^*G$ also carries a Lie algebroid structure and $(AG, A^*G)$ forms a Lie bialgebroid. In the next theorem we show that weak Lie-Filippov bialgebroids are infinitesimal form of Nambu-Lie groupoids. \[nambu-grpd-bialgbd\] Let $(G \rightrightarrows M , \Pi) $ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$. Then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ over $M$. From the Remark \[nlg-bracket-anchor\], we have the space of sections of the bundle $A^*G = (TM)^0 \rightarrow M$ admits a skew-symmetric $n$-bracket $[~, \ldots, ~]$ and there exists a bundle map $$\rho : {\bigwedge}^{n-1}A^*G \rightarrow TM.$$ Let $\alpha_1, \ldots, \alpha_{n-1} \in \Gamma (TM)^0=\Gamma{(A^*G)} $ with $d_A \alpha_i = 0$, for all $i= 1, \ldots ,n-1.$ Let $\tilde{\alpha}_1, \ldots, \tilde{\alpha}_{n-1}$ be, respectively, their extensions to left invariant $1$-forms on $G$ such that $d \tilde{\alpha}_i = 0$, for all $i= 1, \ldots ,n-1.$ Then the conditions $(2)$ and $(3)$ of the Definition \[lie-fill-defn\] of a weak Lie-Filippov algebroid pair follows from the weak Lie-Filippov bialgebroid structure $(TG, T^*G)$ (Note that $G$ is a Nambu-Poisson manifold). Let $f \in C^\infty(M).$ Then observe that $$\begin{aligned} [\tilde {\alpha}_1, \ldots, \tilde{\alpha}_{n-1}, (\beta^*f) \tilde{\alpha}_n ] = (\beta^*f) [\tilde{\alpha}_1, \ldots, \tilde{\alpha}_{n-1}, \tilde{\alpha}_n ] + \Pi^{\sharp}(\tilde{\alpha}_1 \wedge \cdots \wedge \tilde{\alpha}_{n-1})(\beta^*f)\tilde{\alpha}_n.\end{aligned}$$ Since $(\beta^*f) \tilde{\alpha}_n = \widetilde{f \alpha}_n$, by assertion $(2)$ of the Propostion \[coiso-n-bracket\], we get $$\begin{aligned} [\alpha_1,\ldots, \alpha_{n-1}, f \alpha_n] = f [\alpha_1, \ldots, \alpha_n] + \rho (\alpha_1 \wedge \cdots \wedge \alpha_{n-1}) (f) \alpha_n\end{aligned}$$ proving condition (4) of the Definition \[lie-fill-defn\]. Moreover the compatibility condition of the weak Lie-Filippov bialgebroid (condition $(5)$ of the Definition \[lie-fill-defn\]) follows from the observation that for any $\alpha \in \Gamma{(A^*G)} = \Gamma (TM)^0$ and any left invariant extension $\tilde{\alpha} \in \Omega^1(G)$, we have $$\begin{aligned} d_A \alpha = (d \tilde \alpha)|_M.\end{aligned}$$ Thus, $(AG, A^*G)$ is a weak Lie-Filippov bialgebroid of order $n$. If $(G, \Pi)$ is a Nambu-Lie group with Lie algebra $\mathfrak{g}$, the dual vector space $\mathfrak{g}^*$ carries a Filippov algebra structure [@vais]. Moreover the pair $(\mathfrak{g} , \mathfrak{g}^*)$ forms a (weak) Lie-Filippov bialgebra ([@bas-bas-das-muk; @vais]). The Lie-Filippov bialgebra $(\mathfrak{g}, \mathfrak{g}^*)$ is the infinitesimal form of the Nambu-Lie group $(G, \Pi).$ A Lie-Filippov bialgebra $(\mathfrak{g}, \mathfrak{g}^*)$ can also be seen a Lie algebra $\mathfrak{g}$ together with a Filippov algebra structure on the dual vector space $\mathfrak{g}^*$ such that the map $\delta : \mathfrak{g} \rightarrow \bigwedge^n \mathfrak{g}$ dual to the Filippov bracket on $\mathfrak{g}^*$, defines a $1$-cocycle of $\mathfrak{g}$ with respect to the adjoint representation on $\bigwedge^n \mathfrak{g}$. We have seen that given a Nambu-Lie groupoid of order $n$, there is an induced Nambu-Poisson structure on the base manifold (cf. Proposition \[inverse-basenambu\]). On the other hand, given a weak Lie-Filippov bialgebroid, there is an induced Nambu-Poisson structure on the base (cf. Theorem \[wlfb-base-nambu\]). The next proposition compares these Nambu-Poisson structures on the base induced from the Nambu Lie groupoid and its infinitesimal. \[compare-two-NP-structure\] Let $(G \rightrightarrows M , \Pi) $ be a Nambu-Lie groupoid (of order $n$) with associated weak Lie-Filippov bialgebroid $(AG, A^*G).$ Then the induced Nambu structures on $M$ coming from the Nambu-Lie groupoid and the weak Lie-Filippov bialgebroid are related by $$\begin{aligned} \{ ~, \ldots, ~\}_M = (-1)^{n-1}\{ ~, \ldots, ~\}_{(AG, A^*G)} .\end{aligned}$$ For any functions $f_1, \ldots, f_n \in C^\infty(M)$, we have $$\begin{aligned} \{f_1, \ldots, f_n\}_{(AG, A^*G)} =& \Pi^{\sharp}\big|_M (d_A f_1 \wedge \cdots \wedge d_A f_{n-1}) f_n\\ =& \Pi^{\sharp} ( d (\beta^* f_1) \wedge \cdots \wedge d (\beta^* f_{n-1}))\big|_M f_n\\ =& \Pi (\beta^* f_1 , \ldots, \beta^* f_{n-1}, \beta^* f_n)\big|_M\\ =& (-1)^{n-1} \big( \beta^*\{f_1, \ldots, f_n\} \big) \big|_M\\ =& (-1)^{n-1} \{f_1, \ldots, f_n\}_M.\end{aligned}$$ It is known that under some connectedness and simply connectedness assumption, any Lie bialgebra integrates to a Poisson-Lie group [@lu-wein], and any Lie bialgebroid integrates to a Poisson groupoid [@mac-xu2]. These results does not hold in the context of Nambu structures of order $\geqslant 3$. Let $G$ be a connected and simply-connected Lie group with Lie algebra $\mathfrak{g}.$ Given a Lie-Filippov bialgebra structure $(\mathfrak{g}, \mathfrak{g}^*)$ on $\mathfrak{g}$, the $1$-cocycle $\delta : \mathfrak{g} \rightarrow \bigwedge^n \mathfrak{g}$ dual to the Filippov algebra bracket on $\mathfrak{g}^*$ integrates a multiplicative $n$-vector field $\Pi$ on the Lie group. However this $n$-vector field (for $n \geqslant 3$) need not be a Nambu tensor [@vais], that is, need not be locally decomposable. Thus (weak) Lie-Filippov bialgebra does not integrate to a Nambu-Lie group in general. Coisotropic subgroupoids of a Nambu-Lie groupoid ================================================ In this final section, we introduce the notion of coisotropic subgroupoid of a Nambu-Lie groupoid and study the infinitesimal object corresponding to it. Let $(G \rightrightarrows M , \Pi)$ be a Nambu-Lie groupoid of order $n$. Then a subgroupoid $H \rightrightarrows N$ is called a [*coisotropic subgroupoid*]{} if $H$ is a coisotropic submanifold of $G$ with respect to $\Pi$. 1. For $n=2$, that is, when $G \rightrightarrows M$ is a Poisson groupoid, this notion is same as the coisotropic subgroupoid of a Poisson groupoid introduced in [@xu]. 2. Let $(G, \Pi)$ be a Nambu-Lie group. Then a subgroup of $G$ is called coisotropic if it is also a coisotropic submanifold of $G$. Any coisotropic subgroup of $G$ is a coisotropic subgroupoid over a point. 3. Let $(G \rightrightarrows M , \Pi)$ be a Nambu-Lie groupoid. Then by the Proposition \[inverse-basenambu\], there exist an induced Nambu-structure on $M$ for which the source map $\alpha$ is a Nambu-Poisson map. Let $N \hookrightarrow M$ be a coisotropic submanifold of $M$ with respect to this induced Nambu structure. Consider the restriction $G\big|_N := \alpha^{-1}(N) \cap \beta^{-1}(N)$, then $G\big|_N \rightrightarrows N$ is a coisotropic subgroupoid. 4. Let $G \rightrightarrows M$ be a Nambu-Lie groupoid. If the set of all elements of $G$ which has same source and target, is a submanifold of $G$, then it is a coisotropic subgroupoid. Note that, the infinitesimal object corresponding to a Nambu-Lie groupoid $(G\rightrightarrows M , \Pi)$ is the weak Lie-Filippov bialgebroid $(AG, A^*G)$. Therefore it is natural to ask how the Lie algebroid of a coisotropic subgroupoid $H \rightrightarrows N$ is related to the weak Lie-Filippov bialgebroid $(AG, A^*G)$. To answer this question, we introduce a notion of [*coisotropic subalgebroid*]{} of a weak Lie-Filippov bialgebroid and show that infinitesimal forms of coisotropic subgroupoids of a Nambu-Lie groupoid $(G \rightrightarrows M, \Pi)$ appear as coisotropic subalgebroids of the corresponding weak Lie-Filippov bialgebroid $(AG, A^*G)$. \[coiso-subalg\] Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid of order $n$ over $M$. Then a Lie subalgebroid $B \rightarrow N$ of $A\rightarrow M$ is called a [*coisotropic subalgebroid*]{} if the anchor $\rho: \bigwedge^{n-1}A^* \rightarrow TM$ and the $n$-bracket $[~, \ldots, ~]$ on $\Gamma{A^*}$ satisfy the following properties. 1. The anchor $\rho$ maps $\bigwedge^{n-1}B^{0} \rightarrow TN$. 2. If $\alpha_1, \ldots, \alpha_n \in \Gamma{A^*}$ with ${\alpha_i}\big|_N \in B^{0}$ for all $i$, then $[\alpha_1, \ldots, \alpha_n]\big|_N \in B^{0}$. 3. If $\alpha_1, \ldots, \alpha_n \in \Gamma{A^*}$ with ${\alpha_i}\big|_N \in B^{0}$ for all $i$ and $\alpha_n\big|_N = 0$, then $[\alpha_1, \ldots, \alpha_n]\big|_N = 0,$ where $B_x^{0} = \{\gamma \in A_x^* | \gamma (v) = 0, \forall v \in B_x\}$, is the annihilator of $B_x$, $x \in N$. Let $M$ be a Nambu-Poisson manifold, then $(TM, T^*M)$ is a weak Lie-Filippov bialgebroid over $M$. Let $N \hookrightarrow M$ be a coisotropic submanifold. Then from the Proposition \[coiso-n-bracket\], it follows that the tangent bundle $TN \rightarrow N$ is a coisotropic subalgebroid. It is known that (Proposition \[wlfb-base-nambu\], see also [@bas-bas-das-muk]), the base of a weak Lie-Filippov bialgebroid carries a Nambu structure. The next Proposition shows that the base of a coisotropic subalgebroid is a coisotropic submanifold with respect to this induced Nambu structure. Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid over $M$ and $B \rightarrow N$ be a coisotropic subalgebroid. Then $N$ is a coisotropic submanifold of $M$. Let $a : A \rightarrow TM$ denote the anchor of the Lie algebroid $A$ and $\rho : \bigwedge^{n-1}A^* \rightarrow TM$ be the anchor of pair $(A, A^*).$ We first show that, $a^* (TN)^0 \subseteq B^{0}$. This is true because, $ \langle a^*\xi_x, v \rangle = \langle \xi_x, a(v)\rangle = 0$ for $\xi_x \in (TN)_x^0$ and $v \in B_x$. Let $\Pi_{(A, A^*)}$ be the induced Nambu structure on $M$ coming from the weak Lie-Filippov bialgebroid $(A, A^*).$ Then the induced map $\Pi^{\sharp}_{(A, A^*)} : \bigwedge^{n-1}T^*M \rightarrow TM$ is given by $\Pi^{\sharp}_{(A, A^*)} = \rho \circ \bigwedge^{n-1}a^*$. Therefore, for any $\xi_1, \ldots, \xi_{n-1} \in (TN)^0$, we have $$\Pi^{\sharp}_{(A, A^*)} (\xi_1, \ldots, \xi_{n-1}) = \rho (a^*\xi_1, \ldots, a^*\xi_{n-1}) \in TN$$ as $a^*\xi_i \in B^{0}$ and $B$ is a coisotropic subalgebroid. Therefore $N$ is a coisotropic submanifold of $M$. The next proposition shows that the infinitesimal object corresponding to coisotropic subgroupoids are coisotropic subalgebroids. \[coiso-subgrpd-coiso-subalgbd\] Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid with weak Lie-Filippov bialgebroid $(AG, A^*G)$. Let $H \rightrightarrows N$ be a coisotropic subgroupoid of $G \rightrightarrows M$ with Lie algebroid $AH \rightarrow N$. Then $AH \rightarrow N$ is a coisotropic subalgebroid. Since $H \rightrightarrows N$ is a Lie subgroupoid of $G \rightrightarrows M$, therefore $AH \rightarrow N$ is a Lie subalgebroid of $AG \rightarrow M$. We claim that the anchor $\rho = \Pi^{\sharp}\big|_{\bigwedge^{n-1}(TM)^0} = \Pi^{\sharp}\big|_{\bigwedge^{n-1} (A^*G)}$ of the weak Lie-Filippov bialgebroid $(AG, A^*G)$ maps $\bigwedge^{n-1}(AH)^0$ to $TN$. First observe that, for any $x \in N$, $(AH)_x^0 = (TM)_x^0 \cap (TH)_x^0$ and $T_xN = T_xM \cap T_xH$. Therefore, $\rho$ maps $\bigwedge^{n-1} (AH)^0$ to $$\Pi^{\sharp}({\bigwedge}^{n-1} (TM)^0) \cap \Pi^{\sharp}({\bigwedge}^{n-1} (TH)^0) \subseteq TM \cap TH \cong TN,$$ here we have used the fact that $M$ and $H$ are both coisotropic submanifolds of $G$. Let $\alpha_1, \ldots, \alpha_n \in \Gamma{A^*G} = \Gamma{(TM)^0}$ such that $\alpha_i\big|_N \in (AH)^0$, for all $i=1, \ldots, n$. Let $\tilde{\alpha}_1, \ldots, \tilde{\alpha}_n$ be one forms on $G$ extending $\alpha_1, \ldots, \alpha_n$ and are conormal to $H$. Then by the Proposition \[coiso-n-bracket\], the $1$-form $[\tilde{\alpha_1}, \ldots, \tilde{\alpha_n}]$ is conormal to both $M$ and $H$, as $M$ and $H$ are both coisotropic submanifolds of $G$. Therefore, $$[\tilde{\alpha_1}, \ldots, \tilde{\alpha_n}]\big|_N \in (TM)^0 \cap (TH)^0 \cong (AH)^0.$$ Verification of the last condition of the Definition \[coiso-subalg\] is similar. Hence $AH \rightarrow N$ is a coisotropic subalgebroid of $(AG, A^*G).$ Let $(G \rightrightarrows M , \Pi)$ be a Nambu-Lie groupoid and $H \rightrightarrows N$ be a coisotropic subgroupoid. Then $N$ is a coisotropic submanifold of $M$. [BFGM03]{} Samik Basu, Somnath Basu, Apurba Das and Goutam Mukherjee, [[*Nambu structures and associated Bialgebroids*]{}]{}, preprint, arXiv:1502.06533 Jean-Paul Dufour and Nguyen Tien Zung, [[*Poisson structures and their normal forms*]{}]{}, Birkh$\ddot{\text{a}}$user Verlag, Basel-Boston-Berlin (2005). V. T. Filippov, [[*n-Lie algebras*]{}]{}, Sib. Math. Zh., [**26 (6)**]{}, 126-140, (1985). J. Grabowski and G. Marmo, [[*On Filippov algebroid and multiplicative Nambu structures*]{}]{}, Diff. Geom. Appl. [**12**]{} (2000) 35-50. R. Ib$\acute{\text{a}}\tilde{\text{n}}$ez, M. de Le$\acute{\text{o}}$n, J.C. Marrero, E. Padr$\acute{\text{o}}$n, [[*Leibniz algebroid associated with a Nambu-Poisson structure*]{}]{}, J. Phys. A: Math. Gen. [**32**]{} (1999) 8129-8144. R. Ib$\acute{\text{a}}\tilde{\text{n}}$ez, M. de Le$\acute{\text{o}}$n , J.C. Marrero and D. M. de Diego, [[*Reduction of generalized Poisson and Nambu-Poisson manifolds*]{}]{}, Rep. Math. Phys. [**42**]{} (1998) 71-90. Y. Kosmann-Schwarzbach; [[*Exact Gerstenhaber algebras and Lie bialgebroids*]{}]{}, Acta applicandae Mathematicae [[**41**]{}]{}: 153-165, 1995. Jiang-Hua Lu and Alan Weinstein, [[*Poisson Lie groups, dressing transformations, and Bruhat decompositions*]{}]{}, J. Diff. Geom. [**31**]{} (1990) 501-526. Kirill C.H. Mackenzie, [[*Lie groupoids and Lie algebroids in Differential Geometry*]{}]{}, Cambridge University Press, Cambridge, (2005). Kirill C.H. Mackenzie and Ping Xu, [[*Lie bialgebroids and Poisson groupoids*]{}]{}, Duke Math. J. [**73**]{}, (1994) 415-452. Mackenzie, Kirill C. H., and Xu, P.: [[*Integration of Lie bialgebroids*]{}]{}, Topology [**39**]{} (2000) 445-467. Yoichiro Nambu, [[*Generalized Hamiltonian dynamics*]{}]{}, Phys. Rev. D, [**7**]{} (1973) 2405-2412. David Iglesias Ponte, Camille Laurent-Gengoux and Ping Xu, [[*Universal lifting theorem and quasi-Poisson groupoids*]{}]{}, J. Eur. Math. Soc. (JEMS) [**14**]{} (2012), no. 3, 681-731. Leon Takhtajan, [*On foundation of the Generalized Nambu Mechanics*]{}, Comm. Math. Phys. Volume [**160**]{}, Number 2 (1994), 295-315. Izu Vaisman, [[*Lectures on the Geometry of Poisson Manifolds*]{}]{}, Progress in Mathematics, Vol. 118, Birkh$\ddot{\text{a}}$user Verlag, Basel, (1994). Izu Vaisman, [[*Nambu-Lie groups*]{}]{}, J. Lie Theory [**10**]{} (2000), 181-194. Alan Weinstein, [[*Coisotropic calculus and Poisson groupoids*]{}]{}, J. Math. Soc. Japan [**40**]{} (1988) 705-727. Ping Xu, [[*On Poisson groupoids*]{}]{}, Inter. J. math. [**6**]{}, No. 1, 101-124 (1995).

--- abstract: 'It is shown how initial conditions can be appropriately defined for the integration of Lorentz-Dirac equations of motion. The integration is performed [*forward*]{} in time. The theory is applied to the case of the motion of an electron in an intense laser pulse, relevant to nonlinear Compton scattering.' address: - | $^1$Instituto de Física, Universidade de São Paulo,\ C.P.66318, 05315-970, São Paulo, SP, Brazil - | $^2$Institute for Theoretical Physics, University of California,\ Santa Barbara, 93106-4030, USA - | $^3$Institute for Theoretical Atomic and Molecular Physics,\ Harvard-Smithsonian Center for Astrophysics,\ Cambridge, Massachusetts, 02138, USA author: - | M.S. Hussein$^{1}$, M.P. Pato$^{1,2}$, and J.C. Wells$^{2,3\thanks{Present address: Center for Computional Sciences, Oak Ridge National Laboratory, Oak Ridge, Tenessee, 37831-6373, U.S.A.}}$ title: Causal Classical Theory of Radiation Damping --- -20mm The advent of a new generation of extremely high power lasers that uses chirped pulse amplification has put into focus the classical description of the dynamics of relativistic electrons. Under the action of high intensity electromagnetic field, a major ingredient of the dynamics is the electron self-interaction which implies in the damping of the movement caused by the interaction of the charge with its own field. The derivation of the damping force has been reviewed recently[@Luhm] revealing its relativistic origin asociated to the asymmetry introduced by the Doppler effect in the forward and backward emission of radiation. The inclusion of this force in the equation of motion leads to the nonlinear covariant Lorentz-Dirac(LD) equation[@Dirac] for a point charge. Nonlinear effects have been observed in recent experiments with intense-laser relativistic-electron scattering at laser frequencies and field strengths where radiation reaction forces begin to become significant[@Hart; @Bula]. Relativistic nonlinear Thompson scattering has also been observed[@Yuan]. These experiments justify the recent attention the classical electrodynamics theory(CED) has received. A self-consistent classical theoretical treatment of the radiative reaction force would also be useful in simulating future electron accelerators[@HP92]. Further, the kind of study reported upon in this paper may be useful in the quantal treatment[@FLow]. A certain number of problems, conceptual and practical, is known to be associated with LD equation. These difficulties may be traced to the fact that it contains a dependence on the derivative of the acceleration which implies in the necessity of imposing, in order to solve it, an extra condition besides the usual initial conditions on position and velocity of classical mechanics. It has been estabilished by Rohrlich that this condition is given by the asymptotic constraint that at the far future, when the fields vanish, the acceleration should also vanish. The solution which is obtained however, when extrapolated to the instant when the external force is applied violates causality. On the other hand, from the practical point of view, a condition put in the far future is awkward to be implemented specially in a scattering situation. A recent book[@Yagh] has been devoted to this question of causality violation. A clear explanation to its origin was given and it has been shown how it can be solved in the context of the CED. On the other side, attempts has been done in order to replace Rohrlich condition by an equivalent initial condition[@Agui; @Vill]. The purpose of this letter is to apply these modern advances, to the case of the classical description based in LD equation, of the movement of an electron interacting with a short, strong laser pulse. This problem has been recently discussed in the literature Ref. [@Hart], which provided numerical solutions of the Lorentz-Dirac equation. The integration was performed backwards in time so that the unphysical, exponentially growing homogeneous solutions of LD would damp out, resulting in a numerical stable solution. We are going to show that Lorentz-Dirac equation of motion can be integrated forward in time with conditions specified at $t=0$. The idea is to construct the series solution of LD equation. The initial acceleration is then provided by replacing in the series the velocity by their initial value at the instant when the external force is applied. It is easy to show that the solution obtained with this procedure, when extrapolated to the distant future, satisfies Rohrlich condition. However, we still have to cope with the existence of the unphysical runaway solutions which although formally eliminated troubles the process of numerical integration. Then, by combining the recursive use of the series solution with implicit methods of numerical integration we show that the process of integration forward in time can be performed. We write Lorentz-Dirac (LD) equation of motion as $$F_{\mu }^{ext}=a_{\mu }-\epsilon \left( \frac{d^{2}v_{\mu }}{d\tau ^{2}}+v_{\mu }a_{\lambda }a^{\lambda }\right) \label{2}$$ where $v_{\mu }$,$a_{\mu }$ and $F_{\mu }^{ext}$ are, respectively, the four-vector components of the velocity,acceleration and of the external force given explicitly by $$v_{\mu }=\gamma \left( 1,{\bf \beta }\right) , \label{4}$$ $$v^{\mu }=\gamma \left( 1,-{\bf \beta }\right) , \label{4a}$$ $$a_\mu =\frac{dv_\mu }{d\tau }$$ and $$F_{\mu }^{ext}=\gamma \left( {\bf \beta }\cdot {\bf F}_{ext},{\bf F}_{ext}\right) . \label{6}$$ In these equations, $\tau $ is the dimensionless proper time $d\tau =\omega _{0}dt/\gamma $, $\gamma $ is the relativistic factor $\gamma =\frac{1}{\sqrt{1-\beta ^{2}}}$ with ${\bf \beta }=\frac{1}{c}\frac{d{\bf r}}{dt}$ and $\omega _{0}$ is the frequency of the laser pulse with which the electron is interacting. We follow here the same units of Ref. [@Hart]. With these definitions, it can be easily verified that $v_{\mu }a^{\mu }=0$ as it should. The quantity $\varepsilon =\omega _{0}\tau _{0},$ where $\tau _{0}=e^{2}/m_{e}c^{3}=0.626\times 10^{-23}s$ is the Compton time scale. We are assuming that the external force applies at $\tau =0$ and the second term in the right hand side of the equation represents the damping force arising as the charge starts to radiate[@Rohr]. By contracting the Lorentz-Dirac equation with $a_{\mu }$, it is found that $\left( a_{\mu }a^{\mu }\right) \left( \tau \right) =C\exp (2\tau /\epsilon )$ is a solution of the resulting homogeneous equation (no external force) for times greater than $\tau _{0}$. These are the so-called runaway solutions. They are eliminated by imposing the Dirac-Rohrlich (DR) condition $$\lim\limits_{\tau \rightarrow \infty }a_\mu =0 \label{8}$$ when $$\lim\limits_{\tau \rightarrow \infty }F_\mu ^{ext}\left( \tau \right) =0. \label{10}$$ In contrast to the usual initial value problem encountered in classical physics, where all quantities, position, velocity, are fixed at $t=0$, the above asymptotic condition, Eq. (\[8\]) recast the problem into a boundary value one where the $x,v$ are known at $\tau =0$ and $a$ is forced to be zero at $\tau =\infty $, in accordance with the condition on the force, Eq. (\[10\]). A particular situation is that of Ref. [@Hart], in which the problem was solved in the rest frame of the particle at some future time with three final homogeneous conditions. In both situations however, the numerical integration has to be performed backwards in time. Although this procedure eliminates the unphysical runway solutions, it is uncomfortable, from the point of view of applications, to have a condition given at some final time. It is therefore desirable to have an equivalent condition on the acceleration defined at the initial time. For an electron moving in an electromagnetic field the external force is given by $$F_{\mu }^{ext}=-(\partial _{\nu }A_{\mu }-\partial _{\mu }A_{\nu })v^{\nu } \label{12}$$ where the quantity $A_{\mu }$ is the vector potential given in units of $m_{0}c/e$. For a linearly polarized laser pulse, $$A_{\mu }\equiv (\Phi /c,{\bf A}),\;\;\;\;{\bf A}=\hat{x}A_{x}(\phi ),\;\;\;\;\Phi =0 \label{16}$$ which is a function of the invariant phase of the traveling wave, $$\phi =k^{\mu }x_{\mu }(\tau )=x_{0}-z\;, \label{20}$$ where $k^{\mu }=(1,0,0,1)$ is the dimensionless laser wave number. Following Ref. [@Hart] we use $\phi $ as the independent variable to recast the Dirac Lorentz equation as $$\begin{aligned} \frac{d^{2}v_{x}}{d\phi ^{2}} &=&v_{x}\left( \left( \frac{d{\bf v}}{d\phi }\right) ^{2}-\left( \frac{d\gamma }{d\phi }\right) ^{2}\right) +\frac{1}{u}\left( \frac{dv_{x}}{d\phi }\left( \frac{1}{\varepsilon }-\frac{du}{d\phi }\right) +\frac{A}{\varepsilon }G(\phi )sin\phi \right) \label{28} \\ \frac{d^{2}v_{z}}{d\phi ^{2}} &=&v_{z}\left( \left( \frac{d{\bf v}}{d\phi }\right) ^{2}-\left( \frac{d\gamma }{d\phi }\right) ^{2}\right) +\frac{1}{u}\left( \frac{dv_{z}}{d\phi }\left( \frac{1}{\varepsilon }-\frac{du}{d\phi }\right) +\frac{v_{x}}{u}\frac{A}{\varepsilon }G(\phi )sin\phi \right) \label{30}\end{aligned}$$ where the laser pulse electric field $$\frac{dA_{x}(\phi )}{d\phi }=AG(\phi )sin\phi :, \label{32}$$ has been introduced, with $A$ being the maximum amplitude of the pulse, $G(\phi )=e^{-(\phi /\Delta \phi )^{2}}$ is a unit Gaussian envelope of width $\Delta \phi $. Note that $$\left( \frac{d{\bf v}}{d\phi }\right) ^{2}=\left( \frac{dv_{x}}{d\phi }\right) ^{2}+\left( \frac{dv_{z}}{d\phi }\right) ^{2}\;, \label{33a}$$ $$\frac{d\gamma }{d\phi }=\frac{v_{x}}{\gamma }\frac{dv_{x}}{d\phi }+\frac{v_{x}}{\gamma }\frac{dv_{z}}{d\phi } \label{33b}$$ and $$\frac{du}{d\phi }=\frac{d\gamma }{d\phi }-\frac{dv_{z}}{d\phi }. \label{33c}$$ Substituting these relations into Eqs. (\[28\] ) and (\[30\]) we find $$\begin{aligned} \frac{d^{2}v_{x}}{d\phi ^{2}} &=&v_{x}Q+\frac{1}{u}\left( \frac{1}{\varepsilon }\frac{dv_{x}}{d\phi }-\frac{v_{x}}{\gamma }\left( \frac{dv_{x}}{d\phi }\right) ^{2}+\left( 1-\frac{v_{z}}{\gamma }\right) \frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }+\frac{A}{\varepsilon }G(\phi )sin\phi \right) \label{34} \\ \frac{d^{2}v_{z}}{d\phi ^{2}} &=&v_{z}Q+\frac{1}{u}\left( \frac{1}{\varepsilon }\frac{dv_{z}}{d\phi }-\frac{v_{x}}{\gamma }\frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }+\left( 1-\frac{v_{z}}{\gamma }\right) \left( \frac{dv_{z}}{d\phi }\right) ^{2}+\frac{v_{x}}{u}\frac{A}{\varepsilon }G(\phi )sin\phi \right) \label{36}\end{aligned}$$ where $$Q=\left( \frac{dv_{x}}{d\phi }\right) ^{2}\left( 1-\frac{v_{x}^{2}}{\gamma ^{2}}\right) +\left( \frac{dv_{z}}{d\phi }\right) ^{2}\left( 1-\frac{v_{z}^{2}}{\gamma ^{2}}\right) -\frac{2v_{x}v_{z}}{\gamma ^{2}}\frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi } \label{40}$$ and $$\begin{aligned} \gamma &=&\sqrt{1+v_{x}^{2}+v_{z}^{2}} \label{44} \\ u &=&\gamma -u_{z}\;. \label{46}\end{aligned}$$ In Ref. [@Hart], by specifying final homogeneous conditions on the acceleration and the velocity and then integrating [*backward*]{} in time, the solution to these equations were obtained at all times. We now want to show that this problem can also be solved by specifying initial conditions on the motion and integrating [*forward*]{} in time. For the “initial” velocity of our method, we use the final velocity of the backward integration method of [@Hart]. As for the initial acceleration we employ the first terms of the series generated by expanding the equations of motion, in terms of the small quantity $\varepsilon =\omega _{0}\tau _{0}$. To obtain this series we write the two components of the equations of motion (\[34\]) and (\[36\]) as $$\begin{aligned} \frac{dv_{x}}{d\phi } &=&-AG(\phi )sin\phi +\varepsilon \left[ u\left( \frac{d^{2}v_{x}}{d\phi ^{2}}-v_{x}Q\right) +\frac{v_{x}}{\gamma }\left( \frac{dv_{x}}{d\phi }\right) ^{2}-\left( 1-\frac{v_{z}}{\gamma }\right) \frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }\right] \label{50} \\ \frac{dv_{z}}{d\phi } &=&-\frac{v_{x}}{u}AG(\phi )sin\phi +\varepsilon \left[ u\left( \frac{d^{2}v_{z}}{d\phi ^{2}}-v_{z}Q\right) +\frac{v_{x}}{\gamma }\frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }-\left( 1-\frac{v_{z}}{\gamma }\right) \left( \frac{dv_{z}}{d\phi }\right) ^{2}\right] \label{52}\end{aligned}$$ From the above equations we derive the zeroth order for the derivatives of the components of the acceleration $$\begin{aligned} \left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0} &=&-A\frac{d}{d\phi }[G(\phi )sin\phi ] \label{54} \\ \left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0} &=&-A\frac{d}{d\phi }[\frac{v_{x}}{u}G(\phi )sin\phi ] \label{55}\end{aligned}$$ Substituting these relations back into $\left( \ref{50}\right) $ and $\left( \ref{52}\right) $, we obtain the first order approximation for the components of the acceleration $\left( \frac{dv_{x}}{d\phi }\right) _{1},\left( \frac{dv_{z}}{d\phi }\right) _{1}.$ On the other hand, the zeroth order of the second derivative of the components of the acceleration are given by $$\begin{aligned} \left( \frac{d^{3}v_{x}}{d\phi ^{3}}\right) _{0} &=&-A\frac{d^{2}}{d\phi ^{2}}[G(\phi )sin\phi ] \label{56} \\ \left( \frac{d^{3}v_{z}}{d\phi ^{3}}\right) _{0} &=&-A\frac{d^{2}}{d\phi ^{2}}[\frac{v_{x}}{u}AG(\phi )sin\phi ] \label{58}\end{aligned}$$ Taking now the derivatives of $\left( \ref{50}\right) $ and $\left( \ref{52}\right) $ we find the rather lengthy relations $$\begin{aligned} \left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{1} &=&\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}+\varepsilon \{u\left[ \left( \frac{d^{3}v_{x}}{d\phi ^{3}}\right) _{0}-v_{x}Q^{\prime }-\left( \frac{dv_{x}}{d\phi }\right) _{1}Q\right] +u^{\prime }\left[ \left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}-v_{x}Q\right] + \nonumber \\ &&+2\frac{v_{x}}{\gamma }\left( \frac{dv_{x}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}+\frac{1}{\gamma ^{2}}\left[ \gamma \left( \frac{dv_{x}}{d\phi }\right) _{1}-v_{x}\gamma ^{\prime }\right] \left( \frac{dv_{x}}{d\phi }\right) _{1}^{2}-\left( 1-\frac{v_{z}}{\gamma }\right) \nonumber \\ &&\left[ \left( \frac{dv_{x}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0}+\left( \frac{dv_{z}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}\right] +\frac{1}{\gamma ^{2}}\left[ \gamma \left( \frac{dv_{z}}{d\phi }\right) _{1}-v_{z}\gamma ^{\prime }\right] \left( \frac{dv_{x}}{d\phi }\right) _{1}\left( \frac{dv_{z}}{d\phi }\right) _{1}\} \label{60} \\ \left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{1} &=&\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0}+\varepsilon u\left[ \left( \frac{d^{3}v_{z}}{d\phi ^{3}}\right) _{0}-v_{z}Q^{\prime }-\left( \frac{dv_{z}}{d\phi }\right) _{1}Q\right] +u^{\prime }\left[ \left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0}-v_{z}Q\right] + \nonumber \\ &&\frac{v_{x}}{\gamma }\left[ \left( \frac{dv_{x}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0}+\left( \frac{dv_{z}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}\right] +\frac{1}{\gamma ^{2}}\left[ \gamma \left( \frac{dv_{x}}{d\phi }\right) _{1}-v_{x}\gamma ^{\prime }\right] \left( \frac{dv_{x}}{d\phi }\right) _{1}\left( \frac{dv_{z}}{d\phi }\right) _{1} \nonumber \\ &&-2\left( 1-\frac{v_{z}}{\gamma }\right) \left( \frac{dv_{z}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0}+\frac{1}{\gamma ^{2}}\left[ \gamma \left( \frac{dv_{z}}{d\phi }\right) _{1}-v_{z}\gamma ^{\prime }\right] \left( \frac{dv_{x}}{d\phi }\right) _{1}^{2} \label{62}\end{aligned}$$ where $Q^{\prime },\gamma ^{\prime }$ and $u^{\prime }$ are the derivatives of these quantities which can be easily obtained. With these expressions inserted in equations $\left( \ref{50}\right) $ and $\left( \ref{52}\right) ,$ we obtain the next order approximation for the acceleration. This process can be repeated to generate higher order corrections as desired. The forward integration using this above procedure for the initial acceleration was performed using the subroutine Stiffs taken from the Numerical Recipes. The efficiency of the procedure is highly improved when this perturbative expression of the acceleration is recurrently used at each step of the integration process. The results of the numerical calculations are shown in Figs. 1 and 2, respectively, for the transversal and longitudinal components of the momentum. The parameters are $A=1$, $\epsilon =0.05$ and $\Delta \phi =10$. These results are identical to those obtained in Ref. [@Hart] using the backward integration method. In conclusion, it has been shown that LD equation can be integrated forward in time with an appropriate acceleration initial condition. This acceleration is provided by the equation of motion itself, treating the radiation damping term as a perturbation. This peculiar behaviour of LD equation stems from the fact that although the self-interaction term turns it into a third-order differential equation, only its particular solution is physically meaningful: all homogeneous solutions have to be excluded. This constraint makes the solution unique. We have discussed the forward numerical integration procedure in the important case of the nonlinear Compton scattering. The essential feature of the calculations is the crucial role played by the recurrent use of the perturbative series expansion of the acceleration in order to achieve stable and accurate results. We may conclude that the final Dirac-Rohrlich constraint can indeed be replaced by an equivalent initial condition as long as the radiation damping term can be treated as a local perturbation in the iterative solution of the equation of motion. This does not mean that nonlinear effects have been neglected. Indeed, by perturbatively iterating the equation of motion, a local expression for the acceleration is generated, in which the radiation affects the motion as much as it is affected by it. The results presented here clearly show that nonlinear LD equations can still be integrated [*forward*]{} in time if the appropriate, albeit unconventional, initial conditions were utilized. Acknowledgments =============== This work was partly done at Institute for Theoretical Atomic and Molecular Physics-Harvard(ITAMP), Institute for Theoretical Physics(ITP)-Santa Barbara and IFUSP-São Paulo. Partial support was supplied by the CNPq -Brazil, FAPESP-Brazil, The National Science Foundation under Grant No. PHY94-07194 (ITP). The ITAMP is supported by the National Science Foundation. F.V. Hartemann and N.C. Luhmann, Jr., Phys. Rev. Lett. [**74**]{}, 1107 (1995). P.A.M. Dirac, Proc. R. Soc. London, A [**167**]{}, 148 (1938). F.V. Hartemann and A.K. Kerman, Phys. Rev. Lett. [**76**]{}, 624 (1996). C. Bula, K.T. McDonald, E. J. Prebys, C. Bamber, S. Boege, T. Kotseroglou, A. C. Melissinos, D.D. Meyerhofer, W. Ragg, D.L. Burke, R.C. Field, G. Horton-Smith, A. C. Odian, J.E. Spencer, D. Walz, S.C. Berridge, W.M. Bugg, K. Shmakov, and A. W. Weidemann, Phys. Rev. Lett. [**76**]{}, 3116 (1996). S. Chen, A. Maksimchuk, and D. Umstadter, Nature [**396**]{}, 653 (1998). M.S. Hussein, M.P. Pato, and A.K. Kerman, Phys. Rev. A [**46**]{}, 3562 (1992). F.E. Low, Ann. Phys. [**266**]{}, 274 (1998). , Arthur D. Yaghjian, (Springer-Verlag, 1996). J.M. Aguirregabiria, J. Phys. A:Math. Gen. [**30,** ]{}2391 (1997). D. Villaroel, Phys. Rev. A [**55**]{}, 3333 (1997).[** **]{} , F. Rohrlich, (Addison-Wesley, Reading, MA, 1995).  [**Figure Captions:**]{} Fig. 1 The transversal component of the momentum vs. proper time, obtained with the forward integration method. See text for details. Fig. 2 The same for the longitudinal component of the momentum.

--- abstract: | We investigate the redshift dependence of X-ray cluster scaling relations drawn from three hydrodynamic simulations of the $\Lambda$CDM cosmology: a [*Radiative*]{} model that incorporates radiative cooling of the gas, a [*Preheating*]{} model that additionally heats the gas uniformly at high redshift, and a [*Feedback*]{} model that self-consistently heats cold gas in proportion to its local star-formation rate. While all three models are capable of reproducing the observed local - relation, they predict substantially different results at high redshift (to $z=1.5$), with the [*Radiative*]{}, [*Preheating*]{} and [*Feedback*]{} models predicting strongly positive, mildly positive and mildly negative evolution, respectively. The physical explanation for these differences lies in the structure of the intracluster medium. All three models predict significant temperature fluctuations at any given radius due to the presence of cool subclumps and, in the case of the [*Feedback*]{} simulation, reheated gas. The mean gas temperature lies above th e dynamical temperature of the halo for all models at $z=0$, but differs between models at higher redshift with the [*Radiative*]{} model having the lowest mean gaswos temperature at $z=1.5$. We have not attempted to model the scaling relations in a manner that mimics the observational selection effects, nor has a consistent observational picture yet emerged. Nevertheless, evolution of the scaling relations promises to be a powerful probe of the physics of entropy generation in clusters. First indications are that early, widespread heating is favored over an extended period of heating that is associated with galaxy formation. author: - 'Orrarujee Muanwong, Scott T. Kay and Peter A. Thomas' title: 'Evolution of X-ray cluster scaling relations in simulations with radiative cooling and non-gravitational heating' --- Introduction {#sec:introduction} ============ X-ray scaling relations of galaxy clusters, namely the temperature–mass, -$M$, relation and the luminosity–temperature, -, relation, play a pivotal role when using the abundance of clusters to constrain cosmological parameters [@HA91; @WEF93; @Eke96; @VL96; @VL99; @Henry97; @Henry00; @Borgani01; @Pierpaoli01; @Seljak02; @Pierpaoli03; @Viana03; @Allen03; @Henry04]. It is well known, however, that accurate calibration of scaling relations is crucial to avoid a major source of systematic error. For example, the $-M$ relation is widely used by many of these authors to constrain the amplitude of mass fluctuations, conventionally defined using the parameter, $\sigma_8$. Systematic deviations in the normalization of the $-M$ relation, particularly due to how cluster mass is estimated (e.g. see @HMS99) is amplified by the steep slope of the temperature function, leading to large variations in $\sigma_8$ (see @Henry04 for a discussion of recent results). As far as the - relation is concerned, the discrepancies are more prominent as  is highly sensitive to the thermodynamics of the of the inner intracluster medium (ICM), and can yield different values for both normalizations and slopes [@EdS91; @WJF97; @AlF98; @Mar98; @XuW00]. The situation is further complicated by the fact that clusters do not scale self-similarly, as would be the case (approximately) if the only source of heating was via gravitational infall [@Kaiser86]. This makes the problem more difficult to investigate theoretically, although it allows studies of cluster scaling relations to reveal more information on the physics governing the structure of the intracluster medium. The departure from self-similarity can be attributed to an increase in the [*entropy*]{} of the gas that particularly affects low-mass systems [@EH91; @Kaiser91; @Bower97; @TN01; @PCN99; @VB01; @Voit02; @Voit03]. Many theoretical studies have been performed to investigate the effects of various physical processes that can raise the entropy of the gas, based on models involving heating [@ME94; @Balogh99; @KY00; @Low00; @WFN00; @Bower01; @Borgani02] , radiative cooling [@KP97; @Pearce00; @Bryan00; @Muanwong01; @Muanwong02; @DKW02; @WX02], and a combination of the two [@Muanwong02; @KTT03; @Tornatore03; @Valdarnini03; @Borgani04; @Kay04; @McCarthy04]. Measurements of how cluster scaling relations evolve with redshift allow even tighter constraints to be placed on cosmological parameters (and entropy generation models), and observations of cluster properties at high redshift are now starting to become available, owing primarily to the high sensitivity of [*Chandra*]{} and [*XMM–Newton*]{}. From a theoretical point of view, this is an exciting phase as we can now fully exploit the availability of our simulated distant clusters and compare their X-ray properties with real observations. It is therefore timely to investigate further the effects of entropy generation on the evolution of clu ster scaling relations as the available data for high-redshift systems accumulates. In this paper, we will use cosmological hydrodynamical simulations described in @Muanwong02, hereafter MTKP02, and in @Kay04, hereafter KTJP04, to trace the evolution of the cluster population to high redshift ($z=1.5$). Our results will primarily focus on three ([*Radiative*]{}, [*Preheating*]{} and [*Feedback*]{}) models, all able to reproduce the local - relation. The aims of this paper are to determine how the scaling relations evolve with redshift in the three models and to discover what the evolution of scaling relations can teach us about non-gravitational processes occurring in clusters. The rest of this paper is outlined as follows. In Section \[sec:srel\] we introduce the X-ray scaling relations and summarize our present observational knowledge of these quantities. Details of our simulated cluster populations are presented in Section \[sec:sims\]. In Section \[sec:results\] we present our main results, first at $z=0$, where the models are in good agreement with each other and the observations, then as a function of redshift, where the models predict widely different results. We discuss the implications of these differences in Section \[sec:discuss\] and demonstrate that the degree of X-ray evolution is driven by the supply of cold, low entropy gas. Finally, we summarize our conclusions in Section \[sec:conclude\]. X-ray cluster scaling relations {#sec:srel} =============================== @Kaiser86 derived the following relations for temperature $${\mbox{$T_{\rm X}$}}\propto M^{2 \over 3} \, (1+z), \label{eqn:tmrel}$$ and luminosity $$\begin{aligned} {\mbox{$L_{\rm X}$}}&\propto& M^{4 \over 3} \, (1+z)^{7 \over 2} \label{eqn:lmrel} \\ &\propto& {\mbox{$T_{\rm X}$}}^{2} \, (1+z)^{3 \over 2} \label{eqn:ltrel},\end{aligned}$$ assuming the distribution of gas and dark matter in clusters is perfectly self-similar and the X-ray emission is primarily thermal bremsstrahlung radiation. Observed clusters do not form a self-similar population but it is nevertheless convenient to describe their behavior using a generalized power-law form $$Y = C_0(z) \, X^\alpha = Y_0 \, X^\alpha \, (1+z)^{A}, \label{eqn:powerlaw}$$ where $C_0(z)$ and $Y_0$ determine the normalization, $\alpha$ is the slope of the relation (in log-space) and $A$ determines how the relation evolves with redshift. Our main results will focus on the determination of $A$. Observationally, attempts to measure the $-M$ relation at high redshift are currently in their infancy, as they require temperature profiles to be measured so that their mass can be estimated, but initial results are consistent with self-similar evolution ($A \sim 1$, @Maughan05 [@Kotov05]). Measuring the - relation at higher redshift is a somewhat simpler prospect, and has been attempted by many authors [@MS97; @Fairley00; @Holden02; @Novicki02; @Arnaud02; @Vikhlinin02; @Lumb04; @Ettori04; @Maughan05; @Kotov05]. We summarize recent results that adopt a low-density flat cosmology in Figure \[fig:ltevolobs\], attempting to include in the size of the error bars the uncertainty in $A$ due to the choice of local relation (when quoted by the authors). Although the present situation is by no means clear, taking all results at face value generally favors positive evolution ($0 {\mathrel{{{\hbox to 0pt{\lower 3pt\hbox{$\sim$}\hss}}} \raise 2.0pt\hbox{$<$}}}A {\mathrel{{{\hbox to 0pt{\lower 3pt\hbox{$\sim$}\hss}}} \raise 2.0pt\hbox{$<$}}}2$) with the latest results being consistent with sel f-similar evolution ($A=3/2$). Larger samples of high redshift clusters (such as that expected from the [*XMM-Newton*]{} Cluster Survey, @Romer01) will be crucial to accurately constrain the degree of evolution in the - relation. Simulated cluster populations {#sec:sims} ============================= Our results are drawn from three similarly-sized $N$-body/SPH simulations of the $\Lambda$CDM cosmology, which have already been published in MTKP02 and KTJP04. The simulation box in MTKP02 has a comoving side of $100{\mbox{$\, h^{-1}{{\rm {Mpc}}}$}}$ with $160^3$ particles each of gas and dark matter, whose particle masses are set to $2.6\times 10^9$ and $2.1\times 10^{10}{\, h^{-1} {{\, \mbox{M$_\odot$}}}}$, respectively. The box used in KTJP04 is bigger with a side of $120{\mbox{$\, h^{-1}{{\rm {Mpc}}}$}}$ using $256^3$ particles each of gas and dark matter, whose particle masses are $1.3\times 10^9$ and $7.3\times 10^9{\, h^{-1} {{\, \mbox{M$_\odot$}}}}$, respectively. Full details can be found in the articles. The key difference between the simulations is the model used to raise the entropy of the intracluster gas, summarized as follows: 1. A [*Radiative*]{} model where the excess entropy originated from the removal of low entropy gas to form stars, causing higher entropy gas to flow adiabatically into the core from larger radii (MTKP02). 2. A [*Preheating*]{} model where entropy was generated impulsively by uniformly heating the gas by 1.5 keV per particle at $z=4$ (MTKP02). 3. A [*Feedback*]{} model where the entropy of (on average) 10 per cent of cooled gas in high density regions was raised by 1000 keV cm$^2$, mimicking the effects of heating due to stars and active galactic nuclei (KTJP04). These three models differ in the timing and distribution of entropy generation in the intracluster medium. The [*Radiative*]{} model has no explicit feedback of energy but relies on the removal of low-entropy gas via cooling; as such it represents a minimal heating model. The [*Preheating*]{} model contains distributed heating at high redshift such as might occur if entropy generation occurs mainly in low-mass galaxies. By contrast heating in the [*Feedback*]{} model occurs solely in high-density regions. In all our models, there is very little star formation before a redshift of z=4 after which the star-formation rate (sfr) begins to rise rapidly. In the [*Preheating*]{} simulation the sfr is then strongly suppressed, whereas in the other two simulations it peaks at a redshift of z=2 and then declines back down to low values by the present day with a time-variation that matches that of the star-formation history of the Universe. The global baryon fraction in stars (and cold gas) at z=0 is 0.002, 0.076 and 0.127 in the [*Preheating*]{}, [*Radiative*]{} and [*Feedback*]{} simulations, respectively. The largest of these corresponds to a stellar mass density of $\Omega_*$=0.006; thus none of the models has excessive star-formation. These models are far from exhaustive and their precise details should not be taken too seriously. The purpose of this paper is not to examine particular models but to illustrate that the evolution of the X-ray scaling relations can provide a powerful discriminant between different classes of model. Cluster identification and properties ------------------------------------- Clusters were selected at four redshifts ($z=$0, 0.5, 1 & 1.5) using the procedure outlined in MTKP02. They are defined to be spheres of matter, centered on the dark matter density maximum, with total mass $$M_{\Delta} = {4 \over 3} \pi R_{\Delta}^{3} \, \Delta \, \rho_{\rm c0} \, (1+z)^{3}, \label{eqn:mass}$$ where $\rho_{\rm c0}=3H_0^2/8\pi G$ is the critical density at $z=0$. We set $\Delta=500$ as it corresponds to a sufficiently large radius such that the results are not dominated by the core, as well as corresponding approximately to the extent of current X-ray observations. Furthermore, as was shown by @Rowley04, the X-ray properties of simulated clusters within an overdensity of 500 exhibit less scatter than within the virial radius. Our choice of scaling with redshift[^1] is independent of cosmology and would allow the simple power-law scalings to be recovered (equations \[eqn:tmrel\],\[eqn:lmrel\] & \[eqn:ltrel\]) if the clusters were structurally self-similar. We consider scaling relations involving mass, three measures of temperature, and luminosity, for particle properties averaged within $R_{500}$. The mass, $$M_{500}=\sum_i m_i,$$ where the sum runs over all particles, of mass $m_i$. The dynamical temperature, $$kT_{\rm dyn} = \, { \sum_{i,{\rm gas}} m_i k T_i \, + \, \alpha \sum_i {1 \over 2} m_i v_i^2 \over \sum_i m_i}, \label{eqn:tdyn}$$ where $\alpha=(2/3)\mu m_{\rm H} \sim 4.2\times 10^{-16}$ keV for a fully ionized primordial plasma, assuming the ratio of specific heats for a monatomic ideal gas, $\gamma=5/3$, and the mean atomic weight of a zero metalicity gas, $\mu m_{\rm H}=10^{-24} {\rm g}$. The first sum in the numerator runs over all gas particles, of temperature, $T_i$, whereas the second sum runs over particles of all types, of speed $v_i$ as measured in the center of momentum frame of the cluster. We also consider the mass-weighted temperature of hot ($T>10^{5}$K) gas, $$kT_{\rm gas} = { \sum_{i,{\rm hot}} m_i k T_i \over \sum_{i,{\rm hot}} m_i}, \label{eqn:tgas}$$ and we approximate the X-ray temperature of a cluster using the bolometric emission-weighted temperature, $$kT_{\rm bol} = { \sum_{i,{\rm hot}} m_i \rho_i \Lambda_{\rm bol}(T_i,Z) T_i \over \sum_{i,{\rm hot}} m_i \rho_i \Lambda_{\rm bol}(T_i,Z) }, \label{eqn:tbol}$$ where $\rho_i$ is the density and $\Lambda_{\rm bol}$ is the bolometric cooling function used in our simulations [@SD93]; for the [*Radiative*]{} and [*Preheating*]{} runs, $Z=0.3(t/t_0)Z_{\odot}$ (MTKP02), and for the [*Feedback*]{} run, $Z=0.3Z_{\odot}$ (KTJP04). Finally, the X-ray luminosity is approximated by the bolometric emission-weighted luminosity $$L_{\rm bol} = \sum_{i,{\rm hot}} {m_i \rho_i \Lambda_{\rm bol}(T_i,Z) \over (\mu m_{\rm H})^2 }.$$ It has been shown recently that the emission-weighted temperature is not an accurate diagnostic of cluster temperature, overpredicting the [*spectroscopic*]{} temperature by $\sim 20-30$ per cent when the emission is predominantly thermal bremsstrahlung [@Mazzotta04; @Rasia05]. At lower temperatures ($kT<3$keV), line emission from heavy elements makes the problem significantly more complicated [@Vikhlinin05]. The volume sampled by our simulations ($\sim 100{\mbox{$\, h^{-1}{{\rm {Mpc}}}$}}$) means that we have very few clusters with $T>3$keV, and so a more accurate measure of the cluster temperature would require significantly more effort than applying a simple formula to our data. We therefore leave such improvements to future work, when larger samples of simulated clusters are available. It would not affect the conclusions of this paper. Cluster catalogues ------------------ ------------------ --------------------------- ----- ----- ----- ----- Model Relation 0.0 0.5 1.0 1.5 [*Radiative*]{} Total 340 190 85 31 $T_{\rm dyn}-M_{500}$ 330 186 84 31 $T_{\rm gas}-M_{500}$ 332 186 82 31 $T_{\rm bol}-M_{500}$ 319 151 64 24 $L_{\rm bol}-M_{500}$ 317 186 85 31 $L_{\rm bol}-T_{\rm bol}$ 256 95 34 14 [*Preheating*]{} Total 283 147 59 22 $T_{\rm dyn}-M_{500}$ 273 143 56 22 $T_{\rm gas}-M_{500}$ 271 143 56 22 $T_{\rm bol}-M_{500}$ 264 134 53 22 $L_{\rm bol}-M_{500}$ 269 143 59 22 $L_{\rm bol}-T_{\rm bol}$ 190 92 48 14 [*Feedback*]{} Total 342 98 45 13 $T_{\rm dyn}-M_{500}$ 328 96 43 12 $T_{\rm gas}-M_{500}$ 327 89 41 11 $T_{\rm bol}-M_{500}$ 305 90 39 10 $L_{\rm bol}-M_{500}$ 339 98 45 13 $L_{\rm bol}-T_{\rm bol}$ 269 67 32 12 ------------------ --------------------------- ----- ----- ----- ----- : Numbers of clusters at various redshifts[]{data-label="tab:totnumbers"} \[tab:clusters\] Table \[tab:clusters\] lists the numbers of clusters in our catalogues for each of the simulations at all 4 redshifts. The first row for each model gives the total number of clusters in our catalogues, down to a minimum mass, $M_{\rm 500}=1.2\times 10^{13}{\, h^{-1} {{\, \mbox{M$_\odot$}}}}$, corresponding to $\sim 500$ dark matter particles in the [*Radiative*]{} and [*Preheating*]{} simulations, and $\sim 1400$ dark matter particles in the (higher resolution) [*Feedback*]{} simulation. At $z=0$, each model contains around 300 clusters above our mass limit, decreasing by around an order of magnitude by $z=1.5$. We also made a number of additional cuts to the catalogues, specific to each scaling relation. Firstly, we noted a small number of systems that were significantly offset from the mean relation. On inspection, such objects were found to be erroneous as they were subclumps falling into neighbouring clusters. Thus, for each relation, we discarded all objects with $\Delta \log (Y)>0.1$, larger than intrinsic scatter in the $T_{\rm dyn}-M_{\rm 500}$, $T_{\rm gas}-M_{\rm 500}$ and ${T_{\rm bol}-M_{\rm 500}}$ relations; and $\Delta \log (Y)>0.5$ in the $L_{\rm bol}-M_{500}$ and $L_{\rm bol}-T_{\rm bol}$ relations, respectively. Secondly, for the $L_{\rm bol}-T_{\rm bol}$ relation, we made an additional cut in temperature, such that the catalogues were complete in $T_{\rm bol}$ (excluding those clusters classed as outliers). For the [*Radiative*]{} model, the minimum temperatures are $kT_{\rm bol,min}=[0.74,1.0,1.25,1.35]$ keV; for the [*Preheating*]{} model, $kT_{\rm bol,min}=[0.70,0.96,1.1,1.37]$ keV; and for the [*Feedback*]{} model, $kT_{\rm bol,min}=[0.59,1.12,1.31,1.58]$ keV, for $z=[0,0.5,1,1.5]$. The numbers of clusters remaining in each of the relations after these cuts are also listed in Table \[tab:clusters\]. Results {#sec:results} ======= Scaling relations at redshift zero ---------------------------------- Relation Model $\alpha$ $\log C_0(0)$ $\log Y_0$ $A$ --------------------------- ------------------ ---------- --------------- ------------ ------ $T_{\rm dyn}-M_{\rm 500}$ [*Radiative*]{} 0.70 0.34 0.34 1.1 [*Preheating*]{} 0.70 0.33 0.33 1.1 [*Feedback*]{} 0.69 0.33 0.33 1.2 $T_{\rm gas}-M_{\rm 500}$ [*Radiative*]{} 0.61 0.33 0.33 0.9 [*Preheating*]{} 0.61 0.35 0.35 0.9 [*Feedback*]{} 0.61 0.35 0.35 1.1 $T_{\rm bol}-M_{\rm 500}$ [*Radiative*]{} 0.59 0.38 0.37 0.5 [*Preheating*]{} 0.61 0.35 0.35 0.8 [*Feedback*]{} 0.64 0.33 0.33 1.2 $L_{\rm bol}-M_{\rm 500}$ [*Radiative*]{} 1.82 1.36 1.36 3.9 [*Preheating*]{} 1.92 1.40 1.39 3.1 [*Feedback*]{} 2.10 1.40 1.40 3.2 $L_{\rm bol}-T_{\rm bol}$ [*Radiative*]{} 3.06 0.19 0.20 1.9 [*Preheating*]{} 3.05 0.26 0.24 0.7 [*Feedback*]{} 3.13 0.28 0.28 -0.6 : Best-fit scaling relations[]{data-label="tab:evoparams"} \[tab:bestfit\] We first present the scaling relations at $z=0$ as they will form the basis for measuring evolution in the cluster properties with redshift. The parameters $\alpha$ and $C_0(0)$ listed in Table \[tab:bestfit\] are determined from the best least-squares fit to the relation $$\log Y=\log C_0(0) + \alpha\log X,$$ where $X$ and $Y$ represent the appropriate data sets in units of $10^{14} {\, h^{-1} {{\, \mbox{M$_\odot$}}}}$, 1keV and $10^{42}\,h^{-2}\,{\rm erg}\,{\rm s^{-1}}$ for mass, temperature and luminosity, respectively. We will consider each relation in turn. Figure \[fig:tdynmz0\] illustrates the $T_{\rm dyn}-M_{500}$ relation for each of the three simulations at $z=0$, with best-fit relations overplotted as straight lines. The dynamical temperature is dominated by the contribution from the more massive dark matter particles, and so the resulting three relations are almost identical. The measured slope of the relation is $\alpha \sim 0.7$ (Table \[tab:bestfit\]), close to, but slightly larger than the self-similar value ($\alpha=2/3$); this deviation is due to the variation of concentration with cluster mass. When the mass-weighted temperature of hot gas is used instead, the relation becomes flatter than the self-similar prediction, with $\alpha \sim 0.6$. This is expected as the excess entropy generation due to cooling and heating is more effective in lower mass clusters (MTKP02). Shown in Figure \[fig:tbolmz0\] is the $T_{\rm bol}-M_{500}$ relation for each of the 3 models. Cool, dense gas dominates $T_{\rm bol}$ and so this temperature is more susceptible to fluctuations caused by merging substructure, leading to an increase in the scatter when compared to Figure \[fig:tdynmz0\]. Again, the slope is flatter than the self-similar prediction, due to the effects of excess entropy. Differences between the models are larger than for the dynamical temperature but are less than the intrinsic scatter. Finally, we consider relations involving the bolometric luminosity of the cluster. Fitting the relation between luminosity and mass, we find a slope in the range $\alpha \sim 1.8-2.1$, significantly steeper than the self-similar prediction ($\alpha=4/3$). The departure from self-similarity is exacerbated when we plot bolometric luminosity against temperature (Figure \[fig:lbolt0\]). Here, $\alpha \sim 3.1$ in all models, compared to $\alpha=2$ for the self-similar case. The $L_{\rm bol}-T_{\rm bol}$ relations from the three simulations are in reasonable agreement with one another and in good agreement with the observed luminosity–temperature relation (see MTKP02,KTJP04). In summary, all three models successfully generate excess entropy in order to break self-similarity at the level required by the observations at low redshift ($z \sim 0$). Thus, based on the local scaling relations alone, we cannot easily discriminate between the source of the entropy excess in clusters: whether it is mainly due to radiative cooling, additional uniform heating at high redshift (prior to cluster formation) or localized heating from galaxy formation at all redshifts. Evolution of scaling relations with redshift -------------------------------------------- We now examine whether this degeneracy between models in the scaling relations at $z=0$ can be broken by examining the cluster population at higher redshifts ($z=0.5, 1, 1.5$). None of the relations require a significant variation in $\alpha$ with redshift. To make our results easier to interpret, therefore, we use simple power-law relations of the form given in equation \[eqn:powerlaw\] with $\alpha$ fixed at the $z=0$ values given in Table \[tab:bestfit\]. To find the evolution of each relation, we first determine the normalizations, $C_0$, and their corresponding error bars, at each redshift in the same manner as described for redshift zero in Section 4.1 above. We then minimize the $\chi-$squared to obtain parameters $Y_0$ and $A$ as listed in Table \[tab:bestfit\] to fit the relation $$\log C_0 = \log Y_0 + A\log(1+z). \label{eqn:normz}$$ ### Temperature-Mass Evolution In Figure \[fig:tmevol\], we present values of $\log(C_0)$ versus redshift for the three temperature–mass relations, with the best-fit straight line overplotted. For the $T_{\rm dyn}-M_{500}$ relation (upper panel), we find similar evolution parameters for the three models, $A=1.1-1.2$, confirming that including the effects of baryonic physics does not significantly affect cluster dynamics. The slight excess over the self-similar value of $A=1$ is consistent with the changing cluster concentrations. However, both the mass-weighted temperature (middle panel) and especially the emission-weighted temperature (lower panel) show significant variation between the three models. In each case the [*Feedback*]{} simulation approximately follows the scaling found for the dynamical temperature, with the [*Preheating*]{} and the [*Radiative*]{} simulations showing progressively larger deviations below the expected normalization as the redshift increases. The explanation for this lies in the variation of temperature of gas particles within each cluster and how this changes with redshift in the different models. We shall explore this further in Section \[sec:discuss\]. ### Luminosity-Mass Evolution Figure \[fig:lmevol\] illustrates the normalization of the $L_{\rm bol}-M_{500}$ relation versus redshift for all three models. The [*Preheating*]{} and [*Feedback*]{} models evolve almost identically with redshift ($A \sim 3$), but the [*Radiative*]{} run evolves more strongly ($A \sim 4$). These bracket the self-similar value, $A=3.5$, however, this agreement is somewhat coincidental because the slope of the relation at fixed redshift is much steeper than expected ($\alpha\sim1.8$–2.1 rather than 1.3). The reason why the [*Radiative*]{} simulation has steeper evolution is because of enhanced emission from cool gas at high redshift relative that that at low redshift—see discussion in Section \[sec:discuss\]. ### Luminosity-Temperature Evolution Finally, we consider the evolution of the $L_{\rm bol}-T_{\rm bol}$ relation, with the relations at each redshift shown explicitly for each model in Figure \[fig:lbolt4z\], and the variation of normalization with redshift illustrated in Figure \[fig:ltevol\]. It is interesting to note that the values of $A$ are significantly different between all three models: the [*Feedback*]{} model predicts mildly negative evolution ($A=-0.6$), the [*Preheating*]{} mildly positive evolution ($A=0.7$) and the [*Radiative*]{} strongly positive evolution ($A=1.9$). The latter two models straddle the self-similar value ($A=1.5$). The difference in slopes between the [*Feedback*]{} and [*Preheating*]{}runs is driven by the differences in their temperature. The further difference between the [*Preheating*]{} and [*Radiative*]{} runs comes roughly equally from the temperature and luminosity evolution. Discussion {#sec:discuss} ========== In this paper, we have focused on the evolution of cluster scaling relations in three simulations, each adopting a different model for non-gravitational processes that affect the intracluster gas. In the first, [*Radiative*]{} model, the gas could cool radiatively and a significant fraction cooled down to low temperatures and formed stars (MTKP02). In the [*Preheating*]{} model, the same was true, although the gas was additionally heated uniformly and impulsively by 1.5 keV per particle at $z=4$, before cluster formation. In the third, [*Feedback*]{} model, the heating rate was local and quasi-continuous, in proportion to the star-formation rate from cooled gas. All three models are able to generate the required level of excess core entropy in order to reproduce the $L_{\rm X}-T_{\rm X}$ relation at $z=0$ (MTKP02, KTJP04). The most striking result presented in this paper is that the three models predict widely different $L_{\rm X}-T_{\rm X}$ relations at high redshifts. The [*Radiative*]{} model predicts strongly positive evolution, the [*Preheating*]{} model mildly positive evolution and the [*Feedback*]{} model, mildly negative evolution. At this point, it should be stressed that the values of $A$ presented in Table \[tab:bestfit\] should not be taken too seriously. No attempt has been made to convert the bolometric, emission-weighted fluxes used in this paper to observable X-ray fluxes in different instrumental bands. Also, the volume of our simulation boxes is such that we only get a modest number of relatively poor clusters at high redshift. Nevertheless, the qualitative difference between the models is very encouraging and suggests that evolution of X-ray properties may act as a strong discriminant between models in the future. Previous work [e.g.  @Pearce00; @Muanwong01] has made great play of the fact that radiative cooling can remove low-entropy material and lead to a raising of the gas temperature above the virial temperature of the host halo. That effect is reproduced by the [*Radiative*]{} simulation in this paper, but it is interesting to note that the bolometric X-ray temperature exceeds the dynamical temperature of the clusters only at very low redshift, $z{\mathrel{{{\hbox to 0pt{\lower 3pt\hbox{$\sim$}\hss}}} \raise 2.0pt\hbox{$<$}}}0.1$. At higher redshifts it falls below the dynamical temperature and is a factor of 1.6 lower by $z=1.5$. This departure from self-similarity is a consequence of the changing density parameter, $\Omega$, in the concordance  cosmology: at high redshift $\Omega$ is close to unity and structures grow freely; at lower redshifts $\Omega$ falls well below unity and the rate of growth of cosmic structures declines. The behavior of the [*Preheating*]{} simulation is similar, although the relative decline in the ratio of the gas to the dynamical temperature is smaller. The effect cannot, therefore, be due to the cooling of intracluster gas in the cluster cores between a redshift of 1.5 and the present—in the [*Preheating*]{} run very little gas cools below a redshift of 4 and so that cool core gas would still be there today. Instead, we attribute the presence of cool gas to the accretion of low temperature subclumps. Such accretion is a ubiquitous feature of clusters [e.g.  @Rowley04]. Cool gas is seen in maps of clusters at low redshift [@Onuora03] and would be expected to be much more prevalent in clusters at high redshift in the  cosmology. To test this hypothesis, we measured the temperature variation within each cluster as follows. First we averaged properties within 20 spherical annuli out to $R_{500}$, to create smoothed dynamical, $\bar{T}_{\rm dyn}$, and gas temperature, $\bar{T}_{\rm gas}$, profiles. Then we measured the mean deviation (in log space) of the gas temperature from the local dynamical temperature $$\tau={1\over N}\,\sum_i\left(\log_{10}T_i-\log_{10}\bar{T}_{\rm dyn}\right),$$ and the root-mean square deviation of the temperature, $\sigma_T$, from the mean, $$\sigma_T^2={1\over N}\,\sum_i\left(\log_{10}T_i-\log_{10}\bar{T}_{\rm gas}\right)^2,$$ where $N=\sum_i$ and the sum runs over all hot gas particles ($T_i>10^5$K) within $R_{500}$. As an example, Figure \[fig:delt6720.1002\] shows the values of $\tau$ (crosses) and $\sigma_T$ (circles) for each cluster at $z=1$ in the [*Radiative*]{} simulation. This particular example has been chosen simply because the two properties are well-separated and easy to distinguish on the plot. As can be seen, the mean gas temperature of the more massive clusters typically lies below the local dynamical temperature, by as much as 15 per cent at this redshift. However, the dispersion in temperature is much larger, typically a factor of 1.5, so there will be fluctuations both above and below the dynamical temperature. Figure \[fig:delt\] demonstrates visually the evolution of $\tau$ and $\sigma_T$ with redshift. At each redshift, the plot shoes the average value of $\tau$ over all clusters with masses greater than 1.2$\times10^{13}h^{-1}$[[ ]{}]{}. The half-width of the shaded regions represent the average values of $\sigma_T$, divided by 10 for clarity. Concentrating first on the [*Radiative*]{} simulation, it can be seen that the mean cluster temperature increases relative to the virial temperature over time and that the dispersion in temperature decreases. This is consistent with a decreasing amount of substructure within the clusters at lower redshift, although it should be noted that part of the effect is due to the narrower range of cluster masses resolved by the simulations at high redshift, as the average value of $\tau$ decreases with increasing cluster mass. The behavior of the [*Preheating*]{} simulation mimics that of the [*Radiative*]{} one, but with a bias to higher mean temperatures. The [*Feedback*]{} simulation, however, is quite different. It shows a much larger dispersion than the other two, but no bias to low temperatures at high redshift. That is because gas is free to cool down to low temperatures but some of that gas is then heated back up high temperatures by the feedback. At low redshift cooling becomes less important and the [*Feedback*]{} run then shows a slight rise in $\tau$, matching that seen in the other two runs. Our results have two very important implications for observations of high redshift clusters. Firstly, the behavior of the $L_{\rm X}-T_{\rm X}$ relation at high redshift will determine the number of high redshift clusters to be found in surveys such as the ongoing [*XMM-Newton*]{} Cluster Survey [@Romer01] and this will have a significant impact upon their use in probing cosmological parameters. A positive evolution such as that shown by the [*Radiative*]{} simulation will yield many more observable high-redshift clusters than the negative evolution of the [*Feedback*]{} model. As discussed in Section \[sec:srel\] and summarized in Figure \[fig:ltevolobs\], the observational situation is far from clear but does seem to indicate positive evolution. Turning this argument around, our results suggest that observational constraints on the degree of evolution of the $L_{\rm X}-T_{\rm X}$ relation will allow interesting constraints to be placed on the source of entropy generation in clusters, in particular the relative role of cooling and heating and whether most of the heating of the intracluster gas occurred at high redshift (as in the [*Preheating*]{} model) or was a continuous function of redshift (as in the [*Feedback*]{} model). Taking our results at face value with recent observations would suggest that our [*Feedback*]{} model is generating too much excess entropy at $z<1.5$ and that the bulk of the heating must have occurred at higher redshift. However, we stress once again that this result is very tentative. Conclusions {#sec:conclude} =========== The evolution of X-ray cluster scaling relations are a crucial component when constraining cosmological parameters with clusters. Observational studies at low redshift have already shown that the scaling relations deviate from self-similar expectations, attributed to non-gravitational heating and cooling processes, but their redshift dependence is only starting to be explored. In this paper we have investigated the sensitivity of the X-ray scaling relations to the nature of heating processes, using three numerical simulations of the ${{\mbox{$\Lambda$CDM}}}$ cosmology with different heating models. While all three simulations reproduce more or less the same scaling relations at $z=0$ (as they were designed to produce the correct level of excess entropy), they predict significantly different results for the evolution of the - relation to $z=1.5$. In conclusion, our findings strongly suggest that the relative abundance of high and low redshift clusters will place interesting constraints on the nature of non-gravitational entropy generation in clusters. First indications are that an early and widespread preheating of the ICM is to be preferred to an extended period of preheating that is associated with galaxy formation. However much more detailed modeling is required and the observational picture is as yet unclear. The simulations described in this paper were carried out on the Cray-T3E at the Edinburgh Parallel Computing Centre and the COSmology MAchine in Durham as part of the Virgo Consortium investigations into the formation of structure in the Universe. OM is grateful for the hospitality and support of the Astronomy Centre at the University of Sussex where much of work in this paper was carried out, and for financial support from Khon Kaen University. She and PAT also acknowledge support from the Thailand Research Fund and the Commission on Higher Education grant MRG4680129. Allen S. W., Fabian A. C., 1998, , 297, L57 Allen S. W., Schmidt R. W., Fabian A. C., Ebeling H., 2003, , 342, 287 non, D. B., & Navarro, J. F., 1999, Nature, 397, 135 Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1 Arnaud M., Aghanim N., Neumann D. M., 2002, , 389, 1 Balogh M. L., Babul A., Patton D. R., 1999, , 307, 463 Borgani, S., Rosati, P., Tozzi, P., Stanford, S. A., Eisenhardt, P. R., Lidman, C., Holden, B., Della Ceca, R., Norman, C., & Squires, G., 2001, , 561, 13 Borgani S., Governato F., Wadsley J., Menci N., Tozzi P., Quinn T., Stadel J., Lake G., 2002, , 336, 409 Borgani, S., Murante, G., Springel, V., Diaferio, A., Dolag, K., Moscardini, L., Tormen, G., Tornatore, L., Tozzi, P., 2004, , 348, 1078 Bower R. G., 1997, , 288, 355 Bower R. G., Benson A. J., Lacey C. G., Baugh C. M., Cole S., Frenk C. S., 2001, , 325, 497 Bryan G. L., 2000, ApJ, 544, L1 non, D. B., & Navarro, J. F., 1999, Nature, 397, 135 Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1 Davé R., Katz N., Weinberg D. H., 2002, , 579, 23 Edge A. C., Stewart G. C., 1991, , 252, 414 Eke V. R., Cole, S., & Frenk, C. S., 1996, , 282, 263 non, D. B., & Navarro, J. F., 1999, Nature, 397, 135 Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1 Eke V. R., Navarro J. F. N., Frenk C. S., 1998, , 503, 569 Ettori S., Tozzi P., Borgani S., Rosati P., 2004, , 417, 13 Evrard A. E., Henry J. P., 1991, , 383, 95 Fairley, B. W., Jones, L. R., Scharf, C., Ebeling, H., Perlman, E., Horner, D., Wegner, G., & Malkan, M., 2000, , 315, 669 Henry, J. P., 1997, , 489, L1 non, D. B., & Navarro, J. F., 1999, Nature, 397, 135 Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1 Henry, J. P., 2000, , 534, 565 Henry, J. P., 2004, , 609, 603 Henry,non, D. B., & Navarro, J. F., 1999, Nature, 397, 135 Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1 J. P., & Arnaud, K. A., 1991, , 372, 410 Holden B. P., Stanford S. A., Squires G. K., Rosati P., Tozzi P., Eisenhardt P., Spinrad H., 2002, , 124, 33 Horner, D. J., Mushotzky, R. F., & Scharf, C. A., 1999, , 520, 78 Kaiser, N., 1986, , 222, 323 Kaiser, N., 1991, , 383, 104 Kay S. T., Thomas P. A., Theuns T., 2003, , 343, 608non, D. B., & Navarro, J. F., 1999, Nature, 397, 135 Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1 Kay S. T., Thomas P. A., Jenkins A., Pearce F. R., 2004, , 355, 1091 (KTJP04) Knight P. A., Ponman T. J., 1997, , 289, 955 Kotov O., Vikhlinin A., 2005, , submitted (astro-ph/0504233) Kravtsov A. V., Yepes G., 2000, , 318, 227 Loewenstein M., 2000, , 532, 17 Lumb D. H., Bartlett J. G., Romer A. K., Blanchard A., Burke D. J., Collins C. A., Nichol R. C., Giard M., Marty P., Nevalainen J., Sadat R., Vauclair S. C., 2004, , 420, 853 Markevitch M., 1998, , 504, 27 Maughan B. J., Jones L. R., Ebeling H., Scharf C., 2005, , submitted (astro-ph/0503455) Mazzotta P., Rasia E., Moscardini L., Tormen G., 2004, , 354, 10 McCarthy, I. G., Balogh, M. L., Babul, A., Poole, G. B., Horner, D. J., 2004, , 613, 811 Metzler C. A., Evrard A. E., 1994, , 437, 564 Muanwong, O., Thomas, P. A., Kay, S. T., Pearce, F. R., & Couchman, H. M. P., 2001, , 552, L27 Muanwong, O., Thomas, P. A., Kay, S. T., & Pearce, F. R., 2002, , 336, 527 (MTKP02) Mushotzky, R. F., & Scharf, C. A., 1997, , 482, L13 Novicki M. C., Manuela S., Henry J. P., 2002, , 124, 2413 Onuora L., Kay S. T., Thomas P. A., 2003, MNRAS, 341, 1241 Pearce F. R., Thomas P. A., Couchman H. M. P., Edge A. C., 2000, , 317, 1029 Pierpaoli, E., Scott, D., & White, M., 2001, , 325, 77 Pierpaoli, E., Borgani S., Scott D., White M., 2003, , 342, 163 Ponman, T. J, Cannon, D. B., & Navarro, J. F., 1999, Nature, 397, 135 Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1 Romer A. K., Viana P. T. P., Liddle A. R., Mann R. G., 2001, , 547, 594 Rowley D. R., Thomas P. A., Kay S. T., 2004, , 352, 508 Seljak, U., 2002, , 337, 769 Sutherland R. S., Dopita M. A., 1993, , 88, 253 Thomas P. A., Couchman H. M. P., 1992, , 257, 11 Thomas P. A., Muanwong O., Pearce F. R., Couchman H. M. P., Edge A. C., Jenkins A., Onuora L., 2001, , 324, 450 Tornatore L., Borgani S., Springel V., Matteucci F., Menci N., Murante G., 2003, , 342, 1025 Tozzi P., Norman C., 1998, , 546, 63 Valdarnini R., 2003, , 339, 1117 Viana, P. T. P., & Liddle, A. R., 1996, , 281, 323 Viana, P. T. P., & Liddle, A. R., 1999, , 303, 535 Viana P. T. P., Kay S. T., Liddle A. R., Muanwong O., Thomas P. A., 2003, , 346, 319 Vikhlinin A., VanSpeybroeck L., Markevitch M., Forman W. R., Grego L., 2002, , 578, L107 Vikhlinin A., 2005, , submitted (astro-ph/0504098) Voit G. M., Bryan G. L., 2001, , 414, 425 Voit G. M., Bryan G. L., Balogh M. L., Bower R. G., 2002, , 576, 601 Voit G. M., Balogh M. L., Bower R. G., Lacey C. G., Bryan G. L., 2003, , 593, 272 White S. D. M., Efstathiou G., Frenk C. S., 1993, , 262, 1023 White D. A., Jones C., Forman W., 1997, , 292, 419 Wu K. K. S., Fabian A. C., Nulsen P. E. J., 2000, , 318, 889 Xue Y.-J., Wu X.-P., 2000, , 538, 65 Wu X.-P., Xue Y.-J., 2002, , 569, 112 [^1]: Many authors prefer to adopt the redshift scaling of the critical density, $E(z)^2=\Omega_{\rm m}(1+z)^{3} + \Omega_{\Lambda}$ (for a flat universe), rather than the background density, $(1+z)^3$.

--- abstract: 'Spins in molecular magnets can experience both anisotropic exchange interactions and on-site magnetic anisotropy. In this paper we study the effect of exchange anisotropy on the molecular magnetic anisotropy both with and without on-site anisotropy. When both the anisotropies are small, we find that the axial anisotropy parameter $D_M$ in the effective spin Hamiltonian is the sum of the individual contributions due to exchange and on-site anisotropies. We find that even for axial anisotropy of about $15\%$, the low energy spectrum does not correspond to a single parent spin manifold but has intruders states arising from other parent spin. In this case, the low energy spectrum can not be described by an effective Hamiltonian spanning the parent spin space. We study the magnetic susceptibility, specific heat as a function of temperature and magnetization as a function of applied field to characterize the system in this limit. We find that there is synergy between the two anisotropies, particularly for large systems with higher site spins.' author: - 'Sumit Haldar[^1]' - 'Rajamani Raghunathan[^2]' - 'Jean-Pascal Sutter[^3]' - 'S. Ramasesha[^4]' title: 'Modeling Molecular Magnets with Large Exchange and On-Site Anisotropies' --- [***Keywords*** Single Chain Magnets; Anisotropic spin Hamiltonians; On-Site Anisotropy; Low Energy Eigenvalues; Thermodynamic Properties]{} \[sec:introduction\]Introduction ================================ Molecular spin clusters such as single molecule magnets (SMMs) and single chain magnets (SCMs) have been studied extensively over the last few decades [@Sessoli1993; @Gatteschi2003; @Takahashi1994; @Thomas1996; @Sessoli1993a; @Christou2000]. These spin clusters have attracted huge interest from both theoretical and experimental stand points because of the promise they hold for applications such as in memory storage devices, in quantum computations and in information technologies in general [@Tejada2001; @Winpenny2008; @Lehmann2009; @Vincent2012; @Troiani2011; @Leuenberger2001]. The main bottleneck for these applications appears to be the fast relaxation of the magnetization from the fully magnetized to the non magnetized state. This is due to the low blocking temperature, measured as the temperature at which the relaxation time for magnetization, $\tau_R$, is 100s and depends on the energy barrier between two fully and oppositely magnetized states, for the presently known SMMs and SCMs [@Gatteschi2007; @Demir2017]. Current research in this field is focused on enhancing the blocking temperature [@Woodruff2013; @Langley2016]. The energy barrier $\Delta$, between two fully and oppositely magnetized states of an anisotropic spin cluster of spin $S$ is given by $\Delta=|D_M|S^2$ for an integer spin cluster and $|D_M|(S^2-1/4)$ for a half-integer spin cluster. Therefore, there are two routes to enhancing $\Delta$, $(i)$ by increasing $D_M$ and $(ii)$ by increasing $S$. Increasing $D_M$ can be achieved by using magnetic building blocks in unusual coordination number and geometry. Indeed this has been demonstrated for hepta coordinated complexes [@Ruamps2013; @Gogoi2013; @Venkatakrishnan2010; @Bar2015; @Bar2017]. Increasing $S$ can be achieved by using rare earth ions in the high spin state as the building blocks. However, it has been shown by Waldmann [@Waldmann2007] that the magnetic anisotropy of a ferromagnetic assembly of spins is smaller than the anisotropy of individual spins as each spin center with spin $s_i$ only contributes a fraction $$\begin{aligned} \frac{s_i(2s_i-1)}{S(2S-1)}\end{aligned}$$ of the site anisotropy to the anisotropy of the SMM or SCM with total spin is $S$. This result assumes that all the individual magnetic ions have non zero axial anisotropy $d_i$ and zero planar anisotropy $e_i$, and that all the spin centers have the same magnetic axes. Notwithstanding this nuance, the result is illustrative of the fact that the anisotropy of the clusters is smaller than that of individual ions. With $3d$ transition metal complexes, the highest blocking temperature reported is $4.5K$, although the energy barrier $\Delta$ is $62 cm^{-1}$ [@Milios2007]. This could be due to the large off-diagonal anisotropy terms that lead to quantum tunneling of magnetization. The anisotropy can be enhanced by choosing ions of $4d$, $5d$ or $4f$ metals wherein the relativistic effects are large, leading to large spin-orbit interactions [@Tang2006; @Woodruff2013; @Rinehart2011; @Dreiser2013a; @Mironov2003; @Bennett2003]. For example, in the $Dy_4$ systems, the energy barrier is $~692 cm^{-1}$ [@Blagg2013]. However, large quantum tunneling of magnetization leads to small hysteresis loops. In our previous studies [@Haldar2017], we have shown that large magnetic anisotropy of building blocks leads to breaking the spin symmetry. In this event associating a parent spin state to define the $D_M$ and $E_M$ parameters of a cluster is not possible due to intrusion of states from different parent spins within the given spin manifold. In these cases, the Waldmann conclusion that the contribution of the individual anisotropies decreases with increasing total spin of the cluster is no longer valid. The properties of the system will have to be computed from the eigenstates of the full Hamiltonian. The origin of single ion anisotropy as well as anisotropic exchange interactions lie in spin-orbit interactions. Indeed, it is difficult to assume isotropic or simple Heisenberg exchange interactions between spin sites that are highly anisotropic. High nuclearity complexes with large anisotropic interactions are known in a few cases, $[Mn^{III}_6Os^{III}]^{3+}$ cluster has $J_x=-9cm^{-1}, J_y=+17cm^{-1}$ and $J_z=-16.5cm^{-1}$ [@Hoeke2014; @Dreiser2013; @Mironov2003a] and $[Mn^{II}Mo^{III}]$ complex has $J_z=-34cm^{-1}$ and $J_x=J_y=-11cm^{-1}$ [@Mironov2015; @Qian2013]. In this study, we employ a generalized ferromagnetic XYZ model for nearest neighbor spin-spin interactions and on-site anisotropy. Using the full Fock space of the Hamiltonian, we follow the properties such as magnetization, susceptibility and specific heat of spin chains with ferromagnetic interaction and different site spins. In the next section we discuss briefly spin Hamiltonian we have studied and present the numerical approach for obtaining the properties of the model. In the third section, we present the result of a purely anisotropic exchange model. This will be followed by the results on a model with both exchange and site anisotropies in section four. We will end the paper with a discussion of all the results. \[sec:methodology\]Methodology ============================== The basic starting Hamiltonian for studying most magnetic materials is the isotropic Heisenberg exchange model given by $$\begin{aligned} \label{eqn:Heisenberg} \hat{\mathcal{H}}_{Heis}=\sum_{\langle i,j\rangle} J_{ij} \hat{S}_i\cdot \hat{S}_j\end{aligned}$$ where the summation is over nearest neighbors. This model assumes that spin-orbit interactions are weak and hence the exchange constant $J$ associated with the three components of the spin are equal ($J_{ij}^x=J_{ij}^y=J_{ij}^z$). The isotropic model conserves both total $M_s$ and total $S$ and hence we can choose a spin adapted basis such as the valence bond (VB) basis to set up the Hamiltonian matrix. The Rumer-Pauling VB basis is nonorthogonal and hence the Hamiltonian matrix is nonsymmetric. While computing eigenstates of a nonsymmetric matrix is reasonably straight forward, computing properties of the eigenstates in the VB basis is nontrivial. However, the VB eigenstates can be transformed to eigenstates in constant $M_s$ basis and the latter basis being orthonormal is easily amenable to computing properties of the eigenstates. When the spin-orbit interactions are weak, we can include the anisotropy arising from it by adding the site anisotropy term, $$\begin{aligned} \label{eqn:AnisHam} \hat{\mathcal{H}}_{aniso}=\sum_{i} [d_{i,z} \hat{s}_{i,z}^2+d_{i,x} \hat{s}_{i,x}^2+d_{i,y} \hat{s}_{i,y}^2]\end{aligned}$$ ($d_{i,x}$, $d_{i,y}$ and $d_{i,z}$ are local ion anisotropies) and treating it as a perturbation. Usually, it is sufficient to deal with just the site diagonal anisotropy and set $d_{i,x}=d_{i,y}=0$. However, if the local anisotropy axis is not aligned with the global spin axis, then we need to include the off-diagonal site anisotropy terms. For weak on-site anisotropy ($\frac{d}{J} << 1$), we can obtain the splitting of a given total spin state perturbatively by determining the molecular anisotropy parameters $D_M$ and $E_M$ given by the eigenstates of the Hamiltonian in a given spin state $S$ [@Raghunathan2008], $$\begin{aligned} \label{eqn:Molanis} \hat{\mathcal{H}}_{mol}&=&D_M \left( \hat{S}_z^2 - \frac{1}{3}S(S+1) \right)+E_M(\hat{S}_x^2-\hat{S}_y^2)\end{aligned}$$ Spin-orbit interaction can also lead to anisotropy in the exchange Hamiltonian leading to a general $XYZ$ model whose Hamiltonian is given by $$\begin{aligned} \label{eqn:XYZHaml} \hat{\mathcal{H}}_{XYZ}=\sum_{\langle ij \rangle} [J_{ij}^x \hat{s}_i^x\cdot \hat{s}_j^x+J_{ij}^y \hat{s}_i^y\cdot \hat{s}_j^y+J_{ij}^z \hat{s}_i^z\cdot \hat{s}_j^z]\end{aligned}$$ for $J_{ij}^x \neq J_{ij}^y \neq J_{ij}^z$. In this model, there does not exist any spin symmetry and we need to solve the Hamiltonian for its eigenstates in the full Fock basis with no restrictions on total $S$ or $M_s$. In cases where a system has the same exchange constant along x and y directions but different from the exchange constant in the z-direction, we obtain the XXZ model with the Hamiltonian is given by $$\begin{aligned} \label{eqn:XXZHaml} \hat{\mathcal{H}}_{XXZ}=\sum_{\langle ij \rangle} J_{ij}^x [\hat{s}_i^x\cdot \hat{s}_j^x+\hat{s}_i^y\cdot \hat{s}_j^y]+J_{ij}^z \hat{s}_i^z\cdot \hat{s}_j^z\end{aligned}$$ For convenience we write the general XYZ Hamiltonian in eqn. \[eqn:XYZHaml\] as $$\begin{aligned} \label{eqn:generalHaml} \hat{\mathcal{H}}=\sum_{\langle ij \rangle} J [\hat{s}_i^z\cdot \hat{s}_j^z+(\gamma + \delta)\hat{s}_i^x\cdot \hat{s}_j^x+(\gamma - \delta) \hat{s}_i^y\cdot \hat{s}_j^y]\end{aligned}$$ where $J_{ij}^z=J$, $\gamma = \frac{J_{ij}^x+J_{ij}^y}{2J}$ and $\delta = \frac{J_{ij}^x-J_{ij}^y}{2J}$. The deviation of $\frac{J_{ij}^x+J_{ij}^y}{2}$ from $J_{ij}^z$ is then represented by the parameter $\epsilon = 1-\gamma$ and the difference between exchange along x and y directions in normalized units is $\delta$. This model can be solved in the constant $M_s$ basis. Besides exchange anisotropy, a system can also have site anisotropy in which case, the $\hat{\mathcal{H}}_{aniso}$ should be considered together with the respective Hamiltonian, either perturbatively (for weak on site anisotropy) or in the zeroth order Hamiltonian itself. The effect of large anisotropic exchange or large site anisotropy is to mix states with different total spin $S$. Thus, the conventional approach to define molecular anisotropy constants through the effective Hamiltonian (eqn. \[eqn:Molanis\]) fails as the low-lying multiplet states can not be identified as arising from a unique total spin state, as, the total spin of a state is not conserved. In such situations, the approach we have taken is to obtain the thermodynamic properties such as susceptibility $\chi(T)$, magnetization $M(T)$ and specific heat $C_v(T)$ of the system as a function of Hamiltonian parameters. These are computed from the canonical partition function obtained from the full spectrum of the Hamiltonian. The full Fock space of the Hamiltonian is given by $(2s_i+1)^N$, where N is the number of sites in the spin chain. The largest system we have studied corresponds to $s_i=2$ and $N=5$ which spans a Fock space of dimensionality of 3,125. We need to calculate $\langle \langle M_s \rangle \rangle$ for the magnetic properties which is a thermodynamic average of the expectation values in the eigenstates. To obtain the spin expectation value $\langle \hat S^2 \rangle$ in an eigenstate we have computed the spin-spin correlation functions $\langle \hat s_i^z \hat s_j^z \rangle$, $\langle \hat s_i^x \hat s_j^x \rangle$ and $\langle \hat s_i^y \hat s_j^y \rangle$. \[sec:RandD\]Anisotropic Exchange Models ======================================== Here, we discuss the magnetic anisotropy arising only from the exchange anisotropy. In the small exchange anisotropy limit, we first consider the XXZ model and XYZ model with small $\delta$. We will end this section with a discussion of the XYZ models with large anisotropy parameters $\epsilon$ and $\delta$. All the exchange interactions are taken to be ferromagnetic. \[sec:XXZ\]Small Anisotropy models ---------------------------------- In this model we set $\delta$ to zero in eqn. \[eqn:generalHaml\] and study spin chains with site spins $1$, $3/2$ and $2$ in chains of 4 and 5 sites with open boundary condition. We have not considered spin-1/2 system since we wish to study the synergistic effect of anisotropic exchange and on-site anisotropy. The latter exists only for site spin greater than half. The ground state in each case corresponds to $\pm M_s=Ns$ where $N$ is the number of sites and $s$ is the site spin. The total spin of the states is calculated from the eigenstates as expectation value of $\hat S^2$.   \ ------------ ------------- ----------- ----------- -- --------------- ----------- ---------- -- -------------- ----------- ----------- $\epsilon$ $M_s$ $S_{tot}$ Energy $M_s$ $S_{tot}$ Energy $M_s$ $S_{tot}$ Energy   0.10     $\pm 5$     5.00     0     $\pm 7.5$     7.50     0     $\pm 10$     10.00     0   $\pm 4$ 4.99  0.158  $\pm 6.5$ 7.49  0.237  $\pm 9$ 9.99  0.316  $\pm 3$ 4.99  0.282  $\pm 5.5$ 7.49  0.442  $\pm 8$ 9.99  0.601  $\pm 2$ 4.99  0.370  $\pm 4.5$ 7.49  0.612  $\pm 7$ 9.99  0.852  $\pm 1$ 4.99  0.423  0 4.99  0.441  $\pm 3.5$ 7.49  0.749  $\pm 6$ 9.99  1.071  $\pm 2.5$ 7.49  0.852  $\pm 5$ 9.99  1.256  $\pm 1.5$ 7.49  0.921  $\pm 4$ 9.99  1.407  $\pm 0.5$ 7.49  0.955  $\pm 3$ 9.99  1.611  $\pm 2$ 9.99  1.661  $\pm 1$ 9.99  1.673  0 9.99  1.678  0.15     $\pm 5$     5.00     0     $\pm 7.5$     7.50   0     $\pm 10$     10.00     0   $\pm 4$ 4.99   0.236   $\pm 6.5$ 7.49   0.354  $\pm 9$ 9.99   0.472   $\pm 3$ 4.99 0.420 $\pm 5.5$ 7.49 0.658 $\pm 8$ 9.99 0.895 $\pm 2$ 4.98 0.551 $\pm 4.5$ 7.489 0.914 $\pm 7$ 9.991 1.272 $\pm 1$ 4.98 0.630 0 4.97 0.656 $\pm 3.5$ 7.48 1.118 $\pm 6$ 9.98 1.598 $\pm 2.5$ 7.48 1.273 $\pm 5$ 9.98 1.876 $\pm 1.5$ 7.48 1.375 $\pm 0.5$ 7.48 1.426 $\pm 4$ 9.98 2.104 $\pm 3$ 9.98 2.282 $\pm 2$ 9.98 2.408 $\pm 1$ 9.98 2.484 0 9.98 2.510 ------------ ------------- ----------- ----------- -- --------------- ----------- ---------- -- -------------- ----------- ----------- : \[tab:XXZTableFive\]Energy gaps (in units of $J$) from the ground state of the low-lying states lying below the lowest state with $M_s$=0. $M_s$ is conserved and is a good quantum number. The total spin $S_{tot}$ is calculated from the expectation value $\langle \hat S^2 \rangle$ of the state. Intruder states are shown in red. In table \[tab:XXZTableFive\] we present the energy gaps from the ground state of the low-lying states up to first $M_s=0$ state of short spin chains of length up to five spins for different $\epsilon$ values. The table for spin chains of four spins is given in supporting material. We notice from the table that for $\epsilon=0.1$, the lowest energy states of the $s=1$ chains satisfies $E(|M_s|=Ns) < E(|M_s|=Ns-1)...< E(|M_s|=0)$ and the total spin of these states is also very close to $Ns$. In this case we can fit the energy gaps to the Hamiltonian $D_M S_{z}^2$. The diagonal anisotropy of these states is shown in fig. \[fig:DMvsepsln\]. {width="10cm"} In the XXZ model we do not have off-diagonal anisotropy, i.e., $E_M=0$ in the anisotropic Hamiltonian given by eqn. \[eqn:Molanis\]. We note in table \[tab:XXZTableFive\] that for spin chains with $s=3/2$ and $s=2$, there are intruder states within the manifold of $S \simeq 7.5$ and $\simeq 10$ respectively. We also find that as $\epsilon$ is increased to $0.15$, even the $s=1$ spin chain has intruders. Furthermore, for site spin $2$, the intruders within the $S=10$ manifold are from progressively lower total spin states, namely $S=9$, $8$ and $7$. Thus, it is not meaningful anymore to define molecular $D_M$ and $E_M$ parameters. For the $N=4$ chains the intruder states occur in $s=1$ chain for $\epsilon=0.25$ and for $s=3/2$ and $s=2$ chain for $\epsilon=0.20$ (see supporting material). Thus, intruders arise at smaller $\epsilon$ values for longer chains and higher site spin. The $|D_M|$ increases linearly with increase in anisotropy (fig. \[fig:DMvsepsln\]). {width="10cm"} {width="10cm"} We have obtained the thermodynamic properties of these spin chains as a function of temperature and the magnetization as a function of magnetic field at a fixed temperature. We show in fig. \[fig:ChiXX,ChiZZ\], $\chi_{_{xx}}T(=\chi_{_{yy}}T)$ and $\chi_{_{zz}}T$ dependence on temperature for spin chains of five spins for different values of the site spins. Expectedly the susceptibility increases with site spin in all cases. The $\chi_{_{zz}}T$ component is much larger than the $\chi_{_{xx}}T$ component and both show a maxima. The maxima is at a higher temperature for $\chi_{_{xx}}T$ compared to $\chi_{_{zz}}T$ and the $\chi_{_{xx}}T$ maxima is also broader. We also note that $\chi_{_{zz}}T$ is larger than $\chi_{_{xx}}T$ by a factor of between 2 and 3, even though maximum anisotropy $\epsilon$ is only 0.25. Besides the temperature of the maxima also increases with site spin. The $ZZ$ component is larger for large anisotropy while the $XX$ component is smaller at large anisotropy. This is because as $\epsilon$ increases it becomes easier to magnetize along the z-axis, while it becomes harder to magnetize in the x-y plane. This trend is also seen in the magnetization plots as a function of the magnetic field shown in fig. \[fig:MxHx,MzHz\]. We note that the magnetization $\langle M_z \rangle$ increases with $\epsilon$ while $\langle M_x \rangle$ decreases with $\epsilon$ for the same applied field. {width="10cm"} {width="10cm"} The dependence of specific heat, $C_v$, on temperature for different $\epsilon$ values is shown in fig. \[fig:Cvxxz\]. {width="12cm"} We find that for small $\epsilon$, the specific heat shows two peaks, the first peak is narrow and the second peak is broad. This is seen for all site spins. This can be understood from the nature of the full energy spectrum of the Hamiltonian for different $\epsilon$ values fig. \[fig:Histogram\]. We see that there are two successively small gaps in the spectrum below 0.13J for small anisotropy but these gaps shift to much higher energies for large anisotropy. This implies that at small anisotropy, the specific heat first increases with increase in temperature and then drops as thermal energy can not access higher energy states. As the temperature increases further the higher energy states are populated leading to increase in specific heat. Thus, the magnetic specific heat dependence on temperature can be used as a tool to estimate the anisotropy of the chain. {height="10cm" width="12cm"} Introducing small planar anisotropy, $\delta$, does not significantly change the low energy spectrum in table \[tab:xyztable6\] and consequently there is no discernible change in the thermodynamic properties. The main difference is that $M_s$ is also not conserved even for small values of $\delta$.   \ ------------------- ----------------- ----------- --------- -- ----------------- ----------- ---------- -- ----------------- ----------- --------- $\epsilon,\delta$ $M_s$ $S_{tot}$ Energy $M_s$ $S_{tot}$ Energy $M_s$ $S_{tot}$ Energy  0.095,    -4.99, 4.99     4.99     0     -7.49, 7.49    7.49     0     -9.99, 9.99     9.99     0   0.005  -3.99, 3.99   4.998   0.150   -6.49, 6.49   7.49    0.225   -8.997, 8.99   9.99   0.301   -2.99, 2.99   4.997   0.267   -5.49, 5.49   7.49   0.419   -7.995, 7.99   9.99   0.570   -1.94, 2.01   4.995   0.347   -4.48, 4.48   7.49   0.581   -6.992, 6.99   9.99   0.808   -1.04, 1.03   4.993   0.389    ****    4.99   0.423   -3.46, 3.46   7.49   0.709   -5.98, 5.98   9.99   1.015             -2.33, 2.33   7.49   0.805   -4.97, 4.97   9.99   1.189             -1.51, 1.51   7.49   0.865   -3.95, 3.96   9.99   1.332   -0.97, 0.97   7.49   0.928               -2.82, 2.94   9.99   1.435                ****    9.99   1.497                ****    9.99   1.539                ****    9.99   1.552                            ****    9.99   1.636                ****    9.99   1.637   0.15,     -4.95, 4.95     4.99     0     -7.46, 7.46     7.49     0     -9.99, 9.99     9.99     0   0.05  -3.80, 3.80   4.991   0.214   -6.37, 6.37   7.49   0.329   -8.99, 8.99   9.99   0.441    -2.57    4.99   0.352   -5.23, 5.23   7.49   0.605   -7.77, 7.78   9.99   0.832    2.57    4.99   0.403    1.74    4.98   0.448   -3.93, 3.93   7.48   0.817   -3.28, 3.28   7.48   0.992   -6.58, 6.65   9.99   1.266    -1.74    4.98   0.588   -3.19, 3.19   6.48   1.039    ****    4.98   0.590   -4.15, 4.18   7.48   1.200    ****    9.98   1.423    ****    9.98   1.462   -2.67, 2.67   7.49   1.406    ****    9.99   1.732  ****  4.98   0.815   -2.89, 2.89   7.48   1.529  ****  4.98   0.816   -4.32, 4.32   7.48   1.562    ****    9.99   1.769                ****    9.98   2.026                            ****    9.99   2.030  ------------------- ----------------- ----------- --------- -- ----------------- ----------- ---------- -- ----------------- ----------- --------- : \[tab:xyztable6\]Energy gaps from the ground state (in units of $J$) of the low-lying states lying within the manifold of spin $S\simeq ns$. Both $M_s$ and $S$ are not conserved and not good quantum numbers. The total spin $S_{tot}$ is calculated from the expectation value $\langle \hat S^2 \rangle$ of the state. Intruder states are shown in red. $\langle M_s \rangle$ are given for states for which it could be computed. $\langle M_s \rangle$ values are not quoted for the states which show large mixing of different $M_s$ states. \[sec:XYZ\]Large Anisotropy models ---------------------------------- To explore the properties of the spin chains in the large anisotropy limit, we have studied $s=1$, $3/2$ and $2$ models with $\epsilon$ up to $0.75$ and $\delta$ up to $0.15$. In this limit, there are no conserved spin quantities, hence we have studied only thermodynamic properties by computing thermodynamic averages from expectation values in the eigenstates of the Hamiltonian. All the three diagonal components of the susceptibility as a function of temperature are shown in fig. \[fig:ChiXX,chiYY,ChiZZ\]. We find that for large anisotropy $\chi_{_{zz}}T$ increases with $\epsilon$ and $\delta$, while $\chi_{_{xx}}T$ and $\chi_{_{yy}}T$ decreases with $\epsilon$ and $\delta$. $\chi_{_{zz}}T$ shows a smooth maxima for all cases we have studied but $\chi_{_{xx}}T$ and $\chi_{_{yy}}T$ do not show a discernible maxima. The $\chi_{_{zz}}T$ maxima occur at lower temperature than $\chi_{_{xx}}T$ and $\chi_{_{yy}}T$ maxima (when they exist). More significantly $\chi_{_{zz}}T$ is higher for higher anisotropy while $\chi_{_{xx}}T$ and $\chi_{_{yy}}T$ are higher for lower anisotropy. {width="8cm"} {width="8cm"} {width="8cm"} In fig. \[fig:MxHx,MyHy,MzHz\] we show the behaviour of magnetization as a function of the field at $k_BT/J=1$. We find very different behaviour for $M_z$ compared to $M_x$ or $M_y$. The $M_z$ component shows saturation at low magnetic fields. The saturation field decreases with increasing site spin. On the other hand, the $M_x$ and $M_y$ components show saturation only for small anisotropy. For large anisotropy they do not show saturation and show a nearly linear increase in magnetization component over the full range of the applied magnetic field. Furthermore, the magnitude of the magnetization decreases with increasing anisotropy at a given field strength. {width="8cm"} {width="8cm"} {width="8cm"} The specific heat behaviour is similar to the weak anisotropy case, we find a sharp peak at low temperature followed by a broad peak at higher temperatures. At higher anisotropies, we find a single peak in the $C_v$ vs $T$ plot \[fig:Cvxyz\] and the temperature of the peak maxima is higher for higher anisotropy. For a fixed anisotropy, the peak maximum shifts to higher temperature as the site spin increases from $s=1$ to $s=2$. {width="12cm"} \[sec:onsiteans\]Systems with Exchange and On-Site Anisotropies =============================================================== In an earlier paper we discussed the role of on-site single ion anisotropy on the anisotropy of a spin chain. In this section we will discussed the effect of both exchange and on-site anisotropy on the magnetic properties of a spin chain [@Haldar2017]. We have introduced on-site anisotropy ($d/J$) in the eqn. \[eqn:generalHaml\] and studied the spin chains with site spins $s=1$, $3/2$ and $2$ of length of five spins. We have also set $\delta=0$ and study only XXZ models in the presence of site anisotropy. We have taken same on-site anisotropy aligned along the z-axis for all the spins. When the on-site anisotropy is weak, we find that the resultant molecular magnetic anisotropy is nearly a sum of the molecular anisotropy due to on-site anisotropy alone and the molecular anisotropy due to exchange anisotropy alone. Thus, the two anisotropies are additive as seen in fig. \[fig:DMvsepsln\]. This is true up to $\epsilon=0.1$ for all the site spins. In table \[tab:xxztablewithd\],   \ ------------ ------------- ----------- -------- -- --------------- ----------- -------- -- -------------- ----------- -------- $\epsilon$ $M_s$ $S_{tot}$ Energy $M_s$ $S_{tot}$ Energy $M_s$ $S_{tot}$ Energy   0.1     $\pm 5$     5.00     0     $\pm 7.5$     7.50     0     $\pm 10$     10.00     0   $\pm 4$ 4.99 0.412 $\pm 6.5$ 7.49 0.668 $\pm 9$ 9.99 0.9240 $\pm 3$ 4.96 0.726 $\pm 5.5$ 7.47 1.238 $\pm 8$ 9.98 1.749 $\pm 2$ 4.88 0.944 $\pm 4.5$ 7.43 1.708 $\pm 7$ 9.95 2.476 $\pm 1$ 4.78 1.067 0 4.75 1.108 $\pm 3.5$ 7.36 2.080 $\pm 6$ 9.91 3.103 $\pm 2.5$ 7.26 2.351 $\pm 6$ 8.90 3.426 $\pm 5$ 9.84 3.628 $\pm 1.5$ 7.15 2.5270 $\pm 0.5$ 7.08 2.6130 $\pm 4$ 9.75 4.053 $\pm 3$ 9.63 4.377 $\pm 2$ 9.51 4.604 $\pm 1$ 9.41 4.737 0 9.37 4.781 ------------ ------------- ----------- -------- -- --------------- ----------- -------- -- -------------- ----------- -------- : \[tab:xxztablewithd\]Energy gaps (in units of $J$) from the ground state of the low-lying states lying below the lowest state with $M_s$=0 for $d/J=0.1$. $M_s$ is conserved and is a good quantum number. The total spin $S_{tot}$ is calculated from the expectation value of $\langle \hat S^2 \rangle$ of the state. Intruder states are shown in red. we show the low-energy spectrum of the $N=5$ spin chain for $s=1$, $3/2$ and $2$, where both exchange and on-site anisotropies are large. In cases where we can not define the molecular magnetic anisotropy in terms of the parameter $D_M$ of the effective spin Hamiltonian, we follow the system by computing the magnetic susceptibilities, magnetization and specific heat. We have shown in fig. \[fig:ChiXXd,ChiZZd\], the difference in the $\Delta\chi_{_{xx}}T~=~\chi_{_{xx}}T(\epsilon, d \neq 0)-\chi_{_{xx}}T(\epsilon, d=0)$ and $\Delta\chi_{_{zz}}T~=~\chi_{_{zz}}T(\epsilon, d \neq 0)-\chi_{_{zz}}T(\epsilon, d=0)$ of magnetic susceptibility as a function of $d/J$ at $k_BT/J=1$ for different $\epsilon$ values. We find that nonzero $d$ enhances $\Delta\chi_{_{zz}}T$ but decreases $\Delta\chi_{_{xx}}T$ values. In case of site spin $s=1$, the dependence of $\Delta\chi_{_{xx}}T$ and $\Delta\chi_{_{zz}}T$ on site anisotropy is weak and linear. In case of $s=3/2$ and $s=2$ the difference $\Delta\chi_{_{zz}}T$ increases sharply as $d/J$ is increased and for higher $d/J$ it tends to saturate. The saturation is more apparent in the $s=2$ case. $\Delta\chi_{_{xx}}T$ on the other hand decreases with increasing $d/J$. This is because the on-site anisotropy is oriented along the z-axis. This is also the reason why $\Delta\chi_{_{xx}}T$ shows a sharper drop with $d/J$ for larger $\epsilon$ while $\Delta\chi_{_{zz}}T$ shows a sharper rise for larger $\epsilon$. ![\[fig:ChiXXd,ChiZZd\]The effect of on-site anisotropy $d/J$ on (a)$\Delta\chi_{_{xx}}T=[\chi_{_{xx}}T(\epsilon, d \neq 0)-\chi_{_{xx}}T(\epsilon, d=0)]$ at $g\beta H_x/J=0.005$, $k_BT/J=1.0$ and (b) $\Delta\chi_{_{zz}}T=[\chi_{_{zz}}T(\epsilon, d \neq 0)-\chi_{_{zz}}T(\epsilon, d=0)]$ at $g\beta H_z/J=0.005$, $k_BT/J=1.0$ for $\epsilon=0.10$, $0.15$, $0.20$ and $0.25$. Same color code and line type is used for all panels. Also note the sign of $\Delta\chi_{_{xx}}T$ is -ve while $\Delta\chi_{_{zz}}T$ is +ve. ](delPvsdbyJ_ChiXX.pdf){width="10cm"} ![\[fig:ChiXXd,ChiZZd\]The effect of on-site anisotropy $d/J$ on (a)$\Delta\chi_{_{xx}}T=[\chi_{_{xx}}T(\epsilon, d \neq 0)-\chi_{_{xx}}T(\epsilon, d=0)]$ at $g\beta H_x/J=0.005$, $k_BT/J=1.0$ and (b) $\Delta\chi_{_{zz}}T=[\chi_{_{zz}}T(\epsilon, d \neq 0)-\chi_{_{zz}}T(\epsilon, d=0)]$ at $g\beta H_z/J=0.005$, $k_BT/J=1.0$ for $\epsilon=0.10$, $0.15$, $0.20$ and $0.25$. Same color code and line type is used for all panels. Also note the sign of $\Delta\chi_{_{xx}}T$ is -ve while $\Delta\chi_{_{zz}}T$ is +ve. ](delPvsdbyJ_ChiZZ.pdf){width="10cm"} Similarly in fig. \[fig:MxHxd,MzHzd\], we plot $\Delta M_x$ and $\Delta M_z$ for different $s$ and $\epsilon$, as a function of $d/J$. The field strength is $g\mu_BH~=~J/2$. We note that the $\Delta M_x$ decreases sharply with $d/J$ for $s=2$ and large $\epsilon$ while $\Delta M_z$ increases with $d/J$ and saturates for $s=2$ case while in the $s=3/2$ and $s=1$ cases, the saturation does not occur even for $d/J=1.0$. Again $\Delta M_z$ is larger when $\epsilon$ is small while $\Delta M_x$ is larger for large $\epsilon$. ![\[fig:MxHxd,MzHzd\]The effect of on-site anisotropy $d/J$ on (a) $\Delta M_x=M_x(\epsilon, d \neq 0)-M_x(\epsilon, d=0)$ at $g\beta H_x/J=0.25$, $k_BT/J=1.0$ and (b) $\Delta M_z=M_z(\epsilon, d \neq 0)-M_z(\epsilon, d=0)$ at $g\beta H_z/J=0.25$, $k_BT/J=1.0$ for $\epsilon=0.10$, $0.15$, $0.20$ and $0.25$. Same color and line type is used for all panels.](MxvsHx_5site_withdbyJ_Jx_0p9_0p8_0p75.pdf){width="10cm"} ![\[fig:MxHxd,MzHzd\]The effect of on-site anisotropy $d/J$ on (a) $\Delta M_x=M_x(\epsilon, d \neq 0)-M_x(\epsilon, d=0)$ at $g\beta H_x/J=0.25$, $k_BT/J=1.0$ and (b) $\Delta M_z=M_z(\epsilon, d \neq 0)-M_z(\epsilon, d=0)$ at $g\beta H_z/J=0.25$, $k_BT/J=1.0$ for $\epsilon=0.10$, $0.15$, $0.20$ and $0.25$. Same color and line type is used for all panels.](MzvsHz_5site_withdbyJ_Jx_0p9_0p8_0p75.pdf){width="10cm"} The specific heat behaviour is shown in fig. \[fig:Cvxxzd\]. We find that the two peak structure persists for small $d/J$ for $\epsilon=0.1$. However, increasing $d/J$ leads to a single peak. The peak position shifts to higher temperatures as $d/J$ increases and the peak also becomes sharper as $d/J$ increases. This is true for all site spins. ![\[fig:Cvxxzd\]Dependence of specific heat ($C_v$) on temperature ($k_BT/J$) of spin chains with $s=1$, $3/2$ and $s=2$ with systems size $N=5$ for $\epsilon=0.10$ in the presence of $d/J=0.10$, $0.30$, $0.50$ and $0.70$. Same color and line type is used for all panels. ](CvvsT_5site_withdbyJ_Jx_0p9.pdf){width="12cm"} \[sec:conclusion\]Conclusions ============================= Our study of anisotropic ferromagnetic exchange models with site anisotropy shows that for small exchange and site anisotropies, the energy level splitting of the total spin states can be characterized by the axial anisotropy parameter $D_M$ which is a sum of the exchange alone and ion anisotropy alone $D_M$ parameters. For large anisotropic exchange, neither the total spin nor its z-component are conserved and it is not possible to define the molecular anisotropy parameters $D_M$ and $E_M$. The effect of anisotropy is then studied by following thermodynamics properties such as $\chi$, $C_v$ and $M$. This is also true when the on-site anisotropy is large, even in the absence of exchange anisotropy. We find two peak structure in $C_v$ vs $T$ when the exchange is weakly anisotropic. We also find that this feature prevails for weak on-site anisotropy as well. The dual peak structure is more pronounced for smaller on-site spins. In general the effect of anisotropy, as seen form the presence of intruder states from different parent spin state, is more pronounced in the case of higher site spins and longer chain length. The synergy between site anisotropy and exchange anisotropy becomes complicated when both are strong. We observe that the difference in susceptibilities as well as magnetization as a function of the site anisotropy strength for large exchange anisotropy becomes highly nonlinear, particularly for systems with higher site spin. \[sec:ack\]Acknowledgements =========================== SR and JPS acknowledge the support through IFCPAR/CEFIPRA projets. SR also thanks DST for support through different projects and a fellowship and Indian Science Academy for Senior Scientist Position. RR thanks TUE-DST for support. [10]{} Roberta Sessoli, Dante Gatteschi, Hui Lien Tsai, David N. Hendrickson, Ann R. Schake, Sheyi Wang, John B. Vincent, George Christou, and Kirsten Folting. . , 115(5):1804–1816, mar 1993. Dante Gatteschi and Roberta Sessoli. . In [*Angewandte Chemie - International Edition*]{}, volume 42, pages 268–297. WILEY - VCH Verlag, jan 2003. Minoru Takahashi. . , 91(1):1–15, 1994. L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara. . , 383(6596):145–147, sep 1996. R. Sessoli, D. Gatteschi, A. Caneschi, and M. A. Novak. . , 365(6442):141–143, sep 1993. George Christou, Dante Gatteschi, David N. Hendrickson, and Roberta Sessoli. . , 25(11):66–71, nov 2000. J Tejada, E M Chudnovsky, E [Del Barco]{}, J M Hernandez, and T P Spiller. . , 12(01):181–186, 2001. Richard E. P. Winpenny. . , 47(42):7992–7994, 2008. J[ö]{}rg Lehmann, Alejandro Gaita-Ari[ñ]{}o, Eugenio Coronado, and Daniel Loss. . , 19(12):1672–1677, mar 2009. Romain Vincent, Svetlana Klyatskaya, Mario Ruben, Wolfgang Wernsdorfer, and Franck Balestro. , volume 488. Nature Publishing Group, aug 2012. Filippo Troiani and Marco Affronte. , volume 40. The Royal Society of Chemistry, may 2011. Michael N. Leuenberger and Daniel Loss. . , 410(6830):789–793, apr 2001. Dante Gatteschi, Roberta Sessoli, and Jacques Villain. , volume 9780198567. Oxford University Press, mar 2007. Selvan Demir, Miguel I. Gonzalez, Lucy E. Darago, William J. Evans, and Jeffrey R. Long. . , 8(1):2144, dec 2017. Daniel N. Woodruff, Richard E.P. Winpenny, and Richard A. Layfield. , volume 113. American Chemical Society, jul 2013. Stuart K. Langley, Daniel P. Wielechowski, Boujemaa Moubaraki, and Keith S. Murray. . , 52(73):10976–10979, sep 2016. Renaud Ruamps, Luke J. Batchelor, R[é]{}mi Maurice, Nayanmoni Gogoi, Pablo Jim[é]{}nez-Lozano, Nathalie Guih[é]{}ry, Coen Degraaf, Anne Laure Barra, Jean Pascal Sutter, and Talal Mallah. . , 19(3):950–956, jan 2013. Nayanmoni Gogoi, Mehrez Thlijeni, Carine Duhayon, and Jean Pascal Sutter. . , 52(5):2283–2285, mar 2013. Thengarai S. Venkatakrishnan, Shaon Sahoo, Nicolas Br[é]{}fuel, Carine Duhayon, Carley Paulsen, Anne Laure Barra, S. Ramasesha, and Jean Pascal Sutter. . , 132(17):6047–6056, may 2010. Arun Kumar Bar, C[é]{}line Pichon, Nayanmoni Gogoi, Carine Duhayon, S. Ramasesha, and Jean Pascal Sutter. . , 51(17):3616–3619, feb 2015. Arun Kumar Bar, Nayanmoni Gogoi, C[í]{}line Pichon, V. M.L.Durga Prasad Goli, Mehrez Thlijeni, Carine Duhayon, Nicolas Suaud, Nathalie Guih[í]{}ry, Anne Laure Barra, S. Ramasesha, and Jean Pascal Sutter. . , 23(18):4380–4396, mar 2017. Oliver Waldmann. . , 46(24):10035–10037, 2007. Constantinos J. Milios, Alina Vinslava, Wolfgang Wernsdorfer, Stephen Moggach, Simon Parsons, Spyros P. Perlepes, George Christou, and Euan K. Brechin. . , 129(10):2754–2755, 2007. Jinkui Tang, Ian Hewitt, N. T. Madhu, Guillaume Chastanet, Wolfgang Wernsdorfer, Christopher E. Anson, Cristiano Benelli, Roberta Sessoli, and Annie K. Powell. . , 45(11):1729–1733, mar 2006. Jeffrey D. Rinehart and Jeffrey R. Long. . , 2(11):2078, oct 2011. Jan Dreiser, Kasper S. Pedersen, Alexander Schnegg, Karsten Holldack, Joscha Nehrkorn, Marc Sigrist, Philip Tregenna-Piggott, Hannu Mutka, H[ø]{}gni Weihe, Vladimir S. Mironov, Jesper Bendix, and Oliver Waldmann. . , 19(11):3693–3701, mar 2013. Vladimir S. Mironov, Liviu F. Chibotaru, and Arnout Ceulemans. . , 125(32):9750–9760, 2003. Miriam V. Bennett and Jeffrey R. Long. . , 125(9):2394–2395, 2003. Robin J. Blagg, Liviu Ungur, Floriana Tuna, James Speak, Priyanka Comar, David Collison, Wolfgang Wernsdorfer, Eric J.L. McInnes, Liviu F. Chibotaru, and Richard E.P. Winpenny. . , 5(8):673–678, aug 2013. Sumit Haldar, Rajamani Raghunathan, Jean Pascal Sutter, and S. Ramasesha. . , 115(21-22):2849–2859, nov 2017. Veronika Hoeke, Anja Stammler, Hartmut B[ö]{}gge, J[ü]{}rgen Schnack, and Thorsten Glaser. . , 53(1):257–268, jan 2014. Jan Dreiser, Kasper S. Pedersen, Alexander Schnegg, Karsten Holldack, Joscha Nehrkorn, Marc Sigrist, Philip Tregenna-Piggott, Hannu Mutka, H[ø]{}gni Weihe, Vladimir S. Mironov, Jesper Bendix, and Oliver Waldmann. . , 19(11):3693–3701, mar 2013. Vladimir S. Mironov, Liviu F. Chibotaru, and Arnout Ceulemans. . , 125(32):9750–9760, 2003. Vladimir S. Mironov. . , 54(23):11339–11355, dec 2015. Kun Qian, Xing-Cai Huang, Chun Zhou, Xiao-Zeng You, Xin-Yi Wang, and Kim R. Dunbar. . , 135(36):13302–13305, sep 2013. Rajamani Raghunathan, S. Ramasesha, and Diptiman Sen. . , 78(10):104408, sep 2008. [^1]: E-mail: sumithaldar@iisc.ac.in [^2]: E-mail: rajamani@csr.res.in [^3]: E-mail: jean-pascal.sutter@lcc-toulouse.fr [^4]: E-mail: ramasesh@iisc.ac.in

--- author: - | Mark Sellke\ Stanford University bibliography: - 'cbc\_FSP\_update.bib' title: Chasing Convex Bodies Optimally --- Introduction ============ Let $K_1,K_2,\dots, K_N$ be a sequence of convex sets inside $\mathbb R^d$. The *convex body chasing* problem asks an online player starting at $x_0$ to choose online a sequence of points $x_n\in K_n$ minimizing the total cost $\sum_{n=1}^N ||x_n-x_{n-1}||$. The problem is framed in terms of competitive analysis, meaning that the player aims for a finite competitive ratio between his cost and the cost of best offline sequence he could have chosen in hind-sight. Chasing convex bodies was originally proposed in [@friedmanlinial] as a convex version of general set chasing problems on metric spaces, including the famous $k$-server problem. An algorithm with finite competitive ratio was given in the original paper for the already non-trivial $d=2$ case, and was conjectured to exist for larger $d$. The best lower bounds were obtained by essentially using the faces of a hypercube, which shows that the competitive ratio is at least $\sqrt{d}$ in Euclidean space and at least $d$ in $\ell^{\infty}$. Following a lack of progress on the full conjecture, restricted cases such as chasing subspaces were studied (e.g. [@2016chasing]). A notable restricted case is chasing *nested* convex bodies, where the convex sets $K_1\supseteq K_2\supseteq \dots $ are required to decrease. Nested chasing was introduced in [@nestedarechaseable] and solved rather comprehensively in [@ABCGL] and then [@chasingnested2018]. The latter work gave a nearly optimal algorithm for all $\ell^p$ spaces, as well as a memoryless $\min(d,O(\sqrt{d\log N}))$ competitive algorithm for Euclidean spaces based on the *Steiner point* of a convex body. Some time after chasing convex bodies was posed, an equivalent functional variant called *chasing convex functions* was given. In chasing convex functions, the request is a convex function $f_n:\mathbb R^d\to\mathbb R^+$ instead of a convex set. The cost function now consists of two additive parts: the movement cost $\sum_{n=1}^N ||x_n-x_{n-1}||$ from before, and a service cost $\sum_{n=1}^N f_n(x_n)$. Chasing convex functions subsumes chasing convex bodies by considering for any body $K$ the function $f^K$ which is $0$ on $K$ and $+\infty$ off of $K$. At the start of subsection \[subsec:reductions\] we in fact explain why we can use a finite $f^K$. Conversely, convex function chasing in $\mathbb R^d$ can be reduced to convex body chasing in $\mathbb R^{d+1}$ up to a constant factor by alternating requests of the epigraphs $\{(x,y)\in \mathbb R^d\times \mathbb R: y\geq f_n(x)\}$ of the convex functions $f_n$ with the hyperplane $\mathbb R^d\times \{0\}$. The functional perspective opens more possibility for applications. Indeed, chasing convex functions was originally considered for efficiently powering data centers [@datacenters]. More abstractly, it resembles the well-studied framework of online convex optimization (see e.g. [@hazan2016introduction] for an introduction). Instead of aiming for a small additive regret against the best single action, we now aim for a multiplicative guarantee against the best *changing* sequence of actions. The movement costs arise naturally from this point of view: the player’s movement cost is an upper bound for the advantage of seeing $f_n$ before choosing $x_n$, so long as the functions $f_n$ are Lipschitz. Some restricted cases of chasing convex function have also been studied. For example, [@1dSOCO] studies the precise competitive ratio in $1$ dimension, while [@SOCObalanced] shows a dimension-free competitive ratio when the cost functions increase linearly away from their minima. New Results ----------- Recently, in our joint work with S. Bubeck, Y.T. Lee, and Y. Li [@chasingconvex2018] we gave the first algorithm achieving a finite competitive ratio of convex body chasing, which used induction on the dimension. However they obtained an exponential upper bound $2^{O(d)}$, leaving a large gap. We give an upper bound of $d$ for the competitive ratio of chasing convex bodies in a general normed space, which is tight for $\ell^{\infty}$, and nearly matches the $\sqrt{d}$ lower bound for Euclidean space. More fully, our main result is: \[thm:main\] In any $d$-dimensional normed space there exists a $d+1$ competitive algorithm for chasing convex functions and a $d$ competitive algorithm for chasing convex bodies. In the former case, the service cost is $1$-competitive against the total cost of the offline optimum. Moreover in Euclidean space this algorithm is $O(\sqrt{d\log N})$-competitive if there are $N$ requests. The proof is inspired by our joint work with S. Bubeck, B. Klartag, Y.T. Lee, and Y. Li [@chasingnested2018] on chasing nested convex bodies. It is shown there that moving to the new body’s *Steiner point*, a stable center point of any convex body defined long ago in [@steineroriginal], gives total movement at most $d$ starting from the unit ball in $d$ dimensions. It is easy to reduce the general problem to this setting up to a constant factor. (They only gave the result for Euclidean space, but we explain in Subsection \[subsec:generalnormedspace\] why it works in general.) We generalize their argument by defining the functional Steiner point of any convex function, intuitively a stable approximate minimum. We show that for continuous time convex function chasing, following the functional Steiner point of the work function (which encodes the total cost of all requests so far) achieves competitive ratio $d+1$. This easily implies the same result in the discrete time setting. Finally, we show that the same algorithm gives a nearly optimal $\sqrt{d\log N}$ upper bound for Euclidean space, following the same result for nested chasing in [@chasingnested2018]. After this work was completed, we were informed that C.J. Argue, Anupam Gupta, Guru Guruganesh, and Ziye Tang jointly obtained similar results for this problem in [@AGGT]. These works were initially posted to the ArXiv simultaneously. Later we added Section \[sec:unify\] which unifies the two solutions. Problem Setup ============= Notations and Conventions ------------------------- We assume the starting position is $x_0=0$. Because we switch between the continuous and discrete time settings, we use variables the $T,t,s$ to denote real times and $N,n$ to denote integer times. We denote by $\dashint_{x\in X} f(x)dx$ the average value $\frac{\int_{x\in X} f(x)dx}{\int_{x\in X} 1dx}$ of $f(x)$ on $X$. We denote by $B_1$ the unit ball in $\mathbb R^d$, and $\mathbb S^{d-1}$ the unit sphere. Equivalence of Bodies and Functions, and Continuous Time Formulation {#subsec:reductions} -------------------------------------------------------------------- First we argue that chasing convex bodies is a special case of chasing convex functions, even though the functions are require to be finite. The reduction from bodies to functions goes by taking $f_n(x)=f^{K_n}$ for $f^K(x):=3d(x,K)$. Now, we claim that both the player and the offline optimum should always play $x_n\in K_n$ for the sequence $(f_1,f_2,\dots)$ of function requests. The reason is simply that both the player and the offline optimum only improve their performance by moving to the closest point in $K_n$ and back. Indeed, if $\hat x_n\notin K_n$ is a potential choice for $x_n$, this modification incurs a movement cost of $2d(\hat x_n,K_n)$ but reduces the service cost by $3d(\hat x_n,K_n)$. The main part of our proof is more natural in continuous time. In continuous time chasing convex functions, the adversary gives a piecewise continuous family of functions $f_t$, and the player constructs a bounded variation path $(x_t)$ online. (The paths constructed by the algorithm will be continuous, but the optimal path might be piecewise continuous.) The loss of the player is $$\int_{t=0}^T f_t(x_t)+||x'_t||dt.$$ Here the integral of $||x'_t||$ is understood to mean the total variation of the path $x_t$ in the case of singularities or discontinuities. The goal is again to achieve a finite competitive ratio. For the continuous time problem we require that the functions $f_t(x)$ be convex in $x$, piecewise continuous in $t$, and finite everywhere. We now show how to reduce the original discrete time problem to the continuous time problem. \[prop:continuoustodiscrete\] Suppose $f_1,f_2,\dots, f_N$ is a sequence of requests for discete-time chasing convex functions. For $t\in [0,N]$ let $f_t=f_{n}$ for $t\in [n-1,n)$. If a $C$-competitive algorithm for the continuous time requests $f_t$ exists, then a $C$-competitive algorithm for the discrete time requests exists. In fact, the discretized algorithm has lower movement cost and lower service cost. It is easy to see that the continuous and discrete time problems have the same offline optimum value. Given a solution path $x_t$ to the continuous time problem, when the player sees a discrete time request $f_n$ he knows what his continuous time path for $t\in [n-1,n)$ would be, and can simply move to the lowest-cost point on this path defined by $x_{t_n}$ for $t_n=\min_{t\in [n-1,n)} f_n(x_t)$. It is easy to see that the sequence $(x_{t_1},\dots, x_{t_n})$ has both a smaller movement and service cost than the continuous path $(x_t)_{t\in [0,T]}$, hence the result. Outline of the Paper -------------------- In section \[sec:setup\], we recall the Steiner point and define the functional Steiner point. We also dicuss the work function and its concave conjugate. In section \[sec:linearCR\] we establish the $d+1$ competitive ratio for chasing convex functions. For conceptual and notational ease, in these two sections we only consider the Euclidean setting. In section \[sec:extra\], we prove the $O(\sqrt{d\log N})$ upper bound for the Euclidean case and finally explain why the linear estimates hold in any normed space. Functional Steiner Point and Work Function {#sec:setup} ========================================== We recall here the definition of the Steiner point of a convex body, and extend it to convex functions. The Steiner point of a convex body $K$ is the functional Steiner point of the function which is $0$ on $K$ and infinite outside $K$. Given a convex body $K\subseteq \mathbb R^d$, the Steiner point $s(K)$ is defined in the following two equivalent ways: 1. For any vector $v\in {\mathbb{R}}^d$, let $f_K(v)=\arg\max_{x\in K}(v\cdot x) $ be the extremal point in $K$ in direction $v$. Then compute the average of this extremal point for a random vector in the unit ball: $$s(K)=\dashint_{v\in B^{d}_1(0)} f_K(v) dv.$$ 2. For unit vector $\theta\in \mathbb{S}^{d-1}\subset {\mathbb{R}}^d$, let $h_K(\theta)=\max_{x\in K}(\theta\cdot x)$ be the support function for $K$ in direction $\theta$, and compute $$s(K)=d\dashint_{\theta\in \mathbb{S}^{d-1}} h_K(\theta)\theta d\theta.$$ The definitions are equivalent because of the divergence or Gauss-Green theorem and the identity $\nabla h_K(v)=f_K(v)$. The factor $d$ comes from the discrepancy in total measure of the ball and the sphere. Note that for a convex body the functions $h_K$ and $f$ are homogenous so e.g. we could average over the same set in the two definitions. However for the functional Steiner point there is a difference, so we maintain the separation. To define functional Steiner point for a general convex function, we generalize the support function $h_K$ to the concave conjugate, which coincides in the case of convex bodies. Recall that for a convex function $W$, the concave conjugate $W^*(v)$ is the *concave* function defined by $$W^*(v)=\min_w \left(W(w)-v^T w\right)$$ We also define $$v^*=\arg\min_w \left(W(w)-v^T w\right)$$ to be the conjugate point to $v$ with respect to $W_t$. Geometrically $v^*$ is the input with $\nabla W(v^*))=v$. Now we can define the functional Steiner point. Given a convex function $W:\mathbb R^d\to\mathbb R^+$ with $W^*(v)$ finite for all $v\in B_1$, the functional Steiner point $s(W)$ is defined in the following two equivalent ways: 1. $$s(W)=\dashint_{v\in B_1} v^* dv.$$ 2. $$s(W)= -d\dashint_{\theta\in \mathbb{S}^{d-1}} W^*(\theta)\theta d\theta.$$ As with the original Steiner point, these definitions coincide because of the divergence or Gauss-Green theorem and the concave conjugate characterization $\nabla W^*(v) = -v^*.$ Technically, the value $v^*$ might not be well-defined for some $v$ values, but by Alexandrov’s theorem $W^*$ has a second derivative almost everywhere, which implies $v^*$ is well-defined for almost all $v$. To avoid confusion we note that technically, specializing our functional Steiner point to the nested case does not yield the Steiner point of $K_n$ at time $n$. The reason is that the ordinary Steiner point $s(K_n)$ is the functional Steiner point of $f_{K_n}(x):=d(x,K_n)$ while the work function (defined in the next subsection) is $W_n(x)=||x-x_0||.$ Properties of the Work Function ------------------------------- Our algorithm is given by staying at the Steiner point of the work function, so here we give the basic properties of the work function. The work function $W_t(x)$ at any time $t$ is defined as the smallest cost OPT could have at time $t$ while starting at $x_0=0$ and ending up at $x_t=x$. Given a family $\{f_s(x):s\in [0,t]\}$ of functions convex in $x$ and piecewise continuous in $s$, the work function $W_t(x)$ is defined by $$\begin{aligned} W_t(x)=& \inf_{x_s:[0,t]\to\mathbb R^d, x_0=0} \int_0^t f_s(x_s)+||x'_s||ds\\ =& \inf_{x_s:[0,t]\to\mathbb R^d,x_0=0} \Gamma_t(x_s).\end{aligned}$$ Here we allow $x_s$ to be any function of bounded variation with the correct value $x_0=0$, and interpret $\int_0^t ||x'_s||ds$ as the total variation of the path. We call $\Gamma_t(x_s)$ the *cost* of the path $x_s$. Note that our definition allows movement after the final request, so we do not necessarily need to pay $f_t(x)$ in $W_t(x)$ but we have to pay $f_t(y)+||y-x||$ for some $y$. In the case that $f_s(x)$ is piecewise constant in $s$ (which is all we need for the discrete problem), it is rather obvious that the best offline continuous time strategy coincides with the best offline discrete time strategy. For mathematical completeness, we argue that the infimum is attained in the definition above in general. Indeed, given a constant $C$ larger than the infimum, consider the set of paths with total variation at most $C$, starting from $x_0=0$. Such paths can be parametrized instead by their distributional derivatives, which are exactly the vector-valued Radon measures on $[0,t]$ with total variation at most $C$. Such measures form a compact set in the usual e.g. Wasserstein topology, and the total variation is a lower semicontinuous functional, meaning that it only decreases under limits of paths. Therefore, given the infimum value defining the work function, we can take a sequence of paths with value converging to the infimum, and find a subsequential limit in the topology just described. Such a limit must exactly attain the infimum. We will denote by $W^*_t(\cdot)$ the concave conjugate of $W_t$, and $v^*_t$ the position with $\nabla W_t(v^*_t)=v$. We record the following proposition summarizing the properties of the work function. The work function $W_t$ and its concave conjugate $W^*_t$ satisfy: 1. $W_0(x)=||x||.$ 2. $W_t(x)$ is increasing in $t$ and is convex for all fixed $t$. 3. $W^*_0(v)=0$ whenever $|v|\leq 1$. 4. $W^*_t(v)$ is increasing in $t$ and concave for fixed $t$. 5. $W^*_t(v)$ is non-negative and finite whenever $|v|\leq 1.$ It is clear that $W_0(x)=||x||$, and that $W_t(x)$ is increasing in $t$. Convexity holds because the entire optimization problem is convex in path space, and implies that $W^*_t(\cdot)$ is concave. The computation of $W^*_0$ is clear, and since $W_t$ is increasing so is $W^*_t$. $W^*_t(v)$ is finite for all $|v|\leq 1$ because $W_t(0)$ (say) is finite. We also prove the following easy lemma which is key to the upcoming analysis. \[lem:boundondual\] For all $t$ we have: $$\max_{|\theta|\leq 1} W^*_t(\theta)\leq 2\cdot \min_{x}W_t(x)$$ $$\dashint_{\theta\in\mathbb S^{d-1}} W^*_t(\theta) d\theta \leq \min_{x} W_t(x).$$ $$\dashint_{v\in B_1} W^*_t(v) dv \leq \min_{x} W_t(x).$$ Define $$OPT_t=\arg\min_x W_t(x).$$ Since $W^*_t$ is defined as a minimum, we can upper bound $W^*_t$ by plugging in $OPT_t$ to the minimization problem. This gives $$W^*_t(\theta) \leq W_T(OPT_t)-\theta\cdot OPT_t.$$ Since $|OPT_t|\leq |W_t(OPT_t)|$ all claims are clear. Alternatively, since $\min_x W_t(x)=W_t^*(0)$, the latter two claims are just the concavity of $W_t^*$. The next lemma allows us to compute the time derivative of $W^*_t(v)$ for fixed $v$ with $|v|<1$. The proof is an exercise in real analysis which we leave to the appendix. \[lem:derivativeWt\] For any $\delta>0$ suppose $f_s(x)$ is jointly continuous in $(s,x)$ and convex in $x$ for all $(s,x)\in [t,t+\delta)\times \mathbb R^d$. Then for almost all fixed $v$ with $|v|<1$ we have $$\frac{d}{dt} \left(W^*_t(v)\right) = f_t(v^*_t).$$ Proof of Linear Competitive Ratio {#sec:linearCR} ================================= Our algorithm is to take $x_t=s(W_t)$. We call this the continuous time functional Steiner point, and call the discretization obtained from Proposition \[prop:continuoustodiscrete\] the discrete time functional Steiner point. We will use the first (primal) definition of the functional Steiner point to control the functional cost and the second (dual) definition to control the movement cost. Note that the offline optimum we are competing with is exactly $\min_xW_T(x)=W^*_T(0)$. In this section we prove the following main theorem. \[thm:linearCR\] The continuous time functional Steiner point $x_t=s(W_t)$ is $d+1$ competitive in Euclidean space. In particular it satisfies: 1. The movement cost of $x_t$ is $d$-competitive: $$\int_{t=0}^T ||x'_t||dt \leq d\cdot\min_x W_t(x).$$ 2. The service cost of $x_t$ is $1$-competitive: $$\int_{t=0}^T f_t(x_t)dt \leq \min_x W_t(x).$$ \[cor:discretelinearCR\] In $d$-dimensional Euclidean space, the discrete time functional Steiner point is $d+1$ competitive for chasing convex functions and $d$ competitive for chasing convex bodies. This follows from Proposition \[prop:continuoustodiscrete\] and the fact that chasing convex bodies has $0$ service cost. We start with part $1$, which parallels the proof of Theorem 2.1 in [@chasingnested2018]. From the dual definition of $s(W_t)$ and the fact that $W^*_t$ increases with $t$, the total movement is bounded by $$\int_{t=0}^T ||x'_t||dt \leq d\dashint_{\theta\in\mathbb S^{d-1}} W^*_T(\theta) d\theta .$$ By the lemma we just proved we know $d\dashint_{\theta\in\mathbb S^{d-1}} W^*_T(\theta) d\theta \leq d\min_x W_T(x)$. This completes the proof of part $1$. Now we turn to part $2$, estimating the service cost. Using the first definition of the functional Steiner point and convexity of $f_t$, we know that $$f_t(s(W_t)) \leq \dashint_{v\in B_1} f_t(v^*_t)dv .$$ Integrating in time and using Lemma \[lem:derivativeWt\] gives the bound on the service cost: $$\begin{aligned} \int_{t=0}^T f_t(x_t)dt \leq & \dashint_{v\in B_1} \int_{t=0}^T f_t(v_t^*) dtd\theta \\ \leq & \dashint_{v\in B_1}\int_{t=0}^T \frac{d}{dt} \left(W^*_t(v)\right) dt dv\\ \leq & \dashint_{v\in B_1} W^*_T(v)-W^*_0(v) dv\\ = & \dashint_{v\in B_1} W^*_T(v) dv\\ \leq & \min_x W_T(x).\end{aligned}$$ This finishes the proof. Although computing the functional Steiner point might appear to require detailed knowledge of the functions $f_t$, only local data of $f_t$ is essential for the guarantee. Indeed, in the continuous time setting the player can achieve the same guarantee only given access to $f_t(x_t)$ and $\nabla f_t(x_t)$. This is because the player can always lower bound $f_t$ by $$f_t(x)\geq \tilde f_t(x):= \max_x(f_t(x_t)+\nabla f_t(x_t)^T (x-x_t),0).$$ In general, if the player is $C$-competitive against some functions $\tilde f_t$ with $\tilde f_t(x_t)=f_t(x_t)$ and $\tilde f_t(x)\leq f_t(x)$ for all $x$, it is clear that the player is $C$-competitive overall. Because our algorithm yields a continuous path, only this first order information is required to follow the functional Steiner point for $\tilde f_t$, and hence be competitive with $f_t$ as well. Similar remarks apply in the discrete time case, where we are given $f_n(x_{n-1})$ and $\nabla f_n(x_{n-1})$ before choosing $x_n$. The only difference is that we now pay an additive cost of $2Ld$ in the competitive ratio under the assumption that the functions $f_n$ are $L$-Lipschitz. The reason is that here we have to make discontinuous jumps, and we don’t know the function value $f_n(x_n)$ when choosing $x_n$. When $f_n$ are Lipschitz we can bound this error using the movement cost. Nearly Optimal Bound and Arbitrary Normed Space {#sec:extra} =============================================== In this section we prove the two extensions claimed in Theorem \[thm:main\]. First we prove the $O(\sqrt{d\log N})$ upper bound in Euclidean space using concentration of measure, following [@chasingnested2018]. Second, we define the functional Steiner point in any normed space, and explain why it is $d+1$ competitive in this greater generality. Nearly Linear Competitive Ratio for Discrete Requests {#sec:discretetime} ----------------------------------------------------- Here we prove the upper bound $O(\sqrt{d\log N})$ for the discrete time problem. The proof is by concentration of measure on the sphere and follows Theorem 3.3 of [@chasingnested2018]. The intuition is: suppose $\int_{\theta\in\mathbb S^{d-1}}W^*_n(\theta)-W^*_{n-1}(\theta)d\theta$ is significant. Then the movement from $s(W_n)\to s(W_{n+1})$ is an integral of pushes by different $\theta$ vectors in the second definition of functional Steiner point. By concentration of measure, these pushes will mostly decorrelate when the total amount of pushing is non-trivially large. For any $0\leq \varepsilon <1$, the set $$\{{v\in B_1}:v_{i}\geq \varepsilon\}$$ occupies at most $e^{-d \varepsilon^{2}/2}$ fraction of the unit ball. \[lem:hardtomove\] Suppose that $|W^*_n(\theta)-W^*_{n-1}(\theta)|\leq C$ for all $\theta$, and set $$\lambda=\dashint_{v\in B_1} W^*_n(v)-W^*_{n-1}(v)dv.$$ Then we have the estimate $$||s(W_n)-s(W_{n-1})||_2 = O\left(\lambda C\sqrt{d \log\left(\frac{C}{\lambda}\right)}\right).$$ Note that $||s(W_n)-s(W_{n-1})||_2=\sup_{|w|=1}w^T\left(s(W_n)-s(W_{n-1})\right)$. So we fix a unit vector $w$ and estimate the inner product. Setting $g_n(\theta)=W^*_n(\theta)-W^*_{n-1}(\theta)>0$, this inner product is $$w^T\left(s(W_n)-s(W_{n-1})\right)= d\dashint_{\theta\in \mathbb S^{d-1}} g_n(\theta) w^T\theta.$$ Now, $g_n(\theta)\in [0,C]$ for all $\theta$ and $g_n$ has average value $\lambda$. Just given these two constraints, the integral is maximized when $g_n(\theta)=C$ for the set of $\theta$ values most correlated with $w$, and $g_n(\theta)=0$ otherwise. The conclusion now follows easily. For discrete time convex function chasing with requests $f_1,\dots,f_N$, the discrete time functional Steiner point algorithm is $\min(d,O(\sqrt{d\log N})$ competitive. Call the continuous path $(x_t)_{t\in [0,N]}$ so the discrete path is $x_{t_n}$ for $t_n\in [n-1,n)$. As proved in the proposition, we know the movement and service costs for this discrete path are at most that of the continuous path, so we only need to establish the $O(\sqrt{d\log N})$ competitive ratio on the movement of the discrete path.\ Recall that by Lemma \[lem:boundondual\] we have $$\max_{|\theta|\leq 1} W^*_N(\theta)\leq 2\cdot \min_{x}W_N(x).$$ Therefore, we can apply Lemma \[lem:hardtomove\] with $C=2\cdot \min_{x}W_N(x)$ to the movement $|x_{t_n}-x_{t_{n-1}}|$ at each step. Set $\lambda_n=\int_{\theta\in\mathbb S^{d-1}}\left(W^*_{t_n}(\theta)-W^*_{t_{n-1}}(\theta)\right)d\theta$. By Lemma \[lem:hardtomove\] the total movement is at most the maximum value of $$d^{1/2}\sum_{n\leq N} \lambda_n C\sqrt{\log\left(\frac{C}{\lambda_n}\right)}$$ subject to the constraint that the $\lambda_n$ are all non-negative and sum to $\dashint_{\theta\in\mathbb S^{d-1}} W^*_N(\theta)d\theta\leq C$. Because the function $h(x)=x\sqrt{\log(x^{-1})}$ is concave for $x\in [0,1]$, by Jensen’s inequality the maximum is achieved when all $\lambda_n$ are equal. This gives an upper bound $O(C\sqrt{d\log N})$ which means a competitive ratio of $O(\sqrt{d\log N})$ as desired. Non-Euclidean Spaces {#subsec:generalnormedspace} -------------------- By John’s Theorem, all the preceding results for Euclidean space hold in any normed space up to a distortion factor of at most $\sqrt{d}$. In fact the functional Steiner point can be defined in any normed space. As a result, our $d+1$ upper bound holds in this greater generality. We first give the general definition of the ordinary Steiner point. [@steinerlipschitz Chapter 6] Let $X$ be an arbitrary $d$-dimensional normed space, and $K\subseteq X$ a convex body. The Steiner point $s(K)\in X$ is defined in the following two equivalent ways: 1. For any dual vector $v\in X^*$, let $f_K(v)=\arg\max_{x\in K}(\theta\cdot x) $ be the extremal point in $K$ in direction $\theta$. Then compute the average of this extremal point for a random vector in the dual unit ball: $$s(K)=\dashint_{v\in B^*_1} f_K(v) dv.$$ 2. For any dual unit vector $\theta\in \partial B^*_1$, let $h_K(\theta)=\max_{x\in K}(\theta\cdot x)$ be the support function for $K$ in direction $\theta$, and compute $$s(K)=d\dashint_{\theta\in \partial B^*_1} h_K(\theta)n(\theta) d\mu(\theta).$$ Here the measure in the first integral is volume measure, which up to normalization does not depend on any norm properties of $\mathbb R^d$, only the vector space structure. In the second integral, $n(\theta)\in X$ is the outward unit normal to the unit dual ball $B^*_1$ at $\theta$, defined by $||n(\theta)||_X=1$ and $n(\theta)^T\theta = 1$. The surface measure $\mu(\theta)$ is the *cone measure*, which can be sampled by taking a uniformly random $z\in B^*_1$ and normalizing it to $\frac{z}{||z||}.$ See [@steinerlipschitz] chapter $6$ for a careful derivation of essentially this fact, again via the divergence theorem. This definition immediately generalizes the upper bound $d$ for nested chasing [@chasingnested2018] in any normed space. As with the Steiner point for Euclidean space, all that is needed about the functions $f_K$ and $h_K$ is the gradient relationship $\nabla h_K=f_K$. As a result, we can again extend the definition of Steiner point to the functional Steiner point of a convex function. Let $X$ be an arbitrary $d$-dimensional normed space, and $W:X\to\mathbb R^+$ a convex function with $W^*(v)$ finite for all $v$ in the dual unit ball $B^*_1$. The functional Steiner point $s(W)\in X$ is defined in the following two equivalent ways: 1. $$s(W)=\dashint_{v\in B^*_1} v^* dv.$$ 2. $$s(W)= -d\dashint_{\theta\in \partial B^*_1} W^*(\theta)n(\theta) d\mu(\theta).$$ In any $d$-dimensional normed space, the functional Steiner point achieves competitive ratio $d+1$ for chasing convex functions and competitive ratio $d$ for chasing convex bodies. Having defined the proper setup, the proof is completely identical to the Euclidean case. A Unified View on the Functional Steiner Point {#sec:unify} ============================================== We have given a solution to chasing convex bodies using the functional Steiner point of the work function. Concurrently [@AGGT] gave a different solution by taking the Steiner point of a (convex) level set of the work function. In this section (which was added after the initial concurrent posting) we unify these two constructions. In particular we show that Steiner point of any level set of $W_t$ is equal to the functional Steiner point after requesting that level set. For a sufficiently large level set, the final request is vacuous and the definitions exactly coincide. In the context of [@AGGT], this has two consequences. First, their algorithm achieves competitive ratio $O(d)$ in any normed space, even though their proof seems to rely on reflections which are only valid in Euclidean space. Second, it shows that the doubling trick they use is unnecessary since taking an infinitely large level set to begin with exactly recovers the functional Steiner point. Functional Steiner Point as an Online Selector ---------------------------------------------- We start by giving a direct proof that the functional Steiner point in an online selector, i.e. that $s(W_t)\in K_t$. This can be deduced implicitly from the purely functional proof in the main body together with the reduction from functions to sets, but for this section it is useful to write it out explicitly. We also give an interpretation of the functional Steiner point as a curvature center, which also parallels the ordinary Steiner point. For a $1$-Lipschitz convex function $W$, the support set $Supp(W)\subseteq \mathbb R_d$ is the set of points having a subgradient with norm strictly less than $1$. For chasing convex bodies we have $Supp(W_t)\subseteq K_t$. In particular, if $x\notin K_t$ then $\nabla W_t(x)$ is a unit vector pointing from $y\to x$, where $y$ is the last point in $K_t$ for the path $0\to x$ achieving cost $W_t(x)$ and satisfying requests $(K_1,\dots,K_t)$. For $z$ on the line segment $yx$, it is clear by definition of $y$ that $W_t(x)-W_t(z)=||x-z||$. Since $W$ is $1$-Lipschitz, this implies the result. We always have $s(W_t)\in K_t$ for chasing convex bodies. By the primal definition, $s(W_t)$ is a convex combination of points in $Supp(W_t)\subseteq K_t$. A simple induction shows that $|\nabla W_t(x)|=1$ unless $x\in \partial K_s$ for some $s\leq t$. Therefore $Supp(W_t)$ is a measure zero set and $W_t$ is highly non-smooth when chasing convex bodies. However one loses nothing by smoothing the work function by a small mollifier, so these singularities do not cause any meaningful issues. We can be more precise about the statement that $s(W)$ is a convex combination of the points in $Supp(W)$. By applying a change of variables and noting that the Jacobian of the gradient is the Hessian, we obtain that the functional Steiner point $s(W)$ equals $$s(W)=\int_{x\in\mathbb R^d} \det(\nabla^2 W(x)) \cdot x dx.$$ A similar change of variables shows that the Steiner point $s(K)$ is the center of mass of $\partial K$ weighted by the Gaussian curvature, the product of the principal curvatures. This gives an interpretation of the functional Steiner point $s(W)$ as a renormalized Steiner point of $epigraph(W)$, similar to the results in the next subsection. Steiner Points of Level Sets ---------------------------- We now complete the connection to the solution of [@AGGT]. For a chasing convex bodies/functions problem and $R\geq \min_x W_t(x)$, define the (convex) level set $$\Omega_{t,R}=\{x:W_t(x)\leq R\}.$$ Also define $W_t^{K}$ to be the work function at time $t+1$ if $K_{t+1}=K$. \[lem:unify\] We have $Supp(W_t^{\Omega_{t,R}})\subseteq Supp(W_t)$ for any $R\geq \min_x W_t(x)$. \[thm:unify\] For any $R\geq \min_x W_t(x)$ we have $s(\Omega_{t,R})=s(W_t^{\Omega_{t,R}})\in K_t$. If $R$ is large enough that $K_s\subseteq \Omega_{t,R}$ for all $s<t$ then $s(\Omega_{t,R})=s(W_t)$. Because $\Omega_{t,R}$ is a level set, we see that $$W_t^{\Omega_{t,R}}(x)=\Bigg\{\begin{array}{lr} W_t(x), & \text{for } x\in \Omega_{t,R}\\ d(x,\Omega_{t,R})+R, & \text{for } x\notin \Omega_{t,R}\\ \end{array}$$ From this it is easy to see that $Supp(W_t^{\Omega_{t,R}})\subseteq Supp(W_t)$. Indeed, the only possible new support points are on the boundary $\partial\Omega_{t,R}$ of the level set. Fix a boundary point $y\in\partial\Omega_{t,R}$ not already in $Supp(W_t)$. Since $\Omega_{t,R}$ is a level set, we know that $\nabla W_t(y)$ consists only of outward normal vectors to $\Omega_{t,R}$ at $y$ (or, in the normal cone if $y$ is a corner). Since these are subgradients, this means that the derivative of $W_t$ at $y$ in some inward direction $v$ with $-v$ in the normal cone is $-1$. This will remain true in $W_t^{\Omega_{t,R}}$, showing that $y\notin Supp(W_t^{\Omega_{t,R}})$ as claimed. The lemma combined with the primal definition immediately implies $s(W_t^{\Omega_{t,R}})\in K_t$. Also it is clear that for $R$ large we have $W_t=W_t^{\Omega_{t,R}}$. Therefore the only thing to prove is that $s(W_t^{\Omega_{t,R}})=s(\Omega_{t,R})$. This is the heart of our unification. The point is that for $|\theta|=1$, we have $$f_{\Omega_{t,R}}(\theta)\in \arg\max_{x\in \mathbb R^d} (W_t^{\Omega_{t,R}}(x)-x\cdot\theta).$$ Where again $f_K(v)=\arg\max_{x\in K}(v\cdot x).$ Therefore: $$\begin{aligned} s(W_t^{\Omega_{t,R}})=&(-d)\cdot \dashint_{|\theta|=1}W_t^*(\theta)\cdot \theta d\theta \\ =&(-d)\cdot \dashint_{|\theta|=1}\left(W_t(f_{\Omega_{t,R}}(\theta))-f_{\Omega_{t,R}}(\theta)\cdot\theta\right)\cdot \theta d\theta \\ =&(-d)\cdot \dashint_{|\theta|=1}\left(R-f_{\Omega_{t,R}}(\theta)\cdot\theta\right)\cdot \theta d\theta \\ =& d\cdot \dashint_{|\theta|=1}\left(f_{\Omega_{t,R}}(\theta)\cdot\theta\right)\cdot \theta d\theta \\ =&d\cdot \dashint_{|\theta|=1}h_{\Omega_{t,R}}(\theta)\cdot \theta d\theta\\ =& s(\Omega_{t,R}).\end{aligned}$$ In words, the above calculation can described as follows: the support function of a set and the Legendre-Fenchel of a function transform only differ by a function value term. Using a level set of the function makes this term constant, hence vanish under spherically symmetric integration, so the definitions coincide in this case. For a general work function $W_t$, i.e. a $1$-Lipschitz convex function with $W_t(x)-|x|$ positive and bounded, the same proof shows that $\lim_{R\to\infty}s(\Omega_{t,R})=s(W_t)$. Indeed, the above proof relies on the fact that $$\max_{x\in\mathbb R^d} (W_t^{\Omega_{t,R}}(x)-x\cdot\theta)=\max_{x\in\partial\Omega_{t,R}} (W_t^{\Omega_{t,R}}(x)-x\cdot\theta).$$ But the difference always vanishes in the limit $R\to\infty$, so we obtain the result by the bounded convergence theorem. Acknowledgement {#acknowledgement .unnumbered} =============== I thank Sébastien Bubeck, Bo’az Klartag, Yin Tat Lee, and Yuanzhi Li for the introduction to convex body chasing and the Steiner point, and many stimulating discussions. I thank Ethan Jaffe, Felipe Hernandez, and Christian Coester for discussions about properties of the work function. I additionally thank Sébastien for helpful comments on older versions of the paper. The author gratefully acknowledges the support of an NSF graduate fellowship and a Stanford graduate fellowship. Appendix ======== We prove the result for all $v$ where $v^*_t$ is well-defined, that is, where $\nabla^2 W^*_t(v)$ exists. Almost all $v$ satisfy this by Alexandrov’s theorem, and it means $W_t$ is strictly convex at $v^*_t$. First we note that $$\begin{aligned} W^*_{t+\delta}(v)=& \min_{x_s:[0,t]\to\mathbb R^d} \left(\int_{0}^{t+\delta} (f_s(x_s)+||x'_s||ds -v^Tx_t\right)\\ =& \min_{ x_s:[t,t+\delta]\to \mathbb R^d}\left(W_t(x_t)+\int_{t}^{t+\delta}f_s(x_s)+||x'_s||ds - v^Tx_{t+\delta}\right)\\ :=& \min_{ x_s:[t,t+\delta]\to \mathbb R^d} \Gamma(x_s). \end{aligned}$$ For small $\delta>0$, we show $W^*_{t+\delta}(v)=W^*_t(v)+\delta f_t(v^*_t)+o(\delta)$. For the upper bound, we have $$\begin{aligned} W^*_{t+\delta}(v)\leq & W^*_t(v)+\int_t^{t+\delta} f_s(v^*_t)ds\\ = & W^*_t(v)+\delta f_t(v^*_t)+o(\delta)\end{aligned}$$ because we can move optimally to $v^*_t$ and take $x_s$ constant during $s\in [t,t+\delta]$. For the lower bound the idea is that by strict convexity, for small $\delta$ if $x_t$ is far from $v_t^*$, the minimization problem already has too large a value. We have $$W_t(w)\geq W_t(v^*_t)+v^T(w-v^*_t)+f(|w-v^*_t|)$$ for some non-negative convex $f$ with $f(x)$ bounded away from $0$ when $|x|$ is bounded away from $0$. Considering the minimization problem for $W^*_{t+\delta}(v)$, if $x_t=w$ then we have $$\begin{aligned} \Gamma(x_s)\geq & W_t(w)+\int_t^{t+\delta} f_s(x_s)+||x'_s||ds-v^Tx_{t+\delta} \\ \geq & W_t(v^*_t)+v^T(w-v^*_t)+f(|w-v^*_t|) + \int_t^{t+\delta} f_s(x_s)+||x'_s||ds-v^Tx_{t+\delta} \end{aligned}$$ Because $\int_t^{t+\delta} ||x'_s||ds \geq |x_{t+\delta}-x_t| \geq v^T(x_{t+\delta}-x_t)$ and $w=x_t$ we have $$\begin{aligned} \Gamma(x_s)\geq & W_t(v^*_t)-v^Tv^*_t+f(|w-v^*_t|) + \int_t^{t+\delta} f_s(x_s)ds\\ \geq & W_t(v^*_t)-v^Tv^*_t+f(|w-v^*_t|)\\ \geq & W^*_t(v)+f(|w-v^*_t|)\end{aligned}$$ Because $W_{t+\delta}(v)=W_t(v)+O(\delta)$, we see that for $\delta\to 0$ small we must have $|x_t-v_t^*|\to 0$ for any optimal trajectory $x_s$. Additionally, we have $$\int_t^{t+\delta} ||x'_s||ds + v^T(x_t-x_{t+\delta}) \geq (1-|v|)\int_t^{t+\delta} ||x'_s||ds \geq (1-|v|)\max_{s\in [t,t+\delta]} |x_t-x_s|.$$ which implies $|x_t-x_s|=o(1)$ for all optimal trajectories since $|v|<1$. From this we see that all optimal trajectories have $\int_t^{t+\delta} f_s(x_s)ds = \delta(f_t(v^*_t) +o(1)).$ Combining gives the desired lower bound.

--- abstract: | In this paper we present three types of Caputo–Hadamard derivatives of variable fractional order, and study the relations between them. An approximation formula for each fractional operator, using integer-order derivatives only, is obtained, and an estimation for the error is given. At the end we compare the exact fractional derivative of a concrete example with some numerical approximations. **Keywords**: fractional calculus, variable fractional order, Caputo fractional derivative, Hadamard fractional derivative, expansion formulas. **Mathematics Subject Classification 2010**: 26A33, 33F05. author: - | Ricardo Almeida\ `ricardo.almeida@ua.pt` date: '*[Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal]{}*' title: 'Caputo–Hadamard fractional derivatives of variable order' --- Introduction ============ At the same time ordinary calculus was developed for integer-order derivatives in the seventeenth century, L’Hôpital and Leibniz wondered about the notion of derivative of order $n=1/2$. Two centuries later, with the works of Fourier, Riemann, Liouville, Hadamard, Grünwald, etc., numerous definitions trying to generalize the notion of ordinary derivative were developed. This was primarily a theoretical study for mathematicians, with several notions of fractional operators appearing and their properties well studied. In the past decades, with the discovery that processes like conservation mass, viscoelasticity, nanotechnology, signal processing, and several other applications in engineering are better described by fractional derivatives, these discoveries have gained a great importance. Fractional derivatives are nonlocal concepts, and for this reason may be more suitable to translate natural phenomena. In our days, one can find several books and journals dedicated exclusively to fractional calculus theory, not only on the subject of mathematics but also physics, engineering, economics, applied sciences, etc. Due the complexity of dealing with fractional operators, we find different numerical approaches to solve the desired problems. One of those available methods consists in approximating the fractional derivative by an expansion that depends on integer-order derivatives only [@Atanackovic1; @Pooseh4; @Pooseh5]. With this technique in hand, given any problem involving fractional operators, we can simply replace them with the given expansion, obtaining a new problem that depends on integer-order derivatives only. After that, we can apply any known technique to solve it. The main advantage of this procedure is that we do not need higher-order derivatives in order to have a good approximation, in opposite to other methods e.g. the approximation for the Riemann–Liouville fractional derivative [@samko], $${_aD_t^\alpha}x(t)=\sum_{n=0}^\infty \binom{\alpha}{n}\frac{(t-a)^{n-\alpha}}{\Gamma(n+1-\alpha)}x^{(n)}(t)$$ and for the Hadamard fractional derivative we have [@Butzer2] $${_0^HD_t^\alpha}x(t)=\sum_{n=0}^\infty S(\alpha,n)t^n x^{(n)}(t),$$ where $S(\alpha,n)$ is the Stirling function. Fractional calculus of variable order {#sec:FC} ===================================== In this section, we present three new types of fractional operators, which combine the Caputo with the Hadamard fractional derivatives. The order of the derivative is a function $\a$, that depends on time, and when it is constant we prove that the three definitions coincide. To start, we review some concepts on fractional derivative operators with constant order [@Machado; @Miller; @samko]. Let $\alpha$ be a real in the interval $(0,1)$ and $x$ be a real valued function with domain $[a,b]$. The Caputo fractional derivative of $x$ of order $\alpha$ is given by $${_a^CD_t^\alpha}x(t)={_aD_t^\alpha}[x(t)-x(a)],$$ where ${_aD_t^\alpha}$ stands for the Riemann-Liouville fractional derivative: $${_aD_t^\alpha}x(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_a^t(t-\t)^{-\alpha}x(\t)d\t.$$ If $x$ is differentiable, using integration by parts and then differentiating the integral, we obtain an equivalent definition for the Caputo fractional derivative: $${_a^CD_t^\alpha}x(t)=\frac{1}{\Gamma(1-\alpha)}\int_a^t(t-\t)^{-\alpha}x'(\t)d\t.$$ With respect to the Hadamard fractional derivative, we have the following definition: $${_a^HD_t^\alpha}x(t)=\frac{t}{\Gamma(1-\alpha)}\frac{d}{dt}\int_a^t \left(\ln\frac{t}{\tau}\right)^{-\alpha}\frac{x(\tau)}{\tau}d\tau.$$ For example, for $\gamma>0$, if we take the two functions $x(t)=(t-a)^\gamma$ and $y(t)=\left(\ln\frac{t}{a}\right)^\gamma$, then $${_a^CD_t^\alpha}x(t)=\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1-\alpha)} (t-a)^{\gamma-\alpha} \quad \mbox{and} \quad {_a^HD_t^\alpha}y(t)= \frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1-\alpha)} \left(\ln\frac{t}{a}\right)^{\gamma-\alpha}.$$ One natural extension of these two concepts is to define the Caputo–Hadamard fractional derivative of order $\alpha$ [@Gambo5; @Jarad5]: $${_{\,\,\,\,\,a}^{CH}D_t^\alpha}x(t)=\frac{t}{\Gamma(1-\alpha)}\frac{d}{dt}\int_a^t \left(\ln\frac{t}{\tau}\right)^{-\alpha}\frac{x(\tau)-x(a)}{\tau}d\tau,$$ or in an equivalent way, $${_{\,\,\,\,\,a}^{CH}D_t^\alpha}x(t)=\frac{1}{\Gamma(1-\alpha)}\int_a^t \left(\ln\frac{t}{\tau}\right)^{-\alpha}x'(\tau)d\tau.$$ We extend the previous notions by considering the order of the derivative a real valued function $\alpha:[a,b]\to(0,1)$. This is a very recent direction of research, and was studied for the first time in [@SamkoRoss] with respect to the Riemann–Liouville fractional derivative. Since these fractional operators are nonlocal, and contain memory about the past dynamic, it is natural to consider the order of the derivative also variable, depending on the process. In fact, several applications were found and it is nowadays a growing subject that has attracted the attention of a vast community [@Coimbra; @Coimbra2; @Dzherbashyan; @Lin; @Od; @Ramirez2; @Sun]. In this paper we deal with a combined Caputo–Hadamard fractional derivative with fractional variable order. Three different types of operators are given. Let $a,b$ be two reals with $0<a<b$, and $x:[a,b]\to\mathbb R$ be a function. The left Caputo–Hadamard fractional derivative of order $\a$ 1. type 1 is defined by $$\DI x(t)=\frac{1}{\Gamma(1-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{-\a}x'(\t)d\t;$$ 2. type 2 is defined by $$\DII x(t)=\frac{t}{\Gamma(1-\a)}\frac{d}{dt}\left(\int_a^t\left(\ln\frac{t}{\t}\right)^{-\a}\frac{x(\t)-x(a)}{\t}d\t\right);$$ 3. type 3 is defined by $$\DIII x(t)=t\frac{d}{dt}\left(\frac{1}{\Gamma(1-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{-\a}\frac{x(\t)-x(a)}{\t}d\t\right).$$ Analogous definitions for the right fractional operators are given: Let $a,b$ be two reals with $0<a<b$, and $x:[a,b]\to\mathbb R$ be a function. The right Caputo–Hadamard fractional derivative of order $\a$ 1. type 1 is defined by $$\DIR x(t)=\frac{-1}{\Gamma(1-\a)}\int_t^b\left(\ln\frac{\t}{t}\right)^{-\a}x'(\t)d\t;$$ 2. type 2 is defined by $$\DIIR x(t)=\frac{-t}{\Gamma(1-\a)}\frac{d}{dt}\left(\int_t^b\left(\ln\frac{\t}{t}\right)^{-\a}\frac{x(\t)-x(b)}{\t}d\t\right);$$ 3. type 3 is defined by $$\DIIIR x(t)=-t\frac{d}{dt}\left(\frac{1}{\Gamma(1-\a)}\int_t^b\left(\ln\frac{\t}{t}\right)^{-\a}\frac{x(\t)-x(b)}{\t}d\t\right).$$ We will see that these definitions do not coincide. To start, we prove the following Lemma. \[LemmaEx\] Let $\gamma>0$ and consider the function $x(t)=\left(\ln\frac{t}{a}\right)^\gamma$. Then 1. $\DI x(t) =\DS\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1-\a)} \left(\ln\frac{t}{a}\right)^{\gamma-\a}$. 2. $\DII x(t) =\DS\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1-\a)} \left(\ln\frac{t}{a}\right)^{\gamma-\a}$ $\DS\quad -\frac{t\Da\Gamma(\gamma+1)}{\Gamma(\gamma+2-\a)} \left(\ln\frac{t}{a}\right)^{\gamma+1-\a} \left[\ln \left(\ln\frac{t}{a}\right)+\psi(1-\a)-\psi(\gamma+2-\a)\right]$. 3. $\DIII x(t) =\DS\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1-\a)} \left(\ln\frac{t}{a}\right)^{\gamma-\a}$ $\DS\quad -\frac{t\Da\Gamma(\gamma+1)}{\Gamma(\gamma+2-\a)} \left(\ln\frac{t}{a}\right)^{\gamma+1-\a} \left[\ln \left(\ln\frac{t}{a}\right)-\psi(\gamma+2-\a)\right]$. We only prove the first one; the others are proven in a similar way. First, observe that $$\begin{array}{ll} \DI x(t)&=\DS\frac{1}{\Gamma(1-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{-\a}\frac{\gamma}{\t}\left(\ln\frac{\t}{a}\right)^{\gamma-1}d\t\\ &=\DS\frac{\gamma}{\Gamma(1-\a)}\left(\ln\frac{t}{a}\right)^{-\a}\int_a^t\left(1-\frac{\ln\frac{\t}{a}}{\ln\frac{t}{a}}\right)^{-\a} \left(\ln\frac{\t}{a}\right)^{\gamma-1}\frac{d\t}{\t}. \end{array}$$ If we proceed with the change of variables $$\ln\frac{\t}{a}=s\ln\frac{t}{a},$$ we get $$\begin{array}{ll} \DI x(t) &=\DS \frac{\gamma}{\Gamma(1-\a)}\left(\ln\frac{t}{a}\right)^{-\a}\int_0^1 (1-s)^{-\a}s^{\gamma-1}\left(\ln\frac{t}{a}\right)^{\gamma-1} \left(\ln\frac{t}{a}\right) ds\\ &=\DS \frac{\gamma}{\Gamma(1-\a)}\left(\ln\frac{t}{a}\right)^{\gamma-\a}\int_0^1 (1-s)^{-\a}s^{\gamma-1}ds\\ &=\DS \frac{\gamma}{\Gamma(1-\a)}\left(\ln\frac{t}{a}\right)^{\gamma-\a}B(1-\a,\gamma), \end{array}$$ where $B(\cdot,\cdot)$ is the beta function. Using the relation $$B(1-\a,\gamma)=\frac{\Gamma(1-\a)\Gamma(\gamma)}{\Gamma(\gamma+1-\a)},$$ we prove the desired formula. In a similar way we have the following: Let $\gamma>0$ and consider the function $x(t)=\left(\ln\frac{b}{t}\right)^\gamma$. Then 1. $\DIR x(t) =\DS\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1-\a)} \left(\ln\frac{b}{t}\right)^{\gamma-\a}$. 2. $\DIIR x(t) =\DS\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1-\a)} \left(\ln\frac{b}{t}\right)^{\gamma-\a}$ $\DS\quad +\frac{t\Da\Gamma(\gamma+1)}{\Gamma(\gamma+2-\a)} \left(\ln\frac{b}{t}\right)^{\gamma+1-\a} \left[\ln \left(\ln\frac{b}{t}\right)+\psi(1-\a)-\psi(\gamma+2-\a)\right]$. 3. $\DIIIR x(t) =\DS\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1-\a)} \left(\ln\frac{b}{t}\right)^{\gamma-\a}$ $\DS\quad +\frac{t\Da\Gamma(\gamma+1)}{\Gamma(\gamma+2-\a)} \left(\ln\frac{b}{t}\right)^{\gamma+1-\a} \left[\ln \left(\ln\frac{b}{t}\right)-\psi(\gamma+2-\a)\right]$. \[relations\] The following relations hold: $$\DII x(t)=\DI x(t)+\frac{t\Da}{\Gamma(2-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{1-\a}x'(\t)\left[\frac{1}{1-\a}-\ln\left(\ln\frac{t}{\t}\right)\right]d\t$$ and $$\DIII x(t)=\DII x(t)+\frac{t\Da\psi(1-\a)}{\Gamma(1-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{-\a}\frac{x(\t)-x(a)}{\t}d\t.$$ Starting with the formula $$\DII x(t)=\frac{t}{\Gamma(1-\a)}\frac{d}{dt}\left(\int_a^t\left(\ln\frac{t}{\t}\right)^{-\a}\frac{x(\t)-x(a)}{\t}d\t\right),$$ and integrating by parts choosing $$u'(\t)=\left(\ln\frac{t}{\t}\right)^{-\a}\frac{1}{\t} \quad \mbox{and} \quad v(\t)=x(\t)-x(a),$$ we obtain $$\DII x(t)=\frac{t}{\Gamma(1-\a)}\frac{d}{dt}\left(\frac{1}{1-\a}\int_a^t\left(\ln\frac{t}{\t}\right)^{1-\a}x'(\t)d\t\right).$$ Differentiating the product, we prove the formula. The second one follows immediately from the definition. From this result, for an arbitrary function $x$, we see that these three definitions coincide $$\DI x(t)\equiv \DII x(t)\equiv \DIII x(t),$$ only when the order $\a$ or the function $x$ are constant. For what concerns the right fractional operators, we have the two following relations. $$\DIIR x(t)=\DIR x(t)+\frac{t\Da}{\Gamma(2-\a)}\int_t^b\left(\ln\frac{\t}{t}\right)^{1-\a}x'(\t)\left[\frac{1}{1-\a}-\ln\left(\ln\frac{\t}{t}\right)\right]d\t$$ and $$\DIIIR x(t)=\DIIR x(t)-\frac{t\Da\psi(1-\a)}{\Gamma(1-\a)}\int_t^b\left(\ln\frac{\t}{t}\right)^{-\a}\frac{x(\t)-x(b)}{\t}d\t.$$ Let $x$ be of class $C^1$. Then $$\DI x(t)=\DII x(t)=\DIII x(t)=0,$$ at $t=a$. For what concerns $\DI x(t)$, we have $$\left|\DI x(t)\right|\leq\frac{\|x'\|}{\Gamma(1-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{-\a}\frac{1}{\t}\cdot\t d\t.$$ Integrating by parts, we get $$\left|\DI x(t)\right|\leq\frac{\|x'\|}{\Gamma(2-\a)} \left[a\left(\ln\frac{t}{a}\right)^{1-\a}+\int_a^t\left(\ln\frac{t}{\t}\right)^{1-\a}d\t\right],$$ which vanishes at $t=a$. To prove that $\DII x(t)=0$ at $t=a$, using Theorem \[relations\], is enough to prove that $$\int_a^t\left(\ln\frac{t}{\t}\right)^{1-\a}d\t= \int_a^t\left(\ln\frac{t}{\t}\right)^{1-\a}\ln\left(\ln\frac{t}{\t}\right)d\t=0,$$ for $t=a$. With respect to the first integral, is obvious. For the second one, let $$f(\t)=\left(\ln\frac{t}{\t}\right)^{1-\a}\ln\left(\ln\frac{t}{\t}\right), \quad \t\in[a,t[.$$ Since $f(\t)\to0$ as $\t\to t$, we can extend continuously $f$ to the closed interval $[a,t]$ by letting $f(t)=0$. Finding the extremals for $f$, we prove that for all $\t\in[a,t]$, $$\left|f(\t)\right|\leq \max\left\{\left(\ln\frac{t}{a}\right)^{1-\a}\left|\ln\left(\ln\frac{t}{a}\right)\right|, \frac{1}{e(1-\a)}\right\}.$$ With this we prove the second part. The last one is clear, using the second relation in Theorem \[relations\] and integration by parts. Expansion formulas for the Caputo–Hadamard fractional derivatives {#sec:expansion} ================================================================= Define the sequence $(x_k)_{k \in\mathbb N}$ recursively by the formula $$x_1(t)=tx'(t) \quad \mbox{and} \quad x_{k+1}(t)= tx'_k(t), \, k \in\mathbb N.$$ Also, for $k\in\mathbb N$, define the quantities $$\begin{array}{ll} A_k & =\DS \frac{1}{\Gamma(k+1-\a)}\left[1+\sum_{p=n-k+1}^N \frac{\Gamma(\a-n+p)}{\Gamma(\a-k)(p-n+k)!} \right],\\ B_k & = \DS\frac{\Gamma(\a-n+k)}{\Gamma(1-\a)\Gamma(\a)(k-n)!}, \end{array}$$ and the function $$V_k(t)=\int_{a}^{t}\left(\ln\frac{\t}{a}\right)^k x'(\t)d\t.$$ \[teo1\] Let $x:[a,b]\to\mathbb R$ be a function of class $C^{n+1}$, for $n\in\mathbb N$, and fix $N\in\mathbb N$ with $N \geq n$. Then, $$\DI x(t) =\DS\sum_{k=1}^{n}A_k \left(\ln\frac{t}{a}\right)^{k-\a}x_k(t)+\sum_{k=n}^N B_k \left(\ln\frac{t}{a}\right)^{n-k-\a} V_{k-n}(t)+E(t),$$ with $$E(t)\leq \frac{(t-a)\exp((n-\a)^2+n-\a)}{\Gamma(n+1-\a)N^{n-\a}(n-\a)}\left(\ln\frac{t}{a}\right)^{n-\a}\max_{\t\in[a,t]}\left|x'_N(\t)\right|.$$ Integrating by parts the integral $$\DI x(t)=\frac{1}{\Gamma(1-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{-\a}\frac{1}{\t}x_1(\t)d\t,$$ with $$u'(\t)=\left(\ln\frac{t}{\t}\right)^{-\a}\frac{1}{\t} \quad \mbox{and} \quad v(\t)=x_1(\t),$$ we get $$\DI x(t)= \frac{1}{\Gamma(2-\a)}\left(\ln\frac{t}{a}\right)^{1-\a}x_1(a)+\frac{1}{\Gamma(2-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{1-\a}\frac{1}{\t}x_2(\t)d\t.$$ If we proceed integrating by parts $n-1$ more times, the following formula is obtained: $$\DI x(t) =\DS\sum_{k=1}^n \frac{1}{\Gamma(k+1-\a)}\left(\ln\frac{t}{a}\right)^{k-\a}x_k(a)+\frac{1}{\Gamma(n+1-\a)} \int_a^t\left(\ln\frac{t}{\t}\right)^{n-\a}\frac{1}{\t}x_{n+1}(\t)d\t.$$ By the Taylor’s Theorem, we obtain the sum $$\begin{array}{ll} \DS\left(\ln\frac{t}{\t}\right)^{n-\a}&=\DS\left(\ln\frac{t}{a}\right)^{n-\a}\left(1-\frac{\ln\frac{\t}{a}}{\ln\frac{t}{a}}\right)^{n-\a}\\ &=\DS\left(\ln\frac{t}{a}\right)^{n-\a}\sum_{p=0}^N \C (-1)^p \frac{\left(\ln\frac{\t}{a}\right)^p}{\left(\ln\frac{t}{a}\right)^p}+E_1(t), \end{array}$$ where $$E_1(t)=\left(\ln\frac{t}{a}\right)^{n-\a}\sum_{p=N+1}^\infty \C (-1)^p \frac{\left(\ln\frac{\t}{a}\right)^p}{\left(\ln\frac{t}{a}\right)^p}$$ and $$\C (-1)^p=\frac{\Gamma(\a-n+p)}{\Gamma(\a-n)p!}.$$ Using this relation, we get the new formula $$\begin{array}{ll} \DI x(t)&=\DS\sum_{k=1}^n \frac{1}{\Gamma(k+1-\a)}\left(\ln\frac{t}{a}\right)^{k-\a}x_k(a)\\ & \quad \DS +\frac{1}{\Gamma(n+1-\a)}\left(\ln\frac{t}{a}\right)^{n-\a} \sum_{p=0}^N \frac{\Gamma(\a-n+p)}{\Gamma(\a-n)p!\left(\ln\frac{t}{a}\right)^p}\\ & \quad \DS \times\int_a^t \left(\ln\frac{\t}{a}\right)^p\frac{1}{\t}x_{n+1}(\t) d\t+E(t), \end{array}$$ where $$E(t)=\frac{1}{\Gamma(n+1-\a)}\int_a^t E_1(t)\frac{1}{\t}x_{n+1}(\t)d\t.$$ If we split the sum into the first term $p=0$ and the remaining ones $p=1\ldots N$, and use integration by parts taking $$u(\t)=\left(\ln\frac{\t}{a}\right)^p\quad \mbox{and} \quad v'(\t)=\frac{1}{\t}x_{n+1}(\t)=x'_n(\t),$$ we get $$\begin{array}{ll} \DI x(t)&=\DS\sum_{k=1}^{n-1} \frac{1}{\Gamma(k+1-\a)}\left(\ln\frac{t}{a}\right)^{k-\a}x_k(a)+A_n\left(\ln\frac{t}{a}\right)^{n-\a}x_n(t)\\ & \quad \DS +\frac{1}{\Gamma(n-\a)}\left(\ln\frac{t}{a}\right)^{n-1-\a}\sum_{p=1}^N \frac{\Gamma(\a-n+p)}{\Gamma(\a+1-n)(p-1)!\left(\ln\frac{t}{a}\right)^{p-1}}\\ & \quad \DS \times\int_a^t \left(\ln\frac{\t}{a}\right)^{p-1}\frac{1}{\t}x_{n}(\t) d\t+E(t). \end{array}$$ Observe that $$\frac{1}{\t}x_{n}(\t)=x'_{n-1}(\t).$$ Repeating this procedure, i.e., splinting the second sum (first term $p=k$ plus the remaining ones $p=k+1\ldots N$) and integrating by parts the integral that appears in the sum $p=k+1\ldots N$, we obtain the desired the formula. For the error, observe that for $\t\in[a,t]$, we have $$0\leq \frac{\left(\ln\frac{\t}{a}\right)^p}{\left(\ln\frac{t}{a}\right)^p}\leq 1.$$ Thus, $$\begin{array}{ll} \left|E_1(t)\right|&\DS\leq \left(\ln\frac{t}{a}\right)^{n-\a}\sum_{p=N+1}^\infty \frac{\exp((n-\a)^2+n-\a)}{p^{n+1-\a}}\\ &\DS\leq \left(\ln\frac{t}{a}\right)^{n-\a}\int_N^\infty \frac{\exp((n-\a)^2+n-\a)}{p^{n+1-\a}}\, dp\\ &\DS= \left(\ln\frac{t}{a}\right)^{n-\a}\frac{\exp((n-\a)^2+n-\a)}{N^{n-\a}(n-\a)}.\\ \end{array}$$ The rest of the proof follows immediately. Observe that, for $N$ sufficiently large, we have the approximation $$\label{approx1} \DI x(t) \approx\sum_{k=1}^{n}A_k \left(\ln\frac{t}{a}\right)^{k-\a}x_k(t)+\sum_{k=n}^N B_k \left(\ln\frac{t}{a}\right)^{n-k-\a} V_{k-n}(t).$$ \[teo2\] Let $x:[a,b]\to\mathbb R$ be a function of class $C^{n+1}$, for $n\in\mathbb N$, and fix $N\in\mathbb N$ with $N \geq n$. Then, $$\begin{array}{ll} \DII x(t)& =\DS\sum_{k=1}^{n}A_k \left(\ln\frac{t}{a}\right)^{k-\a}x_k(t)+\sum_{k=n}^N B_k \left(\ln\frac{t}{a}\right)^{n-k-\a} V_{k-n}(t) +\frac{t\Da}{\Gamma(2-\a)}\left(\ln\frac{t}{a}\right)^{1-\a}\\ & \quad \DS \times\left[\left(\frac{1}{1-\a}-\ln\left(\ln\frac{t}{a}\right)\right)\sum_{p=0}^N\D\frac{(-1)^p}{ \left(\ln\frac{t}{a}\right)^{p}} V_{p}(t)\right.\\ & \quad \quad \quad \DS \left.+\sum_{p=0}^N\D(-1)^p\sum_{r=1}^N\frac{1}{r \left(\ln\frac{t}{a}\right)^{p+r}} V_{p+r}(t)\right]+E(t), \end{array}$$ with $$\begin{array}{ll} E(t)&\leq \DS \frac{(t-a)\exp((n-\a)^2+n-\a)}{\Gamma(n+1-\a)N^{n-\a}(n-\a)}\left(\ln\frac{t}{a}\right)^{n-\a}\max_{\t\in[a,t]}\left|x'_N(\t)\right|\\ &\quad \DS+\frac{(t-a)t\left|\Da\right|\exp((1-\a)^2+1-\a)}{\Gamma(2-\a)N^{1-\a}(1-\a)}\left(\ln\frac{t}{a}\right)^{1-\a}\max_{\t\in[a,t]}\left|x'(\t)\right|\\ &\quad \times \DS \left|\frac{1}{1-\a}-\ln\left(\ln\frac{t}{a}\right)+\frac{(2t-a)\ln\frac{t}{a}}{N}\right|. \end{array}$$ Recalling Theorem \[relations\], we have $$\DII x(t)=\DI x(t)+\frac{t\Da}{\Gamma(2-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{1-\a}x'(\t)\left[\frac{1}{1-\a}-\ln\left(\ln\frac{t}{\t}\right)\right]d\t,$$ and so we only need to expand the second term in the right side: $$\label{eq1}\frac{t\Da}{\Gamma(2-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{1-\a}x'(\t) \left[\frac{1}{1-\a}-\ln\left(\ln\frac{t}{\t}\right)\right]d\t.$$ On one hand, as was seen in the proof of Theorem \[teo1\], we have $$\left(\ln\frac{t}{\t}\right)^{1-\a}=\left(\ln\frac{t}{a}\right)^{1-\a}\left[\sum_{p=0}^N \D (-1)^p \frac{\left(\ln\frac{\t}{a}\right)^p}{\left(\ln\frac{t}{a}\right)^p}+E_1(t)\right],$$ where $$E_1(t)=\sum_{p=N+1}^\infty \D (-1)^p \frac{\left(\ln\frac{\t}{a}\right)^p}{\left(\ln\frac{t}{a}\right)^p}.$$ On the other hand, $$\begin{array}{ll} \DS\ln\left(\ln\frac{t}{\t}\right)&=\DS\ln\left(\ln\frac{t}{a}\right)+\ln\left(1-\frac{\ln\frac{\t}{a}}{\ln\frac{t}{a}}\right)\\ &=\DS\ln\left(\ln\frac{t}{a}\right)-\sum_{r=1}^N \frac{\left(\ln\frac{\t}{a}\right)^r}{r\left(\ln\frac{t}{a}\right)^r} -E_2(t), \end{array}$$ where $$E_2(t)=\sum_{r=N+1}^\infty \frac{\left(\ln\frac{\t}{a}\right)^r}{r\left(\ln\frac{t}{a}\right)^r}.$$ Substituting these two relations in Eq. , we obtain $$\frac{t\Da}{\Gamma(2-\a)}\left(\ln\frac{t}{a}\right)^{1-\a}$$ $$\times\left[\left(\frac{1}{1-\a}-\ln\left(\ln\frac{t}{a}\right)\right) \sum_{p=0}^N\D\frac{(-1)^p}{\left(\ln\frac{t}{a}\right)^p} V_p(t)+\sum_{p=0}^N\D(-1)^p\sum_{r=1}^N\frac{1}{r\left(\ln\frac{t}{a}\right)^{p+r}} V_{p+r}(t)\right]$$ $$+\frac{t\Da}{\Gamma(2-\a)}\left(\ln\frac{t}{a}\right)^{1-\a}\times\left[\left(\frac{1}{1-\a}-\ln\left(\ln\frac{t}{a}\right)\right) \int_a^t E_1(t)x'(\t)\,d\t+\int_a^t E_1(t)E_2(t)x'(\t)\,d\t\right].$$ We now determine the upper bound for the error. As was seen in proof of Theorem \[teo1\], we have $$\left|E_1(t)\right|\leq \frac{\exp((1-\a)^2+1-\a)}{N^{1-\a}(1-\a)}.$$ On the other hand, $$\int_a^t E_2(t) \, d\t=\sum_{r=N+1}^\infty \frac{1}{r\left(\ln\frac{t}{a}\right)^r}\int_a^t \t \cdot \left(\ln\frac{\t}{a}\right)^r\frac{d\t}{\t}.$$ Integrating by parts, $$\begin{array}{ll} \DS\int_a^t E_2(t) \, d\t & \DS = \sum_{r=N+1}^\infty \frac{1}{r(r+1)}\left[ t\ln\frac{t}{a}- \int_a^t \frac{\left(\ln\frac{\t}{a}\right)^{r+1}}{\left(\ln\frac{t}{a}\right)^{r}}d\t\right]\\ & \DS \leq \sum_{r=N+1}^\infty \frac{1}{r(r+1)}\left[ t\ln\frac{t}{a}+\int_a^t \ln\frac{t}{a} d\t\right]\\ & \DS \leq (2t-a)\ln\frac{t}{a}\int_N^\infty \frac{1}{r^2}\,dr=\frac{(2t-a)\ln\frac{t}{a}}{N}. \end{array}$$ Combining these relations, we prove the result. Finally, we have the expansion formula for the Caputo–Hadamard fractional derivative of order $\a$ type 3. \[teo3\] Let $x:[a,b]\to\mathbb R$ be a function of class $C^{n+1}$, for $n\in\mathbb N$, and fix $N\in\mathbb N$ with $N \geq n$. Then, $$\begin{array}{ll} \DIII x(t)& =\DS\sum_{k=1}^{n}A_k \left(\ln\frac{t}{a}\right)^{k-\a}x_k(t)+\sum_{k=n}^N B_k \left(\ln\frac{t}{a}\right)^{n-k-\a} V_{k-n}(t) +\frac{t\Da}{\Gamma(2-\a)}\left(\ln\frac{t}{a}\right)^{1-\a}\\ & \quad \DS \times\left[\left(\psi(2-\a)-\ln\left(\ln\frac{t}{a}\right)\right)\sum_{p=0}^N\D\frac{(-1)^p}{ \left(\ln\frac{t}{a}\right)^{p}} V_{p}(t)\right.\\ & \quad \quad \quad \DS \left.+\sum_{p=0}^N\D(-1)^p\sum_{r=1}^N\frac{1}{r \left(\ln\frac{t}{a}\right)^{p+r}} V_{p+r}(t)\right]+E(t), \end{array}$$ with $$\begin{array}{ll} E(t)&\leq \DS \frac{(t-a)\exp((n-\a)^2+n-\a)}{\Gamma(n+1-\a)N^{n-\a}(n-\a)}\left(\ln\frac{t}{a}\right)^{n-\a}\max_{\t\in[a,t]}\left|x'_N(\t)\right|\\ &\quad \DS+\frac{(t-a)t\left|\Da\right|\exp((1-\a)^2+1-\a)}{\Gamma(2-\a)N^{1-\a}(1-\a)}\left(\ln\frac{t}{a}\right)^{1-\a}\max_{\t\in[a,t]}\left|x'(\t)\right|\\ &\quad \times \DS \left|\psi(2-\a)-\ln\left(\ln\frac{t}{a}\right)+\frac{(2t-a)\ln\frac{t}{a}}{N}\right|. \end{array}$$ By Theorem \[relations\], we only need to expand the term $$\frac{t\Da\psi(1-\a)}{\Gamma(1-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{-\a}\frac{x(\t)-x(a)}{\t}d\t.$$ Integrating by parts, we get $$\frac{t\Da\psi(1-\a)}{\Gamma(2-\a)}\int_a^t\left(\ln\frac{t}{\t}\right)^{1-\a}x'(\t)d\t.$$ Using Taylor’s Theorem on the term $\left(\ln\frac{t}{\t}\right)^{1-\a}$, combining with Theorem \[teo2\] and using the formula $$\psi(1-\a)+\frac{1}{1-\a}=\psi(2-\a),$$ we prove the result. Similar formulas are obtained for the right Caputo–Hadamard fractional derivatives. Letting, for $k\in\mathbb N$, $$\begin{array}{ll} \overline A_k & =\DS \frac{(-1)^k}{\Gamma(k+1-\a)}\left[1+\sum_{p=n-k+1}^N \frac{\Gamma(\a-n+p)}{\Gamma(\a-k)(p-n+k)!} \right],\\ \overline B_k & = \DS\frac{-\Gamma(\a-n+k)}{\Gamma(1-\a)\Gamma(\a)(k-n)!}, \end{array}$$ and $$\overline V_k(t)=\int_{t}^{b}\left(\ln\frac{b}{\t}\right)^k x'(\t)d\t,$$ we have the next three approximation formulas: $$\DIR x(t)\approx\DS\sum_{k=1}^{n}\overline A_k \left(\ln\frac{b}{t}\right)^{k-\a}x_k(t)+\sum_{k=n}^N \overline B_k \left(\ln\frac{b}{t}\right)^{n-k-\a} \overline V_{k-n}(t),$$ $$\DIIR x(t)\approx\DS\sum_{k=1}^{n}\overline A_k \left(\ln\frac{b}{t}\right)^{k-\a}x_k(t)+\sum_{k=n}^N \overline B_k \left(\ln\frac{b}{t}\right)^{n-k-\a} \overline V_{k-n}(t)+\frac{t\Da}{\Gamma(2-\a)}\left(\ln\frac{b}{t}\right)^{1-\a}$$ $$\times\left[\left(\frac{1}{1-\a}-\ln\left(\ln\frac{b}{t}\right)\right)\sum_{p=0}^N\D\frac{(-1)^p}{ \left(\ln\frac{b}{t}\right)^{p}} \overline V_{p}(t)+\sum_{p=0}^N\D(-1)^p\sum_{r=1}^N\frac{1}{r \left(\ln\frac{b}{t}\right)^{p+r}}\overline V_{p+r}(t)\right],$$ and $$\DIIIR x(t)\approx\DS\sum_{k=1}^{n}\overline A_k \left(\ln\frac{b}{t}\right)^{k-\a}x_k(t)+\sum_{k=n}^N \overline B_k \left(\ln\frac{b}{t}\right)^{n-k-\a} \overline V_{k-n}(t)+\frac{t\Da}{\Gamma(2-\a)}\left(\ln\frac{b}{t}\right)^{1-\a}$$ $$\times\left[\left(\psi(2-\a)-\ln\left(\ln\frac{b}{t}\right)\right)\sum_{p=0}^N\D\frac{(-1)^p}{ \left(\ln\frac{b}{t}\right)^{p}} \overline V_{p}(t)+\sum_{p=0}^N\D(-1)^p\sum_{r=1}^N\frac{1}{r \left(\ln\frac{b}{t}\right)^{p+r}}\overline V_{p+r}(t)\right].$$ Examples {#sec:EX} ======== In this section we test the efficiency of the purposed method, by comparing the exact fractional derivative of the function $$\overline x(t)=\ln t, \quad \mbox{for} \quad t\in[1,5],$$ with some numerical approximations. We fix the order $\a=t/20$, and for the approximations using Theorems \[teo1\], \[teo2\] and \[teo3\], we take $n=1$ and vary $N\in\{10,20,30\}$. The error of approximation at the point $t_0$ is given by the absolute value of the difference between the exact and the approximation at $t=t_0$. Below we give the exact fractional derivatives of $\overline x$, obtained by Lemma \[LemmaEx\] $$\begin{array}{ll} {_1\mathbb{D}_t^\a} \overline x(t) &= \DS \frac{1}{\Gamma\left(2-\a\right)}(\ln t)^{1-\a}\\ {_1D_t^\a} \overline x(t) &= \DS \frac{1}{\Gamma\left(2-\a\right)}(\ln t)^{1-\a} -\frac{t\Da}{\Gamma\left(3-\a\right)}(\ln t)^{2-\a}\\ &\quad \DS\times\left[\ln(\ln t)+\psi\left(1-\a\right) -\psi\left(3-\a\right)\right],\\ {_1\mathcal{D}_t^\a} \overline x(t) &= \DS \frac{1}{\Gamma\left(2-\a\right)}(\ln t)^{1-\a} -\frac{t\Da}{\Gamma\left(3-\a\right)}(\ln t)^{2-\a}\left[\ln(\ln t)-\psi\left(3-\a\right)\right].\\ \end{array}$$ For the right Caputo–Hadamard fractional derivatives, we take $$\overline y(t)=\ln\frac5t, \quad \mbox{for} \quad t\in[1,5],$$ $n=1$ and $N\in\{2,4,6\}$. The fractional derivatives are in this case given by the expressions $$\begin{array}{ll} {_t\mathbb{D}_5^\a} \overline y(t) &= \DS \frac{1}{\Gamma\left(2-\a\right)}\left(\ln \frac5t\right)^{1-\a}\\ {_tD_5^\a} \overline y(t) &= \DS \frac{1}{\Gamma\left(2-\a\right)}\left(\ln \frac5t\right)^{1-\a} +\frac{t\Da}{\Gamma\left(3-\a\right)}\left(\ln \frac5t\right)^{2-\a}\\ &\quad \DS\times\left[\ln\left(\ln \frac5t\right)+\psi\left(1-\a\right) -\psi\left(3-\a\right)\right],\\ {_t\mathcal{D}_5^\a} \overline y(t) &= \DS \frac{1}{\Gamma\left(2-\a\right)}\left(\ln \frac5t\right)^{1-\a} +\frac{t\Da}{\Gamma\left(3-\a\right)}\left(\ln \frac5t\right)^{2-\a}\left[\ln\left(\ln \frac5t\right)-\psi\left(3-\a\right)\right].\\ \end{array}$$ The results are shown in Figure \[fig4\] below. \[IntExp\] Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT–-Fundação para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013. [99]{} T. M. Atanackovic, M. Janev, S. Pilipovic and D. Zorica, An expansion formula for fractional derivatives of variable order, Cent. Eur. J. Phys. [**11**]{} (2013), no. 10, 1350–1360. P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Stirling functions of the second kind in the setting of difference and fractional calculus, Numer. Funct. Anal. Optim. [**24**]{} (2003), no. 7-8, 673-–711. C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys. (8) [**12**]{} (2003), no. 11-12, 692–703. C. F. M. Coimbra, C. M. Soon and M. H. Kobayashi, The variable viscoelasticity operator, Annalen der Physik [**14**]{} (2005), 378–389. M. M. Dzherbashyan and A. B. Nersesian, Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Akad. Nauk Arm. SSR, Mat. [**3**]{} (1968), 3–-29. Y. Y Gambo, F. Jarad, D. Baleanu and T. Abdeljawad, On Caputo modification of the Hadamard fractional derivatives, Adv. Difference Equ. (2014), 2014:10 doi:10.1186/1687–1847–2014–10. F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Difference Equ. (2012), 2012:142 doi:10.1186/1687–1847–2012–142. Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. [**225**]{} (2007), no. 2, 1533–-1552. J. A. T. Machado, M. F. Silva, R. S. Barbosa, I. S. Jesus, C. M. Reis, M. G. Marcos and A. F. Galhano, Some applications of fractional calculus in engineering, Math. Probl. Eng. [**2010**]{} (2010), Art. ID 639801, 34 pp. K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, Wiley, New York, 1993. T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Noether’s theorem for fractional variational problems of variable order, Cent. Eur. J. Phys. [**11**]{} (2013), no. 6, 691–701. S. Pooseh, R. Almeida and D. F. M. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim. [**33**]{} (2012), no. 3, 301–319. S. Pooseh, R. Almeida and D. F. M. Torres, Numerical approximations of fractional derivatives with applications, Asian J. of Control [**15**]{} (2013), no. 3, 698–712. L. E. S. Ramirez and C. F. M. Coimbra, On the variable order dynamics of the nonlinear wake caused by a sedimenting particle, Phys. D [**240**]{} (2011), no. 13, 1111–1118. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, Translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993. S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transform. Spec. Funct. [**1**]{} (1993), no. 4, 277–300. H.G. Sun, W. Chen and Y.Q. Chen, Variable order fractional differential operators in anomalous diffusion modeling, Physica A. [**388**]{} (2009) 4586-–4592

--- abstract: 'In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.' address: - 'Weiyuan Qiu, School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China' - 'Fei Yang, Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China' - 'Yongcheng Yin, Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China' author: - WEIYUAN QIU - FEI YANG - YONGCHENG YIN title: RATIONAL MAPS WHOSE JULIA SETS ARE CANTOR CIRCLES --- Introduction ============ The study of the topological properties of the Julia sets of rational maps is a central problem in complex dynamics. For each degree at least two polynomial with a disconnected Julia set, it was proved that all but countably many components of the Julia set are single points in [@QY]. For rational maps, the Julia sets may exhibit more complex topological structures. Pilgrim and Tan proved that if the Julia set of a hyperbolic (more generally, geometrically finite) rational map is disconnected, then, with the possible exception of finitely many periodic components and their countable collection of preimages, every Julia component is either a point or a Jordan curve [@PT Theorem 1.2]. In this paper, we will consider one class of rational maps whose Julia sets possess simple topological structure: each Julia component is a Jordan curve. A subset of the Riemann sphere $\overline{\mathbb{C}}$ is called a *Cantor set of circles* (sometimes *Cantor circles* in short) if it consists of uncountably many closed Jordan curves which is homeomorphic to $\mathcal{C}\times \mathbb{S}^1$, where $\mathcal {C}$ is the middle third Cantor set and $\mathbb{S}^1$ is the unit circle. The first example of rational map whose Julia set is a Cantor set of circles was discovered by McMullen (see [@Mc $\S$7]). He showed that if $f(z)=z^2+\lambda/z^3$ and $\lambda$ is small enough, then the Julia set of $f$ is a Cantor set of circles. Later, many authors focus on the following family, which is commonly referred as the *McMullen maps*: $$\label{McMullen} g_{\eta}(z)=z^k+\eta/z^l,$$ where $k,l\geq 2$ and $\eta\in\mathbb{C}\setminus\{0\}$ (see [@DLU; @St; @QWY] and the references therein). These special rational maps can be viewed as a perturbation of the simple polynomial $g_0(z)=z^k$ if $\eta$ is small. It is known that when $1/k+1/l<1$, there exists a punched neighborhood $\mathcal{M}$ centered at origin in the parameter space, which is called the *McMullen domain*, such that when $\eta\in\mathcal{M}$, then the Julia set of $g_\eta$ is a Cantor set of circles (see [@Mc $\S$7] for $k=2,l=3$ and [@DLU $\S$3] for the general cases). The following three questions arise naturally: (1) Besides McMullen maps, do there exist any other rational maps whose Julia sets are Cantor circles? (2) If the answer to the first question is yes, what do they look like? Or in other words, can we find specific expressions for them? (3) Can we find out all rational maps whose Julia sets are Cantor circles in some sense? This paper will give affirmative answers to these questions. By quasiconformal surgery, we can obtain many new rational maps after perturbing the immediate super-attracting basin centered at $\infty$ of $g_\eta$ into a geometric one. Fix one of them, then this map is not topologically conjugate to $g_\eta$ on the whole $\overline{\mathbb{C}}$. But they are topologically conjugate to each other on their corresponding Julia sets. In particular, $h_{c,\eta}(z)=\frac{1}{z}\circ (z^k+c)\circ\frac{1}{z}+\eta/z^l$ is an example, where $1/k+1/l<1$ and $c,\eta\in\mathbb{C}\setminus\{0\}$ are both small enough. However, these types of rational maps can be also regarded as the McMullen maps essentially, which are not what we want to find since they can be obtained by doing a surgery only on the Fatou sets of the genuine McMullen maps. So it will be very interesting to find other types of rational maps with Cantor circles Julia sets which are not topologically conjugate to any McMullen maps on their corresponding Julia sets. The existence of of types of rational maps ‘essentially’ different from McMullen maps was known previously (see [@HP $\S\S$1,2]). Here, ‘essentially’ means there exists no topological conjugacy between the Julia sets of McMullen maps and the rational maps whose Julia sets are Cantor circles. In this paper, we will give the specific expressions for these types of rational maps, not only including the cases discussed in [@HP], but also covering all the rational maps whose Julia sets are Cantor circles ‘essentially’ (see Theorem \[this-is-all\]). Let $p\in\{0,1\}$, $n\geq 2$ be an integer and $d_1,\cdots,d_n$ be $n$ positive integers such that $\sum_{i=1}^{n}(1/d_i)<1$. We define $$\label{family} f_{p,d_1,\cdots,d_n}(z)=z^{(-1)^{n-p} d_1}\prod_{i=1}^{n-1}(z^{d_i+d_{i+1}}-a_i^{d_i+d_{i+1}})^{(-1)^{n-i-p}},$$ where $a_1,\cdots,a_{n-1}$ are $n-1$ small complex numbers satisfying $0<|a_1|<\cdots<|a_{n-1}|<1$. In particular, if $n=2$, then $f_{1,d_1,d_2}(z)=z^{d_2}-a_1^{d_1+d_2}/z^{d_1}$ is the McMullen map that has been well studied by many authors. Moreover, $f_{0,d_1,d_2}(z)=z^{d_1}/(z^{d_1+d_2}-a_1^{d_1+d_2})$ is conformally conjugate to the McMullen map $z\mapsto z^{d_1}+\eta/z^{d_2}$ for some $\eta\neq 0$. The degrees of $f_{p,d_1,\cdots,d_n}$ at $0$ and $\infty$ are $d_1$ and $d_n$ respectively and $\text{deg} (f_{p,d_1,\cdots,d_n})=\sum_{i=1}^{n}d_i$. For each element in the family , it is easy to check that $0$ and $\infty$ belong to the Fatou set of $f_{p,d_1,\cdots,d_n}$. Let $D_0$ and $D_\infty$ be the Fatou components containing $0$ and $\infty$ respectively. There are four cases (we use $f$ to replace $f_{p,d_1,\cdots,d_n}$ temporarily): \(1) If $p=1$ and $n$ is odd, then $f(D_0)=D_0$ and $f(D_\infty)=D_\infty$; \(2) If $p=1$ and $n$ is even, then $f(D_0)=D_\infty$ and $f(D_\infty)=D_\infty$; \(3) If $p=0$ and $n$ is odd, then $f(D_0)=D_\infty$ and $f(D_\infty)=D_0$; \(4) If $p=0$ and $n$ is even, then $f(D_0)=D_0$ and $f(D_\infty)=D_0$. Firstly we will find suitable parameters $a_i$ in , where $1\leq i\leq n-1$, such the Julia set of each $f_{p,d_1,\cdots,d_n}$ in the four cases stated above is a Cantor set of circles. \[parameter\] For each given $p\in\{0,1\}$, $n\geq 2$ and $d_1,\cdots,d_n$ satisfying $\sum_{i=1}^{n}(1/d_i)<1$, there exist suitable parameters $a_i$, where $1\leq i\leq n-1$ such that the Julia set of $f_{p,d_1,\cdots,d_n}$ is a Cantor set of circles. The specific value ranges of $a_i$ are given in §\[sec-loc-crit\], where $1\leq i\leq n-1$ (see , and Theorem \[parameter-restate\]). These rational maps can be seen as the perturbations of $z^{d_n}$ or $z^{-d_n}$ (according to whether $p=1$ or 0) since each $a_i$ can be arbitrarily small (see Theorem \[parameter-restate\]). Moreover, it will be shown that if $n\geq 3$, then each $f_{p,d_1,\cdots,d_n}$ is not topologically conjugate to any McMullen maps on their corresponding Julia sets (see Theorem \[no-topo-equiv\]). This means that we have found the specific expressions of rational maps whose Julia sets are Cantor circles which are ‘essentially’ different from McMullen maps. For example, let $p=1$, $n=4$, $d_1=d_2=d_3=d_4=5$ and define $$\label{an-example} f_{1,5,5,5,5}(z)=\frac{(z^{10}-a_1^{10})(z^{10}-a_3^{10})}{z^5(z^{10}-a_2^{10})},$$ where $a_1=0.00025,a_2=0.005$ and $a_3=0.1$. By a straightforward calculation or using Theorem \[parameter-restate\] and Remark \[range-unif\], one can show that the Julia set of $f_{1,5,5,5,5}$ is a Cantor set of circles (see Figure \[Fig\_Cantor-cicle\]). The dynamics on the set of Julia components of $f_{1,5,5,5,5}$ is conjugate to the one-sided shift on four symbols $\Sigma_4:=\{0,1,2,3\}^{\mathbb{N}}$ while the set of Julia components of $g_\eta$ is conjugate to the one-sided shift on only two symbols $\Sigma_{2}:=\{0,1\}^{\mathbb{N}}$. This means that $f_{1,5,5,5,5}$ cannot be topologically conjugate to $g_\eta$ on their corresponding Julia sets. ![The Julia set of $f_{1,5,5,5,5}$ (left picture), which is not topologically conjugate to that of McMullen map $g_\eta(z)=z^3+0.001/z^3$ (right picture). The two Julia sets are both Cantor circles.[]{data-label="Fig_Cantor-cicle"}](Cantor-circle-Julia.png "fig:"){width="65mm"} ![The Julia set of $f_{1,5,5,5,5}$ (left picture), which is not topologically conjugate to that of McMullen map $g_\eta(z)=z^3+0.001/z^3$ (right picture). The two Julia sets are both Cantor circles.[]{data-label="Fig_Cantor-cicle"}](McMullen-Julia-Cantor-circle.png "fig:"){width="65mm"} Note that if the Julia set $J(f)$ of a rational map $f$ is a Cantor set of circles, then there exist no critical points in $J(f)$ since each Julia component is a Jordan closed curve (see Lemma \[no-crit-on-J\]). This means that every periodic Fatou component of $f$ must be attracting or parabolic. In fact, we have following theorem. \[this-is-all\] Let $f$ be a rational map whose Julia set is a Cantor set of circles. Then there exist $p\in\{0,1\}$, positive integers $n\geq 2$, and $d_1,\cdots, d_n$ satisfying $\sum_{i=1}^{n}(1/d_i)<1$ such that $f$ is topologically conjugate to $f_{p,d_1,\cdots,d_n}$ on their corresponding Julia sets for suitable parameters $a_i$, where $1\leq i\leq n-1$. Since the dynamics on the Fatou set can be perturbed freely, it follows from Theorem \[this-is-all\] that we have found ‘all’ the possible rational maps whose Julia sets are Cantor circles. A rational map is *hyperbolic* if all critical points are attracted by attracting periodic orbits. For the regularity of the Julia components of $f_{p,d_1,\cdots, d_n}$, it can be shown that each Julia component of $f_{p,d_1,\cdots,d_n}$ is a quasicircle if $f_{p,d_1,\cdots,d_n}$ is hyperbolic (see Corollary \[Julia-comp\]). If $\eta$ is small enough, then $g_\eta$ is hyperbolic (see [@DLU]). Now we construct some non-hyperbolic rational maps whose Julia sets are Cantor circles. Let $m,n\geq 2$ be two positive integers satisfying $1/m+1/n<1$ and $\lambda\in\mathbb{C}\setminus\{0\}$, we define $$P_\lambda(z)=\frac{\frac{1}{n}((1+z)^n-1)+\lambda^{m+n}z^{m+n}}{1-\lambda^{m+n}z^{m+n}}.$$ It is straightforward to verify that zero is a parabolic fixed point of $P_\lambda$ with multiplier one. We then have the following theorem. \[non-hyper-cantor\] If $0<|\lambda|\leq 1/{(2^{10m}n^3)}$, then $P_\lambda$ is non-hyperbolic and its Julia set is a Cantor set of circles. Inspired by Theorem \[parameter\], we can construct more non-hyperbolic rational maps whose Julia sets are Cantor circles. For simplicity, for each $n\geq 2$, we only consider the case $d_i=n+1$ for every $1\leq i\leq n$. For every $n\geq 2$, we define $$\label{family-para} P_n(z)=A_n\,\frac{(n+1)z^{(-1)^{n+1} (n+1)}}{nz^{n+1}+1}\prod_{i=1}^{n-1}(z^{2n+2}-b_i^{2n+2})^{(-1)^{i-1}}+B_n,$$ where $b_1,\cdots,b_{n-1}$ are $n-1$ small complex numbers satisfying $1>|b_1|>\cdots>|b_{n-1}|>0$ and $$\label{A-B-n} \begin{split} A_n=\frac{1}{1+(2n+2)C_n}\prod_{i=1}^{n-1}(1-b_i^{2n+2})^{(-1)^i}, &~~B_n=\frac{(2n+2)C_n}{1+(2n+2)C_n} \\ \text{and}~C_n=\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}}. & \end{split}$$ The terms $A_n$ and $B_n$ here can guarantee that $P_n(1)=1$ and $P_n'(1)=1$. Namely, $1$ is a parabolic fixed point of $P_n$ with multiplier one (see Lemma \[para-fixed\]). \[parameter-parabolic\] For every $n\geq 2$ and $1\leq i\leq n-1$, if $|b_i|=s^i$ for $0<s\leq 1/(25n^2)$, then $P_n$ is non-hyperbolic and its Julia set is a Cantor set of circles. It can be seen later the dynamics of $P_n$ on their Julia sets are conjugate to that of $f_{1,n+1,\cdots,n+1}$ for $n\geq 2$. One of the differences between their dynamics on the Fatou sets is the super-attracting basin of $f_{1,n+1,\cdots,n+1}$ at $\infty$ is replaced by a parabolic basin of $P_n$. This paper is organized as follows: In §\[sec-loc-crit\], we do some estimates and prove Theorem \[parameter\]. In §\[sec-topo-conj\], we prove Theorem \[this-is-all\]. In §\[sec-para-mcm\], we show that the Julia set of $P_\lambda$ is a Cantor set of circles if $\lambda$ is small enough and prove Theorem \[non-hyper-cantor\]. We will prove Theorem \[parameter-parabolic\] in §\[sec-more-exam\] and leave a key lemma to the last section. 0.2cm *Notation*. We will use the following notations throughout the paper. Let $\mathbb{C}$ be the complex plane and $\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ the Riemann sphere. For $r>0$ and $a\in\mathbb{C}$, let $\mathbb{D}(a,r):=\{z\in\mathbb{C}:|z-a|<r\}$ be the Euclidean disk centered at $a$ with radius $r$. In particular, let $\mathbb{D}_r:=\mathbb{D}(0,r)$ be the disk centered at the origin with radius $r$ and $\mathbb{T}_r:=\partial\mathbb{D}_r$ be the boundary of $\mathbb{D}_r$. As usual, $\mathbb{D}:=\mathbb{D}_1$ and $\mathbb{S}^1:=\mathbb{T}_1$ denote the unit disk and the unit circle, respectively. For $0<r<R<+\infty$, let $\mathbb{A}_{r,R}:=\{z\in\mathbb{C}:r<|z|<R\}$ be the round annulus centered at the origin. Location of the critical points and the hyperbolic case {#sec-loc-crit} ======================================================= First we give some basic and useful estimations. \[very-useful-est\] Let $n\geq 2$ be an integer, $a\in\mathbb{C}\setminus\{0\}$ and $0<\varepsilon<1/2$. $(1)$ If $|z-a|\leq \varepsilon |a|$, then $|z^{n}-a^{n}|\leq ((1+\varepsilon)^{n}-1)\, |a|^{n}$; $(2)$ If $|z^{n}-a^{n}|\leq \varepsilon |a|^{n}$, then $|a/z|^n<1+2\varepsilon$ and $|z-ae^{2\pi i{j}/{n}}|< \varepsilon |a|$ for some $1\leq j\leq n$; $(3)$ If $0<\varepsilon<1/n$, then $n\varepsilon< (1+\varepsilon)^n-1< 3n\varepsilon$ and $n\varepsilon/3< 1-(1-\varepsilon)^n<n\varepsilon$. Let $z=a(1+re^{i\theta})$ for $0\leq r\leq \varepsilon$ and $0\leq\theta<2\pi$, then $$|z^{n}-a^{n}|= |(1+re^{i\theta})^{n}-1|\cdot|a|^{n}\leq ((1+\varepsilon)^{n}-1)\, |a|^{n}.$$ This proves (1). The first statement in (2) follows from $|a/z|^n\leq 1/(1-\varepsilon)<1+2\varepsilon$ if $0<\varepsilon<1/2$. For the second statement, let $z^{n}=a^{n}(1+re^{i\theta})$ for $0\leq r\leq \varepsilon$ and $0\leq\theta<2\pi$, then $z=ae^{2\pi i{j}/{n}}(1+re^{i\theta})^{1/n}$ for some $1\leq j\leq n$ and we have $$|z-ae^{2\pi i{j}/{n}}|=|(1+re^{i\theta})^{1/n}-1|\cdot|a| \leq ((1+\varepsilon)^{1/n}-1)\cdot|a| < \varepsilon |a|$$ if $n\geq 2$. The claim (3) can be proved by using Lagrange’s mean value theorem to $x\mapsto x^n$ on the intervals $[1,1+\varepsilon]$ and $[1-\varepsilon,1]$ respectively. The proof is complete. Fix $n\geq 2$ and let $d_1,\cdots,d_n\geq 2$ be $n$ positive numbers such that $\xi=\sum_{i=1}^{n}(1/d_i)<1$. We use $K\geq 3$ to denote the maximal number among $d_1,\cdots,d_n$. Let $u_1=s_1 K^{-5}$ and $v_1=s_1 K^{-2}$, where $$\label{range-s1} 0<s_1\leq \min\{K^{-5\xi/(1-\xi)},K^{5-2K}\}<1.$$ Let $u_0=s_0^{1+1/d_n+2(1-\xi)/3}$, $v_0=s_0^{1/d_n+(1-\xi)/3}$, where $$\label{range-s0} 0<s_0\leq \min\{2^{-(1-\xi)^{-1}(1+1/d_n-2\xi/3)^{-1}},(4K)^{-3/(1-\xi)},K^{-2K(1+1/d_n+2(1-\xi)/3)^{-1}}\}<1.$$ For $p\in\{0,1\}$, let $|a_{n-1,p}|=v_p^{1/d_{n}}$ and $|a_{i,p}|=u_p^{1/d_{i+1}}|a_{i+1,p}|$ be the $n-1$ parameters in the family $f_{p,d_1,\cdots,d_n}$, where $1\leq i\leq n-2$. Since the cases $p=0$ and $p=1$ can be discussed uniformly in general, we use $s$, $u$, $v$ and $a_i$, respectively, to denote $s_p$, $u_p$, $v_p$ and $a_{i,p}$ for simplicity when the situation is clear, where $1\leq i\leq n-1$. \[esti-a1\] $(1)$ $u^{2/K}\leq K^{-4}$. $(2)$ If $1\leq j\leq i\leq n-1$, then $|a_j/a_i|\leq u^{\frac{i-j}{K}}$. $(3)$ If $p=1$, then    $(s/|a_1|)^{d_1}< su/(2v)=sK^{-3}/2$ and    $(|a_1|/s)^{d_1}v/2>K$. $(4)$ If $p=0$, then    $2Ku/v<s$ and $1/(2Kv)>(2/s)^{1/d_n}$;    $(s/|a_1|)^{d_1}<sv/2<u^{1/2}/2$ and    $(|a_1|/s)^{d_1}u/(2v)>(2/s)^{1/d_n}$. \(1) From and , we have $s_1\leq K^{5-2K}$ and $s_0\leq K^{-2K(1+1/d_n+2(1-\xi)/3)^{-1}}$. This means that $u_1^{2/K}=(s_1 K^{-5})^{2/K}\leq K^{-4}$ and $u_0^{2/K}\leq K^{-4}$. \(2) If $j=i$, then (2) is trivial. Suppose that $1\leq j<i\leq n-1$, then $$|a_j/a_i|= u^{\frac{1}{d_{j+1}}+\cdots+\frac{1}{d_{i}}}\leq u^{\frac{i-j}{K}}$$ since $K\geq d_i$ for $1\leq i\leq n$. This proves (2). \(3) If $p=1$, then $u=s K^{-5}$ and $v=s K^{-2}$. Since $s\leq K^{-5\xi/(1-\xi)}$, we have $s^{1-\xi}K^{5\xi}\leq 1$, so $$s^{1-\frac{1}{d_1}} s^{-(\frac{1}{d_2}+\cdots+\frac{1}{d_n})} K^{5(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{2}{d_n}} 2^{\frac{1}{d_1}} K^{\frac{3}{d_1}}<1.$$ This is equivalent to $s^{1-\frac{1}{d_1}}2^{\frac{1}{d_1}} K^{\frac{3}{d_1}}/|a_1|<1$ since $$|a_1|=u^{\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}}}v^{\frac{1}{d_{n}}}=s^{\frac{1}{d_2}+\cdots+\frac{1}{d_n}}/K^{5(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{2}{d_n}}.$$ So we have $(s/|a_1|)^{d_1}< su/(2v)=sK^{-3}/2$ and is proved. Moreover, (3b) can be derived from (3a) directly since $(|a_1|/s)^{d_1}>2K^3/s=2K/v$. \(4) If $p=0$, then $u=s^{1+1/d_n+2(1-\xi)/3}$, $v=s^{1/d_n+(1-\xi)/3}$. From , we know $4Ks^{(1-\xi)/3}\leq 1$, which means $2Ku/v=2Ks^{1+(1-\xi)/3}<s$. Note that $2^{1+1/d_n}K s^{(1-\xi)/3}<1$, which is equivalent to $1/(2Kv)>(2/s)^{1/d_n}$. This ends the proof of (4a). From , we know that $$\begin{split} 1\geq &~2s^{(1-\xi)(1+1/d_n-2\xi/3)}>2^{\frac{1}{d_1}}s^{(1-\xi)(1+1/d_n-2\xi/3)}\\ = &~ 2^{\frac{1}{d_1}} s^{1-\frac{1}{d_1}}/s^{(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{1}{d_n}(\frac{1}{d_1}+\cdots+\frac{1}{d_{n}})+\frac{2\xi(1-\xi)}{3}}\\ > &~ 2^{\frac{1}{d_1}} s^{1-\frac{1}{d_1}}/s^{(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{1}{d_n}(\frac{1}{d_1}+\cdots+\frac{1}{d_{n}})+\frac{1-\xi}{3}(\frac{1}{d_1}+2(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{1}{d_n})}\\ = &~ s^{1-\frac{1}{d_1}} (2/v)^{\frac{1}{d_1}}/|a_1|. \end{split}$$ This means that $(s/|a_1|)^{d_1}<sv/2=u^{1/2}s^{(1+1/d_n)/2}/2<u^{1/2}/2$. So (4b) holds. The proof of (4c) is similar to (4b). We just need to note that $$1\geq 2s^{(1-\xi)(1+1/d_n-2\xi/3)}>2^{\frac{1}{d_1}(1+\frac{1}{d_n})}s^{(1-\xi)(1+1/d_n-2\xi/3)} > (s/|a_1|)(2v/u)^{\frac{1}{d_1}}(2/s)^{\frac{1}{d_1 d_n}}.$$ This means that $(|a_1|/s)^{d_1}u/(2v)>(2/s)^{1/d_n}$. In the following, we use $f$ to denote $f_{p,d_1,\cdots,d_n}$ for simplicity. Note that $0$ and $\infty$ are critical points of $f$ with multiplicity $d_1$ and $d_n$ respectively, and the degree of $f$ is $\sum_{i=1}^{n}d_i$. Denoting $D_i=d_i+d_{i+1}$, we have $5\leq D_i\leq 2K$, where $1\leq i\leq n-1$. Besides $0$ and $\infty$, the rest of the $\sum_{i=1}^{n-1}D_i$ critical points of $f$ are the solutions of $$\label{solu-crit} (-1)^p \,z\,\frac{f'(z)}{f(z)}=\sum_{i=1}^{n-1}\frac{(-1)^{n-i}D_i z^{D_i}}{z^{D_i}-a_i^{D_i}}+(-1)^n d_1=0.$$ For $1\leq i\leq n-1$, let $\widetilde{CP}_i:=\{\widetilde{w}_{i,j}=r_i a_i \exp(\pi \textup{i}\frac{2j-1}{D_i}):1\leq j\leq D_i\}$ be the collection of $D_i$ points lying on the circle $\mathbb{T}_{r_i|a_i|}$ uniformly, where $r_i=\sqrt[D_i]{d_{i}/d_{i+1}}$. The following lemma shows that the $\sum_{i=1}^{n-1}D_i$ *free* critical points of $f$ are very ‘close’ to $\bigcup_{i=1}^{n-1} \widetilde{CP}_i$. \[crit-close\] For every $\widetilde{w}_{i,j}\in\widetilde{CP}_i$, where $1\leq i\leq n-1$ and $1\leq j\leq D_i$, there exists $w_{i,j}$, which is a solution of , such that $|w_{i,j}-\widetilde{w}_{i,j}|<u^{\frac{2}{K}}|a_i|$. Moreover, $w_{i_1,j_1}= w_{i_2,j_2}$ if and only if $(i_1,j_1)=(i_2,j_2)$. Note that the right side of equation is equivalent to $$\label{solu-crit-111} (-1)^{n-i}\left(\frac{D_i z^{D_i}}{z^{D_i}-a_i^{D_i}}-d_{i}\right)+G_i(z)=0,$$ where $$\label{G_n} G_{i}(z)=\sum_{1\leq j\leq n-1,\,j\neq i}\frac{(-1)^{n-j}D_j z^{D_j}}{z^{D_j}-a_j^{D_j}}+(-1)^n d_1+(-1)^{n-i}d_{i}.$$ After multiplying both sides of by $(z^{D_i}-a_i^{D_i})/d_{i+1}$, where $1\leq i\leq n-1$, we have $$\label{solu-crit-3} (-1)^{n-i}(z^{D_i}+d_{i}a_i^{D_i}/d_{i+1})+(z^{D_i}-a_i^{D_i})\,G_{i}(z)/d_{i+1}=0.$$ Let $\Omega_{i}=\{z:|z^{D_i}+d_{i}a_i^{D_i}/d_{i+1}|\leq \varepsilon\,|a_i|^{D_i}\}$, where $\varepsilon=u^{\frac{2}{K}}$ and $1\leq i\leq n-1$. For every $z\in\Omega_{i}$, since $\varepsilon\leq K^{-4}$ by Lemma \[esti-a1\](1), we have $$\label{estim-0} K^{-1}< d_{i}/d_{i+1}-\varepsilon\leq |z/a_i|^{D_i}\leq d_{i}/d_{i+1}+\varepsilon< K-1<K.$$ This means that $$\label{estim-00} K^{-1}< |a_i/z|^{D_i}< K~~\text{and therefore}~~K^{-1}< |a_i/z|^{5}< K.$$ If $1\leq j<i$ and $z\in\Omega_{i}$, we have $$\label{less-than-1} |{a_j}/{z}|^{D_i}\leq|{a_i}/{z}|^{D_i}|{a_{i-1}}/{a_i}|^{D_i}< K u^{1+d_{i+1}/d_{i}}<1.$$ Therefore, $|{a_j}/{z}|<1$. By the similar argument, it can be shown that $|z/a_j|<1$ if $i<j\leq n-1$ and $z\in\Omega_{i}$. If $1\leq j<i$, by Lemma \[esti-a1\](1) and (2) and , we have $$\label{estim-1} |{a_j}/{z}|^{D_j}\leq|{a_i}/{z}|^{5}|{a_j}/{a_i}|^{5}< K\,\varepsilon^{5(i-j)/2}\leq K^{-9}.$$ Similarly, if $i<j\leq n-1$, we have $$\label{estim-2} |{z}/{a_j}|^{D_j}\leq|{z}/{a_i}|^{5}|{a_i}/{a_j}|^{5}< K\,\varepsilon^{5(j-i)/2}\leq K^{-9}.$$ By definition, we have $$\label{estim-3} \sum_{1\leq j<i}(-1)^{n-j}D_j+(-1)^n d_1+(-1)^{n-i}d_{i}=0.$$ From , (\[estim-1\]), (\[estim-2\]) and , we have $$\begin{split} |G_{i}(z)| = &~ \left|\sum_{1\leq j<i}\frac{(-1)^{n-j}D_j}{1-(a_j/z)^{D_j}}+ \sum_{i< j\leq n-1}\frac{(-1)^{n-j-1}D_j(z/a_j)^{D_j}}{1-(z/a_j)^{D_j}}+(-1)^n d_1+(-1)^{n-i}d_{i}\right|\\ \leq &~ 2\,K\,\left|\sum_{1\leq j<i}\frac{(-1)^{n-j}(a_j/z)^{D_j}}{1-(a_j/z)^{D_j}}+ \sum_{i< j\leq n-1}\frac{(-1)^{n-j-1}(z/a_j)^{D_j}}{1-(z/a_j)^{D_j}}\right|\\ < &~\frac{4 K^2}{1-K^{-9}}\,\sum_{k=1}^{n-1}\varepsilon^{5k/2}< \frac{4 K^2}{1-K^{-9}}\,\frac{\varepsilon^{5/2}}{1-\varepsilon^{5/2}}<5\,K^2\,\varepsilon^{5/2} \end{split}$$ since $\varepsilon^{5/2}\leq K^{-10}$. This means that if $z\in\Omega_{i}$, we have $$|z^{D_i}-a_i^{D_i}|\cdot|\,G_{i}(z)|/d_{i+1}< 3\,K^3\,\varepsilon^{5/2}|a_i|^{D_i} < \varepsilon|a_i|^{D_i}$$ by and Lemma \[esti-a1\](1). From (\[solu-crit-3\]) and by Rouché’s Theorem, there exists a solution $w_{i,j}$ of such that $w_{i,j}\in\Omega_i$ for every $1\leq j\leq D_i$. In particular, $|w_{i,j}-\widetilde{w}_{i,j}|<\varepsilon|a_i|$ by the second statement of Lemma \[very-useful-est\](2). Note that for $1\leq i\leq n-2$, we have $$\label{differ-1} |a_{i+1}|-|a_i|-2\varepsilon|a_i|-2\varepsilon|a_{i+1}|>|a_{i+1}|(1-2\varepsilon-(1+2\varepsilon)K^{-2})>0.$$ By Lemma \[esti-a1\](1) and $r_i=\sqrt[D_i]{d_{i}/d_{i+1}}\leq (K/2)^{1/5}$, we have, $$\label{differ-2} \frac{r_i|a_i|\sin(\pi/D_i)}{\varepsilon|a_i|}\geq K^4(\frac{2}{K})^{1/5}\cdot\frac{2}{\pi}\cdot\frac{\pi}{2K}>K^2>1.$$ This means that $w_{i_1,j_1}= w_{i_2,j_2}$ if and only if $(i_1,j_1)=(i_2,j_2)$. The proof is complete. For $1\leq i\leq n-1$, let $CP_i:=\{w_{i,j}: 1\leq j\leq D_i\}$ be the collection of $D_i$ free critical points of $f$ which lie close to the circle $\mathbb{T}_{r_i|a_i|}$ and denote $CV_i=f(CP_i)$. \[nice-condition\] For every $1\leq i\leq n-1$, there exists an annular neighborhood $A_i$ containing $CP_i\cup\mathbb{T}_{r_i|a_i|}\cup\mathbb{T}_{|a_i|}$, such that If $p=1$, then $f(\overline{A}_i)\subset\mathbb{D}_s$ for odd $n-i$ and $f(\overline{A}_i)\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$ for even $n-i$. In particular, the set of critical values of $f$ satisfies $\bigcup_{i=1}^{n-1}CV_i\subset\mathbb{D}_s \cup \overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$. The disks $\overline{\mathbb{D}}_s$ and $\overline{\mathbb{C}}\setminus\mathbb{D}_{K}$ lie in the Fatou set of $f$ and $f^{-1}(\overline{\mathbb{A}}_{s,K})\subset \mathbb{A}_{s,K}$. $(2)$ If $p=0$, then $f(\overline{A}_i)\subset\mathbb{D}_s$ for even $n-i$ and $f(\overline{A}_i)\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{M}$ for odd $n-i$, where $M=(2/s)^{1/d_n}$. In particular, the set of critical values of $f$ satisfies $\bigcup_{i=1}^{n-1}CV_i\subset\mathbb{D}_s \cup \overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_M$. The disks $\overline{\mathbb{D}}_s$ and $\overline{\mathbb{C}}\setminus\mathbb{D}_M$ lie in the Fatou set of $f$ and $f^{-1}(\overline{\mathbb{A}}_{s,M})\subset \mathbb{A}_{s,M}$. Let $\varepsilon=u^{\frac{2}{K}}\leq K^{-4}$ be the number that appeared in Lemma \[crit-close\]. For every $1\leq i\leq n-1$, define the annulus $$A_i=\{z:(\min\{r_i,1\}-2\varepsilon)|a_i|<|z|<(\max\{r_i,1\}+2\varepsilon)|a_i|\}$$ where $r_i=\sqrt[D_i]{d_{i}/d_{i+1}}$. Obviously, $A_i\supset CP_i\cup\mathbb{T}_{r_i|a_i|}\cup\mathbb{T}_{|a_i|}$. By the definition, we have $$(2/K)^\frac{1}{D_i}\leq \min\{r_i,1\}\leq \max\{r_i,1\}\leq (K/2)^\frac{1}{D_i}.$$ If $z\in\overline{A}_i$, we have $$\label{esti-below} |a_i/z|\leq\frac{1}{(2/K)^\frac{1}{D_i}-2\varepsilon}\leq \frac{(K/2)^\frac{1}{D_i}}{1-2K^{-4}(K/2)^{1/5}}<(K/2)^\frac{1}{D_i}(1+4/K^{19/5}).$$ and $$\label{esti-above} |z/a_i|\leq(K/2)^\frac{1}{D_i}+2\varepsilon\leq (K/2)^\frac{1}{D_i}+2/K^4<(K/2)^\frac{1}{D_i}(1+1/K^3).$$ This means that $$\label{esti-below-new} |a_i/z|^5<(K/2)^\frac{5}{D_i}(1+4/K^{19/5})^{5}<(K/2)\,e^{20/K^{19/5}}< (K/2)\,e^{20/3^{19/5}}<7K/10.$$ and also, $$\label{esti-above-new} |z/a_i|^5<(K/2)^\frac{5}{D_i}(1+1/K^3)^{5}<(K/2)\,e^{5/K^3}< (K/2)\,e^{5/27}<7K/10.$$ Moreover, similar to the argument of and , we have $$\label{esti-other} |a_i/z|^{d_{i}}+|z/a_i|^{d_{i+1}}<7K/5.$$ Recall that $|a_i/a_{i+1}|^{d_{i+1}}=u$ for every $1\leq i\leq n-2$ and $|a_{n-1}|^{d_n}=v$. Let $1\leq i_1\leq i_2\leq n-1$ and $p\in\{0,1\}$, we have $$\label{sequence} \begin{split} \prod_{j=i_1}^{i_2}|a_j|^{(-1)^{n-j-p}D_j}= &~ |a_{i_1}|^{(-1)^{n-i_1-p}d_{i_1}}\,|a_{i_2}|^{(-1)^{n-i_2-p}d_{i_2+1}} \,\prod_{j=i_1}^{i_2-1}\left|\frac{a_{j}}{a_{j+1}}\right|^{(-1)^{n-j-p}d_{j+1}} \\ = &~ |a_{i_1}|^{(-1)^{n-i_1-p}d_{i_1}}\,|a_{i_2}|^{(-1)^{n-i_2-p}d_{i_2+1}}\,u^{\frac{(-1)^{n-i_1-p}-(-1)^{n-i_2-p}}{2}}\\ = &~ \left\{ \begin{array}{ll} (|a_1|^{d_1}u/v)^{(-1)^p} &~~\text{if}~~i_1=1~~\text{and}~~i_2=n-1~~\text{is even} \\ (|a_1|^{-d_1}/v)^{(-1)^p} &~~\text{if}~~i_1=1~~\text{and}~~i_2=n-1~~\text{is odd}. \end{array} \right. \end{split}$$ By and the second equation of , we have $$\label{abs-f-n} \begin{split} &~|f(z)| \\ = &~ |z^{D_i}-a_i^{D_i}|^{(-1)^{n-i-p}}\,|z|^{(-1)^{n-p} d_1}\,\prod_{j=1}^{i-1}|z|^{(-1)^{n-j-p}D_j} \,\prod_{j=i+1}^{n-1}|a_j|^{(-1)^{n-j-p}D_j}\cdot Q_i(z) \\ = &~ |1-(z/a_i)^{D_i}|^{(-1)^{n-i-p}}\,|{z}/{a_i}|^{(-1)^{n-i-p+1}d_{i}}\,|a_{n-1}|^{(-1)^{1-p}d_n} \,u^{\frac{(-1)^{n-i-p}-(-1)^{1-p}}{2}}\cdot Q_i(z) \\ = &~ v^{(-1)^{1-p}}\,u^{\frac{(-1)^{n-i-p}-(-1)^{1-p}}{2}}\,|(a_i/z)^{d_{i}}-(z/a_i)^{d_{i+1}}|^{(-1)^{n-i-p}}\cdot Q_i(z)\\ &~ \left\{ \begin{array}{ll} \leq v^{(-1)^{1-p}}\,u^{\frac{1-(-1)^{1-p}}{2}}\,(|a_i/z|^{d_{i}}+|z/a_i|^{d_{i+1}})\,Q_i(z) &~~\text{if}~~n-i-p~~\text{is even} \\ \geq v^{(-1)^{1-p}}\,u^{\frac{-1-(-1)^{1-p}}{2}}\,(|a_i/z|^{d_{i}}+|z/a_i|^{d_{i+1}})^{-1}\,Q_i(z) &~~\text{if}~~n-i-p~~\text{is odd}, \end{array} \right. \end{split}$$ where $$\label{Q-i} Q_i(z)=\prod_{j=1}^{i-1}\left|1-({a_j}/{z})^{D_j}\right|^{(-1)^{n-j-p}} \prod_{j=i+1}^{n-1}\left|1-({z}/{a_j})^{D_j}\right|^{(-1)^{n-j-p}}.$$ For $1\leq i\leq n-1$, consider $z\in \overline{A}_i$. If $1\leq j<i$, by , we have $$\label{estim-1-2} |{a_j}/{z}|^{D_j}\leq|{a_i}/{z}|^{5}|{a_j}/{a_i}|^{5}< 7K\,\varepsilon^{5(i-j)/2}/10<K^{-9}.$$ If $i<j\leq n-1$, then $$\label{estim-2-2} |{z}/{a_j}|^{D_j}\leq|{z}/{a_i}|^{5}|{a_i}/{a_j}|^{5}< 7K\,\varepsilon^{5(i-j)/2}/10<K^{-9}.$$ by . Since $e^x<1+2x$ if $0<x\leq 1$ and $\varepsilon\leq K^{-4}$, by –, we have $$\label{Q-i-esti-1} Q_i(z)< \prod_{k=1}^{\infty}\left(1+7K\,\varepsilon^{5k/2}/5\right)^2 \leq \exp\left(\frac{14\,K\,\varepsilon^{5/2}/5}{1-\varepsilon^{5/2}}\right)<1+K^{-5}<1.01.$$ and $$\label{Q-i-esti-2} Q_i(z)> \prod_{k=1}^{\infty}\left(1+7K\,\varepsilon^{5k/2}/5\right)^{-2} > 1/1.01 > 0.99.$$ For $p=1$, by Lemma \[esti-a1\](2) and (3a), for every $1\leq i\leq n-1$, if $|z|\leq s$, we have $$\label{last-estima-0} |z^{D_i}/a_i^{D_i}| \leq |s/a_1|^{D_i}|a_1/a_i|^{D_i}\leq (sK^{-3}/2)^{\frac{5}{K}}u^{\frac{5(i-1)}{K}}.$$ If we notice Lemma \[esti-a1\](1), then $$\label{last-estima-00} \sum_{i=1}^{n-1}|z^{D_i}/a_i^{D_i}| \leq \frac{(sK^{-3}/2)^{\frac{5}{K}}}{1-u^{\frac{5}{K}}} \leq\frac{K^{\frac{10}{K}-10}}{1-K^{-10}}<1/200.$$ For $p=0$, by Lemma \[esti-a1\](2) and (4b), for every $1\leq i\leq n-1$, if $|z|\leq s$, we have $$\label{last-estima-0-lp} |z^{D_i}/a_i^{D_i}| \leq |s/a_1|^{D_i}|a_1/a_i|^{D_i}\leq (u^{1/2}/2)^{\frac{5}{K}}u^{\frac{5(i-1)}{K}}.$$ By Lemma \[esti-a1\](1), then $$\label{last-estima-00-lp} \sum_{i=1}^{n-1}|z^{D_i}/a_i^{D_i}| \leq \frac{(u^{1/2}/2)^{\frac{5}{K}}}{1-u^{\frac{5}{K}}} \leq\frac{K^{-5}}{1-K^{-10}}<1/200.$$ Since $(1+2|a|)^{-1}\leq |1+a|^{\pm 1}\leq 1+2|a|$ if $0\leq |a|\leq 1/2$, by and , we know that $$\label{last-estima-1} \prod_{i=1}^{n-1}\left|1-{z^{D_i}}/{a_i^{D_i}}\right|^{(-1)^{n-i-p}} \leq \prod_{i=1}^{n-1}\left(1+2|z/a_i|^{D_i}\right)<e^{1/100}<K.$$ Therefore, $$\label{last-estima-2} \prod_{i=1}^{n-1}\left|1-{z^{D_i}}/{a_i^{D_i}}\right|^{(-1)^{n-i-p}} \geq \prod_{i=1}^{n-1}\left(1+2|z/a_i|^{D_i}\right)^{-1}>e^{-1/100}>1/K.$$ \(1) We first consider the case $p=1$. If $n-i$ is odd, by , and , if $z\in \overline{A}_i$ we have $$\label{bound-f-n-0} |f(z)|\leq v\cdot (7K/5)\cdot 1.01<2Kv<s.$$ If $n-i$ is even, by , and , for $z\in \overline{A}_i$ we have $$\label{bound-f-n-1} |f(z)|\geq (v/u)\cdot (7K/5)^{-1}\cdot 0.99>v/(2Ku)>K.$$ If $n$ is odd, by Lemma \[esti-a1\](3a), and , for every $z$ such that $|z|\leq s$, we have $$|f(z)| =|z|^{d_1} \prod_{i=1}^{n-1}|a_i|^{D_i(-1)^{n-i-1}} \prod_{i=1}^{n-1}\left|1-\frac{z^{D_i}}{a_i^{D_i}}\right|^{(-1)^{n-i-1}} < |s/a_1|^{d_1}vu^{-1}\cdot1.02<s.$$ It follows that $f(\overline{\mathbb{D}}_{s})\subset\mathbb{D}_{s}$ for odd $n$. If $n$ is even and $|z|\leq s$, by Lemma \[esti-a1\](3b), and , we have $$|f(z)|=|a_1/z|^{d_1}v\,\prod_{i=1}^{n-1}\left|1-\frac{z^{D_i}}{a_i^{D_i}}\right|^{(-1)^{n-i-1}} > |a_1/s|^{d_1}v/1.02>K.$$ Therefore $f(\overline{\mathbb{D}}_{s})\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$ for even $n$. Note that $f$ is very ‘close’ to $z\mapsto z^{d_n}$ in the outside of $\mathbb{D}_{K}$ since $|a_i|^{D_i}$ is extremely small, where $1\leq i\leq n-1$. This means that $f$ may exhibit some dynamics of $z\mapsto z^{d_n}$ if $|z|\geq K$. More specifically, by arguments completely similar to those for –, if $|z|\geq K$, then $$\label{bound-lower-out-disk} |f(z)|\geq |z|^{d_n} \prod_{i=1}^{n-1}\left(1+2\frac{|a_i|^{D_i}}{|z|^{D_i}}\right)^{-1}>K.$$ This means that $f(\overline{\mathbb{C}}\setminus\mathbb{D}_{K})\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$. Then we have $f^{-1}(\overline{\mathbb{A}}_{s,K})\subset \mathbb{A}_{s,K}$ for every $n\geq 2$ (see Figure \[Fig\_cantor-gene\]). \(2) Now we consider the case $p=0$. If $n-i$ is even, by , , and Lemma \[esti-a1\](4a), if $z\in \overline{A}_i$ we have $$\label{bound-f-n-0-lp} |f(z)|\leq v^{-1}u\cdot (7K/5)\cdot 1.01<2Ku/v<s.$$ If $n-i$ is odd, by , , and Lemma \[esti-a1\](4a), for $z\in \overline{A}_i$ we have $$\label{bound-f-n-1-lp} |f(z)|\geq v^{-1}\cdot (7K/5)^{-1}\cdot 0.99>1/(2Kv)>M,$$ where $M=(2/s)^{1/d_n}$. If $n$ is even, by Lemma \[esti-a1\](4b), and , for each $z$ such that $|z|\leq s$, we have $$|f(z)| =|z|^{d_1} \prod_{i=1}^{n-1}|a_i|^{D_i(-1)^{n-i}} \prod_{i=1}^{n-1}\left|1-\frac{z^{D_i}}{a_i^{D_i}}\right|^{(-1)^{n-i}} < |s/a_1|^{d_1}v^{-1}\cdot e^{1/100}<s.$$ It follows that $f(\overline{\mathbb{D}}_{s})\subset\mathbb{D}_{s}$ for even $n$. If $n$ is odd and $|z|\leq s$, by Lemma \[esti-a1\](4c), and , we have $$|f(z)|=|a_1/z|^{d_1}uv^{-1}\,\prod_{i=1}^{n-1}\left|1-\frac{z^{D_i}}{a_i^{D_i}}\right|^{(-1)^{n-i}}\geq |a_1/s|^{d_1}uv^{-1}\cdot e^{-1/100} > M.$$ Therefore $f(\overline{\mathbb{D}}_{s})\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{M}$ for odd $n$. If $|z|\geq M$, then $$\label{bound-lower-out-disk-lp} |f(z)| =|z|^{-d_n} \prod_{i=1}^{n-1}\left|1-\frac{a_i^{D_i}}{z^{D_i}}\right|^{(-1)^{n-i}} \leq M^{-d_n} \prod_{i=1}^{n-1}\left(1+\frac{2 |a_i|^{D_i}}{|z|^{D_i}}\right)<2M^{-d_n}=s.$$ This means that $f(\overline{\mathbb{C}}\setminus\mathbb{D}_M)\subset\mathbb{D}_{s}$. Then we have $f^{-1}(\overline{\mathbb{A}}_{s,M})\subset \mathbb{A}_{s,M}$ for every $n\geq 2$. ![Sketch illustrating of the mapping relation of $f_{1,d_1,\cdots,d_n}$, where $n$ is odd and even respectively (from left to right). The small stars denote the critical points and critical values, and the numbers shown at the bottom of the Figures denote the approximate coordinates.[]{data-label="Fig_cantor-gene"}](cantor-gene.pdf){width="130mm"} \[parameter-restate\] If $|a_{n-1}|=(s_1 K^{-2})^{1/d_n}$ and $|a_i|=(s_1 K^{-5})^{1/d_{i+1}}|a_{i+1}|$ for $1\leq i\leq n-2$, where $s_1>0$ is small enough, then the Julia set of $f_{1,d_1,\cdots,d_n}$ is a Cantor set of circles. If $|a_{n-1}|=(s_0^{1/d_n+(1-\xi)/3})^{1/d_n}$ and $|a_i|=(s_0^{1+1/d_n+2(1-\xi)/3})^{1/d_{i+1}}|a_{i+1}|$ for $1\leq i\leq n-2$, where $s_0>0$ is small enough, then the Julia set of $f_{0,d_1,\cdots,d_n}$ is a Cantor set of circles. We only focus on the case $p=1$ since the similar proof can be used to the case $p=0$ by using Lemma \[nice-condition\](2). We also use $f$ to denote $f_{1,d_1,\cdots,d_n}$ for simplicity. Let $U_i$ be the component of $f^{-1}(D)$ containing $a_i$, where $D=\mathbb{D}_s$ if $n-i$ is odd and $D=\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$ if $n-i$ is even. By Lemma \[nice-condition\](1), it follows that the set of critical points $CP_i\subset U_i$ and $U_i$ is a connected domain containing the annulus $A_i$. Moreover, $U_i\cap U_{i+1}=\emptyset$ since $f(U_i)\cap f(U_{i+1})=\emptyset$ by Lemma \[nice-condition\](1), where $1\leq i<n-2$. This means that $U_i\cap U_j=\emptyset$ for different $i,j$. Suppose that $U_i$ has $m_i$ boundary components. Since there are exactly $D_i$ critical points in $U_i$ and $f:U_i\rightarrow D$ is a branched covering with degree $D_i$, then the Riemann-Hurwitz formula tells us $\chi_{U_i}=2-m_i=D_i\chi_{D}-D_i=0$, where $\chi$ denotes the Euler characteristic. This means that $m_i=2$ and therefore $U_i$ is an annulus surrounding the origin for every $1\leq i\leq n-1$. For $1\leq i\leq n-2$, Let $V_{i+1}$ be the annular domain between $U_i$ and $U_{i+1}$. It is easy to see $f:V_{i+1}\rightarrow \mathbb{A}_{s,{K}}$ is a covering map with degree $d_{i+1}$. Note that every component of $f^{-1}(\mathbb{A}_{s,{K}})$ is an annulus since $\mathbb{A}_{s,{K}}$ is double connected and contains no critical values. It follows that there exist two annuli $V_1$ and $V_n$, which lie between $0$ and $U_1$, $U_{n-1}$ and $\infty$ respectively, such that $f:V_1,V_n\rightarrow \mathbb{A}_{s,{K}}$ are covering maps with degree $d_1$ and $d_n$ respectively. In fact, the restriction of $f$ on $\partial U_1$ and $\partial U_{n-1}$ has degree $d_1$ and $d_n$ respectively and there are no critical points in $V_1$ and $V_n$ (see Figure \[Fig\_cantor-gene\]). The Julia set of $f$ is $J=\bigcap_{k\geq 0}f^{-k}(\mathbb{A}_{s,{K}})$. By the construction, the components of $J$ are compact sets nested between $0$ and $\infty$ since each inverse branch $f^{-1}:\mathbb{A}_{s,{K}}\rightarrow V_j$ is conformal for every $0\leq j\leq n$. Since the component of $J$ cannot be a point and $f$ is hyperbolic, every component of $J$ is a Jordan curve (actually quasicircle) by Theorem 1.2 in [@PT]. The dynamics on the set of Julia components of $f$ is isomorphic to the one-sided shift on $n$ symbols $\Sigma_{n}:=\{0,1,\cdots,n-1\}^{\mathbb{N}}$. In particular, $J$ is homeomorphic to $\Sigma_{n}\times\mathbb{S}^1$, which is a Cantor set of circles as desired. This ends the proof of Theorem \[parameter-restate\] and hence Theorem \[parameter\]. \[range-unif\] Since $f$ is hyperbolic, the Julia set of $f$ is also a Cantor set of circles if we perturb some $a_i$ gently, where $1\leq i\leq n-1$. In the first version of our manuscript of this paper, only $d_i=n+1$ for every $1\leq i\leq n$ was considered. In this case, it was shown that for every $n\geq 2$ and $1\leq i\leq n-1$, if $|a_{n-i}|=(\frac{n}{n+1})^{i-1}s^i$ for $0<s\leq 1/10$, then the Julia set of $f_{1,n+1,\cdots,n+1}$ is a Cantor set of circles. \[no-topo-equiv\] Suppose that $a_i$ is chosen as in Theorem \[parameter\] such that the Julia set of $f_{p,d_1,\cdots,d_n}$ is a Cantor set of circles for $n\geq 3$, then $f_{p,d_1,\cdots,d_n}$ is not topologically conjugate to any McMullen maps on their corresponding Julia sets. Since the dynamics on the set of Julia components of $f_{p,d_1,\cdots,d_n}$ is conjugate to the one-sided shift on $n$ symbols $\Sigma_n:=\{0,1,\cdots,n-1\}^{\mathbb{N}}$ and, in particular, the set of Julia components of $g_\eta$ is isomorphic to the one-sided shift on only two symbols $\Sigma_{2}:=\{0,1\}^{\mathbb{N}}$, this means that $f_{p,d_1,\cdots,d_n}$ cannot be topologically conjugate to $g_\eta$ on their corresponding Julia sets if $n\geq 3$. Topological conjugacy between the Cantor circles Julia sets {#sec-topo-conj} =========================================================== In this section, we show that for any given rational map whose Julia set is a Cantor set of circles, there exists a map $f_{p,d_1,\cdots,d_n}$ in such that these two rational maps are topologically conjugate on their corresponding Julia sets. \[no-crit-on-J\] If $f$ is a rational map whose Julia set is a Cantor set of circles. Then there exist no critical points in $J(f)$. Suppose there exists a Julia component $J_0$ of $f$ containing a critical point $c_0$ of $f$ with multiplicity $d$. Then $f$ is not one to one in any small neighborhood of $c_0$. It is known $f(J_0)$ is a Julia component containing $f(c_0)$ [@Be Lemma 5.7.2]. Choose a small topological disk neighborhood $U$ of $f(c_0)$ such that $U\cap f(J_0)$ is a simple curve. The component of $f^{-1}(U)$ containing $c_0$ is mapped onto $U$ in the manner of $d+1$ to one. Note that the component $J'$ of $f^{-1}(U\cap f(J_0))$ containing $c_0$ is connected and contained in $J_0$. However, $J'$ possesses star-like structure and hence is not a simple curve. This contradicts to the assumption that $J_0$ is a Jordan closed curve since $J(f)$ is a Cantor set of circles. We say that a compact set $X\subset \overline{\mathbb{C}}$ *separates* $0$ and $\infty$ if $0$ and $\infty$ lie in the two different components of $\overline{\mathbb{C}}\setminus X$ respectively. Let $X$ and $Y$ be two disjoint compact sets that both separate $0$ and $\infty$ respectively. We say $X\prec Y$ if $X$ is contained in the component of $\overline{\mathbb{C}}\setminus Y$ which contains $0$. Let $A$ be an annulus whose closure separates $0$ and $\infty$, we use $\partial_-A$ and $\partial_+A$ to denote the two components of the boundary of $A$ such that $\partial_-A\prec \partial_+A$. \[this-is-all-resta\] Let $f$ be a rational map whose Julia set is a Cantor set of circles. Then there exist $p\in\{0,1\}$, positive integers $n\geq 2$ and $d_1,\cdots,d_n$ satisfying $\sum_{i=1}^{n}(1/d_i)<1$ such that $f$ is topologically conjugate to $f_{p,d_1,\cdots,d_n}$ on their corresponding Julia sets. Let $J(f)$ be the Julia set of $f$ which is a Cantor set of circles, then every periodic Fatou component of $f$ must be attracting or parabolic by Lemma \[no-crit-on-J\]. We only prove the attracting (hyperbolic) case in detail and explain the parabolic case by using the work of Cui [@Cui]. In the following, we suppose that $f$ is hyperbolic. There exist exactly two simply connected Fatou components of $f$ and all other Fatou components are annuli. Let $\mathcal{D}$ and $\mathcal{A}$ be the collection of simply and doubly connected Fatou components of $f$ respectively. We claim that $f(\mathcal{D})\subset\mathcal{D}$ and there exists an integer $k\geq 1$ such that $f^{\circ k}(A)\in\mathcal{D}$ for every $A\in \mathcal{A}$. The assertion $f(\mathcal{D})\subset\mathcal{D}$ is obvious since the image of a simply connected Fatou component under a rational map is again simply connected. If $f(A_1)=A_2$, where $A_1,A_2\in\mathcal{A}$, then there exists no critical points in $A_1$ by Riemann-Hurwitz’s formula. This means that each $A\in\mathcal{A}$ cannot be periodic since the cycle of every periodic attracting Fatou component must contain at least one critical point. On the other hand, by Sullivan’s theorem, the Fatou components of a rational map cannot be wandering. This completes the proof of claim. Up to a Mobius transformation, we can assume that $0$ and $\infty$, respectively, are belong to the two simply connected Fatou components of $f$, which are denoted by $D_0$ and $D_\infty$. Namely, $\mathcal{D}=\{D_0,D_\infty\}$. Since $f(\mathcal{D})\subset\mathcal{D}$, we first suppose that $f(D_0)=D_0$ and $f(D_\infty)=D_\infty$. Let $f^{-1}(D_0)=D_0\cup A_1\cup\cdots\cup A_m$, where $A_1,\cdots, A_m$ are $m$ annuli separating $0$ and $\infty$ such that $A_{i}\prec A_{i+1}$ for every $1\leq i\leq m-1$. It is easy to see $m\geq 1$. Otherwise, $D_0$ is completely invariant, then $J(f)=\partial D_0$ which contradicts to the assumption that $J(f)$ is a Cantor set of circles. Suppose that $\deg (f|_{D_0}:D_0\rightarrow D_0)=d_1$ and $\deg (f|_{\partial_-A_i}:\partial_-A_i\rightarrow \partial D_0)=d_{2i}$ and $\deg (f|_{\partial_+A_i}:\partial_+A_i\rightarrow \partial D_0)=d_{2i+1}$ for $1\leq i\leq m$. It follows that $\deg(f)=\sum_{j=1}^{2m+1}d_j$. Let $W_1$ be the annular domain between $D_0$ and $A_1$ and $W_i$ be the annular domain between $A_{i-1}$ and $A_i$, where $2\leq i\leq m$. We have $f(W_i)=\overline{\mathbb{C}}\setminus \overline{D}_0$ and $\deg(f|_{W_i}:W_i\rightarrow\overline{\mathbb{C}}\setminus \overline{D}_0)=d_{2i-1}+d_{2i}$. This means that there exists at least one Fatou component $B_i\subsetneq W_i$ such that $f(B_i)=D_\infty$. If there exists $B_i'\neq B_i$ such that $B_i'\subsetneq W_i$ and $f(B_i')=D_\infty$, there must exist one component of $f^{-1}(D_0)$ in $W_i$, which contradicts the assumption that $A_1\cup\cdots\cup A_m$ is the collection of all annular components of $f^{-1}(D_0)$. So there exists exactly one Fatou component $B_i\subsetneq W_i$ such that $f(B_i)=D_\infty$ and $\deg(f|_{B_i}:B_i\rightarrow D_\infty)=d_{2i-1}+d_{2i}$. Similar argument can be used to show that $D_\infty$ is the only component of $f^{-1}(D_\infty)$ lying in the unbounded component of $\overline{\mathbb{C}}\setminus A_m$ which can be mapped onto $D_\infty$. Therefore, $f^{-1}(D_\infty)=B_1\cup\cdots\cup B_m\cup D_\infty$ and $\deg(f|_{D_\infty})=d_{2m+1}$ since $\deg(f)=\sum_{j=1}^{2m+1}d_j$. Denote $\overline{\mathbb{C}}\setminus ({D_0\cup D_\infty})$ by $E$. The preimage $f^{-1}(E)$ consists of $2m+1$ annuli components $E_1,\cdots,E_{2m+1}$ such that $E_i\prec E_{i+1}$ for $1\leq i\leq 2m$. The map $f:E_i\rightarrow E$ is a unramified covering map with degree $d_i$, where $1\leq i\leq 2m+1$ (see Figure \[Fig\_find-conj\]). ![Sketch illustrating of the mapping relation of $f$, where $d_i$, $1\leq i\leq 2m+1$ denote the degrees of the restriction of $f$ on the boundaries of Fatou components.[]{data-label="Fig_find-conj"}](find-conj.pdf){width="120mm"} Let $n=2m+1$ and $p=1$. The assertion $\sum_{i=1}^{n}{1}/{d_i}<1$ follows from Grótzsch’s modulus inequality since each $E_i$ is essentially contained in $E$ and $\text{mod} (E_i)=\text{mod} (E)/d_i$. In the following, we will construct a quasiconformal map $\phi:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ which conjugates the dynamics on the Julia set of $f$ to that of $f_{1,d_1,\cdots,d_n}$. For simplicity, we denote $f_{1,d_1,\cdots,d_n}$ by $F$. Note that $F(0)=0$ and $F(\infty)=\infty$. There exist two simply connected Fatou components $D_0'$ and $D_\infty'$, both are invariant under $F$ such that $0\in D_0'$ and $\infty\in D_\infty'$. From the proof of Theorem \[parameter\], we know that $F^{-1}(D_0')=D_0'\cup A_1'\cup\cdots\cup A_m'$, where $A_1',\cdots, A_m'$ are $m$ annuli separating $0$ and $\infty$ such that $A_{i}'\prec A_{i+1}'$ for every $1\leq i\leq m-1$. Moreover, $\deg (F|_{D_0'}:D_0'\rightarrow D_0')=d_1$ and $\deg (F|_{\partial_-A_i'}:\partial_-A_i'\rightarrow \partial D_0')=d_{2i}$ and $\deg (F|_{\partial_+A_i'}:\partial_+A_i'\rightarrow \partial D_0')=d_{2i+1}$ for $1\leq i\leq m$. Let $W_1'$ be the annular domain between $D_0'$ and $A_1'$ and $W_i'$ be the annular domain between $A_{i-1}'$ and $A_i'$, where $2\leq i\leq m$. There exists exactly one Fatou component $B_i'\subsetneq W_i'$ such that $F(B_i')=D_\infty'$ and $\deg(F|_{B_i'}:B_i'\rightarrow D_\infty')=d_{2i-1}+d_{2i}$. We have $F^{-1}(D_\infty')=B_1'\cup\cdots\cup B_m'\cup D_\infty'$ and $\deg(F|_{D_\infty'})=d_{2m+1}$. Similarly, let $E':=\overline{\mathbb{C}}\setminus ({D_0'\cup D_\infty'})$. There exist $2m+1$ annular components $E_1',\cdots,E_{2m+1}'$ of $F^{-1}(E')$ such that $E_i'\prec E_{i+1}'$ for $1\leq i\leq 2m$. The map $F:E_i'\rightarrow E'$ is a covering with degree $d_i$, where $1\leq i\leq 2m+1$. By a quasiconformal surgery, it can be seen that $\partial D_0,\partial D_\infty,\partial D_0',\partial D_\infty'$ and their preimages are all quasicircles and the dilatation is bounded by a fixed constant. There exists a quasiconformal mapping $\phi_0:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ such that $\phi_0(D_0)=D_0'$ and $\phi_0(D_\infty)=D_\infty'$ hence $\phi_0(\partial D_0)=\partial D_0'$ and $\phi_0(\partial D_\infty)=\partial D_\infty'$. Moreover, $\phi_0$ can be chosen such that $\phi_0\circ f=F\circ\phi_0$ on $\partial D_0\cup \partial D_\infty$. Now we construct a lift $\phi_{E_1}:E_1\rightarrow E_1'$ of $\phi_0:E\rightarrow E'$ as follows. For every $z\in E_1\setminus\partial_- E_1$, we choose a simple curve $\gamma:[0,1]\rightarrow E$ such that $\gamma(1)=f(z)$ and $\gamma(0)=w\in\partial_- E$. Since $f:E_1\rightarrow E$ is a covering map, there exists a unique lift $\widetilde{\gamma}:[0,1]\rightarrow E_1$ of $\gamma$ such that $\widetilde{\gamma}(1)=z$ and $\widetilde{w}:=\widetilde{\gamma}(0)\in\partial_- E_1$. Similarly, since $F:E_1'\rightarrow E'$ is a covering map, there exists a unique lift $\alpha:[0,1]\rightarrow E_1'$ of $\phi_0(\gamma):[0,1]\rightarrow E'$ such that $\alpha(0)=\phi_0(\widetilde{w})$ since $\phi_0\circ f=F\circ\phi_0$ on $\partial D_0=\partial_- E_1$. Define $\phi_{E_1}(z):=\alpha(1)$. We know that $\phi_0\circ f =F\circ \phi_{E_1}$ on $E_1$ and $\phi_{E_1}:E_1\rightarrow E_1'$ is quasiconformal since $f,F$ are both holomorphic covering maps with degree $d_1$ and $\phi_0:E\rightarrow E'$ is quasiconformal. Now some parts of $\phi_1:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ are defined as follows: $\phi_1|_{\overline{D}_0}=\phi_0|_{\overline{D}_0}$, $\phi_1|_{\overline{D}_\infty}=\phi_0|_{\overline{D}_\infty}$ and $\phi_1|_{E_1}=\phi_{E_1}$. Then, $\phi_1\circ f =F\circ \phi_1$ on $\partial E_1$. Similarly, there exists a unique quasiconformal mapping $\phi_{E_{2m+1}}:E_{2m+1}\rightarrow E_{2m+1}'$, which is the lift of $\phi_0:E\rightarrow E'$ such that $\phi_0\circ f =F\circ \phi_{E_{2m+1}}$ on $E_{2m+1}$. Define $\phi_1|_{E_{2m+1}}=\phi_{E_{2m+1}}$. Then, $\phi_1\circ f =F\circ \phi_1$ on $\partial E_{2m+1}$. Unlike the cases of $E_1$ and $E_{2m+1}$, the lift $\phi_{E_i}:E_i\rightarrow E_i'$ of $\phi_0:E\rightarrow E'$ exists but is not unique for $2\leq i\leq 2m$. We first show the existence of $\phi_{E_i}$. Without loss of generality, suppose that $i$ is even. Since $f:\partial_- E_i\rightarrow\partial D_\infty$ and $F:\partial_- E_i'\rightarrow\partial D_\infty'$ are both covering mappings with degree $d_i$, there exists a lift (not unique) $\phi_{E_i}:\partial_- E_i\rightarrow \partial_- E_i'$ of $\phi_0:\partial D_\infty\rightarrow \partial D_\infty'$ such that $\phi_0\circ f =F\circ \phi_{E_i}$ on $\partial_-E_i$. By using the same method of defining $\phi_{E_1}$, there exists a unique lift of $\phi_0:E\rightarrow E'$ defined from $E_i$ to $E_i'$, which we denote also by $\phi_{E_i}$ such that $\phi_0\circ f =F\circ \phi_{E_i}$ on $E_i$. Note that $\phi_{E_i}:E_i\rightarrow E_i'$ is quasiconformal. Define $\phi_1|_{E_{i}}=\phi_{E_i}$. Then, $\phi_0\circ f =F\circ \phi_1$ on $\bigcup_{i=1}^{2m+1}E_i$ and $\phi_1\circ f =F\circ \phi_1$ on $\bigcup_{i=1}^{2m+1}\partial E_i$. In order to unify the notations, let $D_{2i-1}:=B_i$ and $D_{2i}:=A_i$ for $1\leq i\leq m$. Then we have $D_i\prec D_{j}$ for $1\leq i<j\leq 2m$. We need to define $\phi_1$ on $\bigcup_{i=1}^{2m}D_i$. For every $D_i$, where $1\leq i\leq 2m$, its two boundary components $\partial_+ E_i$ and $\partial_- E_{i+1}$ are both quasicircles. Since $\phi_{E_{i}}$ and $\phi_{E_{i+1}}$ are both quasiconformal mappings, the map $\phi_1|_{\partial_+ E_{i}\cup\partial_- E_{i+1}}$ has a quasiconformal extension $\phi_{D_i}:\overline{D}_i\rightarrow \overline{D}_i'$ such that $\phi_{D_i}(D_i)=D_i'$. Now we obtain a quasiconformal mapping $\phi_1:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ defined as $\phi_1|_{E_i}:=\phi_{E_i}$, $\phi_1|_{D_j}=\phi_{D_j}$ and $\phi_1|_{D_0\cup D_\infty}=\phi_0$, where $1\leq i\leq 2m+1$ and $1\leq j\leq 2m$. Next, we define $\phi_2$. First, let $\phi_2|_{D_j}=\phi_1$ for $j\in\{0,1,\cdots,2m,\infty\}$. Then we lift $\phi_1:E\rightarrow E'$ in an appropriate way to obtain $\phi_2:E_i\rightarrow E_i'$ for $1\leq i\leq 2m+1$. Finally, we check the continuity of the resulting map $\phi_2:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$. Now let us make this precise. In order to guarantee the continuity of $\phi_2$ on $D_0\cup E_1$, we need to have $\phi_2|_{\partial_-E_1}=\phi_1$. Then there exists only one way to lift $\phi_1:E\rightarrow E'$ to obtain $\phi_2:E_1\rightarrow E_1'$. In order to guarantee the continuity of the lift $\phi_2$, we need to check the continuity of $\phi_2$ on the boundary $\partial_+E_1$ first. In fact, $\phi_0|_E$ and $\phi_1|_E$ are homotopic to each other and $\phi_1|_{\partial E}=\phi_0|_{\partial E}$, it follows that $\phi_2|_{\partial_+ E_1}=\phi_1|_{\partial_+ E_1}$ since $\phi_2|_{\partial_- E_1}=\phi_1|_{\partial_- E_1}$. This means that $\phi_2$ is continuous on $\partial_+ E_1$. Similarly, we can lift $\phi_1:E\rightarrow E'$ to obtain $\phi_2:E_i\rightarrow E_i'$ for $2\leq i\leq 2m+1$ and guarantee the continuity of $\phi_2$. Above all, the map $\phi_2:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ satisfies (1) $\phi_2$ is quasiconformal and the dilatation $K(\phi_2)=K(\phi_1)$; (2) $\phi_2|_{f^{-1}(D_0\cup D_\infty)}=\phi_1$; (3) $\phi_1\circ f=F\circ\phi_2$ on $\bigcup_{i=1}^{2m+1}E_i$ and hence $\phi_2\circ f=F\circ\phi_2$ on $f^{-2}(\partial D_0\cup \partial D_\infty)$. Suppose we have obtained $\phi_k$ for some $k\geq 1$, then $\phi_{k+1}$ can be defined completely similarly to the process of the derivation of $\phi_2$ from $\phi_1$. Inductively, we can obtain a sequence of quasiconformal mappings $\{\phi_k\}_{k\geq 0}$ such that (1) $K(\phi_k)=K(\phi_1)\geq K(\phi_0)$ for $k\geq 1$; (2) $\phi_{k+1}(z)=\phi_{k}(z)$ for $z\in f^{-k}(D_0\cup D_\infty)$; (3) $\phi_k\circ f=F\circ\phi_k$ on $f^{-k}(\partial D_0\cup \partial D_\infty)$. This means that $\{\phi_k\}_{k\geq 0}$ forms a normal family. Take a convergent subsequence of $\{\phi_k\}_{k\geq 0}$ whose limit we denote by $\phi_\infty$, then $\phi_\infty$ is a quasiconformal mapping satisfying $\phi_\infty\circ f=F\circ\phi_\infty$ on $\bigcup_{k\geq 0} f^{-k}(\partial D_0\cup \partial D_\infty)$. Moreover, $K(\phi_\infty)\leq K(\phi_1)$. Since $\phi_\infty$ is continuous, $\phi_\infty\circ f=F\circ\phi_\infty$ holds on the closure of $\bigcup_{k\geq 0}f^{-k}(\partial D_0\cup \partial D_\infty)$, which is the Julia set of $f$. Therefore $\phi=\phi_\infty$ is the quasiconformal mapping we want to find which conjugates $f$ to $F$ on their corresponding Julia sets. This ends the proof of case $f(D_0)=D_0$ and $f(D_\infty)=D_\infty$. The other three cases: (1) $f(D_0)=D_\infty$, $f(D_\infty)=D_\infty$; (2) $f(D_0)=D_\infty$, $f(D_\infty)=D_0$; and (3) $f(D_0)=D_0$, $f(D_\infty)=D_0$ can be proved completely similarly. If one or both of the components $D_0$ and $D_\infty$ are parabolic, there exists a perturbation $f_\varepsilon$ of $f$ such that $f_\varepsilon$ is hyperbolic and the dynamics of $f_\varepsilon$ are topologically conjugate to that of $f$ on their corresponding Julia sets [@Cui]. Then $f$ has a ‘model’ in since $f_\varepsilon$ always does. This ends the proof of Theorem \[this-is-all-resta\] and hence Theorem \[this-is-all\]. From the proof of Theorem \[this-is-all-resta\] in the hyperbolic case, we have following immediate corollary. \[Julia-comp\] If the parameters $a_i$ are chosen as in Theorem \[parameter\], where $1\leq i\leq n-1$, then each Julia component of $f_{p,d_1,\cdots,d_n}$ is a quasicircle. Non-hyperbolic rational maps whose Julia sets are Cantor circles {#sec-para-mcm} ================================================================ The rational maps $$P_\lambda(z)=\frac{\frac{1}{n}((1+z)^n-1)+\lambda^{m+n}z^{m+n}}{1-\lambda^{m+n}z^{m+n}}$$ where $\lambda\in\mathbb{C}^*=\mathbb{C}\setminus\{0\}$ and $m,n\geq 2$ are both positive integers satisfying $1/m+1/n<1$ can be seen as a perturbation of the parabolic polynomial $$\widetilde{P}(z)=\frac{(1+z)^n-1}{n}.$$ Note that $\widetilde{P}$ has a parabolic fixed point at the origin with multiplier 1 and critical point $-1$ with multiplicity $n-1$. This means that there exists only one bounded and hence simply connected Fatou component of $\widetilde{P}$ in which all points are attracted to the origin. In particular, the Julia set of $\widetilde{P}$ is a Jordan curve with infinitely many cusps. We hope that some properties of $\widetilde{P}$ stated above can be also hold for $P_\lambda$ when $\lambda$ is small. But obviously, there are lots of differences between $P_\lambda$ and $\widetilde{P}$. The degree of $P_\lambda$ is $m+n$ and $P_\lambda(\infty)=-1$. There are $2(m+n)-2$ critical points of $P_\lambda$: $m-1$ at $\infty$, $n-1$ are very close to $-1$ and the remaining $m+n$ critical points lie nearby the circle $\mathbb{T}_{r_0/|\lambda|}$, where $r_0=\sqrt[m+n]{n/m}$ (see Lemma \[crit-close-para\]). In fact, we will see that $P_\lambda$ can be viewed as a ‘parabolic’ McMullen map at the end of this section since $P_\lambda$ is conjugate to some $g_\eta$ on their corresponding Julia sets. Firstly, we show that the fixed parabolic Fatou component of $\widetilde{P}$ contains the Euclidean disk $\mathbb{D}(-\frac{3}{4},\frac{3}{4})$ for every $n\geq 2$ and $P_\lambda$ maps $\mathbb{D}(-\frac{3}{4},\frac{3}{4})$ into itself if $\lambda$ is small enough. \[key-lemma\] $(1)$ For every $n\geq 2$, $\widetilde{P}(\overline{\mathbb{D}}(-\frac{3}{4},\frac{3}{4}))\subset\mathbb{D}(-\frac{3}{4},\frac{3}{4})\cup\{0\}$. $(2)$ If $0<|\lambda|<{1}/{(3n)}$, then $P_\lambda(\overline{\mathbb{D}}(-\frac{3}{4},\frac{3}{4}))\subset\mathbb{D}(-\frac{3}{4},\frac{3}{4})\cup\{0\}$. In particular, $\mathbb{D}(-\frac{3}{4},\frac{3}{4})$ lies in the parabolic Fatou component of $P_\lambda$ with parabolic fixed point $0$. If $z\in\overline{\mathbb{D}}(-\frac{3}{4},\frac{3}{4})$, then $|\widetilde{P}(z)+1/n|=|1+z|^n/n\leq 1/n$. In particular, the inequality sign can be replaced by equality if and only if $z=0$. This ends the proof of (1). The proof of (2) will be divided into two cases: $|z|$ is small and not too small. For every $z=-\frac{3}{4}+\frac{3}{4}e^{i\theta}\in\partial\mathbb{D}(-\frac{3}{4},\frac{3}{4})$, where $-\pi< \theta\leq\pi$, we have $|1+\widetilde{P}(z)|\leq 5/2$ by (1) and $|\lambda z|^{m+n}<1/2$ since $|\lambda|<1/(3n)$. This means that $$|P_\lambda(z)-\widetilde{P}(z)|=\left|\frac{\lambda^{m+n}z^{m+n}(1+\widetilde{P}(z))}{1-\lambda^{m+n}z^{m+n}}\right|\leq 5|\lambda z|^{m+n}.$$ Since $|z|=\frac{3}{4}|1-e^{i\theta}|=\frac{3}{4}|e^{-i\theta/2}-e^{i\theta/2}|=\frac{3}{2}|\sin\frac{\theta}{2}|\leq \frac{3}{4}|\theta|$ and $|\lambda|<1/(3n)$, we have $$\label{P-lambd-est} |P_\lambda(z)-\widetilde{P}(z)|\leq 5\,({|\theta|}/{(4n)})^{m+n}.$$ On the other hand, since $|\sin\theta|\geq \frac{2}{\pi}|\theta|$ if $|\theta|\leq\frac{\pi}{2}$, we have $$\label{P-est-0} \begin{split} |\widetilde{P}(z)+{3}/{4}| = &~ \left| \frac{(\frac{1}{4}+\frac{3}{4}e^{i\theta})^n-1}{n}+\frac{3}{4}\right| \leq \frac{\left|\frac{1}{4}+\frac{3}{4}e^{i\theta}\right|^n-1}{n}+\frac{3}{4} \\ = &~ \frac{(1-\frac{3}{4}\sin^2\frac{\theta}{2})^{n/2}-1}{n}+\frac{3}{4} \leq \frac{(1-\frac{3\theta^2}{4\pi^2})^{n/2}-1}{n}+\frac{3}{4}. \end{split}$$ If $|\theta|<2\pi/n$, then $\frac{3\theta^2}{4\pi^2}<\frac{2}{n}$. By Lemma \[very-useful-est\](3), we have $$\label{P-est} |\widetilde{P}(z)+{3}/{4}|\leq -\frac{\frac{n}{2}\cdot\frac{3\theta^2}{4\pi^2}}{3n}+\frac{3}{4}=\frac{3}{4}-\frac{\theta^2}{8\pi^2}.$$ Therefore, combining and , it follows that if $|\theta|<2\pi/n$, then $$|P_\lambda(z)+3/4| \leq |\widetilde{P}(z)+{3}/{4}|+|P_\lambda(z)-\widetilde{P}(z)|\leq \frac{3}{4}-\frac{\theta^2}{8\pi^2}+5\,(\frac{|\theta|}{4n})^{m+n}\leq 3/4.$$ If $2\pi/n\leq |\theta|\leq \pi$, from and , we know that $$\label{P-est-2} |\widetilde{P}(z)+{3}/{4}|\leq \frac{3}{4}-\frac{1}{2 n^2}.$$ From and , it follows that if $2\pi/n\leq |\theta|\leq \pi$, then $$|P_\lambda(z)+3/4| \leq \frac{3}{4}-\frac{1}{2 n^2}+5\,(\frac{|\theta|}{4n})^{m+n}< 3/4.$$ Therefore, we have shown that $|P_\lambda(z)+\frac{3}{4}|\leq \frac{3}{4}$ for every $z\in\partial\mathbb{D}(-\frac{3}{4},\frac{3}{4})$ and $|P_\lambda(z)+\frac{3}{4}|= \frac{3}{4}$ if and only if $z=0$. The proof is complete. As in the procedure in §2, now we locate the free critical points of $P_\lambda$. By a direct calculation, the bounded $m+2n-1$ critical points of $P_\lambda$ are the solutions of $$\label{crit-P-lamb} (1+z)^{n-1}+\lambda^{m+n}z^{m+n-1}\{(1+m/n)[(1+z)^n+n-1]-z(1+z)^{n-1}\}=0.$$ \[nice-cond-para\] If $0<|\lambda|<{1}/{(3n)}$, then there are $n-1$ critical points of $P_\lambda$ in $\mathbb{D}(-1,|\lambda|)\subsetneq \mathbb{D}(-\frac{3}{4},\frac{3}{4})$. If $|z+1|\leq |\lambda|<\frac{1}{3n}$, then $|z|\cdot|1+z|^{n-1}\leq (1+|\lambda|)|\lambda|^{n-1}<1$ and $$(1+m/n)\,|(1+z)^n+n-1|\leq (1+m/n)(|\lambda|^{n}+n-1)<m+n.$$ This means that if $|z+1|\leq |\lambda|$, then $$\begin{split} &~ \left|\lambda^{m+n}z^{m+n-1}\{(1+m/n)[(1+z)^n+n-1]-z(1+z)^{n-1}\}\right|\\ < &~|\lambda|^{n-1}\cdot |\lambda z|^{m-1}|\lambda|^2|z|^n(m+n+1) < |\lambda|^{n-1}\cdot (2n)^{1-m}(9n^2)^{-1}e^{1/3}(m+n+1)\\ < &~|\lambda|^{n-1}\cdot (m+n-1)/(2n)^{m+1}<|\lambda|^{n-1}. \end{split}$$ By Rouché’s Theorem, the proof is completed. Let $\widetilde{CP}:=\{\widetilde{w}_{j}=\frac{r_0}{\lambda} \exp(\pi i\frac{2j-1}{m+n}):1\leq j\leq m+n\}$ be the collection of the zeros of $m\lambda^{m+n}z^{m+n}+n=0$, where $r_0=\sqrt[m+n]{n/m}$. Since $h(x)=x^{1/x},x>0$ has maximal value $e^{1/e}<3/2$ at $x=e$, we have $$2/3<1/\sqrt[m]{m}<r_0<\sqrt[n]{n}<3/2.$$ The following lemma shows that the remaining $m+n$ critical points of $P_\lambda$ are very ‘close’ to $\widetilde{CP}$. \[crit-close-para\] If $0<|\lambda|<{1}/{(2^m n^2)}$, then has a solution $w_j$ such that $|w_j-\widetilde{w}_j|<2(m+n)/m$, where $1\leq j\leq m+n$. Moreover, $w_i= w_j$ if and only if $i=j$. Dividing $(1+z)^{n-1}$ on both sides of , we have $$1+\lambda^{m+n}z^{m+n-1}\left(\frac{m}{n}z+\frac{m+n}{n}\left(1+\frac{n-1}{(1+z)^{n-1}}\right)\right)=0.$$ Or, in more useful form $$\label{use-ful} \frac{n}{m\lambda^{m+n}}+z^{m+n}+\frac{(m+n)z^{m+n-1}}{m}\left(1+\frac{n-1}{(1+z)^{n-1}}\right)=0.$$ Let $\Omega=\{z:|z^{m+n}+\frac{n}{m}\lambda^{-(m+n)}|\leq \beta|\lambda|\cdot\frac{n}{m}|\lambda|^{-(m+n)}\}$, where $\beta=\frac{2(m+n)}{mr_0}<\frac{3(m+n)}{m}$. If $z\in\Omega$, then $|\lambda^{m+n}z^{m+n}+\frac{n}{m}|<\beta|\lambda|\cdot\frac{n}{m}$ and $|z-\widetilde{w}_j|<\beta r_0$ for some $1\leq j\leq 2n$ by Lemma \[very-useful-est\](2). If $z\in\Omega$ and $0<|\lambda|<{1}/{(2^m n^2)}$, we have $$\label{P-estima} \frac{n-1}{|1+z|^{n-1}}<\frac{n-1}{((|\lambda|^{-1}-\beta)r_0-1)^{n-1}} <\frac{n-1}{(2^{m+1}n^2/3-3-2n/m)^{n-1}}<\frac{1}{15}$$ and $$\label{P-estima-1} \beta|\lambda|\leq\frac{2(m+n)}{2^m n^2\cdot mr_0}<\frac{3}{2^m n}\left(\frac{1}{m}+\frac{1}{n}\right)<\frac{1}{4}, \text{~~therefore~~} \frac{1+\beta|\lambda|}{2(1-\beta|\lambda|)}<\frac{5}{6}.$$ Therefore, if $z\in\Omega$ and $0<|\lambda|<{1}/{(2^m n^2)}$, from and , we have $$\begin{split} &~ \left|\frac{(m+n)z^{m+n-1}}{m}\left(1+\frac{n-1}{(1+z)^{n-1}}\right)\right| = \frac{m+n}{m|\lambda|^{m+n}}\left|\frac{\lambda^{m+n}z^{m+n}}{z}\left(1+\frac{n-1}{(1+z)^{n-1}}\right)\right|\\ < &~ \frac{m+n}{m|\lambda|^{m+n}}~\frac{(\beta|\lambda|+1)n/m}{r_0(1/|\lambda|-\beta)}\cdot \frac{16}{15} = \frac{n\beta|\lambda|}{m|\lambda|^{m+n}}\,\frac{1+\beta|\lambda|}{2(1-\beta|\lambda|)}\cdot \frac{16}{15} <\frac{n\beta|\lambda|}{m|\lambda|^{m+n}}. \end{split}$$ Applying Rouché’s Theorem to and then using Lemma \[very-useful-est\](2), the proof of the first assertion is completed. By means of the same argument as , if $0<|\lambda|<{1}/{(2^m n^2)}$, we have $$\frac{(r_0/|\lambda|)\cdot\sin(\pi/(m+n))}{2(m+n)/m}\geq \frac{mr_0}{(m+n)^2|\lambda|}>\frac{2^{m+1}m}{3(m/n+1)^2}>1.$$ This means that $w_i= w_j$ if and only if $i=j$. The proof is complete. Let $CP:=\{w_j:1\leq j\leq m+n\}$ be the $m+n$ critical points of $P_\lambda$ lying near the circle $\mathbb{T}_{r_0/|\lambda|}$ and $CV:=\{P_\lambda(w_j):1\leq j\leq m+n\}$. Let $CP_{-1}$ be the collection of $n-1$ critical points of $P_\lambda$ near $-1$ (see Lemma \[nice-cond-para\]) and $CV_{-1}=\{P_\lambda(z):z\in CP_{-1}\}$. Let $T_0$ be the Fatou component of $P_\lambda$ containing the attracting petal at the origin and $U:=\mathbb{D}(-\frac{3}{4},\frac{3}{4})$. By Lemmas \[key-lemma\](2) and \[nice-cond-para\], we know that $CP_{-1}\cup CV_{-1}\subset U\subset T_0$. Since $P_\lambda(\infty)=-1$, it follows that there exists a neighborhood of $\infty$ such that $P_\lambda$ maps it to a neighborhood of $-1$. Let $T_\infty$ be the Fatou component such that $\infty\in T_\infty$ and $U_0,U_\infty$ be the component of $P_\lambda^{-1}(U)$ such that $0\in\overline{U}_0$ and $\infty\in U_\infty$. Obviously, we have $U\subset U_0\subset T_0$ and $U_\infty\subset T_\infty$. \[Nice-cond-para-McM\] If $0<|\lambda|\leq {1}/{(2^{10m} n^3)}$, there exists an annular neighborhood $A_1$ of $CP$ containing $\mathbb{T}_{1/|\lambda|}\cup CP$ such that $P_\lambda(A_1)\subset \overline{U'}_\infty\subset U_\infty$, where $U'_\infty$ is a neighborhood of $\infty$. It is known from Lemma \[crit-close-para\] that $CP$ is ‘almost’ lying uniformly on the circle $\mathbb{T}_{r_0/|\lambda|}$ and all the finite poles of $P_\lambda$ lie on the circle $\mathbb{T}_{1/|\lambda|}$. Define the annulus $$A_1=\{z:1/(2|\lambda|)<|z|<2/|\lambda|\}.$$ Note that $$\frac{r_0}{|\lambda|}+\frac{2(m+n)}{m}<\frac{3}{2|\lambda|}+2+\frac{2n}{m}<\frac{2}{|\lambda|}$$ and $$\frac{r_0}{|\lambda|}-\frac{2(m+n)}{m}>\frac{2}{3|\lambda|}-2-\frac{2n}{m}>\frac{1}{2|\lambda|}.$$ We have $\mathbb{T}_{1/|\lambda|}\cup CP\subset A_1$ by Lemma \[crit-close-para\]. If $z\in A_1$ and $|\lambda|\leq\frac{1}{2^{10m} n^3}$, then $$\label{import-1} |P_\lambda(z)+1|\geq \frac{(|z|-1)^n}{n(|\lambda z|^{m+n}+1)}\geq\frac{(\frac{1}{2|\lambda|}-1)^n}{n(2^{m+n}+1)}=\frac{(1-2|\lambda|)^n}{2^n n|\lambda|^n(2^{m+n}+1)}>\frac{2}{|\lambda|^{1+\frac{n}{m}}}+1.$$ In fact, $$\frac{(1-2|\lambda|)^n}{2^{m+n}+1}>\frac{(1-\frac{2}{2^{10m}n^3})^n}{2^{m+n}+1}>\frac{0.9}{2^{m+n}+1}>\frac{1}{2^{m+n+1}}+2^n n|\lambda|^n.$$ This means that follows by $$2^{m+2n+2}\,n\,|\lambda|^n\leq |\lambda|^{1+n/m}.$$ This is true because $|\lambda|\leq\frac{1}{2^{10m} n^3}$. Now we have proved that if $z\in A_1$ and $|\lambda|\leq\frac{1}{2^{10m} n^3}$, then $|P_\lambda(z)|>\frac{2}{|\lambda|^{1+{n}/{m}}}$. On the other hand, if $|z|\geq \frac{2}{|\lambda|^{1+{n}/{m}}}$, then $$|P_\lambda(z)+1|\leq \frac{(|z|+1)^n+1}{|\lambda z|^{m+n}-1}\leq \frac{(1+|z|^{-1})^n+|z|^{-n}}{2^m-|z|^{-n}}<\frac{1}{2}.$$ This means that $P_\lambda(z)\in\mathbb{D}(-1,\frac{1}{2})\subset U$. Let $U'_\infty$ be the component of $P_\lambda^{-1}(\mathbb{D}(-1,\frac{1}{2}))$ containing $\{z:|z|\geq \frac{2}{|\lambda|^{1+{n}/{m}}}\}$, it follows that $P_\lambda(A_1)\subset \overline{U'}_\infty\subset U_\infty$ (see Figure \[Fig\_para-map\]). ![Sketch illustrating of the mapping relation of $P_\lambda$. The small pentagons denote the critical points.[]{data-label="Fig_para-map"}](parab-cantor.pdf){width="130mm"} *Proof of Theorem \[non-hyper-cantor\]*. For every $\lambda$ such that $0<|\lambda|\leq {1}/{(2^{10m} n^3)}$, let $A:=\overline{\mathbb{C}}\setminus (U\cup U'_\infty)$. Since $P_\lambda:U'_\infty\rightarrow\mathbb{D}(-1,\frac{1}{2})$ is proper with degree $m$, it follows that $U'_\infty$ is simply connected and $A$ is an annulus. Note that $P_\lambda^{-1}(U'_\infty)$ is an annulus since there are $m+n$ critical points in $P_\lambda^{-1}(U'_\infty)$ and on which the degree of $P_\lambda$ is $m+n$. This means that $P_\lambda^{-1}(A)$ consists of two disjoint annuli $I_1$ and $I_2$ and $I_1\cup I_2\subset A$. The degree of the restriction of $P_\lambda$ on $I_1$ and $I_2$ are $m$ and $n$ respectively. The following argument is very similar to that of Theorem \[parameter\]. The Julia set of $P_\lambda$ is $J_\lambda=\bigcap_{k\geq 0}P_\lambda^{-k}(A)$. By the construction, the components of $J_n$ are compact sets nested between $-1$ and $\infty$ since $P_\lambda^{-1}:A\rightarrow I_j$ is conformal for $j=1$ or $2$. Since the component of $J_n$ cannot be a point and the proof of Theorem 1.2 in [@PT] can also be applied to geometrically finite rational maps (see [@PT $\S$9] and [@TY]), we know that every component of $J_n$ is a Jordan curve. The dynamics of $P_\lambda$ on the set of Julia components is isomorphic to the one-sided shift on $2$ symbols $\Sigma_{2}:=\{0,1\}^{\mathbb{N}}$. In particular, $J_\lambda$ is homeomorphic to $\Sigma_{2}\times\mathbb{S}^1$, which is a Cantor set of circles as claimed. $\square$ From the proof of Theorem \[non-hyper-cantor\] and Theorem \[this-is-all-resta\], we know that the dynamics on the Julia set of $P_\lambda$ is conjugate to that of some $g_\eta$ with the form . Therefore, we can view $P_\lambda$ as a ‘parabolic’ McMullen map since the only difference is the super-attracting basin and its preimages of $g_\eta$ have been replaced by a fixed parabolic basin and its preimages of $P_\lambda$ (see Figure \[Fig\_C-C-C\]). ![The Julia set of $P_\lambda$, where $m=3,n=2$ and $\lambda$ is small enough such that $J_\lambda$ is a Cantor set of circles. All the Fatou components of $P_\lambda$ are iterated onto the fixed parabolic component (the ‘cauliflower’ in the center of this figure) with parabolic fixed point 1.[]{data-label="Fig_C-C-C"}](Cantor_Circle_Cauliflower.png){width="80mm"} More Non-hyperbolic Examples {#sec-more-exam} ============================ In this section, we will construct more non-hyperbolic rational maps whose Julia sets are Cantor circles but they are not included by the previous section. Inspired by Theorem \[parameter\], for every $n\geq 2$, we define $$\label{family-para-restate} P_n(z)=A_n\,\frac{(n+1)z^{(-1)^{n+1} (n+1)}}{nz^{n+1}+1}\prod_{i=1}^{n-1}(z^{2n+2}-b_i^{2n+2})^{(-1)^{i-1}}+B_n,$$ where $|b_i|=s^i$ for some $0<s\leq 1/(25n^2)$ and $$\label{A-B-n-restate} A_n=\frac{1}{1+(2n+2)C_n}\prod_{i=1}^{n-1}(1-b_i^{2n+2})^{(-1)^i},~~B_n=\frac{(2n+2)C_n}{1+(2n+2)C_n}~~\text{and}~~ C_n=\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}}.$$ \[para-fixed\] $(1)$ $P_n(1)=1$ and $P_n'(1)=1$. $(2)$ $1-s^{2n+1}/(n+1)<|A_n|<1+s^{2n+1}/(n+1)$ and $|B_n|<s^{2n+1}/(3n+3)$. It is easy to see $P_n(1)=1$ by a straightforward calculation. Note that $$\label{solu-crit-parabolic} F_n(z):=\frac{zP_n'(z)}{P_n(z)-B_n}=\sum_{i=1}^{n-1}\frac{(-1)^{i-1}(2n+2)z^{2n+2}}{z^{2n+2}-b_i^{2n+2}}+(-1)^{n+1} (n+1)-\frac{n(n+1)z^{n+1}}{nz^{n+1}+1}.$$ This means that $$\label{A-B-equation-1} \begin{split} &~ \frac{P_n'(1)}{P_n(1)-B_n} \\ = &~ (2n+2)\,\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}}+(2n+2)\,\sum_{i=1}^{n-1}(-1)^{i-1}+(-1)^{n+1} (n+1)-n\\ = &~ (2n+2)\,\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}}+1:=(2n+2)C_n+1. \end{split}$$ Therefore, we have $$\label{A-B-equation-2} P_n'(1)=(1-B_n)((2n+2)C_n+1)=1.$$ It follows that $1$ is a parabolic fixed point of $P_n$. This completes the proof of (1). For (2), since $|1-b_i^{2n+2}|^{-1}\leq 1+2|b_1|^{2n+2}$ for $1\leq i\leq n-1$ and $0<s\leq 1/(25n^2)\leq 1/100$, then $$\label{C-n-estim} \begin{split} &~ (2n+2)|C_n|< (2n+2)\,(1+2|b_1|^{2n+2})\sum_{i=1}^{n-1}|b_i|^{2n+2}\\ \leq &~ \frac{(2n+2)(1+2s^{2n+2})s^{2n+2}}{1-s^{2n+2}}<\frac{s^{2n+1}}{4n+4}. \end{split}$$ We have $$\label{B-n-estim} |B_n|=\left|\frac{(2n+2)C_n}{1+(2n+2)C_n}\right|<(2n+2)|C_n|(1+(4n+4)|C_n|)<\frac{s^{2n+1}}{3n+3}$$ and $$\label{A-n-estim} |A_n|<(1+(4n+4)|C_n|)\prod_{i=1}^{n-1}(1+2|b_i|^{2n+2})<(1+\frac{s^{2n+1}}{2n+2})(1+5s^{2n+2})<1+\frac{s^{2n+1}}{n+1}.$$ Moreover, we have $$\label{A-n-estim-lower} |A_n|>(1-(2n+2)|C_n|)\prod_{i=1}^{n-1}(1-|b_i|^{2n+2})>(1-\frac{s^{2n+1}}{4n+4})(1-\frac{s^{2n+2}}{1-s^{2n+2}})>1-\frac{s^{2n+1}}{n+1}.$$ The proof is complete. Let us first explain some ideas behind the construction. For $n\geq 2$, define $\widetilde{Q}(z)=(z^{n+1}+n)/(n+1)$ and $\varphi(z)=1/z$, then $Q(z):=\varphi\circ\widetilde{Q}\circ\varphi^{-1}(z)=(n+1)z^{n+1}/(nz^{n+1}+1)$ satisfies: $\infty$ is a critical point of $Q$ with multiplicity $n$ which is attracted to the parabolic fixed point $1$. Since $\{b_i\}_{1\leq i\leq n-1}$ are very small, the rational map $P_n$ can be viewed as a small perturbation of $Q$. The terms $A_n$ and $B_n$ here guarantee that $1$ is always a parabolic fixed point of $P_n$ (see Lemma \[para-fixed\]). It can be shown that $P_n$ maps an annular neighborhood of $\mathbb{T}_{|b_i|}$ into $T_0$ or $T_\infty$ according to whether $i$ is odd or even, where $T_0$ and $T_\infty$ denote the Fatou components containing $0$ and $\infty$ respectively (see Lemma \[lemma-want\]). The Fatou component $T_\infty$ is always parabolic while $T_0$ is attracting or mapped to $T_\infty$ according to whether $n$ is odd or even. The proof of Theorem \[parameter-parabolic\] will based on the mixed arguments as in the previous 2 sections. If $|z|\leq 1$, then $|\widetilde{Q}(z)|\leq 1$. This means that the fixed parabolic Fatou component of $\widetilde{Q}$ contains the unit disk for every $n\geq 2$. Therefore, the parabolic Fatou component of $Q$ contains the exterior of the closed unit disk $\overline{\mathbb{C}}\setminus \overline{\mathbb{D}}$. Although the polynomial $Q$ has been perturbed into $P_n$, we still have following \[key-lemma-complex\] $P_n(\overline{\mathbb{C}}\setminus \mathbb{D})\subset(\overline{\mathbb{C}}\setminus \overline{\mathbb{D}})\cup\{1\}$. In particular, the disk $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ lies in the parabolic Fatou component of $P_n$ with parabolic fixed point $1$. The proof of Lemma \[key-lemma-complex\] is very subtle, and will be delayed to next section. \[sum-of\] For $n\geq 2$ and $1\leq i\leq n-1$, then $$\label{sum-equa} \sum_{1\leq j<i}(-1)^j +\sum_{i<j\leq n-1}(-1)^{j-1}+\frac{1+(-1)^{n+1}}{2}=0.$$ The argument is based on several cases shown in Table \[Tab\_number\]. $\sum_{1\leqslant j<i}(-1)^j$ $\sum_{i<j\leqslant n-1}(-1)^{j-1}$ $(1+(-1)^{n+1})/{2}$ ---------- ---------- ------------------------------- ------------------------------------- ---------------------- odd $n$ odd $i$ $0$ $-1$ $1$ even $i$ $-1$ $0$ $1$ even $n$ odd $i$ $0$ $0$ $0$ even $i$ $-1$ $1$ $0$ : The proof of Lemma \[sum-of\].[]{data-label="Tab_number"} 0.2cm -0.5cm As before, we first locate the critical points of $P_n$. Note that $0$ and $\infty$ are both critical points of $P_n$ with multiplicity $n$ and the degree of $P_n$ is $n^2+n$. The remaining $2(n^2-1)$ critical points of $P_n$ are the solutions of $F_n(z)=0$ (see equation ). For $1\leq i\leq n-1$, let $\widetilde{CP}_i:=\{\widetilde{w}_{i,j}=b_i \exp(\pi \textup{i}\frac{2j-1}{2n+2}):1\leq j\leq 2n+2\}$ be the collection of $2n+2$ points lying on $\mathbb{T}_{|b_i|}$ uniformly. The following lemma is similar to Lemmas \[crit-close\] and \[crit-close-para\]. \[crit-close-Parameter\] For every $\widetilde{w}_{i,j}\in\widetilde{CP}_i$, where $1\leq i\leq n-1$ and $1\leq j\leq 2n+2$, there exists $w_{i,j}$, which is a solution of $F_n(z)=0$, such that $|w_{i,j}-\widetilde{w}_{i,j}|<s^{n+1/2}|b_i|$. Moreover, $w_{i_1,j_1}= w_{i_2,j_2}$ if and only if $(i_1,j_1)=(i_2,j_2)$. Note that $F_n(z)=0$ is equivalent to $$\label{solu-crit-2} \sum_{i=1}^{n-1}(-1)^{i-1}\frac{z^{2n+2}+b_i^{2n+2}}{z^{2n+2}-b_i^{2n+2}}+\frac{1+(-1)^{n+1}}{2}-\frac{nz^{n+1}}{nz^{n+1}+1}=0.$$ Timing $z^{2n+2}-b_i^{2n+2}$ on both sides of (\[solu-crit-2\]), where $1\leq i\leq n-1$, we have $$\label{solu-crit-3-parabolic} (-1)^{i-1}(z^{2n+2}+b_i^{2n+2})+(z^{2n+2}-b_i^{2n+2})\,G_{i}(z)=0,$$ where $$\label{G_n-parabolic} G_{i}(z)=\sum_{1\leq j \leq n-1,\,j\neq i}(-1)^{j-1}\frac{z^{2n+2}+b_j^{2n+2}}{z^{2n+2}-b_j^{2n+2}}+\frac{1+(-1)^{n+1}}{2}-\frac{nz^{n+1}}{nz^{n+1}+1}.$$ Let $\Omega_{i}=\{z:|z^{2n+2}+b_i^{2n+2}|\leq s^{n+1/2}|b_i|^{2n+2}\}$, where $1\leq i\leq n-1$. If $z\in\Omega_i$, then $|z|^{n+1}\leq (1+s^{n+1/2})|b_i|^{n+1}\leq(1+s^{n+1/2})s^{n+1}$ by Lemma \[very-useful-est\](2). So $$\left|\frac{nz^{n+1}}{nz^{n+1}+1}\right|\leq \frac{n(1+s^{n+1/2})s^{n+1}}{1-n(1+s^{n+1/2})s^{n+1}} \leq \frac{(1+100^{-5/2})s^{n+1/2}/5}{1-(1+100^{-5/2})100^{-5/2}/5}<0.3 \, s^{n+1/2}$$ since $s\leq 1/(25n^2)\leq 1/100$. For every $z\in\Omega_{i}$, if $1\leq j<i$, we have $$\label{estim-1-new} |{z}/{b_j}|^{2n+2}=|{z}/{b_i}|^{2n+2}|{b_i}/{b_j}|^{2n+2}< (1+s^{n+1/2})\,s^{(2n+2)(i-j)}.$$ If $i<j\leq n-1$, by the first statement of Lemma \[very-useful-est\](2), we have $$\label{estim-2-new} |{b_j}/{z}|^{2n+2}=|{b_i}/{z}|^{2n+2}|{b_j}/{b_i}|^{2n+2}\leq (1+2\cdot s^{n+1/2})\,s^{(2n+2)(j-i)}.$$ From (\[estim-1-new\]), (\[estim-2-new\]) and Lemma \[sum-of\], we have $$\label{bound-neww} \begin{split} &~ \left|G_{i}(z)+\frac{nz^{n+1}}{nz^{n+1}+1}\right|\\ = &~ \left|\sum_{1\leq j<i}(-1)^{j}\frac{1+(z/b_j)^{2n+2}}{1-(z/b_j)^{2n+2}}+ \sum_{i< j\leq n-1}(-1)^{j-1}\frac{1+(b_j/z)^{2n+2}}{1-(b_j/z)^{2n+2}}+\frac{1+(-1)^{n+1}}{2}\right|\\ < &~ 3\cdot(1+2\cdot s^{n+1/2})\,\left(\sum_{1\leq j<i}s^{(2n+2)(i-j)}+\sum_{i< j\leq n-1} s^{(2n+2)(j-i)}\right)\\ < &~6\cdot(1+2\cdot s^{n+1/2})^2\,s^{2n+2}. \end{split}$$ The first inequality in (\[bound-neww\]) follows from the inequality $2x/(1-x)\leq 3x$ if $x<1/3$ (Here $x\leq (1+2\cdot s^{n+1/2})\,s^{2n+2}<10^{-10}$). So we have $$\label{bound-parabolic} |G_{i}(z)|< ~6\cdot(1+2\cdot s^{n+1/2})^2\,s^{2n+2}+0.3 \, s^{n+1/2} < 0.4 \,s^{n+1/2}.$$ Therefore, if $z\in\Omega_{i}$, then $$|z^{2n+2}-b_i^{2n+2}|\cdot|\,G_{i}(z)|< (2+s^{n+1/2})|b_i|^{2n+2}\cdot 0.4 \,s^{n+1/2} < s^{n+1/2}|b_i|^{2n+2}.$$ From (\[solu-crit-3-parabolic\]) and by Rouché’s Theorem, there exists a solution $w_{i,j}$ of $F_n(z)=0$ such that $w_{i,j}\in\Omega_i$ for every $1\leq j\leq 2n+2$. In particular, $|w_{i,j}-\widetilde{w}_{i,j}|<s^{n+1/2}|b_i|$ by the second statement of Lemma \[very-useful-est\](2). The assertion $w_{i_1,j_1}= w_{i_2,j_2}$ if and only if $(i_1,j_1)=(i_2,j_2)$ can be verified similarly as and . The proof is complete. For $1\leq i\leq n-1$, let $CP_i:=\{w_{i,j}: 1\leq j\leq 2n+2\}$ be the collection of critical points of $P_n$ which lie close to the circle $\mathbb{T}_{|b_i|}$. \[lemma-want\] There exist $n-1$ annuli $\{A_i\}_{i=1}^{n-1}$ satisfying $A_{n-1}\prec \cdots\prec A_1$ and two simply connected domain $U_0$ and $U_\infty$ which contains $0$ and $\infty$ respectively, such that $(1)$ $U_\infty\supset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ and $P_n(\overline{U}_\infty)\subset U_\infty\cup\{1\}$; $(2)$ $A_i\supset\mathbb{T}_{|b_i|}\cup CP_i$, $P_n(\overline{A}_i)\subset U_0$ for odd $i$ and $P_n(\overline{A}_i)\subset U_\infty$ for even $i$; $(3)$ $P_n(\overline{U}_0)\subset U_\infty$ for even $n$ and $P_n(\overline{U}_0)\subset U_0$ for odd $n$. Let $U_\infty:=\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ be the exterior of the closed unit disk. Then (1) is obvious if we apply Lemma \[key-lemma-complex\]. Let $\varepsilon=s^{n+1/2}$ and $A_i=\mathbb{A}_{|b_i|(1-2\varepsilon),|b_i|(1+2\varepsilon)}$. From , we know that $$\label{abs-f-n-yang} |R_n(z)|: = \left|\frac{P_n(z)-B_n}{A_n}\cdot\frac{nz^{n+1}+1}{n+1}\right| = |z|^{(-1)^{n+1} (n+1)}\,|z^{2n+2}-b_i^{2n+2}|^{(-1)^{i-1}}H_i(z),$$ where $$\label{H-i-yang} H_i(z)=\prod_{j=1}^{i-1}|b_j|^{(2n+2)(-1)^{j-1}} \prod_{j=i+1}^{n-1}|z|^{(2n+2)(-1)^{j-1}}\cdot Q_i(z)$$ and $$\label{Q-i-yang} Q_i(z)=\prod_{j=1}^{i-1}\left|1-({z}/{b_j})^{2n+2}\right|^{(-1)^{j-1}} \prod_{j=i+1}^{n-1}\left|1-({b_j}/{z})^{2n+2}\right|^{(-1)^{j-1}}.$$ If $z\in \overline{A}_i$, where $1\leq i\leq n-1$, we have $$\label{Q-i-esti-1-yang} Q_i(z)< \prod_{j=1}^{i-1}\left(1+3\,|{b_i}/{b_j}|^{2n+2}\right) \prod_{j=i+1}^{n-1}\left(1+3\,|{b_j}/{b_i}|^{2n+2}\right) < (1+6s^{2n+2})^2$$ and $$\label{Q-i-esti-2-yang} Q_i(z)> \prod_{j=1}^{i-1}\left(1+3\,|{b_i}/{b_j}|^{2n+2}\right)^{-1} \prod_{j=i+1}^{n-1}\left(1+3\,|{b_j}/{b_i}|^{2n+2}\right)^{-1} > (1+6s^{2n+2})^{-2}.$$ Note that $\varepsilon=s^{n+1/2}\leq (5n)^{-2n-1}\leq 10^{-5}$. If $n$ is even and $1\leq i\leq n-1$ is odd, then for $z\in \overline{A}_i$, we have $$\begin{split} |R_n(z)| = &~ \frac{|z^{2n+2}-b_i^{2n+2}|}{|z|^{n+1}}\,\frac{1}{s^{(i-1)(n+1)}}\, Q_i(z) < \frac{|b_i|^{n+1}(1+(1+2\varepsilon)^{2n+2})}{(1-2\varepsilon)^{n+1}}\,\frac{(1+6s^{2n+2})^2}{s^{(i-1)(n+1)}}\\ = &~\frac{1+(1+2\varepsilon)^{2n+2}}{(1-2\varepsilon)^{n+1}}(1+6s^{2n+2})^2 s^{n+1}<2.1\cdot s^{n+1}. \end{split}$$ If $n$ and $1\leq i\leq n-1$ are both even, then for $z\in \overline{A}_i$, we have $$|R_n(z)|=\frac{|b_{i-1}|^{2n+2}|z|^{2n+2}}{|z|^{n+1}|z^{2n+2}-b_i^{2n+2}|}\,\frac{1}{s^{(i-2)(n+1)}}\, Q_i(z) > \frac{(1-2\varepsilon)^{n+1}}{1+(1+2\varepsilon)^{2n+2}}\,(1-6s^{2n+2})^2 > 0.49.$$ This means that if $n$ is even and $1\leq i\leq n-1$ is odd, for $z\in \overline{A}_i$, we have $$\begin{split} &~ |P_n(z)|<\left|\frac{2.1\cdot s^{n+1}\cdot (n+1)\,A_n}{nz^{n+1}+1}\right|+|B_n| \\ \leq &~ \frac{2.1\,(s^{n+1/2}/5)\cdot(1+s^{2n+1}/(n+1))}{1-n(1+2\varepsilon)s^{n+1}}+\frac{s^{2n+1}}{3n+3}<s^{n+1/2} \end{split}$$ by Lemma \[para-fixed\](2). If $n$ and $1\leq i\leq n-1$ are both even, then for $z\in \overline{A}_i$, we have $$\begin{split} |P_n(z)| > &~ \left|\frac{0.49(n+1)A_n}{nz^{n+1}+1}\right|-|B_n| \\ \geq &~ \frac{0.49(n+1)(1-s^{2n+1}/(n+1))}{1+n(1+2\varepsilon)s^{n+1}}-\frac{s^{2n+1}}{3n+3}>\frac{n+1}{3}\geq 1. \end{split}$$ By the completely similar arguments, one can show that if $n$ is odd, for $z\in \overline{A}_i$, we have $$\label{bound-f-n-3-parabolic} |P_n(z)|<s^{n+1/2} \text{~for odd~} i \text{~and~} |P_n(z)|>1 \text{~for even~} i.$$ Let $U_0=\mathbb{D}_r$, where $r=s^{n+1/2}$. This proves (2). If $n$ is odd, for every $z$ such that $|z|\leq s^{n+1/2}$, we have $$\begin{split} |P_n(z)| \leq & \left|\frac{(n+1)A_n}{nz^{n+1}+1}\right|\,|z|^{n+1} \prod_{i=1}^{n-1}|b_i|^{(2n+2)(-1)^{i-1}} \prod_{i=1}^{n-1}\left|1-\frac{z^{2n+2}}{b_i^{2n+2}}\right|^{(-1)^{i-1}}+|B_n|\\ \leq & \frac{(n+1)(1+s^{2n+1}/(n+1))}{1-ns^{n^2+n/2}}\,s^{3(n+1)/2}\prod_{i=1}^{n-1}\left(1+2\frac{|z|^{2n+2}}{|b_i|^{2n+2}}\right)+\frac{s^{2n+1}}{3n+3}<s^{n+1/2}. \end{split}$$ It follows that $P_n(\overline{\mathbb{D}}_r)\subset\mathbb{D}_r$ for odd $n$, where $r=s^{n+1/2}$. If $n$ is even, then $P_n$ maps a neighborhood of $0$ to that of $\infty$. For every $z$ such that $|z|\leq s^{n+1/2}$, we have $$\label{bound-lower-in-disk-even-parabolic-lp} \begin{split} |P_n(z)| \geq &~\frac{(n+1)\,s^{-(n+1)/2}\,(1-s^{2n+1}/(n+1))}{1+ns^{n^2+n/2}} \prod_{i=1}^{n-1}\left(1-2\frac{|z|^{2n+2}}{|b_i|^{2n+2}}\right)-\frac{s^{2n+1}}{3n+3} \\ > &~ n>1. \end{split}$$ This ends the proof of (3). The proof is complete. ![The Julia set of $P_3$, which is a Cantor set of circles. The parameter $s$ is chosen small enough. The gray parts in the Figure denote the Fatou components which are iterated to the attracting Fatou component containing the origin, while the white parts denote the Fatou components iterated to the parabolic Fatou component whose boundary contains the parabolic fixed point $1$. Some equipotentials of Fatou coordinate have been drawn in the parabolic Fatou component and its preimages. Figure range: $[-1.6,1.6]\times[-1.2,1.2]$.[]{data-label="Fig_C-C-F"}](Cantor_Circle_Parabolic_Finally.png){width="100mm"} *Proof of Theorem \[parameter-parabolic\]*. Let $A:=\overline{\mathbb{C}}\setminus (U_0\cup U_\infty)$. The Julia set of $P_n$ is equal to $\bigcap_{k\geq 0}P_n^{-k}(A)$. Note that $P_n$ is geometrically finite. The argument is completely similar to the proofs of Theorems \[parameter\] and \[non-hyper-cantor\]. The set of Julia components of $P_n$ is isomorphic to the one-sided shift on $n$ symbols $\Sigma_{n}:=\{0,1,\cdots,n-1\}^{\mathbb{N}}$. In particular, the Julia set of $P_n$ is homeomorphic to $\Sigma_{n}\times\mathbb{S}^1$, which is a Cantor set of circles, as desired (see Figure \[Fig\_C-C-F\]). We omit the details here. $\square$ Proof of Lemma \[key-lemma-complex\] {#sec-key-lemma} ==================================== This section will be devote to proving Lemma \[key-lemma-complex\], which is the key ingredient in the proof of Lemma \[lemma-want\] and hence in Theorem \[parameter-parabolic\]. Let $\widetilde{R}(z)=1/P_n(1/z)$, then Lemma \[key-lemma-complex\] reduces to proving $\widetilde{R}(\overline{\mathbb{D}})\subset \mathbb{D}\cup\{1\}$. Let $w=z^{n+1}$, by a straightforward calculation, we have $$R(w):=\widetilde{R}(z)=\frac{w+n}{n+1}\cdot\frac{1}{S(w)},$$ where $$\label{S-w-ori} \begin{split} S(w)= &~ A_n\,\prod_{i=1}^{n-1}(1-b_i^{2n+2}w^2)^{(-1)^{i-1}}+\frac{w+n}{n+1}B_n \\ = &~ 1+\frac{w-1}{1+(2n+2)C_n}\left(\frac{H(w)-1}{w-1}+2C_n\right) \end{split}$$ and $$H(w)=\prod_{i=1}^{n-1}(1-b_i^{2n+2})^{(-1)^{i}}\prod_{i=1}^{n-1}(1-b_i^{2n+2}w^2)^{(-1)^{i-1}}.$$ Since $H(1)=1$, it follows that $H'(1)$ is a finite number. In fact, $$\label{I_w} I(w):=\frac{H'(w)}{H(w)}=-2w\,\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}w^2}.$$ We know that $I(1)=H'(1)=-2C_n$. For every small enough $w-1$, we can write $S(w)$ as $$\label{S-w} S(w)=1+\frac{(w-1)^2}{1+(2n+2)C_n}\cdot \frac{\frac{H(w)-1}{w-1}+2C_n}{w-1}=:1+\frac{(w-1)^2}{1+(2n+2)C_n}\cdot\Phi(w),$$ where $$\label{Phi-w} \Phi(w)=\sum_{k\geq 2}\frac{H^{(k)}(1)}{k!}(w-1)^{k-2}.$$ The next step is to estimate $H^{(k)}(1)$ for every $k\geq 2$. For every $k\geq 1$, let $$Y_k(w)=\sum_{i=1}^{n-1}(-1)^{i-1}\left(\frac{b_i^{2n+2}}{1-b_i^{2n+2}w^2}\right)^k.$$ In particular, $Y_1(1)=C_n$ and $$Y_k'(w)=2kw\,Y_{k+1}(w).$$ If $|w|=1$, we have $$|Y_k(w)|\leq \left|\frac{b_1^{2n+2}}{1-b_1^{2n+2}}\right|^k \left(1+\sum_{i=2}^{n-1}\left|\frac{b_i^{2n+2}(1-b_1^{2n+2})}{b_1^{2n+2}(1-b_i^{2n+2})}\right|^k\right) \leq \frac{11}{10}\,\left|\frac{b_1^{2n+2}}{1-b_1^{2n+2}}\right|^k.$$ Similarly, we have $|Y_k(w)|\geq \frac{9}{10}|{b_1^{2n+2}}/{(1-b_1^{2n+2})}|^k$. This means that $$\label{Y_k} \left|\frac{Y_{k+1}(w)}{Y_k(w)}\right|\leq \frac{11}{9}\left|\frac{b_1^{2n+2}}{1-b_1^{2n+2}}\right|\leq 2s^{2n+2}<1/2.$$ We first claim that $|I^{(k)}(1)|\leq 2^{k+1}k!|C_n|$ for every $k\geq 0$. Since $I^{(0)}(w)=-2wY_1(w)$ and $I^{(1)}(w)=-2Y_1(w)-4w^2Y_2(w)$, it can be proved inductively that $I^{(k)}(w)$ can be written as $$\label{expansion} I^{(k)}(w)=\sum_{j=1}^{2^k}Q_{k,j}(w)=\sum_{j=1}^{2^k}P_{k,j}(w)Y_{k,j}(w),$$ where $P_{k,j}(w)$ is a polynomial with degree at most $k+1$ and $Y_{k,j}=Y_l$ for some $1\leq l\leq k+1$. Note that some terms $Q_{k,j}$ may be equal to zero (the degree of corresponding polynomial $P_{k,j}$ is regarded as $-\infty$) and the formula can be simplified, but what we need is this ‘long’ expansion. In particular, without loss of generality, for $1\leq j\leq 2^k$, we require further that $$\label{dera} P_{k+1,2j-1}(w)Y_{k+1,2j-1}(w)=P_{k,j}'(w)Y_{k,j}(w)~~\text{and}~~P_{k+1,2j}(w)Y_{k+1,2j}(w)=P_{k,j}(w)Y_{k,j}'(w).$$ Since $\deg (P_{k,j})\leq k+1$ and $Y_{k,j}=Y_l$ for some $1\leq l\leq k+1$, it follows that $$\label{deri-leq} \begin{split} &~ |P_{k+1,2j-1}(1)Y_{k+1,2j-1}(1)|+|P_{k+1,2j}(1)Y_{k+1,2j}(1)|\\ = &~ |P_{k,j}'(1)Y_{l}(1)|+|P_{k,j}(1)Y_{l}'(1)|\\ \leq &~ (k+1)|P_{k,j}(1)Y_{l}(1)|+2(k+1)|P_{k,j}(1)Y_{l+1}(1)|\\ \leq &~ 2(k+1)|P_{k,j}(1)Y_{k,j}(1)| \end{split}$$ since $|Y_{l+1}(1)/Y_{l}(1)|\leq 1/2$ for every $l\geq 1$ by . Denote $||I^{(k)}(1)||:=\sum_{j=1}^{2^k}|P_{k,j}(1)Y_{k,j}(1)|$, we have $||I^{(k)}(1)||\leq 2k||I^{(k-1)}(1)||$. This means that $$\label{bound-I-k} |I^{(k)}(1)|\leq ||I^{(k)}(1)||\leq 2^k k!||I^{(0)}(1)||=2^{k+1}k!|C_n|.$$ This proves the claim $|I^{(k)}(1)|\leq 2^{k+1}k!|C_n|$ for every $k\geq 0$. Secondly, we check by induction that $|H^{(k)}(1)|\leq 4^k k!|C_n|$ for $k\geq 1$. For $k=1$, we have $|H'(1)|=2|C_n|< 4|C_n|$. Assume that $|H^{(i)}(1)|\leq 4^i i!|C_n|$ for every $1\leq i\leq k$. By , we have $H'(w)=H(w)I(w)$. So $$\label{Deri-H-k} \begin{split} |H^{(k+1)}(1)| \leq &~ |I^{(k)}(1)|+\sum_{i=1}^{k}\frac{k!}{i!(k-i)!}|H^{(i)}(1)|\cdot|I^{(k-i)}(1)|\\ \leq &~ 2^{k+1}k!|C_n|(1+2^{k+1}|C_n|)\leq 4^{k+1} (k+1)!|C_n| \end{split}$$ since $|I^{(k-i)}(1)|\leq 2^{k-i+1}(k-i)!|C_n|$ and $|H^{(i)}(1)|\leq 4^i i!|C_n|$ for every $1\leq i\leq k$. If $w=e^{i\theta}$ for $|\theta|\leq 1/20$, then $|w-1|<|\theta|\leq 1/20$. By and , we have $$\label{Phi-w-est} |\Phi(w)|\leq \sum_{k\geq 2}4^k|C_n|(1/20)^{k-2}\leq 16|C_n|\sum_{k\geq 0}5^{-k}=20|C_n|.$$ By and , it follows that $$|S(w)|\geq 1-\frac{\theta^2}{1-(2n+2)|C_n|}20|C_n|\geq 1-\frac{s^{2n+1}}{n+1}\theta^2$$ since $n\geq 2$ and $|C_n|<s^{2n+1}/(8(n+1)^2)$ by . On the other hand, if $w=e^{i\theta}$ for $0\leq|\theta|\leq \pi$, then $$\label{Q-z-est} \left|\frac{w+n}{n+1}\right|=\left(1-\frac{4n}{(n+1)^2}\sin^2\frac{\theta}{2}\right)^{1/2}\leq \left(1-\frac{4n}{\pi^2(n+1)^2}\theta^2\right)^{1/2}\leq 1-\frac{2n}{(n+1)^2\pi^2}\theta^2$$ since $(1-x)^{1/2}\leq 1-x/2$ for $0\leq x< 1$. This means that if $w=e^{i\theta}$ for $|\theta|\leq 1/20$, then $$|R(w)|\leq (1-\frac{2n}{(n+1)^2\pi^2}\theta^2)(1-\frac{s^{2n+1}}{n+1}\theta^2)^{-1}\leq 1.$$ Moreover, $|R(w)|=1$ if and only if $w=1$. If $w=e^{i\theta}$ for $|\theta|> 1/20$, by and Lemma \[para-fixed\](2), we have $$\label{S-w-est} |S(w)|\geq (1-\frac{s^{2n+1}}{n+1})\prod_{i=1}^{n-1}(1-|b_i|^{2n+2})-\frac{s^{2n+1}}{3n+3}\geq 1-\frac{3s^{2n+1}}{n+1}.$$ By and , we have $$|R(w)|\leq (1-\frac{2}{20^2 (n+1)\pi^2})(1-\frac{3s^{2n+1}}{n+1})^{-1}< 1.$$ It follows that $R(w)$ maps the boundary of the unit disk into the unit disk except at $w=1$. Since $R(w)\neq \infty$ if $|w|\leq 1$, we know that $R(\overline{\mathbb{D}})\subset\mathbb{D}\cup\{1\}$. Therefore, $\widetilde{R}(\overline{\mathbb{D}})\subset\mathbb{D}\cup\{1\}$ and $\widetilde{R}$ maps $\{z\in\mathbb{C}:z^{n+1}=1\}$ onto 1. This ends the proof of Lemma \[key-lemma-complex\]. 0.2cm *Acknowledgements.* The authors would like to thank Guizhen Cui for discussions and the referees for their careful reading and comments. The first author was supported by the National Natural Science Foundation of China under grant No.11271074, and the third author was supported by the National Natural Science Foundation of China under grant No.11231009. [99]{} A. Beardon, *Iteration of rational functions*. Grad. Texts Math., **132**, Springer, New York, 1991. G. Cui, Dynamics of rational maps, topology, deformation and bifurcation. Preprint, May 2002 (Early version: Geometrically finite rational maps with given combinatorics, 1997). R. Devaney, D. Look and D. Uminsky, The Escape Trichotomy for Singularly Perturbed Rational Maps. *Indiana Univ. Math. J.*, **54**(6) (2005), 1621-1634. P. Haïssinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets. *Rev. Math. Iberoam.*, **28**(4) (2012), 1025-1034. C. McMullen, Automorphisms of rational maps. In *Holomorphic Functions and Moduli I*, Math. Sci. Res. Inst. Publ. **10**, Springer, New York, 1988. K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets. *Astérisque*, **261** (2000), 349-383. W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps. *Adv. Math.*, **229**(4) (2012), 2525-2577. W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets. *Sci. China Ser. A*, **52**(1) (2009), 45-65. N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^\ell$. *Conform. Geom. Dyn.*, **10** (2006) 159-183. L. Tan and Y. Yin, Local connectivity of the Julia sets for geometrically finite rational maps. *Sci. China Ser. A*, **39**(1) (1996), 39-47.

--- address: - 'Konkoly Observatory, Budapest, HUNGARY' - 'Kapteyn Institute, Groningen, The NETHERLANDS' - 'Physics Department, University of Florida, Gainesville, FL, USA' -   author: - Zoltán Kolláth - 'Jean-Philippe Beaulieu' - 'J. Robert Buchler & Phil Yecko' title: '[ASTROPHYSICAL JOURNAL LETTERS, in press, accepted May 18, 1998]{}   Nonlinear Beat Cepheid Models' --- \#1 2 2 The Fourier analysis of the observational data of the [*beat Cepheid*]{} light curves and radial velocities shows constant power in two basic frequencies and in their linear combinations which indicates that the stars pulsate in two modes (or more if resonances are involved). Since the beginning of theoretical Cepheid studies in the early 1960s numerical hydrodynamical attempts at modelling the phenomenon of beat pulsation have failed, and beat Cepheids have been a bane of stellar pulsation theory. In Cepheids energy is carried through the pulsating envelope to the surface by radiation transport as well as by turbulent convection (TC). Even though convection can transport almost all the energy in the hydrogen partial ionization region, this convection is inefficient in the sense that it only mildly affects the structure of the envelope. It was thus generally thought that convection, while important for providing a red edge to the instability strip, would play a minor role the appearance of the nonlinear pulsation. Purely radiative models did indeed give good overall agreement with the observed light and radial velocities. However, recently it has become increasingly clear that there are a number of severe problems with radiative models (Buchler 1998), in addition to their inability to account for beat behavior. We have recently implemented in our hydrodynamics codes a one dimensional model diffusion equation for turbulent energy (Yecko, Kolláth & Buchler 1998) similar to those advocated by Stellingwerf (1982), Kuhfuss (1986), Gehmeyr & Winkler (1992) and Bono & Stellingwerf (1994). In contrast to these authors, however, we have developed additional tools that allow us to find beat behavior without having to rely on very time-consuming and sometimes inconclusive hydrodynamic integrations to determine if a model undergoes stable, or steady beat pulsations. These are (a) a linear stability analysis which yields the frequencies and growth rates of [*all modes*]{}, (b) a relaxation method (based on the general algorithm of Stellingwerf with the modifications of Kovács & Buchler, 1987) to obtain nonlinear periodic pulsations (limit cycles) when they exist, (c) a stability analysis of the limit cycles that gives their (Floquet) stability exponents. The 1D turbulent diffusion equation, and the concomitant eddy viscosity, the turbulent pressure and the convective and turbulent fluxes contain (seven) order unity parameters that need to be calibrated through a comparison to observations. In a first paper (Yecko, Kolláth & Buchler 1998) in which we performed a broad survey of the linear properties of TC Cepheid models we found that of these the mixing length, the strengths of the convective flux and of the eddy viscosity play a dominant role and that broad combinations of these three parameters exist that give agreement with the observed widths of both the fundamental and first overtone instability strips. In this Letter we show that the inclusion of TC produces pulsating beat Cepheid models that satisfy the observational constraints, in particular those of period ratios, of modal pulsation amplitudes and of their ratios. Furthermore the models are very robust with respect to the numerical and physical parameters. Our discovery of beat Cepheid models has been partially serendipitous. When we started to investigate the nonlinear pulsations of a typical Small Magellanic Cloud Cepheid model ($M\ngth =\ngth 4.0M_\odot$, $L\ngth =\ngth 1100L_\odot$, = 5900K, $X\ngth =\ngth 0.73$ and $Z\ngth =\ngth 0.004$) with the the turbulent convective hydrocode, we encountered beat pulsations that appeared steady. We use the OPAL opacities of Iglesias & Rogers (1996) combined with those of Alexander & Ferguson (1994). The values of the TC parameters – for a definition cf. Yecko et al. (1998) – are $\alpha_c\ngth =\ngth 3$, $\alpha_\Lambda \ngth =\ngth0.41$, $\alpha_p\ngth =\ngth 0.667$, $\alpha_t\ngth =\ngth 1$, $\alpha_D\ngth =\ngth 4$, $\alpha_s\ngth =\ngth 0.75$, $\alpha_\nu \ngth =\ngth 1.2$. The steadiness of these beat pulsations was confirmed when several nonlinear hydrodynamics calculations, each initiated with a different admixture of fundamental and first overtone eigenvectors, converged towards the same final steady beat pulsational state. This convergence could be corroborated when we extracted the slowly varying amplitudes with the help of a time-dependent Fourier decomposition, and plotted the resulting phase portraits ($A_0(t)$ vs. $A_1(t)$) that are shown in Fig. 1 where all initializations are seen to converge toward a fixed point located at $A_0 \ngth =\ngth 0.0104$ and $A_1\ngth =\ngth 0.0200$ (These radial displacement amplitudes assume the eigenvectors to be normalized to unity at the stellar surface, $\delta r/r_*\ngth =\ngth 1$). While the observed transient behavior of the models provides a conclusive proof of the presence of steady beat pulsations, it is important to explain and describe the behavior on a more fundamental level. The phase portrait of Fig. 1 is very similar to those found for nonresonant mode interaction on the basis of amplitude equations. (Buchler & Kovács 1986, 1987, hereafter BK86 and BK87). We show here that indeed the nonlinear behavior of the hydrodynamical model pulsations can be understood very simply that way. The amplitudes of the two nonresonantly interacting modes obey remarkably simple equations $$\begin{aligned} \quad\quad {dA_0 \over dt} = & A_0 \th (\kappa_0 - q_{00} A_0^2 - q_{01} A_1^2) \nonumber \quad\quad\quad\quad\quad (1a)\\ \quad\quad {dA_1\over dt} = & A_1 \th (\kappa_1 - q_{10} A_0^2 - q_{11} A_1^2) \nonumber \quad\quad\quad\quad\quad (1b) \end{aligned}$$ These amplitude equations are ‘normal forms’ and are therefore generic for any dynamical system in which two modes interact nonresonantly. The assumptions underlying these amplitude equations are satisfied for Cepheids: (a) The lowest modes (fundamental and first overtone here) are weakly nonadiabatic, i.e. the ratios of linear growth rates $\kappa$ to periods are small, a condition that is readily confirmed by our linear stability analysis; (b) The pulsations are weakly nonlinear which allows a truncation of the amplitude equations in the lowest permissible (third) order; weak nonlinearity can be established by comparing the linear and nonlinear periods which differ less than a tenth of a percent. Furthermore both the nonlinear self-saturation coefficients $q_{00}$ and $q_{11}$ as well as the cross-coupling coefficients $q_{01}$ and $q_{10}$ have always been found to be positive in Cepheid models so that amplitude saturation can occur in third order, and it is sufficient to keep terms up to cubic in the amplitudes. (c) In the range of interest there is no important low order resonance between the fundamental and the first overtone modes, and possibly a higher mode. In the following discussion we look at the regime where both modes are linearly unstable, $\kappa_0>0$ and $\kappa_1> 0$. Eqs. (1) then have two [*single mode*]{} fixed points. The amplitude of the single mode fundamental (0) fixed point is $A_0=\sqrt{\kappa_0/q_{00}}$ and its coefficient for stability to first overtone perturbations is $\bar\kappa_{1(0)} = \kappa_1 - q_{10} A_0^2 \th$. In our notation a positive coefficient implies growth and thus instability. The corresponding first overtone (1) limit cycle amplitude is $A_1=\sqrt{\kappa_1/q_{11}}$ and its coefficient of stability to fundamental perturbations is $\bar\kappa_{0(1)} = \kappa_0 - q_{01} A_1^2 \th$. The $\bar\kappa$’s, when multiplied by the periods $P_k$ of their limit cycles, are equal to the corresponding Floquet exponents (Buchler, Moskalik & Kovács 1991). Eqs. (1) can also have [*a double mode fixed point*]{} whose amplitudes satisfy $A_{0\th DM}^2 = \bar\kappa_{0(1)} q_{11}/ {\cal D} < A_0^2$, $A_{1\th DM}^2 = \bar\kappa_{1(0)} q_{00}/ {\cal D} < A_1^2$, where ${\cal D} = q_{00} q_{11}$ – $q_{01} q_{10}$. This fixed point exists provided $A_{0\th DM}^2>0$ and $A_{1\th DM}^2>0$. Then, if ${\cal D} <$ 0, the double mode limit cycle is unstable. Stable pulsations occur either in the fundamental [*or*]{} first overtone, and the pulsational mode is determined by the evolutionary history of the model (hysteresis). On the other hand, if ${\cal D}>0$, the double mode fixed point is stable, and steady double mode pulsations occur (BK86). One can show that these conditions are equivalent to requiring $\bar\kappa_{0(1)}>0$ and $\bar\kappa_{1(0)}>0$, conditions which also imply that both [*single mode*]{} limit cycles (fundamental and first overtone) are individually unstable. This validates Stellingwerf’s (1975) suggestion that the simultaneous instability of the fundamental and the first overtone leads to steady beat pulsations (in the absence of a resonance). It provides an economical tool to search for double mode behavior, because we can now relatively easily compute single mode limit cycles and their stability. As a further confirmation that the nonresonant scenario applies to the pulsating Cepheid model, we have determined the coefficients of Eqs. (1) as in BK87 by fitting time-dependent solutions of these equations to the temporal variation of the amplitudes in their approach to the limit cycle as shown in Fig. 1. The fitted trajectories in the phase portrait are practically undistinguishable from the hydro results, confirming the applicability and accuracy of the amplitude equation formalism and the absence of any relevant resonances. We mention in passing that the expression ‘double mode Cepheids’ is often used cavalierly for beat Cepheids. Since no additional, resonant overtone is involved in the beat pulsations, the latter are thus truly double mode pulsations. With the relaxation code we are able to compute both the fundamental and the first overtone limit cycles with their respective amplitudes and Floquet stability exponents $\lambda_{1(0)} = P_0 \bar\kappa_{1(0)}$ and $\lambda_{0(1)} = P_1 \bar\kappa_{0(1)}$. The above discussion then shows that from these four quantities we can extract the four nonlinear $q_{jk}$ coefficients when we have already computed the linear periods and growth rates. The values we obtain this way for this beat Cepheid model agree quite well with those that we obtain from the fit described in the previous paragraph. Note that these two determinations rely on independent numerical hydrodynamical input, the first on two periodic limit cycles (that are linearly unstable), the second on transient evolution toward the stable double mode pulsation. In order to investigate the robustness of the observed beat behavior we now explore the pulsational behavior of a [*sequence*]{} of Cepheid models in which the effective temperature of the equilibrium modes of the sequence varies from =6200K to 5800K. Such a sequence is approximately along an evolutionary path. The eddy viscosity parameter $\alpha_\nu$ is treated as an additional variable parameter to explore the effect of TC on the behavior. In Fig. 2 the stability coefficients of the sequence are plotted versus , with open/filled circles for those of the fundamental/overtone single mode cycles. The curves are labelled with the corresponding strengths $\alpha_\nu$ of the eddy viscosity. As discussed above we expect double mode behavior where both Floquet exponents are positive. (The stability exponents due to perturbations with other modes are always smaller in this sequence and are therefore irrelevant here). For the low value of $\alpha_\nu \ngth =\ngth 0.5$ (dotted lines) the two stability coefficients are never positive simultaneously, thus excluding double-mode behavior. On the other hand, for $\alpha_\nu = 1.2$ a double mode region appears between $\sim$ 5875 – 5915K and for $\alpha_\nu \ngth =\ngth 2.0$ this broadens to $\sim$ 5965 – 6050K. How does turbulent convection bring about double mode behavior? Fig. 2 shows that, in the region of interest, an increase in the turbulent eddy viscosity causes a rapid decrease in the stability of the fundamental limit cycle ($\bar\kappa_{1(0)}$, filled circles), but an increase in that of the first overtone limit cycle ($\bar\kappa_{0(1)}$, open circles). This description, though, does not tell us whether it is the effect of TC on the linear $\kappa$’s or on the nonlinear $q$’s, or on both, that is responsible for the beat pulsations. Table 1 shows the variation with $\alpha_\nu$ of the relevant model quantities, viz. the linear growth rates, the nonlinear coupling coefficients, the discriminant ${\cal D}= q_{00} q_{11} - q_{01}q_{10}$, the amplitude of the fundamental cycle and its stability coefficient with respect to overtone perturbations, and the same for the first overtone. The [*necessary*]{} condition for stable double mode pulsations, viz. ${\cal D} >$ 0, is never found to be satisfied in radiative models. In these models the cross-coupling always dominates over the self-saturation coefficients. Table 1 shows that an increase in the strength of the eddy viscosity causes $q_{00}$ and $q_{11}$ to increase faster than $q_{01}$ and $q_{10}$. It is therefore the resultant change in the sign of ${\cal D}$ that makes double mode behavior possible for sufficiently large $\alpha_\nu$. The condition for beat behavior is thus seen to be rather subtle in that it involves the effects of convection beyond the linear regime for which it seems difficult to give a ‘simple’ physical explanation. Fig. 3 gives the overall modal selection picture in the $\alpha_\nu$ –  plane. The linear edges of the instability region ($\kappa_0\ngth =\ngth 0$ and $\kappa_1\ngth =\ngth 0$) are shown as dashed lines. By computing the fundamental and first overtone limit cycles for a number of $\alpha_\nu$ and  values, by interpolation, we can obtain $\bar\kappa_{0(1)}$ or $\bar\kappa_{1(0)}$ as a function of $\alpha_\nu$ and , and in particular the loci where they vanish. The solid curves give the nonlinear pulsation edges and are marked ORE and FBE. It is straightforward to show that if the two linear growth rates vanish at the same point, the four curves will intersect in a single point on this diagram, that we label [*critical point*]{}. The curve marked OBE is the linear blue edge of the first overtone mode and it coincides with the overtone nonlinear blue edge up to and on the left of the critical point. The linear fundamental blue edge becomes also the fundamental blue edge above the critical point. Above the line ORE we have $\bar\kappa_{0(1)} \ngth > \ngth 0$ and the first overtone limit cycle is unstable. Below the line FBE the quantity $\bar\kappa_{1(0)}\ngth > \ngth 0$ and the fundamental limit cycle is unstable. Thus in the region marked ‘dm’ both single mode limit cycles are unstable, and this is the region of double-mode pulsation. In the small triangular region at the bottom, on the other hand, both limit cycles are stable, and [*either*]{} fundamental or first overtone limit cycles can occur. [ ]{} In summary, stable first overtone pulsations occur in the dotted region, delineated by the lines OBE and ORE. The fundamental limit cycle is stable in the region marked by open squares, delineated by FBE and FRE (not shown on the far right). This figure makes it particularly evident how TC favors double mode pulsations and why all efforts with radiative codes have failed in modelling beat Cepheids. We have seen that when TC effects are sufficiently large then the Cepheids should run into the double mode regime in both their crossings of the instability strip. Furthermore, as a Cepheid crosses the double mode regime redward, say, the first overtone amplitude should gradually go to zero while the fundamental amplitude increases from zero to the value it attains as a fundamental mode Cepheid (BK86). The question arises whether this nonresonant scenario that is derived from the amplitude equations is in agreement with the observations. The four SMC beat Cepheids from the EROS survey (analyzed by Beaulieu and reproduced in Buchler 1998) all have the same amplitude ratios, $A_0/A_1 \sim 0.45$, a priori in disagreement with the nonresonant scenario shown in Fig. 1. of BK86 that suggests that Cepheids with all amplitudes ratios should occur. In Fig. 4 we display the behavior of the component modal amplitudes of the beat Cepheid models for the $\alpha_\nu\ngth =\ngth 1.2$ sequence of Fig. 2. The amplitudes of the stable single mode limit cycles are shown as solid lines with solid dots for the fundamental and open dots for the first overtone, and as dashed lines where they are unstable. The fundamental and first overtone component amplitudes of the stable double mode pulsators are shown as solid squares and open diamonds, respectively. It is seen that although the modal amplitudes do indeed vary continuously throughout as the double mode regime is traversed, the behavior is very rapid near the cooler side. The reason for this unexpected behavior is that the q’s are [*not*]{} constant in this sequence, and what is more, they vary in such a way that ${\cal D}$ happens go through zero around 5850K. It is the presence of this nearby pole that causes a change in the curvature of $A_0$. According to Fig. 4 it is therefore much more likely to find beat Cepheids in the slowly varying regime where the ratio $A_0/A_1\approxlt 0.5$. The computed behavior of the modal amplitudes is thus in agreement with the observed SMC Cepheids, and the nonresonant scenario is consistent with the observations. We have demonstrated that turbulent convection leads naturally to beat behavior in Cepheids, which does not occur with purely radiative models. The reason is that the nonlinear effects of TC dissipation can create a region in which both the fundamental and the first overtone cycles are unstable, and the model undergoes stable double mode pulsations. At a more basic level the amplitude equation formalism shows that turbulent convection modifies the nonlinear coupling between the fundamental and first overtone modes in such as way as to allow beat behavior. The development of a relaxation code (TC) to find periodic pulsations, and a Floquet stability analysis of these limit cycles has made the search quite efficient, and a broader survey of beat Cepheids, with wide ranges of metallicities is in progress. This will also search for beat Cepheid models that pulsate in the first and second overtones. [ This research has been supported by the Hungarian OTKA (T-026031), AKP (96/2-412 2,1) and by the NSF (AST95–28338) at UF. Two of us (JPB) and (ZK) thank the French Académie des Sciences for financial support. ]{} [rrrrrrrrrrrr]{}  $\alpha_\nu$ &$\kappa_0$   &$\kappa_1$    &$q_{00}$ &$q_{01}$ &$q_{10}$ &$q_{11}$  &${\cal D}$ &$A_0$   &$\bar\kappa_{0(1)}$   &$A_1$   &$\bar\kappa_{1(0)}$  \ 0.5 &2.382e-3 &1.124e-2 &1.199 &3.532 & 4.898 &13.311 &–1.343 & 4.458e-2 &1.510e-3 &2.906e-2 &–6.012e-4\ 1.0 & 2.169e-3 & 8.636e-3 & 1.815 & 3.865 & 6.645 & 14.924 & 1.407 & 3.457e-2 & 6.959e-4 & 2.406e-2 & –6.752e-5\ 1.2 &2.082e-3 &7.582e-3 &2.179 &4.200 & 7.554 &16.001 &3.140 & 3.091e-2 &3.650e-4&2.177e-2 &9.169e-5\ 1.5 &1.947e-3 &5.983e-3 &2.893 &4.917 & 9.212 & 18.172 & 7.269 & 2.595e-2 & –2.188e-4 & 1.814e-2 & 3.285e-4\ 2.0 &1.720e-3 &3.282e-3 &4.685 &7.194 & 13.039 &24.229 &19.704 & 1.916e-2 &–1.506e-3 &1.164e-2 & 7.458e-4\ Alexander, D. R., Ferguson, J. W. 1994, ApJ 437, 879 Bono, G., Stellingwerf, R.F. 1994, ApJ Suppl 93, 233–269 Buchler, J.R. 1998, in [*A Half Century of Stellar Pulsation Interpretations: A Tribute to Arthur N. Cox*]{}, ed. P.A. Bradley and J.A. Guzik, ASP Conf. Ser. 135, 220 Buchler, J.R., Kolláth, Z., Beaulieu, J.P. , Goupil. M.J. 1996, ApJ Letters, 462, L83 Buchler, J.R., , Kovács, G. 1986, ApJ 308, 661, \[BK86\]; 1987, ibid. 318, 232 \[BK87\] Buchler, J.R., Moskalik, P., Kovács, G. 1991, ApJ 380, 185. Gehmeyr, M. , Winkler, K.-H. A. 1992, AA 253, 92–100; ibid. 253, 101–112 Iglesias, C. A.& Rogers, F. J. 1996, ApJ 464, 943 Kovács, G. , Buchler, J.R. 1987, ApJ 324, 1026. Kuhfuss, R. 1986, AA 160, 116 Stellingwerf, R.F. 1975, ApJ 199, 705 Stellingwerf, R.F. 1982, ApJ 262, 330 Yecko, P., Kolláth Z., Buchler, J. R. 1998, A&A, in press

--- abstract: | We investigate the relationship between galaxies and metal-line absorption systems in a large-scale cosmological simulation with galaxy formation. Our detailed treatment of metal enrichment and non-equilibrium calculation of oxygen species allow us, for the first time, to carry out quantitative calculations of the cross-correlations between galaxies and absorbers. We find the following: (1) The cross-correlation strength depends weakly on the absorption strength but strongly on the luminosity of the galaxy. (2) The correlation distance increases monotonically with luminosity from $\sim 0.5-1h^{-1}$Mpc for $0.1L_*$ galaxies to $\sim 3-5h^{-1}$Mpc for $L_*$ galaxies. (3) The correlation distance has a complicated dependence on absorber strength, with a luminosity-dependent peak. (4) Only 15% of absorbers lie near $\ge L_\mathrm{z,*}$ galaxies. The remaining 85%, then, must arise “near” lower-luminosity galaxies, though, the positions of those galaxies is not well-correlated with the absorbers. This may point to pollution of intergalactic gas predominantly by smaller galaxies. (5) There is a subtle trend that for $\gtrsim 0.5L_\mathrm{z,*}$ galaxies, there is a positive correlation between absorber strength and galaxy luminosity in the sense that stronger absorbers have a slightly higher probability of finding such a large galaxy at a given projection distance. For less luminous galaxies, there seems to be a negative correlation between luminosity and absorber strength. author: - 'Rajib Ganguly, Renyue Cen, Taotao Fang and Kenneth Sembach' title: Correlations between Absorbers and Galaxies at Low Redshift --- Introduction ============ Cosmological hydrodynamic simulations have shown that most of the so-called missing baryons [@fhp98] may be in a filamentary network of Warm-Hot Intergalactic Medium [WHIM; @co99; @d01]. Visual inspection of simulations suggests that the WHIM is spatially correlated with galaxies. This is consistent with the observed large-scale structure of galaxies, as well as the physical expectation that both galaxies and intergalactic medium (IGM) are subject to the dominant gravitational force of dark matter which tends to lead to such large-scale structures [@z70]. Moreover, the WHIM may provide a primary conduit for matter and energy exchanges between galaxies and the IGM. Thus, a detailed understanding of the WHIM may shed useful light on galaxy formation [e.g., @cno05]. The $\lambda \lambda$1032, 1038 absorption line doublet in the spectra of low-redshift QSOs provides a valuable probe of the WHIM [e.g., @l07; @sd07; @t08]. In this [*Letter*]{} we use cosmological hydrodynamic simulations to make predictions of the correlations between absorbers and galaxies, which may be used for detailed comparisons with upcoming [*Hubble Space Telescope Cosmic Origins Spectrograph*]{} observations. Simulation and Construction of Absorber and Galaxy Catalogs =========================================================== We use the simulation from @co06 and @cf06, to make predictions of the relationship between WHIM absorption and the presence of galaxies. This cold dark matter simulation assumes: $\Omega_M = 0.31$, $\Omega_b = 0.048$, $\Omega_\Lambda=0.69$, $\sigma_8 = 0.89$, $H_0 = 100 h=69$[ [km s]{}$^{-1}$]{}Mpc$^{-1}$, $n_s = 0.97$, co-moving box size $85 h^{-1}$Mpc and $1024^3$ cells. The simulation follows star formation using a physically-motivated prescription and includes feedback processes from star formation to the IGM in the form of UV radiation and galactic superwinds carrying energy and metal-enriched gas [@co06]. In star-formation sites, “star particles” are produced at each time step, which typically have a mass of $10^6$M$_\odot$. Galaxies are identified post facto using the HOP grouping scheme [@Eisenstein98 see Nagamine et al. 2001 for details] on these particles. This grouping scheme provides a catalog of galaxies containing 3D positions, peculiar velocities, stellar masses and ages. The catalog consists of 33,887 galaxies. Stellar population synthesis models from @bc03 are used to compute luminosities in all SDSS bands. For the purposes of this paper, we focus on the Sloan z-band, which is centered at 9100Å, and has a width of 1200Å [@sdss]. For galaxies, the flux in this band is most tightly correlated (of all the Sloan bands) with the total stellar mass [e.g., @kauffmann03]. We find that the luminosities of field galaxies in this simulation follow a Schechter function down to $\sim10^{-3.5} L_\mathrm{z,*}$($L_\mathrm{z,*} = 3 \times 10^{11}\,L_\odot$, Ganguly et al. 2008, in preparation). We extract a [$100 \times 100$]{} grid of sight-lines uniformly separated by $850 h^{-1}$kpc. For each sight-line, a synthetic spectrum of $\lambda$1032 absorption is generated, taking into account effects due to peculiar velocities and thermal broadening. For each sight-line, we decompose the spectrum into individual Gaussian-broadened components with an algorithm similar to [autovp]{} [@autovp]. Finally, we associate grouped components into systems; this is important since, physically, it does not serve our purpose to treat individual components separately when comparing to the locations of galaxies. \[A single physical system like a galactic disk/halo is often composed of multiple components. This is a result of complicated velocity structures and the clumpiness of gas. Historically, older surveys did not have the spectral dispersion needed to resolve individual components. Therefore, it is desirable to re-group these components back to physically independent systems.\] We accomplish this by computing the one-dimensional two-point correlation function of components and identify a characteristic velocity scale. We find that, on small scales, absorption-line components are correlated out to a velocity separation of $\sim\!\!\pm300$[ [km s]{}$^{-1}$]{}, which we adopt as the characteristic velocity interval to identify systems. For each system, we record the integrated flux-weighted centroid redshift, column density, and $\lambda$1032 rest-frame equivalent width. A total of $\sim$180,000 components were found to be grouped into $\sim$21,000 systems with $W_\lambda(\lambda1032) \ge 1$mÅ. Analysis & Results ================== We consider the probability that an absorber of a given equivalent width or column density lies within some distance of a galaxy with a certain luminosity. We take three approaches to address this question, but we must first tackle the problem of assigning galaxies to absorbers. Due to peculiar velocity effects, the exact 3D separation between an absorber and a galaxy cannot be precisely known, although the projected distance in the sky plane can be directly measured. Thus, we simply limit absorber-galaxy associations to within the physically motivated line-of-sight separation of $1000$[ [km s]{}$^{-1}$]{}, corresponding approximately to the velocity dispersion of clusters of galaxies. Approach \#1: For each absorber, we find the most luminous galaxy within a projected distance of 1h$^{-1}$Mpc or 3h$^{-1}$Mpc (and within the aforementioned velocity separation). Figure \[fig:1\] shows the cumulative fraction of absorbers that have a galaxy of luminosity $>L_\mathrm{z}$ within this cylindrical volume. We show this distribution for absorbers of different integrated column density cuts (upper panel) or $\lambda$1032 equivalent width cuts (lower panel). For comparison, we also do the same exercise for a sample of galaxies placed randomly in the box, but with the same luminosity function and the same total number of galaxies. Approach \#2: We relax the projected distance requirement, and find the closest galaxy with luminosity $\ge f L_\mathrm{z,*}$(with four different $f=0.03,0.1,0.3,1$). Figure \[fig:2\] shows the cumulative fraction of absorbers as a function of projected distance. As with Figure \[fig:1\], we show different column density (upper panel) and equivalent width cuts (lower panel). In addition, we show the different cuts in galaxy luminosity, and we repeat the exercise with a sample of randomly placed galaxies. Note that Figures \[fig:1\] and \[fig:2\] are complementary. A vertical cut in Figure \[fig:2\] at 1h$^{-1}$Mpc or 3h$^{-1}$Mpc, with the cumulative fraction of absorbers plotted against luminosity reproduces Figure \[fig:1\]. Approach \#3; For each absorber, we determine the number of galaxies that lie within the 3h$^{-1}$Mpc radius, 2000[ [km s]{}$^{-1}$]{} deep cylindrical volume. In Table \[tab:ngal\], we list the fraction of these absorbers that have more than n galaxies ($n=0,1,...$) in that volume. We do this for two subsamples of absorbers, those with $\lambda1032$ equivalent width larger than 30mÅ (typical detection limit for HST/STIS and FUSE spectra) and 10mÅ(expected limit for HST/COS observations). In addition to the equivalent width cuts in the absorber sample, we also make luminosity cuts in the galaxy sample. Current catalogs are typically able to reach galaxy luminosities of 0.1$L_*$ for statistically interesting volumes [e.g., @s06]. Taking Figures \[fig:1\] and \[fig:2\] together, we first address the question of which galaxies, and on what scales, are correlated (if any) with WHIM absorption. In Figure \[fig:1\], at luminosities fainter than $0.3-0.5 L_\mathrm{z,*}$, the curves lie below the equivalent curves for randomly-placed galaxies. That is, finding a lower luminosity galaxy near an absorber is less probable than if galaxies were uncorrelated with the absorbers for $D=1-3h^{-1}$Mpc. This merely implies that the correlation distance, defined to be the distance within which there is positive enhancement of pairs compared to random distributions, is smaller than $1h^{-1}$Mpc for these low luminosity galaxies, consistent with results summarized in Table 2 (see below). At higher luminosities, the curves are above the random galaxies. This implies that galaxies with $\gtrsim0.3 L_\mathrm{z,*}$ are correlated with WHIM absorption with the correlation distance larger than $3h^{-1}$Mpc, again consistent with results summarized in Table 2. This same information may be seen, in a different way, in Figure \[fig:2\], as the families of curves for fainter galaxies lie below equivalent ones for random galaxies, but the converse is true for higher luminosity galaxies. A perhaps somewhat counter-intuitive result in Figure 1 is that an absorber is more likely to find a faint galaxy within the $D=1h^{-1}$Mpc cylinder, if the galaxies were randomly distributed, and $\sim 30\%$ of absorbers do not find any galaxy to the faintest limit simulated. This is due to the fact that the galaxies themselves are more strongly clustered than absorbers among themselves or between galaxies and absorbers. Therefore, absorbers will have a lower probability of finding neighboring galaxies than when the latter are randomly distributed, beyond the cross-correlation distance. An interesting feature immediately visible from Figures \[fig:1\] and \[fig:2\] is that the association between galaxies and absorbers depends weakly on either the column density or $\lambda$1032 equivalent width. This suggests that, while overall galaxies and absorbers are correlated on small scales, the physical properties of absorbers (e.g., strength, kinematics, number of components) themselves do not display any tight correspondence with nearby galaxies. It seems that the strengths of absorbers do not provide useful indicators for the properties of nearby galaxies. Physically, it suggests that absorbers may arise in the vicinities of galaxies in a wide variety of ways through complex feedback and thermodynamic processes. Complex interactions involving gravity-induced shocks, feedback and photoionization appear to have erased or smoothed out any potential trend with respect to column density or equivalent width. This finding is in accord with observations [e.g., @p06]. From Figure \[fig:2\], it appears that the distance out to which absorbers are correlated with galaxies is a function of both galaxy luminosity and absorber strength (even though the mere presence of a galaxy is not tightly correlated with absorber strength as from Figure \[fig:1\]). Comparison of the black curves in Figure \[fig:2\], showing the results of uncorrelated absorber and galaxy positions, with the equivalent families of the curves for the simulated galaxies and absorbers shows that there is typically a distance beyond which it is more probable to find an uncorrelated galaxy. This distance changes depending on the galaxy luminosity and the absorption strength. In the extreme case, there is no distance at which the strongest absorbers are correlated with the least luminous galaxies. In Table \[tab:distlum\], we list the correlation distances as a function of galaxy luminosity and absorption strength. We note two interesting features from the table: (1) For a given equivalent width limit, the correlation distance is a monotonic function of the limiting luminosity. (2) However, for a given limiting luminosity, the correlation distance is not a monotonic function of equivalent width limit. This may also point toward the eclectic nature of the absorbers as mentioned above. It is interesting, however, to consider the number of galaxies that may be responsible for producing absorption. While the intragroup/intracluster medium (ICM) is typically too hot ($T\sim10^6$K) for to survive in appreciable quantities, the interfaces between warm, denser, photoionized clouds of temperature $T\sim 10^4$K and the ICM are potential locations for production. Examples of such interfaces include the boundaries between Milky Way high-velocity clouds and the hot Galactic corona [e.g., @fox05; @sem03]. From Table \[tab:ngal\], we find that 85% of , regardless of absorption equivalent width, do not lie near $L_\mathrm{z,*}$galaxies. Furthermore, the remaining 15% have at most 3 nearby $L_\mathrm{z,*}$, comparable the Local Group. This is not surprising given that $L_\mathrm{z,*}$ are not common. However, 99% of absorbers do lie near galaxies of lower luminosity, even if the presence of those galaxies is not correlated over what is expected from randomly placed galaxies. Figure \[fig:2\] shows that $\le$20% of absorbers should find an $L_\mathrm{z,*}$ galaxy within a projected distance of $5 \mathrm{h}^{-1}$Mpc. Of course, one would not likely associate absorbers with $L_\mathrm{z,*}$ galaxies at such large projected separations, since, for example, one would have already found a nearby $\ge 0.03L_\mathrm{z,*}$ galaxy within 1h$^{-1}$Mpc, or $\ge 0.1L_\mathrm{z,*}$ galaxy at closer distance with comparable probability. In any case, it seems unlikely that $L_\mathrm{z,*}$ galaxies at such remote distances are responsible for creating the absorbers. The rapid rise of probability in Figure \[fig:2\] from $D_\mathrm{min}=0$ to $D_\mathrm{min}\sim 1 \mathrm{h}^{-1}$Mpc from galaxies $\ge 0.03-0.1L_\mathrm{z,*}$ may reflect a ubiquitous physical connection between absorbers and these relatively small galaxies, perhaps a result of galactic superwinds being able to transport metals to a distance of $\le 1 \mathrm{h}^{-1}$Mpc from these galaxies. This, however, does not necessarily exclude larger galaxies from being able to do the same. The slower rise of probability in Figure \[fig:2\] from $D_\mathrm{min}=0$ to $D_\mathrm{min}\sim 5 \mathrm{h}^{-1}$Mpc for galaxies $\ge 0.3-1L_\mathrm{z,*}$ may be a result of the intrinsic correlation of large galaxies and small galaxies on these scales. These more detailed issues will be examined subsequently elsewhere. Our results appear to be in broad agreement with observations [e.g., @s06; @s07; @t06]. We take a more detailed look at the dependencies of probability on the column density or equivalent width. Closer examination of the curves in Figure \[fig:1\] (lower-right corner of both panels) and in Figure \[fig:2\] (solid curves, lower-left corner of lower panel) reveals that the probability of finding a $\sim L_\mathrm{z,*}$ galaxy is higher for absorbers with a higher column density equivalent width. In particular, $N$()$\ge 10^{14}$cm$^{-2}$ or $W_\lambda\ge 100$ mÅ absorbers deviate noticeably from the weaker absorbers. This is a reversal of the trends from other parts of those figures. For smaller galaxies, e.g., $\ge 0.3L_\mathrm{z,*}$ galaxies (the set of dotted curves in the lower panel of Figure \[fig:2\]), the trend is considerably weaker although still visible. For still smaller galaxies, the trend is reversed, with weaker absorbers having a higher probability than stronger ones. These results seem to suggest that these very strong absorbers tend to be produced in richer, high density environments where the probability of finding massive galaxies is enhanced (we will examine this physical link elsewhere). The fact that weaker absorbers have a higher probability of finding a galaxy than stronger absorbers, for galaxies less luminous than $\sim 0.2-0.3L_\mathrm{z,*}$(Figure \[fig:1\]), once again suggests that these relatively weaker absorbers ($W_\lambda\le 50$mÅ) are probably produced by galaxies of $\sim 0.1L_\mathrm{z,*}$, not by more luminous galaxies. This is consistent with the trend in Figure \[fig:2\] above. Conclusions =========== We investigate the relationship between galaxies and metal-line absorption systems in a large-scale cosmological simulation with galaxy formation included. Our detailed treatment of metal enrichment and non-equilibrium calculation of oxygen species allow us, for the first time, to carry out quantitative calculations of the cross-correlations between galaxies and absorbers. We examine the the cross-correlations between absorbers and galaxies as a function of projection distance, with the line-of-sight velocity separation between an absorber and a galaxy constrained to within $\pm 1000$km/s. Here are some major findings: (1) The cross-correlation strength depends only weakly on the strength of the absorber but strongly on the luminosity of the galaxy. This result suggests that absorbers are produced ubiquitously and their physical/thermal properties and history vary widely, presumably due to the combined effects of gravitational shocks, feedback, photo-ionization and cooling processes. (2) The correlation length, however, does depend on both the galaxy luminosity and on the absorber strength from $\sim 0.5-1h^{-1}$Mpc for $0.1L_*$ galaxies to $\sim 3-5h^{-1}$Mpc for $L_*$ galaxies. While the dependence on luminosity is monotonic, the dependence on limiting equivalent width appears to peak at some luminosity-dependent value and then falls. (3) Only 15% of absorbers lie near $\ge L_\mathrm{z,*}$ galaxies. Thus the remaining 85% must be produced by gas ejected from fainter galaxies. The positions of lower-luminosity galaxies is not well correlated with absorbers (i.e. in comparison with randomly-placed galaxies). This may point toward pollution of intracluster gas by many galaxies, rather than a single high-luminosity galaxy. (4) For $\gtrsim 0.5L_\mathrm{z,*}$ galaxies, there is a positive correlation between absorber strength and galaxy luminosity (stronger absorbers have a slightly higher probability if finding such a large galaxy at a given projection distance). The reverse seems true for less luminous galaxies. On average, these results indicate that very strong absorbers tend to be produced in richer, high density environments where it is more likely to find massive galaxies. The spatial resolution of our simulation is $\sim 80h^{-1}$kpc. While this is adequate for resolving large galaxies, it becomes marginal for galaxies in halos of total mass less than $\sim 10^{11} M_\odot$. Therefore, some of the smaller galaxies of luminosities $0.01-0.03L_\mathrm{z,*}$ may be significantly affected and their abundances underestimated. In addition, systems that would have been produced from these under-resolved galaxies may be absent. As a result, one should treat the cross-correlation strength between absorbers and the low-luminosity galaxies as a lower bound. But we hope that the preliminary results presented here will provide a useful framework for comparison with upcoming imaging campaigns of galaxies in the field of quasars and spectroscopic observations with the [*Cosmic Origins Spectrograph*]{}. Bruzual, G., & Charlot, S. 2003, , 344, 1000 , R., & [Fang]{}, T. 2006, ApJ, 650, 573 , R., & [Ostriker]{}, J. P. 1999, , 514, 1 Cen, R., Nagamine, K., & Ostriker, J. P. 2005, , 635, 86 , R., & [Ostriker]{}, J.P. 2006, ApJ, 650, 560 Davé, R., Hernquist, L., Weinberg, D. H., & Katz, N. 1997, , 477, 21 Davé, R., et al. 2001, ApJ, 552, 473 Eisenstein, D. J. and Hut, P. 1998, , 498, 137 Fox, A. J., Wakker, B. P., Savage, B. D., Tripp, T. M., Sembach, K. R., & Bland-Hawthorn, J. 2005, , 630, 332 Fukugita, M., Ichikawa, T., Gunn, J. E., Doi, M., Shimasaku, K., & Schneider, D. P. 1996, , 111, 1748 Fukugita, M., Hogan, C.J., & Peebles, P.J.E. 1998, ApJ, 503, 518 Kauffmann, G., et al. 2003, , 341, 33 Lehner, N., Savage, B. D., Richter, P., Sembach, K. R., Tripp, T. M., & Wakker, B. P. 2007, , 658, 680 Prochaska, J.X., Chen, H.-W., Howk, J.C., Weiner, B.J., & Mulchaey, J.S. 2004, ApJ, 647, 718 Prochaska, J.X., Weiner, B.J., Chen, H.-W., & Mulchaey, J.S. 2006, ApJ, 643, 680 Sembach, K. R., et al. 2003, , 146, 165 Shull, J. M., & Danforth, C. W. 2007, , in press (arXiv:0709.4030v1) Stocke, J.T., Penton, S.V., Danforth, C.W., Shull, J.M., Tumlinson, J., & McLin, K.M. 2006, ApJ, 641, 217 Stocke, J.T., Danforth, C.W., Shull, J.M., & Penton, S.V. 2007, arXiv:0708.4362 Tripp, T.M., Aracil, B., Bowen, D.V., & Jenkins, E.B. 2006, ApJ, 643, L77 Tripp, T. M., Sembach, K. R., Bowen, D. V., Savage, B. D., Jenkins, E. B., Lehner, N., & Richter, P. 2008, , in press (arXiv:0706.1214v1) Zel’dovich, Y.B. 1970, A&A, 5, 84

--- abstract: 'In this paper, we collect various structural results to determine when an integral homology $3$–sphere bounds an acyclic smooth $4$–manifold, and when this can be upgraded to a Stein embedding. In a different direction we study whether smooth embedding of connected sums of lens spaces in $\mathbb{C}^2$ can be upgraded to a Stein embedding, and determined that this never happens.' address: - | Department of Mathematics\ Georgia Institute of Technology\ Atlanta\ Georgia - | Department of Mathematics\ University of Alabama\ Tuscaloosa\ Alabama author: - 'John B. Etnyre' - Bülent Tosun title: Homology spheres bounding acyclic smooth manifolds and symplectic fillings --- Introduction ============ The problem of embedding one manifold into another has a long, rich history, and proved to be tremendously important for answering various geometric and topological problems. The starting point is the Whitney Embedding Theorem: every compact $n$–dimensional manifold can be smoothly embedded in $\mathbb {R}^{2n}$. In this paper we will focus on smooth embeddings of 3–manifolds into $\R^4$ and embeddings that bound a convex symplectic domain in $(\R^4, \omega_{std})$. One easily sees that given such an embedding of a (rational) homology sphere, it must bound a (rational) homology ball. Thus much of the paper is focused on constructing or obstructing such homology balls. Smooth embeddings ----------------- In this setting, an improvement on the Whitney Embedding Theorem, due to Hirsch [@Hirsch:embedding] (also see Rokhlin [@Rohlin:3manembedding] and Wall [@Wall:embedding]), proves that every $3$–manifold embeds in $\mathbb{R}^5$ smoothly. In the smooth category this is the optimal result that works for all $3$–manifolds; for example, it follows from a work of Rokhlin that the Poincaré homology sphere $P$ cannot be embeded in $\mathbb{R}^4$ smoothly. On the other hand in the topological category one can always find embeddings into $\R^4$ for any integral homology sphere by Freedman’s work [@Freedman:4manifolds]. Combining the works of Rokhlin and Freedman for $P$ yields an important phenomena in $4$–manifold topology: there exists a closed oriented non-smoothable $4$–manifold — the so called $E_8$ manifold. In other words, the question of [*when does a $3$–manifold embeds in $\mathbb{R}^4$ smoothly*]{} is an important question from the point of smooth $4$–manifold topology. This is indeed one of the question in the Kirby’s problem list (Problem $3.20$) [@Kirby:problemlist]. Since the seminal work of Rokhlin in 1952, there has been a great deal of progress towards understanding this question. On the constructive side, Casson-Harrer [@CH], Stern, and Fickle [@Fickle] have found many infinite families of integral homology spheres that embeds in $\mathbb{R}^4$. On the other hand techniques and invariants, mainly springing from Floer and gauge theories, and symplectic geometry [@FS:R; @Manolescu:T; @OS:grading], have been developed to obstruct smooth embeddings of $3$–manifolds into $\mathbb{R}^4$. It is fair to say that despite these advances and lots of work done in the last seven decades, it is still unclear, for example, which Brieskorn homology spheres embed in $\mathbb{R}^4$ smoothly and which do not. A weaker question is whether an integral homology sphere can arise as the boundary of an acyclic $4$–manifold? Note that a homology sphere that embeds in $\mathbb{R}^4$ necessarily bounds an acyclic manifold, and hence is homology cobordant to the $3$–sphere. Thus a homology cobordism invariant could help to find restrictions, and plenty of such powerful invariants has been developed. For example, for odd $n$, $\Sigma(2,3,6n-1)$ and $\Sigma(2,3,6n+1)$ have non-vanishing Rokhlin invariant. For even $n$, $\Sigma(2,3,6n-1)$ has $R=1$, where $R$ is the invariant of Fintushel and Stern, [@FS:R]. Hence none of these families of homology spheres can arise as the boundary of an acyclic manifold. On the other hand, for $\Sigma(2,3,12k+1)$ all the known homology cobordism invariants vanish. Indeed, it is known that $\Sigma(2,3,13)$ [@AKi] and $\Sigma(2,3,25)$ [@Fickle] bound contractible manifolds of Mazur type. Motivated by the questions and progress mentioned above and view towards their symplectic analogue, we would like to consider some particular constructions of three manifolds bounding acyclic manifolds. Our first result is the following, which follows by adapting a method of Fickle. \[main1\] Let $K$ be a knot in the boundary of an acyclic, respectively rationally acyclic, $4$–manifold $W$ which has a genus one Seifert surface $F$ with primitive element $[b]\in H_1(F)$ such that the curve $b$ is slice in $W.$ If $b$ has $F$–framing $s$, then the homology sphere obtained by $\frac{1}{(s\pm 1)}$ Dehn surgery on $K$ bounds an acyclic, respectively rationally acyclic, $4$–manifold. Fickle [@Fickle] proved this theorem under the assumption that $\partial W$ was $S^3$ and $b$ was an unknot, but under these stronger hypothesis he was able to conclude that the homology sphere bounds a contractible manifold. Fintushel and Stern conjectured, see [@Fickle], the above theorem for $\frac{1}{k(s\pm 1)}$ Dehn surgery on $K$, for any $k\geq 0$. So the above theorem can be seen to verify their conjecture in the $k=1$ case. As noted by Fickle, if the conjecture of Fintushel and Stern is true then all the $\Sigma(2,3,12k+1)$ will bound acyclic manifolds since they can be realized by $-1/2k$ surgery on the right handed trefoil knot that bounds a Seifert surface containing an unknot for which the surface gives framing $-1$. Notice that if $b$ is as in the theorem, then the Seifert surface $F$ can be thought of as obtained by taking a disk around a point on $b$, attaching a 1–handle along $b$ (twisting $s$ times) and then attaching another 1–handle $h$ along some other curve. The proof of Theorem \[main1\] will clearly show that $F$ does not have to be embedded, but just ribbon immersed so that cutting $h$ along a co-core to the handle will result in a surface that is “ribbon isotopic" to an annulus. By ribbon isotopic, we mean there is an 1-parameter family of ribbon immersions between the two surfaces, where we also allow a ribbon immersion to have isolated tangencies between the boundary of the surface and an interior point of the surface. Consider the (zero twisted) $\pm$ Whitehead double $W_\pm(K_p)$ of $K_p$ from Figure \[exampleKp\]. In [@Cha07], Cha showed that $K_p$ is rationally slice. That is $K_p$ bounds a slice disk in some rational homology $B^4$ with boundary $S^3$. (Notice that $K_1$ is the figure eight knot originally shown to be rationally slice by Fintushel and Stern [@FintushelStern84].) Thus Theorem \[main1\] shows that $\pm 1$ surgery on $W_\pm(K_p)$ bounds a rationally acyclic 4–manifolds. This is easy to see as a Seifert surface for $W_\pm(K_p)$ can be made by taking a zero twisting ribbon along $K_p$ and plumbing a $\pm$ Hopf band to it. [exampleKp]{} (44.5, 51)[$-p$]{} (48,6)[$p$]{} Moreover, from Fickle’s original version of the theorem, $\pm\frac12$ surgery on $W_\pm(K_p)$ bounds a contractible manifold. We can generalize this example as follows. Given a knot $K$, we denote by $R_m(K)$ the $m$-twisted ribbon of $K$. That is take an annulus with core $K$ such that its boundary components link $m$ times. Now denote by $P(K_1, K_2, m_1, m_2)$ the plumbing of $R_{m_1}(K_1)$ and $R_{m_2}(K_2)$. If the $K_i$ are rationally slice then $\frac{1}{m_i\pm 1}$ surgery on $P(K_1, K_2, m_1, m_2)$ yields a manifold bounding a rationally acyclic manifolds; moreover, if the $K_i$ are slice in some acyclic manifold, then the result of these surgeries will bound an acyclic manifold. [**Symplectic embeddings.**]{} Another way to build examples of integral homology spheres that bound contractible manifolds is via the following construction. Let $K$ be a slice knot in the boundary of a contractible manifold $W$ (e.g. $W=B^4$), then $\frac{1}{m}$ Dehn surgery along $K$ bounds a contractible manifold. This is easily seen by removing a neighborhood of the slice disk from W (yielding a manifold with boundary 0 surgery on $K$) and attaching a 2–handle to a meridian of $K$ with framing $-m$. With this construction one can find examples of three manifolds modeled on not just Seifert geometry, for example $\Sigma(2,3,13)$ is the result of $1$ surgery on Stevedore’s knot $6_1$ but also hyperbolic geometry, for example the boundary of the Mazur cork is the result of $1$ surgery on the pretzel knot $P(-3,3,-3)$, which is also known as $\overline {9}_{46}$. See Figure \[fig:smooth\]. ![On the left is the 3-manifold $Y_{m,n}$ described as a smooth $\frac{1}{m}$ surgery on the slice knot $P(3,-3,-n)$ for $n\geq 3$. On the right is the contractible Mazur-type manifold $W_{m,n}$ with $\partial W_{m,n}\cong Y_{m,n}$. Note the $m=1,~n=3$ case yields the original Mazur manifolds (with reversed orientation).[]{data-label="fig:smooth"}](RegSlicetoContractible.pdf){width="16cm"} We ask the question of when $\frac{1}{m}$ surgery on a slice knot produces a [*Stein contractible*]{} manifold. Here there is an interesting asymmetry not seen in the smooth case. \[regslice\] Let $L$ be a Legendrian knot in $(S^3, \xi_{std})$ that bounds a regular Lagrangian disc in $(B^4, w_{std})$. Contact $(1+\frac{1}{m})$ surgery on $L$ (so this is smooth $\frac{1}{m}$ surgery) is the boundary of a contractible Stein manifolds if and only if $m>0$. This result points out an interesting angle on a relevant question in low dimensional contact and symplectic geometry: which compact contractible 4-manifolds admit a Stein structure? In [@MT:pseudoconvex] the second author and Mark found the first example of a contractible manifold without Stein structures with either orientation. This manifold is a Mazur-type manifold with boundary the Brieskorn homology sphere $\Sigma(2,3,13)$. A recent conjecture of Gompf remarkably predicts that Brieskorn homology sphere $\Sigma(p,q,r)$ can never bound acyclic Stein manifolds. It is an easy observation that $\Sigma(2,3,13)$ is the result of smooth $1$ surgery along the stevedore’s knot $6_1$. The knot $6_1$ is not Lagrangian slice, and indeed if Gompf conjecture is true, then by Theorem \[regslice\] $\Sigma(2,3,13)$ can never be obtained as a smooth $\frac{1}{n}$ surgery on a Lagrangian slice knot for any natural number $n$. Motivated by this example, Theorem \[regslice\], and Gompf’s conjecture we make the following weaker conjecture. No non-trivial Brieskorn homology sphere $\Sigma(p,q,r)$ can be obtained as smooth $\frac{1}{n}$ surgery on a regular Lagrangian slice knot. On the other hand as in Figure  \[fig:smooth\] we list a family of slice knots, that are regular Lagrangian slice because they bound decomposable Lagrangian discs and by [@CET] decomposable Lagrangian cobordisms/fillings are regular. We explicitly draw the contractible Stein manifolds these surgeries bound in Figure \[fig:stein\]. ![Stein contractible manifold with $\partial X_{m,n}\cong Y_{m,n}$.[]{data-label="fig:stein"}](RegSlicetoContractibleStein.pdf){width="12cm"} A related embedding question is the following: when does a lens space L(p,q) embeds in $\mathbb{R}^4$ or $S^4$? Two trivial lens spaces, $S^3$ and $S^1\times S^2$ obviously have such embeddings. On the other hand, Hantzsche in $1938$ [@Hantzsche] proved, by using some elementary algebraic topology that if a $3$–manifold $Y$ embeds in $S^4$, then the torsion part of $H_1(Y)$ must be of the form $G \oplus G$ for some finite abelian group $G$. Therefore a lens space $L(p,q)$ for $|p|>1$ never embeds in $S^4$ or $\mathbb{R}^4$. For punctured lens spaces, however the situation is different. By combining the works of Epstein [@Epstein] and Zeeman [@Zeeman], we know that, a punctured lens space $L(p,q)\setminus B^3$ embeds in $\mathbb{R}^4$ if and only if $p>1$ is odd. Note that given such an embedding a neighborhood of $L(p,q)\setminus B^3$ in $\mathbb{R}^4$ is simply $(L(p,q)\setminus B^3)\times [-1,1]$ a rational homology ball with boundary $L(p,q)\# L(p,p-q)$ (recall $-L(p,q)$ is the same manifold as $L(p,p-q)$). One way to see an embedding of $L(p,q)\# L(p,p-q)$ into $S^4$ is as follows: First, it is an easy observation that if $K$ is a doubly slice knot (that is there exists a smooth unknotted sphere $S\subset S^4$ such that $S\cap S^3=K$), then its double branched cover $\Sigma_2(K)$ embeds in $S^4$ smoothly. Moreover by a known result of Zeeman $K \# m(K)$ is a doubly slice knot for any knot $K$ (here $m(K)$ is the mirror of $K$). It is a classic fact that $L(p,q)$ is a double branched cover over the the 2-bridge knot $K(p,q)$ (this is exactly where we need $p$ to be odd, as otherwise $K(p,q)$ is a link). In particular $L(p,q)\#L(p,p-q)$, being double branched cover of doubly slice knot $K(p,q)\#m(K(p,q))$, embeds in $S^4$ smoothly. On the other hand, Fintushel-Stern [@FS:lensspace] and independently Gilmer-Livingston [@GilmerLivingston] showed this is all that could happen. That is they proved that $L(p,q)\#L(p,q')$ embeds in $S^4$ if and only if $L(p,q')=L(p,p-q)$ and $p$ is odd. In particular for $p$ odd, $L(p,q)\#L(p,p-q)$ bounds a rational homology ball in $\mathbb{R}^4$. A natural question in this case is to ask whether any of this smooth rational homology balls can be upgraded to be Symplectic or Stein submanifold of $\mathbb{C}^2$. We prove that this is impossible. \[main2\] No contact structure on $L(p,q)\#L(p,p-q)$ has a symplectic filling by a rational homology ball. In particular, $L(p,q)\#L(p,p-q)$ cannot embed in $\mathbb{C}^2$ as the boundary of exact symplectic submanifold in $\mathbb{C}^2$. Donald [@Donald] generalized Fintushel-Stern and Gilmer-Livingston’s construction further to show that for $L=\#_{i=1}^h L(p_i,q_i)$, the manifold $L$ embeds smoothly in $\mathbb{R}^4$ if and only if there exists $Y$ such that $L\cong Y\#-Y$. Our proof of Theorem \[main2\] applies to this generalization to prove none of the sums of lens spaces which embed in $\mathbb{R}^4$ smoothly can bound an exact symplectic manifold in $\mathbb{C}^2.$ To prove this theorem we need a preliminary result of independent interest. \[prop\] If a symplectic filling $X$ of a lens space $L(p,q)$ is a rational homology ball, then the induce contact structure on $L(p,q)$ is a universally tight contact structure $\xi_{std}$. Recall that every lens space admits a unique contact structure $\xi_{std}$ that is tight when pulled back the covering space $S^3$. Here we are not considering an orientation on $\xi_{std}$ when we say it is unique. On some lens spaces the two orientations on $\xi_{std}$ give the same oriented contact structure and on some they are different. After completing a draft of this paper, the authors discovered that this result was previously proven by Fossati [@Fossati19pre] and Golla and Starkston [@GollaStarkston19pre]. As the proof we had is considerably different we decided to present it here. [**Acknowledgements:**]{} We are grateful to Agniva Roy for pointing out the work of Fossati and of Golla and Starkston. The first author was partially supported by NSF grant DMS-1906414. Part of the article was written during the second author’s research stay in Montreal in Fall 2019. This research visit was supported in part by funding from the Simons Foundation and the Centre de Recherches Mathmatiques, through the Simons-CRM scholar-in-residence program. The second author is grateful to CRM and CIRGET, and in particular to Steve Boyer for their wonderful hospitality. The second author was also supported in part by a grant from the Simons Foundation (636841, BT) Bounding acyclic manifolds ========================== We now prove Theorem \[main1\]. The proof largely follows Fickle argument from [@Fickle], but we repeat it here for the readers convince (and to popularize Fickle’s beautiful argument) and to note where changes can be made to prove our theorem. Suppose the manifold $\partial W$ is given by a surgery diagram $D$. Then the knot $K$ can be represented as in Figure \[TheKnot\]. There we see in grey the ribbon surface $F$ with boundary $K$ and the curve $b$ on the surface. [TheKnot]{} (48, 68)[$D$]{} (103, 15)[$K$]{} (62, 15)[$b$]{} (11, 10)[$-s+1$]{} (90, 33)[$0$]{} The result of $\frac{1}{s-1}$ surgery on $K$ is obtained by doing $0$ surgery on $K$ and $(-s+1)$ surgery on a meridian as shown in Figure \[TheKnot\]. (The argument for $\frac{1}{s+1}$ surgery is analogous and left to the reader.) Now part of $b$ is the core of one of the 1–handles making up $F$. So we can handle slide $b$ and the associated 1–handle over the $(-s+1)$ framed unknot to arrive at the left hand picture in Figure \[Step1\]. Then one may isotope the resulting diagram to get to the right hand side of Figure \[Step1\]. [Step1]{} (30, 26)[$D$]{} (84, 26)[$D$]{} (48, 11)[$0$]{} (100, 11)[$0$]{} (11, 6)[$-s+1$]{} (63, 6)[$-s+1$]{} (14.5, 17.5)[$1$]{} We now claim the left hand picture in Figure \[Step2\] is the same manifold as the right hand side of Figure \[Step1\]. To see this notice that the green part of the left hand side of Figure \[Step2\] consists of two 0-framed knots. Sliding one over the other and using the new 0-framed unknot to cancel the non-slid component results in the right hand side of Figure \[Step1\]. Before moving forward we discuss the strategy of the remainder of the proof. The left hand side of Figure \[Step2\] represents the 3-manifold $M$ obtained from $\partial W$ by doing $\frac{1}{s-1}$ surgery on $K$. We will take $[0,1]\times M$ and attach a 2–handle to $\{1\}\times M$ to get a 4-manifold $X$ with upper boundary $M'$ so that $M'$ is obtained from $W$ by removing a slice disk $D$ for $b$. Since $W$ is acyclic, the complement of $D$ will be a homology $S^1\times D^3$. Let $W'$ denote this manifold. Attaching $X$ upside down to $W'$ (that is attaching a 2–handle to $W'$) to get a 4–manifold $W''$ with boundary $-M$. Since $-M$ is a homology sphere, we can easily see that $W''$ is acyclic. Thus $-W''$ is an acyclic filling of $M$. Now to see we can attach the 2–handle to $[0,1]\times M$ as described above, we just add a 0-framed meridian to the new knot unknot on the left hand side of Figure \[Step2\]. This will result in the diagram on the right hand side of Figure \[Step2\]. [Step2]{} (32, 29)[$D$]{} (87, 29)[$D$]{} (50, 11)[$0$]{} (97, 11)[$0$]{} (45.5, 19)[$0$]{} (26.5, 17.5)[$0$]{} (11, 6.5)[$-s+1$]{} (70, 6)[$-s+1$]{} We are left to see that the right hand side of Figure \[Step2\] is the boundary of $W$ with the slice disk for $b$ removed. To see this notice that the two green curves in Figure \[Step2\] co-bound an embedded annulus with zero twisting (the grey in the figure) and one boundary component links the $(-s+1)$ framed unknot and the other does not. Sliding the former over the latter results in the left hand diagram in Figure \[Step3\]. Cancelling the two unknots from the diagram results in the right hand side of Figure \[Step3\] which is clearly equivalent to removing the slice disk $D$ for $b$ from $W$. [Step3]{} (34, 31)[$D$]{} (78, 31)[$D$]{} (26, 3.5)[$0$]{} (26.5, 17.5)[$0$]{} (11, 3)[$-s+1$]{} (92, 6)[$0$]{} Stein fillings ============== We begin this section by proving Theorem \[regslice\] concerning smooth $\frac 1 m$ surgery on a Lagrangian slice knot. We begin by recalling a result from [@CET] that says contact $(r)$ surgery on a Legendrian knot $L$ for $r\in(0,1]$ is strongly symplectically fillable if and only if $L$ is Lagrangian slice and $r=1$. Thus $(1+1/m)$ contact surgery for $m<0$ will never be fillable, much less fillable by a contractible Stein manifold. We now turn to the $m>0$ case and start by a particularly helpful visualization of the knot $L$ (here and forward $L$ stands both for the knot type and Legendrain knot that realizing the knot type that bounds the regular Lagrangian disk). By [@CET Theorem $1.9$, Theorem $1.10$], we can find a handle presentation of the 4-ball $B^4$ made of one 0–handle, and $n$ cancelling Weinstein $1$– and $2$–handle pairs, and a maximum Thurston-Bennequin unknot in the boundary of the 0–handle that is disjoint from $1$– and $2$–handles such that when the $1$– and $2$–handle cancellations are done the unknot becomes $L$. See Figure \[Handle\]. [Handle2]{} (56, 2)[$L$]{} Now smooth $1/m$ surgery on $L$ can also be achieved by smooth $0$ surgery (which corresponds to taking the complement of the slice disk) on $L$ followed by smooth $-m$ surgery on its meridian. As the proof of Theorem 1.1 in [@CET] shows, removing a neighborhood of the Lagrangian disk $L$ bounds from $B^4$ gives a Stein manifold with boundary $(+1)$ contact surgery on $L$ (that is smooth $0$ surgery on $L$). Now since the meridian to $L$ can clearly be realized by an unknot with Thurston-Bennequin invariant $-1$, we can stabilize it as necessary and attach a Stein $2$–handle to it to get a contractible Stein manifold bounding $(1+1/m)$ contact surgery on $L$ for any $m>1$. For the $m=1$ case we must argue differently. One may use Legendrian Reidemeister moves to show that in any diagram for $L$ as described above the $2$–handles pass through $L$ as shown on the left hand side of Figure \[normalform\]. [NormalForm]{} (74, 7.75)[$-1$]{} Smoothly doing contact $(1+1/1)$–surgery on $L$ (that is smooth $1$ surgery) is smoothly equivalent to replacing the left hand side of Figure \[normalform\] with the right hand side and changing the framings on the strands by subtracting their linking squared with $L$. Now notice that if we realize the right hand side of Figure \[normalform\] by concatenating $n$ copies of either diagram in Figures \[sandz\] (where $n$ is the number of red strands in Figure \[normalform\]) then the Thurston-Bennequin invariant of each knot involved in Figure \[normalform\] is reduced by the linking squared with $L$. Thus we obtain a Stein diagram for the result of $(2)$ contact surgery on $L$. [sandz]{} Notice that the diagram clearly describes an acyclic 4–manifolds and moreover the presentation for its fundamental group is the same as for the presentation for the fundamental group of $B^4$ given by the original diagram. Thus the 4–manifolds is contractible. We now turn to the proof that connected sums of lens spaces can never have acyclic symplectic fillings, but first prove Proposition \[prop\] that says any contact structure on a lens space that is symplectically filled by a rational homology ball must be universally tight. Let $X$ be a rational homology ball symplectic filling of $L(p,q)$. We show the induces contact structure must be the universally tight contact structure $\xi_{std}$. This will follow from unpacking recent work of Menke [@menke2018jsjtype] where he studies exact symplectic fillings of a contact $3$–manifold that contains a [*mixed torus*]{}. We start with the set-up. Honda [@Honda:classification1] and Giroux [@Giroux:classification] have classified tight contact structures on lens spaces. We review the statement of Honda in terms of the Farey tessellation. We use notation and terminology that is now standard, but see see [@Honda:classification1] for details. Consider a minimal path in the Farey graph that starts at $-p/q$ and moves counterclockwise to $0$. To each edge in this path, except for the first and last edge, assign a sign. Each such assignment gives a tight contact structure on $L(p,q)$ and each tight contact structures comes from such an assignment. If one assigns only $+$’s or only $-$’s to the edges then the contact structure is universally tight, and these two contact structures have the same underlying plane field, but with opposite orientations. We call this plane field (with either orientation) the the universally tight structure $\xi_{std}$ on $L(p,q)$. All the other contact structures are virtually overtwisted, that is they are tight structures on $L(p,q)$ but become overtwisted when pulled to some finite cover. The fact that at some point in the path describing a virtually overtwisted contact structure the sign must change is exactly the same as saying a Heegaard torus for $L(p,q)$ satisfies Menke’s mixed torus condition. \[mixed\] Let $(Y, \xi)$ denote closed, co-oriented contact $3$–manifold and let $(W, \omega)$ be its strong (resp. exact) symplectic filling. If $(Y,\xi)$ contains a mixed torus $T$, then there exists a (possibly diconnected) symplectic manifold $(W', \omega')$ such that: - $(W', \omega')$ is a strong (rep. exact) symplectic filling of its boundary $(Y',\xi')$. - $\partial W'$ is obtained from $\partial W$ by cutting along $T$ and gluing in two solid tori. - $W$ can be recovered from $W'$ by symplectic round $1$–handle attachment. In our case we have $X$ filling $L(p,q)$. Suppose the contact structure on $L(p,q)$ is virtually overtwisted. The theorem above now gives a symplectic manifold $X'$ two which a round 1–handle can be attached to recover $X$; moreover, $\partial X'$ is a union of two lens spaces or $S^1\times S^2$. However, Menke’s more detailed description of $\partial X'$ shows that $S^1\times S^2$ is not possible. We digress for a moment to see why this last statement is true. When one attaches a round 1–handle, on the level of the boundary, one cuts along the torus $T$ and then glues in two solid tori. Menke gives the following algorithm to determine the meridional slope for these tori. That $T$ is a mixed torus means there is a path in the Farey graph with three vertices having slope $r_1, r_2,$ and $r_3$, each is counterclockwise of the pervious one and there is an edge from $r_i$ to $r_{i+1}$ for $i=1,2$. The torus $T$ has slope $r_2$ and the signs on the edges are opposite. Now let $(r_3,r_1)$ denote slopes on the Farey graph that are (strictly) counterclockwise of $r_3$ and (strictly) clockwise of $r_1$. Any slope in $(r_3,r_1)$ with an edge to $r_2$ is a possible meridional slope for the glued in tori, and these are the only possible slopes. Now since our $r_i$ are between $-p/q$ and $0$ we note that if there was an edge from $r_2$ to $-p/q$ or $0$ then $r_2$ could not be part of a minimal path form $-p/q$ to $0$ that changed sign at $r_2$. Thus when we glue in the solid tori corresponding to the round 1–handle attachment, they will not have meridional slope $0$ or $-p/q$ and thus we cannot get $S^1\times S^2$ factors. The manifold $X'$ is either connected or disconnected. We notice that it cannot be connected because it is know that any contact structure on a lens space is planar [@Schoenenberger05], and Theorem 1.2 from [@Etnyre:planar] says any filling of a contact structure supported by a planar open book must have connected boundary. Thus we know that $X'$ is, in fact, disconnected. So $X'=X'_1\cup X'_2$ with $\partial X'_i$ a lens space. The Mayer–Vietoris sequence for the the decomposition of $X'$ into $X'_1\cup X'_2$ (glued along an $S^1\times D^2$ in their boundaries) shows that $H_1$ of $X_1'$ or $X_2'$ has rank 1, while both of their higher Betti numbers are 0. But now the long exact sequence for the pair $(X'_i,\partial X'_i)$ implies that $b_1$ must be 0 for both the $X_i'$. This contradiction shows that a symplectic manifold which is rational homology ball and with boundary $L(p,q)$ must necessarily induce the universally tight contact structure on the boundary. The statement about embeddings follows directly from the statement about symplectic fillings. To prove that result let $X$ be an exact symplectic filling of $L(p,q)\# L(p,p-q)$ that is also a rational homology ball. Observe that there is an embedded sphere in $\partial X$ as it is reducible. Eliashberg’s result in [@CE Theorem $16.7$] says that $X$ is obtained from another symplectic manifold with convex boundary by attaching a 1–handle. Thus $X\cong X_1\natural X_2$ where $X_1$ and $X_2$ are exact symplectic manifolds with $\partial X_1=L(p,q)$ and $\partial X_2=L(p,p-q)$ or $X\cong X'\cup (\text{1--handle})$ where $X'$ is symplectic 4-manifold with the disconnected boundary $\partial X'\cong L(p,q)\sqcup L(p,p-q)$. As argued above in the proof of Proposition \[prop\] it is not possible to have $X'$ with disconnected boundary being lens spaces and we must be in the case $X\cong X_1\natural X_2$; moreover, since $X$ is a rational homology balls, so are the $X_i$. Moreover, since $X_1$ and $X_2$ are symplectic filling of their boundaries, they induce tight contact structures on $L(p,q)$ and $L(p,p-q)$), respectively. Proposition \[prop\] says that these tight contact structures must be, the unique up to changing orientation, universally tight contact structures $\xi_{std}$ on $L(p,q)$ and $\xi'_{std}$ on $L(p,p-q)$. Thus we have that $X_1$ and $X_2$ are rational homology balls, and are exact symplectic fillings of $(L(p,q), \xi_{std})$, and $(L(p,p-q), \xi'_{std})$, respectively. In [@Lisca08 Corollary 1.2(d)] Lisca classified all such fillings. According to Lisca’s classification, symplectic rational homology ball fillings of $(L(p,q), \xi_{std})$ are possible exactly when $(p,q)=(m^2, mh-1)$ for some $m$ and $h$ co-prime natural numbers, and similarly for $(L(p,p-q), \xi'_{std})$ exactly when $(p,p-q)=(m^2, mk-1)$ for $m$ and $k$ co-prime natural numbers. Now simple calculation shows that, the only possible value for $m$ satisfying these equations is $m=2$. In particular, we get that $p=4$, but then we must have $\{q, p-q\}=\{1,3\}$, and 3 cannot be written as $2k-1$, for $k$ co-prime to $2$. Thus there is no such $X$. [10]{} Selman Akbulut and Robion Kirby, *Mazur manifolds.*, The Michigan Mathematical Journal **26** (1979), no. 3, 259–284. Andrew Casson and John Harer, *Some homology lens spaces which bound rational homology balls*, Pacific Journal of Mathematics **96** (1981), no. 1, 23–36. Jae Choon Cha, *The structure of the rational concordance group of knots*, Mem. Amer. Math. Soc. **189** (2007), no. 885, x+95. [MR ]{}[2343079]{} K. Cieliebak and Y. Eliashberg, *From [S]{}tein to [W]{}einstein and [B]{}ack – [S]{}ymplectic [G]{}eometry of [A]{}ffine [C]{}omplex [M]{}anifolds*, Colloquium Publications, vol. 59, Amer. Math. Soc., 2012. James Conway, John B Etnyre, and B[ü]{}lent Tosun, *[Symplectic Fillings, Contact Surgeries, and Lagrangian Disks]{}*, International Mathematics Research Notices (2019), rny291. Andrew Donald, *Embedding [S]{}eifert manifolds in [$S^4$]{}*, Trans. Amer. Math. Soc. **367** (2015), no. 1, 559–595. [MR ]{}[3271270]{} D. B. A. Epstein, *Embedding punctured manifolds*, Proc. Amer. Math. Soc. **16** (1965), 175–176. [MR ]{}[208606]{} John B. Etnyre, *Planar [O]{}pen [B]{}ook [D]{}ecompositions and [C]{}ontact [S]{}tructures*, IMRN **79** (2004), 4255–4267. Henry Clay Fickle, *Knots, [${\bf Z}$]{}-homology [$3$]{}-spheres and contractible [$4$]{}-manifolds*, Houston J. Math. **10** (1984), no. 4, 467–493. [MR ]{}[774711]{} Ronald Fintushel and Ronald Stern, *Rational homology cobordisms of spherical space forms*, Topology **26** (1987), no. 3, 385–393. [MR ]{}[899056]{} Ronald Fintushel and Ronald J. Stern, *A [$\mu$]{}-invariant one homology [$3$]{}-sphere that bounds an orientable rational ball*, Four-manifold theory ([D]{}urham, [N]{}.[H]{}., 1982), Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 265–268. [MR ]{}[780582]{} [to3em]{}, *Pseudofree orbifolds*, Ann. of Math. (2) **122** (1985), no. 2, 335–364. [MR ]{}[808222]{} Edoardo Fossati, *Topological constraints for stein fillings of tight structures on lens spaces*, 2019. Michael Hartley Freedman, *The topology of four-dimensional manifolds*, J. Differential Geometry **17** (1982), no. 3, 357–453. [MR ]{}[679066]{} Patrick M. Gilmer and Charles Livingston, *On embedding [$3$]{}-manifolds in [$4$]{}-space*, Topology **22** (1983), no. 3, 241–252. [MR ]{}[710099]{} Emmanuel Giroux, *Structures de contact en dimension trois et bifurcations des feuilletages de surfaces*, Invent. Math. **141** (2000), no. 3, 615–689. [MR ]{}[1779622]{} Marco Golla and Laura Starkston, *The symplectic isotopy problem for rational cuspidal curves*, 2019. W. Hantzsche, *Einlagerung von [M]{}annigfaltigkeiten in euklidische [R]{}äume*, Math. Z. **43** (1938), no. 1, 38–58. [MR ]{}[1545714]{} Morris W. Hirsch, *On imbedding differentiable manifolds in euclidean space*, Ann. of Math. (2) **73** (1961), 566–571. [MR ]{}[124915]{} Ko Honda, *On the [C]{}lassification of [T]{}ight [C]{}ontact [S]{}tructures [I]{}*, Geom. Topol. **4** (2000), 309–368. Rob Kirby, *Problems in low dimensional manifold theory*, Algebraic and geometric topology ([P]{}roc. [S]{}ympos. [P]{}ure [M]{}ath., [S]{}tanford [U]{}niv., [S]{}tanford, [C]{}alif., 1976), [P]{}art 2, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 273–312. [MR ]{}[520548]{} Paolo Lisca, *On symplectic fillings of lens spaces*, Trans. Amer. Math. Soc. **360** (2008), no. 2, 765–799. [MR ]{}[2346471]{} Ciprian Manolescu, *Pin(2)-equivariant [S]{}eiberg-[W]{}itten [F]{}loer homology and the triangulation conjecture*, J. Amer. Math. Soc. **29** (2016), no. 1, 147–176. [MR ]{}[3402697]{} Thomas E. Mark and Bülent Tosun, *Obstructing pseudoconvex embeddings and contractible [S]{}tein fillings for [B]{}rieskorn spheres*, Adv. Math. **335** (2018), 878–895. [MR ]{}[3836681]{} Michael Menke, *A jsj-type decomposition theorem for symplectic fillings*, 2018. Peter Ozsv[á]{}th and Zolt[á]{}n Szab[ó]{}, *Absolutely graded floer homologies and intersection forms for four-manifolds with boundary*. V. A. Rohlin, *The embedding of non-orientable three-manifolds into five-dimensional [E]{}uclidean space*, Dokl. Akad. Nauk SSSR **160** (1965), 549–551. [MR ]{}[0184246]{} Stephan Schönenberger, *Planar open books and symplectic fillings*, Ph.D. thesis, University of Pennsylvania, 2005. C. T. C. Wall, *All [$3$]{}-manifolds imbed in [$5$]{}-space*, Bull. Amer. Math. Soc. **71** (1965), 564–567. [MR ]{}[175139]{} E. C. Zeeman, *Twisting spun knots*, Trans. Amer. Math. Soc. **115** (1965), 471–495. [MR ]{}[195085]{}

--- abstract: 'Physical implementations of quantum information processing devices are generally not unique, and we are faced with the problem of choosing the best implementation. Here, we consider the sensitivity of quantum devices to variations in their different components. To measure this, we adopt a quantum metrological approach, and find that the sensitivity of a device to variations in a component has a particularly simple general form. We use the concept of cost functions to establish a general practical criterion to decide between two different physical implementations of the same quantum device consisting of a variety of components. We give two practical examples of sensitivities of quantum devices to variations in beam splitter transmitivities: the KLM and Reverse nonlinear sign gates for linear optical quantum computing with photonic qubits, and the enhanced optical Bell detectors by Grice and Ewert & van Loock. We briefly compare the sensitivity to the diamond distance and find that the latter is less suited for studying the behaviour of components embedded within the larger quantum device.' author: - 'Mark J. Kandula' - Pieter Kok title: Analysis of quantum information processors using quantum metrology --- Introduction ============ Quantum technologies promise dramatic improvements in computation, sensing, and communication, and many efforts are underway to develop it into a mature technology. One of the general challenges is that quantum devices typically need to be extraordinarily precise. We know from quantum fault tolerance theory that models with uncorrelated gate, propagation, and measurement errors may have an error rate of $0.75$% per element [@Raussendorf07], and it is not known whether more forgiving thresholds exist for equally realistic error models. The tolerances in quantum communication devices are likely less severe, but quantum sensing models are again known to be very susceptible to imperfections in the implementation [@Dobrzanski12]. This means that these quantum devices must be fabricated to a very high standard. The precision of a device is usually specified in terms of the *fidelity*, which measures how much the actual output state of a device deviates from the intended (ideal) output state. In cases where the ideal output state is pure, the fidelity can be interpreted as the probability of mistaking the actual output state for the ideal output state [@Uhlmann76]. It was shown by Myerson *et al.* that a single-shot readout of a qubit in an ion trap can be read out with $99.99$% fidelity [@Lucas08], and single- and two-qubit gates can achieve fidelities of $99.99$% and $99.9$%, respectively [@Lucas16]. In other implementations, similar fidelities have been achieved [@Ghosh13]. A general method for calculating the fidelity of quantum operations was given by Pedersen *et al.* [@Molmer07], and there is a sizeable literature on measuring deviations from ideal operations using various mathematical techniques [@Cui14; @Plenio15; @Kong17]. However, it was pointed out among others by Sanders, Wallman, and Sanders [@Sanders16] that the average gate fidelity is problematic when it comes to assessing the quality of a quantum gate for quantum computing. They show that the gate error rate can be dramatically higher than the fault tolerant threshold even when the average gate fidelity is 99%. In other words, the average fate fidelity is too optimistic. Even when fault tolerance is not our main concern, such as in various quantum communication protocols, the average gate fidelity may not be the most suitable figure of merit. There are often multiple ways to implement a device, sometimes with dramatically different susceptibilities to variations in the device’s components [@Crickmore16]. Given additional constraints such as costs, it is not clear *a priori* how an array of fidelities associated with different component variations should be combined into a single number that can be used to identify the best way to implement the device. Another technical complication is that the fidelity is a *function* of device parameters, rather than a single number. In order to obtain a meaningful value for the fidelity, we must choose some non-zero deviation of the device parameters since for zero deviations the fidelity will by definition be equal to unity. This choice of deviation introduces a level of arbitrariness into the metric that we wish to avoid. Instead, we want a single number for each component (operating perfectly) that indicates the sensitivity of the device to deviations in that component. ![Decomposition of a quantum device into unitary components, specifically highlighting the component $u_j$. The horizontal lines may be qubits or optical modes, depending on the implementation. a) A gate $\ket{\psi_{\rm out}} = \mathscr{U}_g\ket{\psi_{\rm in}}$ with input state $\ket{\psi_{\rm in},A}$ and post-selected on a projection on $\ket{D}$; b) a quantum measurement device that has no output state, but gives a classical output “$m$”, indicated by the state $\ket{D_m}$. Entangling the input $\ket{\psi_{\rm in}}$ with an auxiliary system prior to the device’s operation ($\ket{\Psi_{\rm in}}$) allows us to apply the techniques for a) to measurements.[]{data-label="fig:device"}](./devices.pdf){width="8.5cm"} In this paper, we propose a method for testing the sensitivity of quantum devices that is not based on the fidelity. We use a method from quantum metrology to approach this problem, where the output state of the quantum device carries information about the characteristics of the device’s components. This leads to the definition of the *sensitivity* of the device to variation in a component, and for a multi-component device we will obtain a sensitivity matrix. Together with a cost function for the different components, this sensitivity matrix provides a clear metric for the performance of different architectures for the same quantum device. This paper is organised as follows: in section \[sec:sensitivity\] we introduce the [sensitivity]{} for a device component. To realise this, we divide quantum devices into two categories, namely gates and measurement devices. The latter differs from the former in that there is no output state to the device. In section \[sec:examples\] we demonstrate how the sensitivity works for two incarnations of the nonlinear sign gate in linear optical quantum computing [@klm01; @Crickmore16], and for two implementations of the enhanced optical Bell measurement [@Grice11; @Ewert14]. In section \[sec:design\] we bring together the sensitivities for different components into a single metric that tests different implementations of a quantum device. In section \[sec:diamond\] we briefly comment on the relation between our sensitivity and the diamond norm. We conclude our discussion in section \[sec:conclusions\]. Defining the [Sensitivity]{} for components of quantum devices {#sec:sensitivity} ============================================================== We wish to consider the component sensitivity of two kinds of quantum devices. First, we consider quantum gates that have an input and an output, and which may be based on post-selection of auxiliary quantum systems (e.g., qubits or photons). This situation is depicted in Fig. \[fig:device\]a. As an example of this type of device, we will consider the nonlinear sign gate of linear optical quantum computing with photonic qubits [@Crickmore16; @klm01]. Second, we consider complex detection devices that use quantum gates to implement the desired observable. In this situation there is no surviving quantum state that can be used to track variations in components. To remedy this, we use entangled input states that allow us to define the action of a measurement in terms of a surviving quantum state [@kok01b]. This situation is depicted in Fig. \[fig:device\]b. As an example of this type of device, we will consider enhanced Bell measurements [@Grice11; @Ewert14]. Quantum gates ------------- Consider a quantum gate $g$ described by the unitary evolution $\ket{\psi_{\rm out}} = \mathscr{U}_g\ket{\psi_{\rm in}}$, where the evolution can be post-selected using an auxiliary input state $\ket{A}$ and a detected state $\ket{D}$ (see Fig. \[fig:device\]). The detected state may be one of a family of states that herald a successful gate. The intervening evolution can often be decomposed in terms of $N$ smaller unitary operations $U = \prod_{j=1}^N u_j$. These $u_j$ are the physical components of the quantum device generated by a Hamiltonian $H_j$: $$\begin{aligned} u_j = \exp(-i\theta_j H_j)\, ,\end{aligned}$$ with $\theta_j$ the component parameter whose value determines the gate operation. In general, the practical gate operation is more accurately described by a completely positive map that allows for imperfections in the device, but here we are interested in the ideal device and how deviations in the components affect the gate. While a more general discussion is certainly possible, it would also obscure some of the more intuitive aspects of this work. After normalisation, the output of the device can be written as $$\begin{aligned} \label{eq:gate} \ket{\psi_{\rm out}} = \frac{1}{\sqrt{p}} \braket{D|U|\psi_{\rm in},A}\, ,\end{aligned}$$ where $p = \norm{\braket{D|U|\psi_{\rm in},A}}^2$ is the probability of success of the quantum device that implements the operation $\mathscr{U}_g$. Suppose we are interested in the $j^{\rm th}$ component of the device, denoted by $u_j$. Define $$\begin{aligned} V_j = \prod_{k=1}^{j-1} u_k \qquad\text{and}\qquad W_j = \prod_{k=j+1}^{N} u_k\, . \end{aligned}$$ Then we can decompose the output state as (see Fig. \[fig:device\]a): $$\begin{aligned} \ket{\psi_{\rm out}} = \frac{1}{\sqrt{p}} \braket{D|W_j u_j V_j|\psi_{\rm in},A}\, .\end{aligned}$$ We can treat the sensitivity of the device to variations in $u_j$ as an estimation problem of the parameter $\theta_j$ that characterises the component $u_j$. To this end we use the output state $\ket{\psi_{\rm out}}$ as the basis for the estimation procedure. This state is already post-selected on the correct measurement outcome $\Pi_D \equiv \ket{D}\bra{D}$. This is consistent with the operation of the gate, where the quantum computer trusts that upon getting the measurement outcome “$D$” the gate does what it is supposed to do. Fortunately, we do not explicitly have to perform a complicated estimation procedure. Instead, we can calculate the average amount of information about $\theta_j$ that is contained in the output state $\ket{\psi_{\rm out}}$. If the output state is very sensitive to variations in $\theta_j$ (the aspect we are trying to capture), then it must by definition vary strongly when the value of $\theta_j$ changes. The variation of the output state with $\theta_j$ is quantified by the quantum Fisher information $\smash{I_Q^{(j)}}$, according to [@braunstein95] $$\begin{aligned} I_Q^{(j)} = \braket{\partial_j{\psi}_{\rm out}|\partial_j{\psi}_{\rm out}} - \Abs{\braket{\psi_{\rm out}|\partial_j{\psi}_{\rm out}}}^2\, ,\end{aligned}$$ where $\partial_j$ is the partial derivative with respect to $\theta_j$. When we define $$\begin{aligned} \ket{\phi_{\rm in}} = u_j V_j \ket{\psi_{\rm in},A} ~\text{and}~ \ket{\phi_{\rm out}} = W_j^\dagger \ket{\psi_{\rm out},D} \, ,\end{aligned}$$ the derivative of the output state is compactly written as $$\begin{aligned} \ket{\smash{\partial_j{\psi}_{\rm out}}} = \frac{-i}{\sqrt{p}} \braket{D|W_j H_j|\phi_{\rm in}} +\frac12 \left( \partial_j \log p \right) \ket{\psi_{\rm out}}\, ,\end{aligned}$$ where $H_j$ is the generator of translations in $\theta_j$. We can then explicitly calculate the quantum Fisher information. First we calculate $$\begin{aligned} \braket{\partial_j{\psi}_{\rm out}|\partial_j{\psi}_{\rm out}} = ~ & \frac{1}{p} \Norm{\braket{D|W_j H_j|\phi_{\rm in}}}^2 + \frac14 \left( \partial_j \log p \right)^2 \cr & - \frac{\partial_j \log p}{\sqrt{p}}\, \operatorname{Im}\braket{\phi_{\rm in}|H_j|\phi_{\rm out}}\, ,\end{aligned}$$ and $$\begin{aligned} \braket{\psi_{\rm out}|\partial_j{\psi}_{\rm out}} = \frac{-i}{\sqrt{p}} \braket{\phi_{\rm out}|H_j|\phi_{\rm in}} +\frac12 \left( \partial_j \log p \right) \, .\end{aligned}$$ From this, we find that $$\begin{aligned} I_Q^{(j)} = \frac{1}{p} \Norm{\Braket{D|W_j H_j|\phi_{\rm in}}}^2 - \frac{1}{p} \Abs{\Braket{\phi_{\rm out}|H_j|\phi_{\rm in}}}^2\, .\end{aligned}$$ We can clean up this expression by inserting a resolution of the identity ${\mathbb{I}}= \smash{\ket{\psi_{\rm out}}\bra{\psi_{\rm out}} + \sum_k \ket{k}\bra{k}}$ in the first term of $I_Q^{(j)}$, where the orthonormal states $\ket{k}$ complete $\ket{\psi_{\rm out}}$ to form an orthonormal basis of the output Hilbert space. We find that $$\begin{aligned} I_Q^{(j)} = \frac{1}{p} \sum_{k} \Abs{\Braket{\phi_k| H_j|\phi_{\rm in}}}^2\, ,\end{aligned}$$ where we defined $\ket{\phi_k}\equiv W_j^\dagger \ket{k,D}$. We can understand this expression as the quadratic sum over the weak values of the generator $H_j$ given the input state $\ket{\psi_{\rm in},A}$ and the output states $\ket{k,D}$ that are *orthogonal* to the intended output state $\ket{\psi_{\rm out},D}$. The success probability $p$ of the quantum device is a common factor in $\smash{I_Q^{(j)}}$ and does not play a role in the determination of the component sensitivity of a device (although it is important to include this factor when comparing the sensitivity of components in *different* devices with different $p$). In general, $p$ changes when the component $u_j$ changes, and this can in principle be used in an estimation procedure of $\theta_j$. However, we post-select the state on the detection outcome $\Pi_D$, which means we have already discarded the information about the success rate of the quantum device. This is consistent with the normal operation of the gate $\mathscr{U}_g$. The sensitivity $S_j$ of the quantum device to components $u_j$ can now be defined as $$\begin{aligned} \label{eq:sensitivity} S_j \equiv \sum_{k} \abs{\braket{\phi_k| H_j|\phi_{\rm in}}}^2\, .\end{aligned}$$ While this is an elegant theoretical expression that gives a clear intuitive meaning for $S_j$, for practical purposes it may be beneficial to express $S_j$ instead as $$\begin{aligned} S_j = \braket{\phi_{\rm in}|H_j K_D H_j |\phi_{\rm in}} - \abs{\braket{\phi_{\rm out}|H_j|\phi_{\rm in}}}^2\, ,\end{aligned}$$ using $K_D = W_j^\dagger ({\mathbb{I}}\otimes \Pi_D) W_j$. This expression does not require the construction of the complementary basis states $\ket{k}$. It also holds for gates that rely on higher rank post-selection described by projectors $\Pi_D$, such as for example the double heralding procedure for creating entangled networks [@Barrett05]. To determine a general, non-state-specific sensitivity of a device, we can average $S_j$ over all possible input states. Alternatively, we can take as a standard input state an equal superposition of the eigenstates of $\mathscr{U}_g$, which is computationally much more straightforward. Quantum measurement devices --------------------------- Next, we consider quantum measurement devices, as shown in Fig \[fig:device\]b. The situation is slightly more complicated than the sensitivity for gate components, since there are typically multiple detection outcomes $m$, corresponding to projections onto $\ket{D_m}$ (which in turn are generally projections onto a subspace of the output space). The corresponding surviving quantum state $\ket{\smash{\psi_{\rm out}^{(m)}}}$ is defined by $$\begin{aligned} \ket{\smash{\psi_{\rm out}^{(m)}}} = \frac{1}{\sqrt{p_m}} \braket{D_m|U|\Psi_{\rm in},A}\, ,\end{aligned}$$ where the input state $\ket{\Psi_{\rm in}}$ is a maximally entangled state that allows us to relate the measurement outcome to an output state that can be used to define the sensitivity: $$\begin{aligned} \ket{\Psi_{\rm in}} = \frac{1}{\sqrt{d}} \sum_k \ket{B_k,B_k}\, ,\end{aligned}$$ with $d$ the dimension of the input state space of the measurement device. The states $\ket{B_k}$ are the orthonormal eigenstates of the observable measured in the measurement device [@kok01b]. To calculate the sensitivity of the $j^{\rm th}$ component of the measurement device, parameterised by $\theta_j$, we again calculate the quantum Fisher information of $\theta_j$ in the output state $\ket{\smash{\psi_{\rm out}^{(m)}}}$. Clearly, this will be different for different outcomes $m$, and we define the quantum Fisher information $\smash{I_Q^{(j,m)}}$ for each component $j$ and measurement outcome $m$. The total quantum Fisher information for the $j^{\rm th}$ component is then the weighted sum over all measurement outcomes $$\begin{aligned} I_Q^{(j)} = \sum_{m\neq m_f} p_m \, I_Q^{(j,m)}\, .\end{aligned}$$ One subtlety that we will encounter in the next section is that sometimes there are outcomes $m_f$ of the measurement device that indicate the measurement has *failed* to produce a useful outcome. There may still be information in $\ket{\smash{\psi_{\rm out}^{(m)}}}$, but since these outcomes (and any post-selection based on these outcomes) are discarded in normal operation of the device, deviations in $\ket{\smash{\psi_{\rm out}^{(m_f)}}}$ have no effect on the device operation and we must *not* include $\smash{I_Q^{(j,m_f)}}$ in the calculation of the sensitivity. Proceeding with the calculation of $\smash{I_Q^{(j,m)}}$, we use that $$\begin{aligned} I_Q^{(j,m)} = \Braket{\partial_j{\psi}^{(m)}_{\rm out}|\partial_j{\psi}^{(m)}_{\rm out}} - \Abs{\Braket{\psi^{(m)}_{\rm out}|\partial_j{\psi}^{(m)}_{\rm out}}}^2\, .\end{aligned}$$ Following the same method as in the previous section, we find that $$\begin{aligned} I_Q^{(j,m)} = ~& \frac{1}{p_m} \Braket{\phi_{\rm in}| H_j W_j^\dagger \left(\Pi_m \otimes {\mathbb{I}}\right) W_j H_j|\phi_{\rm in}} \cr & - \Abs{\Braket{\phi_{\rm out}^{(m)}|H_j|\phi_{\rm in}}}^2\, ,\end{aligned}$$ where $\Pi_m$ is the projector onto the subspace associated with the state $\ket{D_m}$, which may have rank greater than one, and the states $\ket{\smash{\phi_{\rm out}^{(m)}}}$ and $\ket{\phi_{\rm in}}$ are defined as $$\begin{aligned} \ket{\phi_{\rm in}} = u_j V_j \ket{\Psi_{\rm in},A} ~\text{and}~ \ket{\psi_{\rm out}^{(m)}} = W_j^\dagger \ket{\psi_{\rm out}^{(m)},D_m} .\end{aligned}$$ The unitary evolutions $u_j$, $V_j$ and $W_j$ are defined as in Fig. \[fig:device\]b. We can insert a resolution of the identity into the first term: $$\begin{aligned} {\mathbb{I}}= \ket{\psi_{\rm out}^{(m)}}\bra{\psi_{\rm out}^{(m)}} + \sum_k \ket{\xi_k^{(m)}}\bra{\xi_k^{(m)}}\, ,\end{aligned}$$ for some orthonormal set $\{ \ket{\smash{\xi_k^{(m)}}} \}$ that span the subspace ${\mathbb{I}}- \ket{\smash{\psi_{\rm out}^{(m)}}}\bra{\smash{\psi_{\rm out}^{(m)}}}$, and this leads to $$\begin{aligned} I_Q^{(j,m)} = \frac{1}{p_m} \sum_k \Abs{\Braket{\phi^{(m)}_k | H_j | \phi_{\rm in}}}^2\, ,\end{aligned}$$ with $\ket{\smash{\phi^{(m)}_k}} = W_j^\dagger \ket{\smash{\xi_k^{(m)},D_m}}$. The sensitivity then becomes $$\begin{aligned} S_j & = \sum_{k=1}^{d-1} \sum_{m\neq m_f} \Abs{\Braket{\phi^{(m)}_k | H_j | \phi_{\rm in}}}^2\cr & = \sum_{k=1}^{d-1} \sum_{m\neq m_f} \braket{\phi_{\rm in} | H_j \left( \widetilde{\Pi}_m \otimes {\mathbb{I}}\right) H_j |\phi_{\rm in}}\, ,\end{aligned}$$ where $\widetilde{\Pi}_m \equiv W_j^\dagger \Pi_m W_j$. Two explicit examples of the beam splitter sensitivity of optical Bell measurements are given in section \[sec:Bell\]. One may notice that the sensitivity, as measured by the quantum Fisher information, carries units of the inverse-squared of $\theta_j$. When the $\theta_j$ refer to different components in a device, these units may be different, and a straight comparison will not be possible. Indeed, this is a key problem in constructing a single metric for different implementations of a quantum device, and we will return to this issue in section \[sec:design\]. Variations in sources and detectors ----------------------------------- The discussion so far has been restricted to unitary elements in a quantum device, but in practice we will also have to include variations in the auxiliary input states and the detectors. These can be included as unitary evolutions also. For example, an auxiliary input state $\ket{A}$ may be transformed into a different input state $\ket{A(\bm\theta_A)}$ according to a unitary evolution $$\begin{aligned} \ket{A(\bm\theta_A)} = u_A(\bm\theta_A) \ket{A}\end{aligned}$$ with $$\begin{aligned} u_A(\bm\theta) = \exp\left( -\frac{i}{\hbar} \bm{H}_A \cdot \bm\theta_A \right)\, ,\end{aligned}$$ where $\bm\theta_A$ is a vector of parameters associated with the evolution from $\ket{A}$ to $\ket{A(\bm\theta_A)}$ generated by $\bm{H}_A$. In the ideal case, $\bm\theta_A=0$ and we recover the original auxiliary state. The sensitivity of the device to the auxiliary input state with respect to a particular variation $\theta_{A,j}$, the $j^{\rm th}$ entry of $\bm\theta_A$, is then defined as $$\begin{aligned} S_{A,j} = \braket{\psi_{\rm in},A| H_{A,j} U^\dagger ({\mathbb{I}}\otimes \Pi_D) U H_{A,j} |\psi_{\rm in},A}\, ,\end{aligned}$$ where $H_{A,j}$ is the $j^{\rm th}$ entry of $\bm{H}_A$, and $U = W_j u_j V_j$ is the unitary evolution defined in Eq. (\[eq:gate\]) and Fig. \[fig:device\]. Often, a physical imperfection in the source will lead to auxiliary input states that are mixed. However, due to the convex nature of the quantum Fisher information, we need only consider the effect of unitary deviations from the pure ancilla state. Similarly, for imperfect detectors there is a way of calculating the sensitivity using the technique developed above. In the most general terms, an imperfect detector does not measure the exact observable $M_D$, but instead some rotated observable $$\begin{aligned} M_D(\bm\theta_D) = u_D M_D u_D^\dagger\, ,\end{aligned}$$ where $u_D = \exp(-i \bm{H}_D \cdot\bm\theta_D/\hbar)$ is the unitary evolution that rotates the eigenbasis of $M_D$ to the eigenbasis of $M_D(\bm\theta_D)$. The sensitivity of the device to detector imperfections with respect to a particular variation $\theta_{D,j}$ is then defined as $$\begin{aligned} S_{D,j} = \braket{\psi_{\rm in},A| U^\dagger ({\mathbb{I}}\otimes H_{D,j} \Pi_D H_{D,j} ) U |\psi_{\rm in},A}\, ,\end{aligned}$$ where $H_{D,j}$ is the $j^{\rm th}$ entry of $\bm{H}_D$. This allows for a complete analysis of which variations in the observable are most detrimental to the device operation. Stochastic noise ---------------- So far, we have considered sensitivities to systematic errors in the device components. A natural question is whether we can include stochastic noise in the analysis of our quantum devices. Stochastic noise models generally describe the situation where some of the parameters $\theta_j$ are fluctuating over time, rather than offset by some amount from the ideal value. It is important to note that the sensitivity defined here is an intrinsic property of each device component, and does not change with the type of deviation from the ideal values of the parameters, systematic or stochastic. Therefore, the device component analysis presented in this section is complete. The different types of errors and noise that a device exhibits is included instead in the metric used to decide between implementations. We will return to this aspect in section \[sec:design\]. ![The nonlinear sign (NS) circuit for linear optical quantum computing. There are multiple versions of this circuit that are equivalent in terms of the success probability, number of optical elements, auxiliary photons and detectors, but they exhibit inequivalent behaviour in the presence of variations in the components. Here, we consider the KLM NS gate (top) introduced in Ref. [@klm01], and the Reverse NS gate (bottom) introduced in Re. [@Crickmore16].[]{data-label="fig:ns"}](./NS-circuit.pdf "fig:"){width="6cm"}   Examples {#sec:examples} ======== To demonstrate the sensitivity measure, we consider several examples. We calculate the sensitivities to variations in the beam splitters in nonlinear sign (NS) gates for photonic linear optical quantum computing. These devices fall in the category of quantum gates. Next, we calculate the sensitivities to variations in the beam splitters in two types of optical Bell detectors, which fall in the category of quantum measurement devices. We compare the sensitivities to the device fidelity in order to show that the sensitivity behaves as expected. The dependence of optical gates on their constituent optical components has been studied before [@glancy02; @ralph02a; @lund03; @rohde05a; @rohde05b; @rohde05c; @Clements16], but to our knowledge a unifying model for comparing implementations in arbitrary quantum device architectures has not yet been proposed. The Nonlinear Sign gate in LOQC ------------------------------- The NS gate is a key component in the original proposal for linear optical quantum computing with photonic qubits by Knill, Laflamme and Milburn in 2001 [@klm01]. It is a probabilistic gate that implements the unitary evolution $$\begin{aligned} \alpha\ket{0} + \beta\ket{1}+\gamma\ket{2} \to \alpha\ket{0} + \beta\ket{1} - \gamma\ket{2}\, ,\end{aligned}$$ with $\ket{n}$ denoting single mode optical Fock states. The success probability is one quarter, and there are several inequivalent ways to implement the NS gate, two of which are shown in Fig. \[fig:ns\]. The beam splitter values in the two implementations are different, and we reported in Ref. [@Crickmore16] that the gate operation in the presence of variations in the beam splitter reflectivities depends strongly on the implementation. ![Sensitivities ${S}_{\rm KLM}$ and ${S}_{\rm Reverse}$ of the beam splitters in the KLM NS gate (top) and the Reverse NS gate (bottom), averaged over all input states. The units along the vertical axis are technically rad$^{-2}$, but are not important for the comparison of similar components in a gate. These results are consistent with the detailed analysis in Ref. [@Crickmore16].[]{data-label="fig:ns_sensitivity"}](./KLM.pdf "fig:"){width="4cm"} ![Sensitivities ${S}_{\rm KLM}$ and ${S}_{\rm Reverse}$ of the beam splitters in the KLM NS gate (top) and the Reverse NS gate (bottom), averaged over all input states. The units along the vertical axis are technically rad$^{-2}$, but are not important for the comparison of similar components in a gate. These results are consistent with the detailed analysis in Ref. [@Crickmore16].[]{data-label="fig:ns_sensitivity"}](./KLM_fidelity.pdf "fig:"){width="4.5cm"}\ ![Sensitivities ${S}_{\rm KLM}$ and ${S}_{\rm Reverse}$ of the beam splitters in the KLM NS gate (top) and the Reverse NS gate (bottom), averaged over all input states. The units along the vertical axis are technically rad$^{-2}$, but are not important for the comparison of similar components in a gate. These results are consistent with the detailed analysis in Ref. [@Crickmore16].[]{data-label="fig:ns_sensitivity"}](./Reverse.pdf "fig:"){width="4cm"} ![Sensitivities ${S}_{\rm KLM}$ and ${S}_{\rm Reverse}$ of the beam splitters in the KLM NS gate (top) and the Reverse NS gate (bottom), averaged over all input states. The units along the vertical axis are technically rad$^{-2}$, but are not important for the comparison of similar components in a gate. These results are consistent with the detailed analysis in Ref. [@Crickmore16].[]{data-label="fig:ns_sensitivity"}](./Reverse_fidelity.pdf "fig:"){width="4.5cm"} For both the KLM and Reverse NS gate we choose the following description of the beam splitter operation acting on modes $a_1$ and $a_2$: $$\begin{aligned} \begin{pmatrix} \hat{b}_1 \\ \hat{b}_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{a}_1 \\ \hat{a}_2 \end{pmatrix} ,\end{aligned}$$ where hats denote mode operators and $\theta$ is the defining parameter of the beam splitter. For the KLM NS gate, the three beam splitter parameters $\theta_j$ are $$\begin{aligned} \label{eq:nsklmtheta} \theta_1 & = \arccos \left( \frac{1}{4-2\sqrt{2}} \right) \, , \cr \theta_2 & = \arccos \left( 3-2\sqrt{2} \right) \, ,\cr \theta_3 & = -\theta_1\, ,\end{aligned}$$ whereas for the Reverse NS gate, the three beam splitter parameters $\xi_j$ are $$\begin{aligned} \xi_1 & = \arctan \left( \sqrt[4]{8} \right) \, , \cr \xi_2 & = \pi - \arctan \left( \frac{\sqrt{16\sqrt{2}-13}}{7} \right) \,, \cr \xi_3 & = -\xi_1\, .\end{aligned}$$ Both implementations use the same number of auxiliary photons and detections. We calculate the sensitivity of the two NS gates to the various beam splitters. The results are shown in Fig. \[fig:ns\_sensitivity\]. Clearly, the sensitivity to variations in in the KLM NS gate is much greater than the sensitivity to variations in and . This is reflected in the average fidelity of the output state of the KLM NS gate. By contrast, the sensitivity to variations in in the Reverse NS gate is much smaller than the sensitivity to variations in and . Again, this is borne out in the average fidelities of the output state of the Reverse NS gate. Enhanced linear optical Bell state detectors {#sec:Bell} -------------------------------------------- Optical Bell measurements are an important tool in optical quantum information processing. They are used in a variety of applications, including teleportation [@bennett93; @bouwmeester97; @kok00], optical quantum computing [@Browne05], and quantum repeater proposals [@briegel98; @Dur99; @Kok03b]. Originally, the optical Bell detector was introduced by Weinfurter [@Weinfurter94] and Braunstein and Mann [@Braunstein95b], and both schemes have a success probability of one half. In particular, these Bell detectors are capable of identifying the Bell states $$\begin{aligned} \ket{\smash{\Psi^\pm}} \equiv \frac{\ket{H,V} \pm \ket{V,H}}{\sqrt{2}}\, , \end{aligned}$$ while they are completely incapable of distinguishing between the Bell states $$\begin{aligned} \ket{\smash{\Phi^\pm}} \equiv \frac{\ket{H,H} \pm \ket{V,V}}{\sqrt{2}}\, ,\end{aligned}$$ where $\ket{H}$ and $\ket{V}$ denote horizontally and vertically polarised photons, respectively. It was proved by Vaidman and Yoran [@Vaidman99] and Lütkenhaus, Calsamiglia, and Suominen [@Lutkenhaus99] that optical Bell detectors without auxiliary photons have an upper bound of one half on the success probability. This severely increases the overhead of any practical application relying on these Bell detectors, since provisions must be made to ensure a failed Bell measurement does not negatively affect the operation of the quantum device (e.g., see the solution provided by the original proposal for linear optical quantum computing by Knill, Laflamme, and Milburn [@klm01]). ![The entanglement-assisted Bell detection circuits of Grice (top) and Ewert & van Loock (bottom). The Grice detector takes as input two photonic polarisation qubits and a polarisation Bell state $(\ket{H,H}+\ket{V,V})/\sqrt{2}$ in the auxiliary input. Every mode carries a polarisation degree of freedom. The Ewert & van Loock Bell detector operates on dual-rail photonic qubits with auxiliary input states $(\ket{2,0}+\ket{0,2})/\sqrt{2}$, but here we translated it to a polarisation implementation: each mode carries a polarisation degree of freedom. In this paper we consider only the sensitivity of the Bell detection circuits to beam splitter variations.[]{data-label="fig:bellcircuits"}](./Bell-Detectors.pdf){width="7.5cm"} A modification of the optical Bell detector was proposed by Grice [@Grice11], and Ewert & Van Loock [@Ewert14], who suggested employing auxiliary photons to help distinguish between the remaining Bell states $\ket{\smash{\Phi^\pm}}$. They showed that using one or two photon pairs increases the success probability to three quarters, and more generally, the use of $n$ photons leads to a success probability of $$\begin{aligned} \nonumber p_{\rm Grice}(n) = 1 - \frac{1}{n+2} \quad\text{and}\quad p_{\rm EvL}(n) = 1 - \frac{1}{2^{n/2}}\, .\end{aligned}$$ The circuits for the Grice and Ewert & Van Loock Bell detectors using two and four auxiliary photons, respectively, is given in Fig. \[fig:bellcircuits\]. To obtain a success probability of three quarters, the Grice circuit requires a two-photon input state of the form $$\begin{aligned} \ket{\smash{\Phi^+}} = \frac{\ket{H,H}+\ket{V,V}}{\sqrt{2}}\, ,\end{aligned}$$ while the Ewert & van Loock circuit requires two two-photon input states in modes $c$ and $d$ of the form $$\begin{aligned} \ket{\Upsilon} = \frac{\ket{2H}+\ket{2V}}{\sqrt{2}}\, .\end{aligned}$$ Note that while the Ewert & van Loock circuit requires twice as many auxiliary photons to achieve the same success probability of three quarters as Grice’s circuit, for higher success probabilities the Ewert & van Loock family of circuits is more efficient. ![The beam splitter sensitivities of the Bell detection circuits of Grice, compared with the fidelities for the different beam splitter variations. Beam splitters BS1 and BS2 are entirely equivalent, as are BS3 and BS4. The sensitivities for BS1 and BS2 are 5.2, while the sensitivities for BS1 and BS2 are 3.4. The fidelities (right) reflect this sensitivity.[]{data-label="fig:gricecircuits_sensitivity"}](./Grice.pdf "fig:"){width="4cm"} ![The beam splitter sensitivities of the Bell detection circuits of Grice, compared with the fidelities for the different beam splitter variations. Beam splitters BS1 and BS2 are entirely equivalent, as are BS3 and BS4. The sensitivities for BS1 and BS2 are 5.2, while the sensitivities for BS1 and BS2 are 3.4. The fidelities (right) reflect this sensitivity.[]{data-label="fig:gricecircuits_sensitivity"}](./Grice_fidelity.pdf "fig:"){width="4.5cm"} We calculate the sensitivity of the beam splitters in the Grice circuit, shown in Fig. \[fig:gricecircuits\_sensitivity\]. We choose as input to the quantum measurement device in Fig. \[fig:device\]b the state $$\begin{aligned} \label{eq:input} \ket{\smash{\Psi_{\rm in}}} = ~ & \frac{1}{2} \ket{\smash{\Phi^+,\Phi^+}} + \frac{1}{2}\ket{\smash{\Phi^-,\Phi^-}} + \frac{1}{2}\ket{\smash{\Psi^+,\Psi^+}} \cr & + \frac{1}{2}\ket{\smash{\Psi^-,\Psi^-}}\, ,\end{aligned}$$ and we calculate the fidelity of the output state of the device with the expected Bell state due to the measurement outcome. The first two beam splitters that the input photons encounter ( and ) exhibit a significantly lower sensitivity than the last two beam splitters ( and ). This is corroborated by the fidelity of the output state. For the Ewert & van Loock circuit we perform similar calculations, with the same input state in Eq. (\[eq:input\]), and we find that it is the first beam splitter () that exhibits the greatest sensitivity. As expected, due to the symmetry of the circuit, beamsplitters and have the same sensitivity. The fidelity plots again confirm the sensitivities (see Fig. \[fig:ewertcircuits\_sensitivity\]). It is tempting to make a judgement, based on the above analysis, which NS gate or Bell detector is more suitable for implementation. However, in our examples we have considered only the sensitivities of the beam splitters, and we have not included variations in path lengths (or phases), auxiliary input states, or detector imperfections. These must all be taken into account before a value judgement can be made about a particular quantum device or gate implementation. However, it is already clear that more resources (i.e., time spent in alignment, or money spent on high-quality components) should be devoted to and in the Grice circuit, and, more dramatically, to in the Ewert & van Loock circuit. Device analysis based on the Sensitivity {#sec:design} ======================================== The discussion has so far been restricted to sensitivities of individual components. However, quantum devices, including those in the previous section, typically consist of multiple components, and we would like to have some sense of which components are more critical than others to the device’s operation. In general it will be hard to compare sensitivities of different types of components. For example, how should we compare the numerical values for the sensitivity of the device to a beam splitter reflectivity (dimensionless) with the sensitivity to a path length variation (units of length)? To solve this problem, we borrow the concept of cost functions from estimation theory. It will also provide us with a single metric that allows us to choose between different implementations of the same quantum device. In this section we will introduce the concept of cost functions, show how they apply to quantum device analysis, and give examples for the NS gates and Bell detectors. ![The beam splitter sensitivities of the Bell detection circuits of Ewert & van Loock, compared with the fidelities for the different beam splitter variations. Beam splitters BS2 and BS3 are equivalent. The sensitivity for BS1 is 12, while the sensitivities for BS2 and BS3 are 3.6. Again, the fidelities (right) reflect this sensitivity. In particular, the large sensitivity of the device to variations in BS1 is clear in the fidelity plot.[]{data-label="fig:ewertcircuits_sensitivity"}](./Ewert.pdf "fig:"){width="4cm"} ![The beam splitter sensitivities of the Bell detection circuits of Ewert & van Loock, compared with the fidelities for the different beam splitter variations. Beam splitters BS2 and BS3 are equivalent. The sensitivity for BS1 is 12, while the sensitivities for BS2 and BS3 are 3.6. Again, the fidelities (right) reflect this sensitivity. In particular, the large sensitivity of the device to variations in BS1 is clear in the fidelity plot.[]{data-label="fig:ewertcircuits_sensitivity"}](./Ewert_fidelity.pdf "fig:"){width="4.5cm"} Cost functions and their construction ------------------------------------- Multiple components in a device will lead to a multi-parameter estimation problem in our device analysis. Let the output state $\ket{\psi_{\rm out}}$ of a device (or $\ket{\smash{\psi_{\rm out}^{(m)}}}$ for measurement devices) depend on an array of parameters corresponding to the different device components: $$\begin{aligned} \bm\theta = (\theta_1,\ldots,\theta_M)\, ,\end{aligned}$$ where $M$ denotes the number of different components in the quantum device. In quantum metrology, for each output state $\ket{\smash{\psi_{\rm out}}}$ we assign a quantum Fisher information *matrix*—defined on the parameter space of $\bm\theta$—that can be described by $$\begin{aligned} \label{eq:eouhgdfk} [I_Q]_{jk} = ~ & 4\operatorname{Re}\left[ \Braket{\partial_j{\psi}_{\rm out}|\partial_k{\psi}_{\rm out}} \right] \cr & - 4\operatorname{Re}\left[\Braket{\partial_j\psi_{\rm out}|{\psi}_{\rm out}}\braket{\psi_{\rm out}|\partial_k{\psi}_{\rm out}}\right]\, ,\end{aligned}$$ where $\partial_j$ and $\partial_k$ are derivatives with respect to $\theta_j$ and $\theta_k$, respectively. Calculating the derivatives as before, we obtain the matrix elements $$\begin{aligned} [I_Q]_{jk} = ~& \frac{4}{p}\,\operatorname{Re}[ \Bra{\psi_{\rm in},A} V_j^\dagger u_j^\dagger H_j W_j^\dagger ({\mathbb{I}}-\ket{\psi_{\rm out}}\bra{\psi_{\rm out}}) \cr & \qquad\otimes \Pi_D\, W_k H_k u_k V_k \ket{\psi_{\rm in},A}] \, .\end{aligned}$$ Once we have a quantum Fisher information matrix, a bound on the covariance matrix of $\bm\theta$ can be defined: $$\begin{aligned} \label{eq:qcrbmult} \operatorname{Cov}(\bm\theta) \geq I_Q^{-1}\, .\end{aligned}$$ This is the famous quantum Cramér-Rao bound for multiple parameters [@Helstrom73], defined in the sense that $\operatorname{Cov}(\bm\theta) - \smash{I_Q^{-1}}$ is a positive definite matrix. The bound is tight if and only if the generators $H_j$ are co-measurable [@Ragy16]. The smaller the covariances, the better we can estimate variations in $\bm\theta$ from the output state, and the more sensitive the implementation is to variations in $\bm\theta$. The Sensitivity, which in Eq. (\[eq:sensitivity\]) was proportional to the quantum Fisher information, now becomes a Sensitivity matrix. In order to compare the sensitivities to different components $\theta_j$ and $\theta_k$, we need physically motived unit scales $\Delta\theta_j$ and $\Delta\theta_k$. For example, suppose that the beam splitter in the KLM NS gate can be manufactured to much higher precision than and , perhaps due to the different values of $\theta_j$ in Eq. (\[eq:nsklmtheta\]). Then a natural choice for the unit scale is the manufacturing tolerance $\Delta\theta_j$. Consequently, even though $S_2$ is larger than $S_1$ and $S_3$ (see Fig. \[fig:ns\_sensitivity\]), we could be in the position that the dimensionless product of the tolerance-squared $(\Delta\theta_j)^2$ and the sensitivity $S_j$ indicates that will have a lower impact on the device performance than and : $$\begin{aligned} {(\Delta\theta_2)^2} {S_2} < {(\Delta\theta_1)^2} {S_1} = {(\Delta\theta_3)^2}{S_3}\, .\end{aligned}$$ The cost function for a single parameter can then be given as $(\Delta\theta_j)^2$. In the context of covariance bounds, a real, symmetric, positive semi-definite cost matrix $R$ is introduced such that Eq. (\[eq:qcrbmult\]) becomes the simple inequality $$\begin{aligned} \label{eq:bhu043we} \Tr{R \operatorname{Cov}(\bm\theta)} \geq \Tr{R\, I_Q^{-1}}\, .\end{aligned}$$ For our purpose of quantum device analysis, we are interested in minimising the sensitivity, rather than minimising the covariance of $\bm\theta$. We can achieve this by letting $R = \bm\Sigma(\bm\theta)$, where $\bm\Sigma(\bm\theta)$ is the covariance matrix associated with the fluctuations due to manufacturing tolerances (which is different from $\operatorname{Cov}(\bm\theta)$ in the Cramér-Rao bound). In the case where components exhibit stochastic fluctuations of the parameter, the diagonal elements of $\bm\Sigma(\bm\theta)$ are the variances $(\Delta\theta_j)^2$ of the fluctuations. The dimensionless measure for the total sensitivity is then $$\begin{aligned} \tr{R S} = \sum_{j,k=1}^M \bm\Sigma(\bm\theta)_{jk}\, S_{kj}\, .\end{aligned}$$ When a component is subject to both manufacturing tolerances and stochastic fluctuations, we may combine their effects according to the standard rule $$\begin{aligned} (\Delta\theta_j)^2 = (\Delta_{\rm man}\theta_j)^2 + (\Delta_{\rm st}\theta_j)^2\, ,\end{aligned}$$ where $\Delta_{\rm man}$ and $\Delta_{\rm st}$ denote manufacturing and stochastic errors, respectively. This formula can easily be extended to include more types of noise in the components, including correlated noise. In addition, we can include the economical cost for each component as a multiplier. Such a choice will favour devices with fewer components, *ceteris paribus*. Now consider two quantum device implementations, $\mathscr{I}_1$ and $\mathscr{I}_2$. We can calculate the sensitivity matrices $S(\mathscr{I}_1)$ and $S(\mathscr{I}_2)$ for these implementations according to Eq. (\[eq:eouhgdfk\]) and $S = pI_Q$. Given two cost functions $R_1$ and $R_2$, we say that implementation $\mathscr{I}_1$ is better than implementation $\mathscr{I}_2$ if $$\begin{aligned} \label{eq:decision} \Tr{R_1\, S(\mathscr{I}_1)} < \Tr{R_2\, S(\mathscr{I}_2)}.\end{aligned}$$ Note that the decision criterion in Eq. (\[eq:decision\]) depends on the choice of cost functions. For example, for the NS gate implementation we can choose values of $(\Delta\theta_1,\Delta\theta_2,\Delta\theta_3)$ and $(\Delta\xi_1,\Delta\xi_2,\Delta\xi_3)$ such that either the KLM NS gate or the Reverse NS gate is a easier to implement experimentally. The practically achievable tolerances are, not surprisingly, an integral part of the device analysis. In general, we are forced to pick two different cost functions $R_1$ and $R_2$ when the components of the two implementations differ, and we must make sure that $R_1$ and $R_2$ are constructed using the same criteria. In other words, the beam splitter variations in $\mathscr{I}_1$ should be constructed according to the same physical principles as the beam splitter variations in $\mathscr{I}_2$, even though they may not result in identical numerical values (c.f., our example of the KLM and Reverse NS gates above). This gives a natural sense in which devices with more or harder to fabricate components are likely to perform worse than simpler, easier to fabricate devices. When all components (of the same type) have identical manufacturing tolerances and are independent of the other components, the cost function may be chosen as the identity matrix. Finally, one may ask what the statistical interpretation of Eq. (\[eq:bhu043we\]) means for the task of differentiating between proposed implementations of a quantum device, and hence how we should interpret Eq. (\[eq:decision\]). When the Cramér-Rao bound in Eq. (\[eq:bhu043we\]) is achievable, we can extract the full quantum Fisher information’s worth from measurements on the output state $\ket{\psi_{\rm out}}$ or $\ket{\smash{\psi_{\rm out}^{(m)}}}$. The sensitivity therefore immediately leads to observable effects. On the other hand, the multi-parameter Cramér-Rao bound cannot be saturated in general, and this leads to the question whether the decision condition in Eq. (\[eq:decision\]) should be modified (e.g., along the lines of Ref. [@Holevo82], using right-logarithmic derivatives). However, we should remember that we do not actually wish to estimate the parameters $\bm\theta$, but merely seek a measure of sensitivity for the *quantum state* (which will typically be used in further processing) given a cost matrix $R$. This is exactly what the multi-parameter quantum Fisher information—and therefore the sensitivity—provides. Example of cost functions for Bell detectors -------------------------------------------- We can make a simple comparison between the Grice Bell detector and the Ewert & van Loock Bell detector, where we take into account only the cost of beam splitter variations. While the Grice Bell detector employs more beam splitters, the beam splitters in the Ewert & van Loock Bell detector have a higher sensitivity. Comparing the implementations with a simple identity matrix cost function $R = {\mathbb{I}}$ (since we have no reason to believe that there is a different cost or tolerance for these beam splitters), we find that the Ewert & van Loock Bell detector has an overall sensitivity of $\smash{\tr{S(\mathscr{I}_{\rm EvL})}} = 19.25$, and the Grice Bell detector has an overall beam splitter sensitivity of $\smash{\tr{S(\mathscr{I}_{\rm Grice})}} = 17.19$. While the difference is not large, this shows that when we ignore path length differences and other imperfections, the design with lower sensitivities to the beam splitter variations is in this case marginally preferable to the design with fewer beam splitters. When a device has more components than there are degrees of freedom in the output state, the quantum Fisher information matrix may become singular. In that case we cannot evaluate Eq. (\[eq:decision\]), and the method presented here does not provide a value for the overall sensitivity (and this device implementation may allow for significant simplifications!). However, our method will still be able to provide valuable information about the relative importance of variations in the constituent components, and even when the sensitivity matrix is not singular, a full device analysis must always include the consideration of the individual elements. This will inform us which components will give the greatest benefits in the precision of the device when extra resources are spent on improvements. Relation to the Diamond Distance {#sec:diamond} ================================ We have compared the metrological approach to quantum device characterisation to the average gate fidelity. However, another important metric for judging the quality of a quantum device is the diamond distance. Here we briefly review some key properties of the diamond distance and explore how it can be used in device analysis. The diamond distance is a unitarily invariant quantity for measuring the distance between two general quantum channels [@Kitaev97]. In the case of quantum information processors, the diamond distance tells us how much an actual gate transformation deviates from the intended gate transformation, and it is closely related to the error rate of the gate [@Sanders16]. Given the Schatten 1-norm $\norm[1]{\cdot}$, the diamond norm of a quantum channel $\mathscr{E}$ is defined as [@Kitaev97] $$\begin{aligned} \norm[$\diamond$]{\mathscr{E}} \equiv \sup \left\{ \norm[1]{(\mathscr{E}\otimes{\mathbb{I}})(\rho)} ; \norm[1]{\rho} \leq 1 \right\}\, ,\end{aligned}$$ which leads to a diamond distance between two channels $\mathscr{E}$ and $\mathscr{E}'$ given by $$\begin{aligned} d_\diamond (\mathscr{E},\mathscr{E}') \equiv \frac12 \Norm[$\diamond$]{\mathscr{E}-\mathscr{E}'}\, .\end{aligned}$$ It was shown in Refs. [@Wang13; @Wang18] that the diamond distance for unitary channels $u_j$ and $u_j'$ is upper bounded by the operator norm $$\begin{aligned} \label{eq:vnuw90eofsdj} d_\diamond (u_j,u_j') \leq \norm{u_j-u_j'}\, .\end{aligned}$$ For the types of unitary transformations discussed here, with $u_j = \exp(-i\theta_j H_j)$ and $u_j' = \exp(-i\theta_j' H_j)$, and only small difference between the parameters $\abs{\theta_j - \theta_j'}$, the diamond norm is very close to the operator norm $\norm{u_j - u_j'}$ [@Campbell17; @Campbell18]. To see this, we first define the ground state $\ket{m}$ and the maximum eigenvalue state $\ket{M}$ of $H_j$. Without loss of generality (by fixing a global phase) the eigenvalues of $\ket{m}$ and $\ket{M}$ are $-\mu$ and $+\mu$, respectively ($\mu>0$). We can then evaluate the operator norm as $$\begin{aligned} \label{eq:b739wrfbd} \Norm{u_j - u_j'} = \mu\Abs{\theta_j' - \theta_j} \, .\end{aligned}$$ Next, we show that in addition to the upper bound in Eq. (\[eq:vnuw90eofsdj\]), we can construct a lower bound that approaches the upper bound in the limit of vanishing $\abs{\theta_j - \theta_j'}$. By definition, $$\begin{aligned} \label{eq:bhg90worsj} \Norm[$\diamond$]{u_j - u_j'} \geq \Norm[1]{(u_j\otimes {\mathbb{I}}) \rho (u_j^\dagger\otimes {\mathbb{I}}) - (u_j'\otimes {\mathbb{I}}) \rho ({u_j'}^\dagger\otimes {\mathbb{I}})}\end{aligned}$$ for any quantum state $\rho$. Since any $\rho$ will provide a valid lower bound we are looking to choose $\rho$ judiciously, such that it maximises the lower bound. Let $\rho = \ket{\psi}\bra{\psi} \otimes {\mathbb{I}}$, with $$\begin{aligned} \label{eq:bn5t0whrfks} \ket{\psi} = \frac{\ket{m}+\ket{M}}{\sqrt{2}}\, .\end{aligned}$$ We then obtain [@Campbell18] $$\begin{aligned} \label{eq:b902wesd} \Norm[$\diamond$]{u_j - u_j'} \geq 2 \mu\Abs{\theta_j' - \theta_j} .\end{aligned}$$ Combining the upper and lower bounds in Eqs. (\[eq:b739wrfbd\]) and (\[eq:b902wesd\]) to the diamond distance, and defining $\delta\theta_j = \abs{\theta_j' - \theta_j}$ (assuming $\theta_j$ is the ideal value), this leads to $$\begin{aligned} \label{eq:bn8049wesojidfknv} d_\diamond (u_j,u_j') = \mu\, \delta\theta_j\, .\end{aligned}$$ This is the diamond distance between $u_j$ and $u_j'$, which depends again on the deviation of $\theta_j'$ away from $\theta_j$. It can be interpreted as the maximum possible angle between two quantum states $\ket{\psi_0}$ and $\ket{\psi(\delta\theta_j)} \equiv \exp(-i\delta\theta_j H_j)\ket{\psi_0}$, where the maximisation is over the input states $\ket{\psi_0}$. Our choice for $\rho$ in Eq. (\[eq:bn5t0whrfks\]) achieves this maximum. For pure states, the angle between quantum states is known as the statistical distance between those states [@Wootters81]. The square of the derivative of the statistical distance with respect to the angle is equal to the quantum Fisher information for that angle. Therefore, the diamond distance for a device component—translated into a Fisher information—provides us with the largest amount of information about the component parameter $\theta_j$ that can be extracted in a measurement procedure, since the diamond distance involves a maximisation over the input states of the component. By contrast, the sensitivity presented in this paper does not involve such a maximisation, but instead uses the quantum state of the system as produced at the point just before the component $u_j$. As such, the sensitivity allows us to consider each component within the context of the device as a whole, rather than as a stand-alone device for which we seek the worst-case behaviour. The diamond distance not distinguish between similar elements in a device, e.g., the beam splitters in the different implementations of the NS gates or the Bell detectors. The diamond distance for individual components therefore cannot be used to identify sensitivity bottlenecks in a quantum device. Finally, note that we have assumed a maximum eigenvalue $\mu>0$ for the Hamiltonian $H_j$. In our examples involving optical modes, such Hamiltonians are typically unbounded. In our metrological approach we circumvent this problem by averaging the sensitivity over the relevant input state space. For example, the NS gate acts on at most two photons, and the averaging is performed relative to the Haar measure on the space of zero, one and two photons in the input state. For the diamond distance to be meaningful, a similar state space truncation must be employed. Conclusions {#sec:conclusions} =========== The construction of quantum information processing devices is a challenging technical problem, and reducing sources of errors is an essential element of it. We introduced a method for comparing different physical implementations of a quantum information processing device, including composite quantum gates and detectors, in terms of the sensitivity of the device to variations in its components. Our method is based on the amount of information about a component parameter present in the output state. This is measured by the quantum Fisher information. For the examples considered here, we show that this method is consistent with the predictions based on the fidelity of the device given variations in the components. The benefit of our method over the fidelity method is that we can collect the combined effect of variations in all components into a single overall sensitivity metric based on cost functions that match our design requirements. Furthermore, the sensitivity takes into account the context of each component in a quantum device, as opposed to the diamond distance, which captures the worst-case behaviour of each component. For each quantum device, quantum architecture designers should consider as many different implementations as possible, and carry out a sensitivity analysis along the lines of the discussion presented here. This may be a lengthy task, and it is currently an open question whether we can construct guiding design principles that provide shortcuts to this task. Such principles will likely be highly implementation dependent. Acknowledgements {#acknowledgements .unnumbered} ================ The authors thank Earl T. Campbell for valuable discussions about the diamond distance and its relation to the Fisher information. This research was funded in part by EPSRC’s Quantum Communications Hub `EP/M013472/1`. [45]{}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 [****, ()](\doibase 10.1088/1367-2630/9/6/199) [****, ()](\doibase 10.1038/ncomms2067) [****,  ()](\doibase 10.1016/0034-4877(76)90060-4) [****,  ()](\doibase 10.1103/PhysRevLett.100.200502) [****,  ()](\doibase 10.1103/PhysRevLett.117.060504) [****,  ()](\doibase 10.1103/PhysRevA.87.022309) [****,  ()](\doibase 10.1016/j.physleta.2007.02.069) [****,  ()](\doibase 10.1140/epjd/e2014-50354-5) [****,  ()](\doibase 10.1103/PhysRevA.91.042129) @noop [****,  ()]{} [**** (), 10.1088/1367-2630/18/1/012002](\doibase 10.1088/1367-2630/18/1/012002) [****,  ()](\doibase 10.1103/PhysRevA.94.022326) [****,  ()](\doibase 10.1038/35051009) [****, ()](\doibase 10.1103/PhysRevA.84.042331) [****,  ()](\doibase 10.1103/PhysRevLett.113.140403) [****,  ()](\doibase 10.1103/PhysRevA.63.033812) [****, ()](\doibase 10.1006/aphy.1996.0040) [****,  ()](\doibase 10.1103/PhysRevA.71.060310) [****,  ()](\doibase 10.1103/PhysRevA.65.062317) [****,  ()](\doibase 10.1103/PhysRevA.65.012314) [****,  ()](\doibase 10.1103/PhysRevA.68.022313) [****,  ()](\doibase 10.1103/PhysRevA.72.032306) [****,  ()](\doibase 10.1103/PhysRevA.72.052332) [****, ()](\doibase 10.1103/PhysRevA.71.032320) @noop [ ()]{},  [****,  ()](\doibase 10.1103/PhysRevLett.70.1895) [****,  ()](\doibase 10.1038/37539) [****,  ()](\doibase 10.1103/PhysRevA.61.042304) [****,  ()](\doibase 10.1103/PhysRevLett.95.010501) [****,  ()](\doibase 10.1103/PhysRevLett.81.5932) [****,  ()](\doibase 10.1103/PhysRevA.59.169) [****,  ()](\doibase 10.1103/PhysRevA.68.022301) [****,  ()](\doibase 10.1238/Physica.Topical.076a00203) [****, ()](\doibase 10.1103/PhysRevA.51.R1727) [****, ()](\doibase 10.1103/PhysRevA.59.116) [****,  ()](\doibase 10.1103/PhysRevA.59.3295) [****,  ()](\doibase 10.1007/BF00687093) [****,  ()](\doibase 10.1103/PhysRevA.94.052108) @noop [**]{} (, ) @noop [****,  ()]{} [****, ()](\doibase 10.1103/PhysRevLett.111.130504) [****,  ()](\doibase 10.1088/1367-2630/17/4/043004) [****,  ()](\doibase 10.1103/PhysRevA.95.042306) [****,  ()](\doibase 10.1088/2058-9565/1/1/015007) [****, ()](\doibase 10.1103/PhysRevD.23.357)

\ Valerie Coffman,$^{(1)}$ Joydip Kundu,$^{(2)}$ and William K. Wootters$^{(3)}$\ Abstract {#abstract .unnumbered} -------- [Consider three qubits A, B, and C which may be entangled with each other. We show that there is a trade-off between A’s entanglement with B and its entanglement with C. This relation is expressed in terms of a measure of entanglement called the “tangle,” which is related to the entanglement of formation. Specifically, we show that the tangle between A and B, plus the tangle between A and C, cannot be greater than the tangle between A and the pair BC. This inequality is as strong as it could be, in the sense that for any values of the tangles satisfying the corresponding equality, one can find a quantum state consistent with those values. Further exploration of this result leads to a definition of the “three-way tangle” of the system, which is invariant under permutations of the qubits. ]{} PACS numbers: 03.65.Bz, 89.70.+c Quantum entanglement has rightly been the subject of much study in recent years as a potential resource for communication and information processing. As with other resources such as free energy and information, one would like to have a quantitative theory of entanglement giving specific rules about how it can and cannot be manipulated; indeed such a theory has begun to be developed. The first step in building the theory has been to quantify entanglement itself. In the last few years a number of entanglement measures for bipartite states have been introduced and analyzed \[1–7\], the one most relevant to the present work being the “entanglement of formation” [@entmeasures], which is intended to quantify the amount of quantum communication required to create a given state. In the present paper we draw on some of the earlier work on entanglement of formation [@Hill; @Wootters] in order to explore another basic quantitative question: To what extent does entanglement between two objects limit their entanglements with other objects? Unlike classical correlations, quantum entanglement cannot be freely shared among many objects. For example, given a triple of spin-$\frac{1}{2}$ particles A, B and C, if particle A is fully entangled with particle B, [*e.g.*]{}, if they are in the singlet state $\frac{1}{\sqrt{2}}(|\hspace{-0.7mm}\uparrow\downarrow\rangle - |\hspace{-0.7mm}\downarrow\uparrow\rangle )$, then particle A cannot be simultaneously entangled with particle C. (If A were entangled with C, then the pair AB would also be entangled with C and would therefore have a mixed-state density matrix, whereas the singlet state is pure.) One expects that a less extreme form of this restriction should also hold: if A is [*partly*]{} entangled with B, then A can have only a limited entanglement with C. The first goal of this paper is to verify this intuition and express it quantitatively. We will see that the restriction on the sharing of entanglement takes a particularly elegant form in terms of a measure of entanglement called the “tangle,” which is closely related to the entanglement of formation. Further analysis of this result will lead us naturally to a quantity that measures a three-way entanglement of the system and is invariant under all permutations of the particles [@multi]. The present work is related to recent work on the characterization of multiparticle states in terms of invariants under local transformations \[9–12\]; indeed, both the tangle and our measure of three-way entanglement are invariants in this sense. Our work is also related to research exploring the connection between entanglement and cloning \[13–17\]. An example along these lines was studied by Bruß, who asked, in the case of a singlet pair AB, to what extent particle B’s entanglement with particle A can be shared symmetrically and isotropically with a third particle, for a purpose such as teleportation where isotropy is desired [@Bruss]. Our investigation is similar in spirit to that of Bruß but has a different focus in that we are looking for a general law governing the splitting or sharing of entanglement; thus, for example, we make no assumptions about the symmetry of the system. Some of the results presented here have been mentioned in a recent paper by one of us [@Royal], but the proofs and most of the details and observations have not been previously published. In this paper we confine our attention to binary quantum objects (qubits) such as spin-$\frac{1}{2}$ particles—we will use the generic basis labels $|0\rangle$ and $|1\rangle$ rather than $|\hspace{-0.1cm}\uparrow\rangle$ and $|\hspace{-0.1cm}\downarrow\rangle$—but the same questions could be raised for larger objects. We begin by defining the tangle. Let A and B be a pair of qubits, and let the density matrix of the pair be $\rho_{AB}$, which may be pure or mixed. We define the “spin-flipped” density matrix to be $$\tilde{\rho}_{AB} = (\sigma_y \otimes \sigma_y) \rho^*_{AB} (\sigma_y \otimes \sigma_y),$$ where the asterisk denotes complex conjugation in the standard basis $\{|00\rangle$, $|01\rangle$, $|10\rangle, |11\rangle\}$ and $\sigma_y$ expressed in the same basis is the matrix ${\left(\begin{array}{cc}0&{-i}\\i&0\end{array}\right)}$. As both $\rho_{AB}$ and $\tilde{\rho}_{AB}$ are positive operators, it follows that the product $\rho_{AB} \tilde{\rho}_{AB}$, though non-Hermitian, also has only real and non-negative eigenvalues. Let the square roots of these eigenvalues, in decreasing order, be $\lambda_1$, $\lambda_2$, $\lambda_3$, and $\lambda_4$. Then the tangle of the density matrix $\rho_{AB}$ is defined as $$\tau_{AB} = [{\rm max}\{\lambda_1 - \lambda_2 -\lambda_3 -\lambda_4,0\}]^2. \label{tau}$$ For the special case in which the state of AB is pure, the matrix $\rho_{AB} \tilde{\rho}_{AB}$ has only one non-zero eigenvalue, and one can show that $\tau_{AB}=4\det \rho_{A}$, where $\rho_{A}$ is the density matrix of qubit A, that is, the trace of $\rho_{AB}$ over qubit B. It is by no means obvious from the definition that the tangle is a measure of entanglement for mixed states. This interpretation comes from previous work, in which a specific connection is established between the tangle and the entanglement of formation of a pair of qubits [@Wootters]. For the purpose of this paper it is sufficient to note that $\tau = 0$ corresponds to an unentangled state, $\tau = 1$ corresponds to a completely entangled state, and the entanglement of formation is a monotonically increasing function of $\tau$.[^1] (The earlier work did not define the tangle [*per se*]{} but rather the “concurrence,” which is simply the square root of the tangle. It is with some hesitation that we introduce the new term “tangle” here instead of speaking of the square of the concurrence. However, for our present purpose the tangle does seem to be the more natural measure to use, and the discussion is significantly simplified by introducing this term.) At present, the tangle is defined only for a pair of qubits, not for higher-dimensional systems. We now turn to the first problem of this paper: given a pure state of three qubits A, B, and C, how is the tangle between A and B related to the tangle between A and C? For this special case—a pure state of three qubits—the formula for the tangle simplifies: each pair of qubits, being entangled with only one other qubit in a joint pure state, is described by a density matrix having at most two non-zero eigenvalues. It follows that the product $\rho_{AB} \tilde{\rho}_{AB}$ also has only two non-zero eigenvalues. We can use this fact and Eq. (\[tau\]) to write the following inequality for the tangle $\tau_{AB}$ between A and B. $$\begin{aligned} \tau_{AB} & = & (\lambda_1 - \lambda_2)^2 = {\lambda_1}^2 + {\lambda_2}^2 - 2\lambda_1 \lambda_2 \nonumber\\ & = & {\rm Tr} (\rho_{AB}\tilde{\rho}_{AB}) - 2\lambda_1 \lambda_2 \le {\rm Tr} (\rho_{AB}\tilde{\rho}_{AB}). \label{2lambdas}\end{aligned}$$ Here $\rho_{AB}$ is the density matrix of the pair AB, obtained from the original pure state by tracing over qubit C. Eq. (\[2lambdas\]) and the analogous equation for $\tau_{AC}$ allow us to bound the sum $\tau_{AB} + \tau_{AC}$: $$\tau_{AB} + \tau_{AC} \le {\rm Tr} (\rho_{AB}\tilde{\rho}_{AB}) + {\rm Tr} (\rho_{AC}\tilde{\rho}_{AC}). \label{tautau}$$ The next paragraph is devoted to evaluating the right-hand side of this inequality. Let us express the pure state $|\xi\rangle$ of the three-qubit system in the standard basis $\{|ijk\rangle \}$, where each index takes the values 0 and 1: $$|\xi\rangle = \sum_{ijk} a_{ijk}|ijk \rangle.$$ In terms of the coefficients $a_{ijk}$, we can write ${\rm Tr} (\rho_{AB}\tilde{\rho}_{AB})$ as $${\rm Tr} (\rho_{AB}\tilde{\rho}_{AB}) =\sum a_{ijk}a^*_{mnk}\epsilon_{mm^\prime}\epsilon_{nn^\prime} a^*_{m^\prime n^\prime p}a_{i^\prime j^\prime p} \epsilon_{i^\prime i}\epsilon_{j^\prime j},$$ where $\epsilon_{01}=-\epsilon_{10}=1$ and $\epsilon_{00}=\epsilon_{11}=0$ and the sum is over all the indices. We now replace the product $\epsilon_{nn^\prime}\epsilon_{j^\prime j}$ with the equivalent expression $\delta_{nj^\prime}\delta_{n^\prime j}- \delta_{nj}\delta_{n^\prime j^\prime}$, and in the first of the two resulting terms (that is, the one associated with $\delta_{nj^\prime}\delta_{n^\prime j}$) we perform a similar substitution for $\epsilon_{mm^\prime}\epsilon_{i^\prime i}$. These substitutions directly give us $${\rm Tr} (\rho_{AB}\tilde{\rho}_{AB}) = 2\det {\rho_A} -{\rm Tr} (\rho_B^2) + {\rm Tr} (\rho_C^2), \label{dettr}$$ where $\rho_A$, $\rho_B$, and $\rho_C$ are the 2x2 density matrices of the individual qubits. Because each of these matrices has unit trace, we can rewrite Eq. (\[dettr\]) as $${\rm Tr} (\rho_{AB}\tilde{\rho}_{AB}) = 2(\det{\rho_A} + \det {\rho_B} - \det {\rho_C}).$$ By symmetry we must also have $${\rm Tr} (\rho_{AC}\tilde{\rho}_{AC}) = 2(\det {\rho_A} + \det {\rho_C} - \det {\rho_B}).$$ Summing these last two equations, we finally get a simple expression for the right-hand side of Eq. (\[tautau\]), namely, $${\rm Tr} (\rho_{AB}\tilde{\rho}_{AB}) + {\rm Tr} (\rho_{AC}\tilde{\rho}_{AC}) = 4\det {\rho_A}. \label{trdet}$$ Eqs. (\[tautau\]) and (\[trdet\]) give us our first main result: $$\tau_{AB}+\tau_{AC} \le 4\det\rho_A. \label{first}$$ We can interpret the right-hand side of Eq. (\[first\]) as follows. Thinking of the pair BC as a single object, it makes sense to speak of the tangle between qubit A and the pair BC, because, even though the state space of BC is four-dimensional, only two of those dimensions are necessary to express the pure state $|\xi\rangle$ of ABC. (The two necessary dimensions are those spanned by the two eigenstates of $\rho_{BC}$ that have non-zero eigenvalues. That there are only two such eigenvalues follows from the fact that A is only a qubit.) We may thus treat A and BC, at least for this purpose, as a pair of qubits, and so the tangle is well defined. In fact this tangle $\tau_{A(BC)}$ is simply $4\det{\rho_A}$ as we have mentioned before. We can therefore rewrite our result as $$\tau_{AB}+\tau_{AC} \le \tau_{A(BC)}. \label{3taus}$$ Informally, Eq. (\[3taus\]) can be expressed as follows. Qubit A has a certain amount of entanglement with the pair BC. This amount bounds A’s entanglement with qubits B and C taken individually, and the part of the entanglement that is devoted to qubit B (as measured by the tangle) is not available to qubit C. We will say more shortly about the case of three qubits in a pure state, but at this point it is worth mentioning a generalization to mixed states. If ABC is in a mixed state $\rho$, then $\tau_{A(BC)}$ is not defined, because all four dimensions of BC might be involved, but we can define a related quantity $\tau^{min}_{A(BC)}$ via the following prescription. Consider all possible pure-state decompositions of the state $\rho$, that is, all sets $\{(\psi_i,p_i)\}$ such that $\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$. For each of these decompositions, one can compute the average value of $\tau_{A(BC)}$: $$\langle \tau_{A(BC)} \rangle = \sum_i p_i \tau_{A(BC)}(\psi_i).$$ The minimum of this average over all decompositions of $\rho$ is what we take to be $\tau^{min}_{A(BC)}(\rho)$. The following analogue of Eq. (\[3taus\]) then holds for mixed states: $$\tau_{AB}+\tau_{AC} \le \tau^{min}_{A(BC)}. \label{mixed}$$ To prove this, consider the pure states $|\psi_i\rangle$ belonging to an optimal decomposition of $\rho$, that is, a decomposition that minimizes $\langle\tau_{A(BC)}\rangle$. We can write our basic inequality, Eq. (\[3taus\]), for each such pure state and then average both sides of the inequality over the whole decomposition. The right-hand side of the resulting inequality is $\tau^{min}_{A(BC)}(\rho)$, which is what we want on the right-hand side. On the left-hand side we have two terms: (i) the average of the tangle between A and B over a set of mixed states whose [*average*]{} is $\rho_{AB}$ ([*i.e.*]{}, ${\rm Tr}_C (\rho)$), and (ii) the average of the tangle between A and C over a set of mixed states whose average is $\rho_{AC}$. It is a fact that the tangle is a [*convex*]{} function on the set of density matrices.[^2] That is, the average of the tangles is greater than or equal to the tangle of the average. In this case the tangles of the averages are $\tau_{AB} = \tau(\rho_{AB})$ and $\tau_{AC}=\tau(\rho_{AC})$. The sum of these two tangles must thus be less than or equal to $\tau^{min}_{A(BC)}(\rho)$, which is what we wanted to prove. Returning to the case of pure states, one may wonder how tight the inequality (\[3taus\]) is. Could one find, for example, a more stringent bound of the same form, based on a different measure of entanglement? To address this question, consider the following pure state of ABC: $$|\phi\rangle = \alpha |100\rangle + \beta |010\rangle + \gamma |001\rangle, \label{phi}$$ where the three positions in the kets refer to qubits A, B, and C in that order. For this state, one finds that $\tau_{AB}=4|\alpha|^2|\beta|^2$, $\tau_{AC}=4|\alpha|^2|\gamma|^2$, and $\tau_{A(BC)}=4|\alpha|^2(|\beta|^2+|\gamma|^2)$. Thus the inequality (\[3taus\]) becomes in this case an equality: $\tau_{AB}+\tau_{AC} = \tau_{A(BC)}$. This example shows that for any values of the tangles satisfying this equality, there is a quantum state that is consistent with those values. Now let $T(\tau)$ be a monotonically increasing function of $\tau$ that we might propose as an alternative measure of entanglement. For simplicity let us assume that $T(0)=0$ and $T(1)=1$. Because of the above example, $T$ could satisfy the inequality $T_{AB} + T_{AC} \le T_{A(BC)}$ only if $T(x) + T(y) \le T(x+y)$ for all non-negative $x$ and $y$ such that $x+y \le 1$. Suppose $T$ has this property. Then could there exist some quantum state for which $T_{AB} + T_{AC} = T_{A(BC)}$ but $\tau_{AB}+\tau_{AC} < \tau_{A(BC)}$? That is, could $T$ yield an equality for some state for which $\tau$ gives only an inequality? The answer is no, because if $\tau_{AB}+\tau_{AC} < \tau_{A(BC)}$, then $T_{AB} + T_{AC}$ = $T(\tau_{AB}) + T(\tau_{AC}) \le T(\tau_{AB}+\tau_{AC}) < T(\tau_{A(BC)}) = T_{A(BC)}$. Moreover, the only way $T$ can match $\tau$ in those cases where $\tau$ gives an equality is for $T$ to be equal to $\tau$. In this sense, $\tau$ is an optimal measure of entanglement with respect to the inequality given in Eq. (\[3taus\]). Note, however, that the above argument applies only to functions of $\tau$. There could in principle be other measures of entanglement that are not functions of $\tau$ that could make an equal claim to optimality. The entanglement of formation is a function of $\tau$, but it is a concave function and therefore does not satisfy an inequality of the form of Eq. (\[3taus\]). Consider, for example, the state $\frac{1}{\sqrt{2}} |100\rangle + \frac{1}{2}|010\rangle + \frac{1}{2}|001\rangle$. One finds that the relevant entanglements of formation are $E_{AB} = 0.601$, $E_{AC} = 0.601$, and $E_{A(BC)} = 1$. Thus, contrary to what one might expect, the sum of the entanglements of formation between A and the separate qubits B and C is greater than 1 “ebit,” despite the fact that one can reasonably regard 1 ebit as the entanglement capacity of a single qubit. This is not a paradox; it simply shows us that entanglement of formation does not exhibit this particular kind of additivity. (This sense of “additivity” should not be confused with the additivity of entanglement when one combines pairs to make larger systems[@Wootters]. It is not known whether entanglement of formation satisfies the latter notion of additivity.) We have just seen that there are some states for which the inequality (\[3taus\]) becomes an equality. Of course there are other states for which the inequality is strict. As we will see, it turns out to be very interesting to consider the [*difference*]{} between the two sides of Eq. (\[3taus\]). This difference can be thought of as the amount of entanglement between A and BC that [*cannot*]{} be accounted for by the entanglements of A with B and C separately. In the following paragraphs we refer to this quantity as the “residual tangle.” Let the system ABC be in a pure state $|\xi\rangle$, and as before, let the components of $|\xi\rangle$ in the standard basis be $a_{ijk}$: $$|\xi\rangle = \sum_{ijk}a_{ijk}|ijk\rangle.$$ According to Eqs. (\[2lambdas\]) and (\[trdet\]) and the discussion following Eq. (\[first\]), the residual tangle is equal to $$\tau_{A(BC)} - \tau_{AB} - \tau_{AC} = 2(\lambda^{AB}_1\lambda^{AB}_2 + \lambda^{AC}_1\lambda^{AC}_2),$$ where $\lambda^{AB}_1$ and $\lambda^{AB}_2$ are the square roots of the two eigenvalues of $\rho_{AB}{\tilde{\rho}}_{AB}$, and $\lambda^{AC}_1$ and $\lambda^{AC}_2$ are defined similarly. We now derive an explicit expression for the residual tangle in terms of the coefficients $a_{ijk}$. We focus first on the product $\lambda^{AB}_1\lambda^{AB}_2$. This product can almost be interpreted as the square root of the determinant of $\rho_{AB}{\tilde{\rho}}_{AB}$. But $\rho_{AB}{\tilde{\rho}}_{AB}$ is an operator acting on a four-dimensional space, and two of its eigenvalues are zero; so its determinant is also zero. However, if we consider the action of $\rho_{AB}{\tilde{\rho}}_{AB}$ only on its [*range*]{}, then $\lambda^{AB}_1\lambda^{AB}_2$ [*will be*]{} the square root of the determinant of this restricted transformation. The range of $\rho_{AB}{\tilde{\rho}}_{AB}$ is spanned by the two vectors $|v_0\rangle = \sum_{ij}a_{ij0}|ij0\rangle$ and $|v_1\rangle = \sum_{ij}a_{ij1}|ij1\rangle$. (These vectors also span the range of $\rho_{AB}$.) To examine the action of $\rho_{AB}{\tilde{\rho}}_{AB}$ on this subspace, we consider its effect on vectors of the form $x|v_1\rangle +y|v_2\rangle \equiv {\left(\begin{array}{c}x\\y\end{array}\right)}$. This effect is given by $${\left(\begin{array}{c}x^\prime \\ y^\prime\end{array}\right)} = R {\left(\begin{array}{c}x\\y\end{array}\right)},$$ where $R$ is a 2x2 matrix. The product $\lambda^{AB}_1\lambda^{AB}_2$ is the square root of the determinant of $R$. One finds that $$R_{ij} = \sum a_{klj}a^*_{mni}\sigma_{mp} \sigma_{nq}a^*_{pqr}a_{str}\sigma_{sk}\sigma_{tl},$$ where the sum is over all repeated indices. (We have ordered the factors so as to suggest the expression $\rho_{AB}{\tilde{\rho}}_{AB}$ from which $R$ is derived.) Taking the determinant of $R$ involves somewhat tedious but straightforward algebra, with the following result: $$\lambda^{AB}_1\lambda^{AB}_2 = \sqrt{\det R} = |d_1 - 2d_2 + 4d_3|,$$ where $$\begin{aligned} d_1 & = & a^2_{000}a^2_{111}+a^2_{001}a^2_{110}+ a^2_{010}a^2_{101}+a^2_{100}a^2_{011}; \nonumber \\ d_2 & = & a_{000}a_{111}a_{011}a_{100} + a_{000}a_{111}a_{101}a_{010} + a_{000}a_{111}a_{110}a_{001} \\ & + & a_{011}a_{100}a_{101}a_{010} + a_{011}a_{100}a_{110}a_{001} + a_{101}a_{010}a_{110}a_{001}; \nonumber \\ d_3 & = & a_{000}a_{110}a_{101}a_{011} + a_{111}a_{001}a_{010}a_{100}. \nonumber \end{aligned}$$ We can get a mental picture of this expression by imagining the eight coefficients $a_{ijk}$ attached to the corners of a cube. Then each term appearing in $d_1$, $d_2$, or $d_3$ is a product of four of the coefficients $a_{ijk}$ such that the “center of mass” of the four is at the center of the cube. Such configurations fall into three classes: those in which the four coefficients lie on a body diagonal and each one is used twice ($d_1$); those in which they lie on a diagonal plane ($d_2$), and those in which they lie on the vertices of a tetrahedron ($d_3$). Within each category, all the possible configurations are given the same weight. This picture immediately yields an interesting fact: the quantity $\lambda^{AB}_1\lambda^{AB}_2$ is invariant under permutations of the qubits. (A permutation of qubits corresponds to a reflection or rotation of the cube, but each $d_i$ is invariant under such actions.) This means in particular that we need not carry out a separate calculation to find $\lambda^{AC}_1\lambda^{AC}_2$, since we know we will get the same result. We can now therefore write down an expression for the residual tangle: $$\tau_{A(BC)} - \tau_{AB} - \tau_{AC} = 4|d_1 - 2d_2 + 4d_3|. \label{residual}$$ Note that the residual tangle does not depend on which qubit one takes as the “focus” of the construction. In our calculations we have focused on entanglements with qubit A, but if we had chosen qubit B instead, we would have found that $$\tau_{B(CA)} - \tau_{BC} - \tau_{BA} = 4|d_1 - 2d_2 + 4d_3|.$$ The residual tangle thus represents a collective property of the three qubits that is unchanged by permutations; it is really a kind of three-way tangle. If we call this quantity $\tau_{ABC}$, we can summarize the main results of this paper in the following equation. $$\tau_{A(BC)} = \tau_{AB} + \tau_{AC} + \tau_{ABC}. \label{final}$$ In words, the tangle of A with BC is the sum of its tangle with B, its tangle with C, and the essential three-way tangle of the triple. As an example, consider the Greenberger-Horne-Zeilinger state $\frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$ [@GHZ]. For this state the tangle of each qubit with the rest of the system is 1, the three-way tangle is also 1, and all the pairwise tangles are zero (the qubits in each pair are classically correlated but not entangled). Thus Eq. (\[final\]) in this case becomes $1 = 0 + 0 + 1$. Finally, we note that the expression for $\tau_{ABC}$ in terms of $d_1$, $d_2$, and $d_3$ \[Eq. (\[residual\])\] can be rewritten, after a little more algebra, in a more standard form: $$\tau_{ABC} = 2\bigg|\sum a_{ijk}a_{i^\prime j^\prime m} a_{npk^\prime}a_{n^\prime p^\prime m^\prime} \epsilon_{ii^\prime}\epsilon_{jj^\prime}\epsilon_{kk^\prime} \epsilon_{mm^\prime}\epsilon_{nn^\prime}\epsilon_{pp^\prime} \bigg|,$$ where the sum is over all the indices. This form does not immediately reveal the invariance of $\tau_{ABC}$ under permutations of the qubits, but the invariance is there nonetheless. It would be very interesting to know which of the results of this paper generalize to larger objects or to larger collections of objects. At this point it is not clear how one might begin to generalize this approach to qutrits or higher dimensional objects, because the spin-flip operation seems peculiar to qubits. On the other hand, it appears very likely that at least some of these results can be extended to larger collections of qubits. The one solid piece of evidence we can offer is the existence of a generalization of the state $|\phi\rangle$ of Eq. (\[phi\]) to $n$ qubits: $$|\phi\rangle = \alpha_1|100\ldots0\rangle+\alpha_2|010\ldots0\rangle+ \alpha_3|001\ldots0\rangle+\cdots+\alpha_n|000...1\rangle.$$ One can show that for this state, the following equality holds. $$\tau_{12}+\tau_{13}+\cdots+\tau_{1n} = \tau_{1(23\ldots n)},$$ where the qubits are now labeled by numbers rather than letters. We are willing to conjecture that the corresponding inequality, analogous to Eq. (\[3taus\]), is valid for all pure states of $n$ qubits. This work was supported in part by the National Science Foundation’s Research Experiences for Undergraduates program. WKW is grateful for the support and hospitality of the Isaac Newton Institute. We would also like to thank Tony Sudbery and Noah Linden for discussions that helped streamline the derivation of Eq. (\[first\]). [99]{} C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, [*Phys. Rev. A*]{} [**53**]{}, 2046 (1996). C. H. Bennett, D. P. DiVincenzo, J. Smolin, and W. K. Wootters, [*Phys. Rev. A*]{} [**54**]{}, 3824 (1996). V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, [*Phys. Rev. Lett.*]{} [**78**]{}, 2275 (1997); V. Vedral, M. B. Plenio, K. Jacobs, and P. L. Knight, [*Phys. Rev. A*]{} [**56**]{}, 4452 (1997); V. Vedral and M. B. Plenio, [*Phys. Rev. A*]{} [**57**]{}, 1619 (1998). M. Horodecki, P. Horodecki, and R. Horodecki, [*Phys. Rev. Lett.*]{} [**80**]{}, 5239 (1998). E. M. Rains, quant-ph/9809078, quant-ph/9809082. S. Hill and W. K. Wootters, [*Phys. Rev. Lett.*]{} [**78**]{}, 5022 (1997). W. K. Wootters, [*Phys. Rev. Lett.*]{} [**80**]{}, 2245 (1998). The problem of quantifying multiparticle entanglement has been discussed, for example, in M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, [*Phys. Rev. A*]{} [**57**]{}, 4075 (1998) and A. Thapliyal, [*Phys. Rev. A*]{} [**59**]{}, 3336 (1999). N. Linden and S. Popescu, quant-ph/9711017; N. Linden, S. Popescu, and A. Sudbery, quant-ph/9801076. J. Schleinz and G. Mahler, [*Phys. Lett. A*]{} [**224**]{}, 39 (1996). M. Grassl, M. R[ö]{}tteler, and T. Beth, quant-ph/9712040. J. Kempe, quant-ph/9902036. V. Bužek, V. Vedral, M. B. Plenio, P. L. Knight, and M. H. Hillery, [*Phys. Rev. A*]{} [**55**]{}, 3327 (1997). N. Cerf, quant-ph/9805024. D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin, [*Phys. Rev. A*]{} [**57**]{}, 2368 (1998). A. Karlsson and M. Bourennane, [*Phys. Rev. A*]{} [**58**]{}, 4394 (1998). M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, [*Phys. Rev. A*]{} [**59**]{}, 156 (1999). D. Bruß, quant-ph/9902023. W. K. Wootters, [*Phil. Trans. R. Soc. Lond. A*]{} [**356**]{}, 1717 (1998). D. M. Greenberger, M. Horne, and A. Zeilinger, in [*Bell’s Theorem, Quantum Theory, and Conceptions of the Universe*]{}, ed. M. Kafatos (Kluwer 1989). [^1]: The entanglement of formation is given by $E=h(\frac{1}{2} + \frac{1}{2}\sqrt{1-\tau})$, where $h$ is the binary entropy function $h(x)= -x\log x -(1-x)\log (1-x)$. [^2]: It follows from Ref. [@Wootters] that the concurrence is a convex non-negative function on the set of density matrices for two qubits. The tangle, being the square of the concurrence, is therefore also convex.

--- abstract: 'The effective transparency of rare-gas clusters, post-interaction with an extreme ultraviolet (XUV) pump pulse, is predicted by using an atomistic hybrid quantum-classical molecular dynamics model. We find there is an intensity range for which an XUV probe pulse has no lasting effect on the average charge state of a cluster after being saturated by an XUV pump pulse: the cluster is effectively transparent to the probe pulse. The intensity range for which this phenomena occurs increases with cluster size, and thus is amenable to experimental verification. We present predictions for clusters at the peak of the laser pulse profile, as well as the expected experimental time-of-flight signal integrated over the laser profile. Since our model uses only atomic photoionization rates, significant experimental deviations from our predictions would provide evidence for modified ionization potentials due to plasma effects.' author: - | Rishi Pandit$^1$, Kasey Barrington$^1$, Thomas Teague$^1$,\ Zachary Hartwick$^1$, Nicolas Bigaouette$^2$, Lora Ramunno$^2$, Edward Ackad$^1$ bibliography: - 'photobleaching.bib' title: 'Effective Transparency in the XUV: A Pump-Probe Test of Atomistic Laser-Cluster Models' --- The extreme ultraviolet (XUV) regime has the simplest interaction between ultra-intense laser pulses and matter, primarily through photoionization. When a nanoscopic dense clump of matter (cluster) is irradiated, secondary ionization events then take place such as collisional ionization. Clusters have solid density but their inter-cluster distance is so large that clusters do not interact with each other, thus they bridge the gap between the gas and solid phases of matter. Experimental and theoretical laser-cluster interaction studies in the XUV regime are simpler to interpret than at other wavelengths, and it is thus an ideal regime to further test detailed, atomistic models of laser-cluster interactions [@Arbeiter2011; @Ziaja2013]. At longer wavelengths, even low intensity pulses have efficient processes to transfer energy from the pulse to the free electrons, heating the electron plasma (termed inverse Bremsstrahlung heating, IBH) along the axis of the laser’s polarization [@Peltz2014; @Schutte2016]. At shorter wavelengths, the photoionization that occurs is from the inner shell electrons and leads to subsequent Auger ionization [@Hoener2008; @Thomas2009; @Fennel2010; @PhysRevLett.112.183401; @Tachibana2015]. Thus, XUV pulses – which through photoionization only access valence shell electrons, and where IBH is negligible for intensities $<10^{16}$ W/cm$^2$ – present the ideal regime for experiments to probe the degree to which the ionization potential may be modified by the plasma environment [@Gets2006; @PhysRevLett.93.043402; @PhysRevA.76.043203; @Ziaja2013HED; @PhysRevA.82.013201; @Iwayama2015]. In this letter, we report on the finding that the ionization in XUV-cluster interaction can become effectively *saturated*. In our model, which uses only atomic ionization potentials, this occurs when a cluster is irradiated with an XUV pulse above a saturation intensity. Additional pulses irradiating the cluster leave no *net* effect on the ionization or total energy; thus the cluster is effectively transparent to the probe pulse. Multiple models of laser-cluster interaction exist, and new experiments are needed to allow the community to distinguish between the different models [@Ziaja2013HED]. Atomistic cluster models to date fall into two primary categories: those with collisional processes beyond single step collisional ionization from the valence shell (atomistic augmented collisional model, AACM) and those with enhanced photoionization processes arising from ionization potentials that are lowered below the atomic ionization potentials due to the presence of the nanoplasma environment (atomistic augmented photoionization model, AAPM). Our prediction of effective transparency for pump-probe XUV laser cluster interaction was determined using an AACM, and uses only well-established atomic phenomena. Thus an experimental verification of effective transparency would place an upper bound on the significance of enhanced photoionization mechanisms. On the other hand, the failure of the AACM to correctly predict experimental outcomes would be evidence in favor of an enhanced photoionization mechanism (such as, eg, electron screening or barrier suppression). Thus this letter presents a proposal for an experiment. The schema to distinguish the two models is as follows. A pump pulse irradiates a cluster at an intensity that is above the saturation intensity predicted by the AACM, where atomic ionization potentials are used. Above the saturation intensity, the average ion charge state (AICS, the total charge divided by the number of ions since only ions are detectable) of the cluster in the calculation is independent of pump intensity. This occurs when the intensity is high enough that all possible photoionizations have occurred, but not high enough for significant multiphoton photoionization or IBH. The AACM predicts that the cluster would then be effectively saturated, and would not increase its AICS if subsequently irradiated by a probe pulse. However, if the electromagnetic fields of the nanoplasma sufficiently perturb the atomic ionization potentials so that single-photon ionization from deeper states becomes possible, effective transparency would not be detected in an experiment. The strength of the nanoplasma perturbation would have to be a function of the plasma density. If a probe pulse irradiates the cluster after only a short delay, while the cluster is still dense, the different models will strongly disagree. The nanoplasma perturbation, if large, would allow the cluster to further ionize due to the lowered ionization potentials. With increasing delay of the probe pulse the cluster’s density decreases and so does the nanoplasma’s perturbation. Thus, the two models predict different trends as a function of pump-probe delay. This methodology is complimentary to previous proposals [@Ziaja2013] with the advantage that the effect is enhanced by the cluster size distribution. All previous work in this area has neglected the role of collisional excitation which is known to play a dominant role in the ionization in the XUV [@Muller2015]. Our implementation of an AACM is a hybrid approach wherein the particles are treated as classical charge distributions whose motion is solved by molecular dynamics. The ionization rates are determined from a mix of experimental (when available) and theoretical cross-sections in the gas phase [@AckadPRL]. During ionization, the perturbation on the ions due to the cluster environment (the nanoplasma) has been shown to be well represented by our Local Ionization Threshold (LIT) model [@ionization_approx], which maintains the use of atomic ionization potentials. Using the LIT model we include single- and multi-photon ionization, collisional ionization, augmented collisional ionization (ACI) [@AckadPRL], and many-body recombination [@RecombAckad]. The model has been successful in reproducing the laser-cluster experimental signals [@Ackad_cluster_expansion_XUV], including experiments where Auger ionization is dominant [@RecombAckad]. In AACMs, collisional ionization beyond a single step process is considered. The standard ionization channels are augmented to include the possibility of collisional excitation, so called augmented collisional ionization (ACI) [@AckadPRL]. A bound electron can first be promoted from the ground state to an excited state by a collision of an already ionized electron. Subsequently, this excited electron can be ionized by being promoted from the excited state to the continuum through a second collision. While the *whole trip* can be energetically the same, breaking the process up into two steps reduces the energy required for each transition. This allows an electron with less kinetic energy (compared with single-step ionization) to execute the process. In a nanoplasma, the energy distribution is, on average, Maxwellian and thus there are many more electrons with enough energy to excite an atom than there are who can ionize an atom directly [@Ackad_cluster_expansion_XUV]. This ionization pathway leads to higher charge states in the cluster and collisionally reduced photoabsorption (CRP) where clusters absorb less photons due to fast collisional ionization removing target ions [@Ackad_cluster_expansion_XUV]. The current work includes one and two photon ionization given by the rate, $$\frac{dN}{dt} = \left(\frac{I}{E_{ph}}\right) \sigma^{(1)} + \left(\frac{I}{E_{ph}}\right)^2 \sigma^{(2)}$$ where $I$ is the intensity of the laser, $E_{ph}$ is the photon energy and $\sigma^{(n)}$ is the $n$-th order photoionization process. The values of $\sigma^{(1)}=5.0\times 10^{-18}$ cm$^2$ [@Argon_photo_neutral] and $\sigma^{(2)}= 10^{-50}$ cm$^4$/s (taken as an upper limit from reference [@McKenna2004]) were used. The higher $\sigma^{(2)}$ is, the smaller the range of intensities in which effective saturation will occur, and thus taking an upper limit gives a conservative estimate of the saturation effect. To show the saturation effect in our AACM model, we solved the interaction of argon clusters (Ar$_{147}$) irradiated by two XUV pulses at $\lambda =33$ nm (37.6 eV) 25-fs apart. Both pulses have a full-width-at-half-maximum of 10 fs. Although short pulses increase the probability of multiphoton ionization (which undermines our signal), they also allow the probe pulse to irradiate the cluster while the density is still high (which enhances the likelihood of nanoplasma perturbations which must depend on the plasma density). The number density of the ions at the peak of the pump pulse is around $3.99\times10^{-3}$ bohr$^{-3}$ (where the distance of the furthest ion is used as the radius of the spherical volume) while at the peak of the probe pulse the density is $3.84\times10^{-3}$ bohr$^{-3}$ , a percent difference of about 3.85%. Further, the plasma number-density of the cluster at the same radius is $1.27\times10^{-3}$ bohr$^{-3}$ at the peak of the pump and $1.58\times10^{-3}$ bohr$^{-3}$ at the peak of the probe. Thus, the effects of an enhanced photoionization mechanism will be most pronounced during the probe pulse and would decrease as the pulse delay increases. The specific XUV-wavelength was chosen to be above the singly ionized ionization potential for argon (27.6 eV) and below any significant inner-ionization thresholds. An intensity scan was then performed for the pump pulse. The AICS after 500 fs of the start of the pump pulse is used as a measure of the overall ionization of the cluster, and the cluster is considered to be at the focus of the laser pulse. The average is taken over all ions; ions containing classically bound electrons have their charges decreased accordingly. The solid red curve in Fig. \[fig\_intensity\_vs\_charge\_states\] shows the AICS vs pump intensity when an Ar$_{147}$ cluster is irradiated only by a pump pulse. As pump intensity is increased from $10^{12}$ W/cm$^2$ to $10^{17}$ W/cm$^2$, the AICS starts to increase very gradually. At around $10^{14}$ W/cm$^2$ the AICS increases dramatically until, at an intensity of about $10^{15}$ W/cm$^2$, the AICS becomes saturated around AICS=3.5. This is what we call the “saturation intensity”. Further increasing the pump intensity, only marginally increases the AICS until after the AICS plateau, around $10^{16}$ W/cm$^2$. This small increase is due to multiphoton ionization and IBH. At an intensity of $10^{17}$ W/cm$^2$ AICS reaches about 4.9. Even at this intensity, more than 50% of the ionization is due to collisional ionization, almost exclusively through ACI. At the saturation intensity ACI accounts for well above 90% of all ionizations. To demonstrate effective saturation, a pump probe setup is modeled showing that the probe pulse has almost no effect on the AICS. The pump pulse is fixed at $2.5\times10^{15}$ W/cm$^2$, just above the saturation intensity. The intensity of the subsequent probe pulse (25 fs later) is scanned from $10^{12}$ W/cm$^2$ to $10^{17}$ W/cm$^2$ as depicted at the bottom right illustration of Fig. \[fig\_intensity\_vs\_charge\_states\]. The blue short-dashed curve in Fig. \[fig\_intensity\_vs\_charge\_states\]) shows AICS versus probe intensity, where the AICS is measured 500 fs after the start of the pump pulse. We find that the AICS begins and remains saturated until the intensity of the probe pulse exceeds about $10^{16}$ W/cm$^2$. This is when the probe pulse reaches sufficient intensity for IBH to become significant. Below this intensity, the additional probe pulse does not meaningfully increase the AICS from what it was after the pump; this is the basis for terming the phenomenon effective transparency, since it is as if the cluster were transparent to the probe pulse. Why is there a plateau in the AICS? An analysis of the charge state distribution verses time shows that the irradiation of the cluster by the pump pulse at the saturation intensity ionizes all possible targets via photoionization and collisional ionization. Thus, during the pulse there are no more targets to further photoionize [@Rishi_coming]. This is the high intensity limit to the previously observed CRP [@Ackad_cluster_expansion_XUV]. Without any targets the probe pulse does not contribute to the AICS. The result is thus the same saturated AICS, both with and without the probe pulse. Conceptually, this is where any enhanced photoionization mechanisms would play a significant role. The probe pulse is irradiating a dense nanoplasma and, according to ionization potential (IP) lowering models, would allow for the photoionization of ions well beyond Ar$^{1+}$ and thus change the final AICS significantly [@PhysRevLett.93.043402; @PhysRevA.82.013201]. Our simulations further show that the effective transparency phenomenon is fairly insensitive to the change of the delay time from 15 fs to around 150 fs. Noticeable deviations occur only when the delay time is $>$200 fs. This insensitivity would be a verifiable trend in the experimental data only if no significant ionization potential lowering occurs. As the density of the cluster decreases with the cluster’s disintegration, one would expect the IP lowering effect must also decrease. It would tend to zero as the density becomes that of a gas, since no IP lowering mechanism has been observed in gas [@PhysRevLett.93.043402; @georgescu:043203; @murphy:203401; @PhysRevA.83.043203; @Ziaja2013HED]. IP lowering effects would thus be sensitive to pump-probe delay time. If a lack of sensitivity to the delay time (within the 15-150 fs range for the aforementioned parameters) were found experimentally, it would place constraints on how strong IP lowering contributes to the total ionization. Artificially turning off the probe pulse’s electric field, allowing only direct photoionization the cluster (no IBH), shows that the end of the AICS plateau is due almost exclusively to IBH (short-dashed green curve in figure \[fig\_intensity\_vs\_charge\_states\]). We now consider calculations that correspond more directly to what an experiment would detect. In any cluster beam, there is a log-normal distribution of cluster sizes. Thus, we examine the effect of cluster size on saturation intensity, and the intensity range over which the AICS remains constant. In the range of parameters examined, the saturation intensity $I_{sat}$ decreases as the cluster size increases (shown as the red plus signs in Fig. \[sat\_intensity\] where the line is drawn to aid the eye). This makes intuitive sense since the larger clusters absorb the same amount of energy *per ion* as the smaller clusters. However, the amount of energy needed for an electron to escape the cluster remains the same [@Arbeiter2011]. Thus, larger clusters absorb more *total* energy (than smaller clusters) at the same intensity. It thus takes less intensity to effectively saturate the cluster’s ionization channels. It should be noted that the trend ends once the cluster’s size becomes large enough that the pulse is significantly depleted by the photoabsorption ($N \ge 2057$). ![(Color Online) The saturation intensity (minimum intensity needed to saturate the cluster) $I_{sat}$ as a function of the cluster size is shown as the (red) pluses for pump pulse duration of 10 fs at $\lambda=33$ nm. The intensity range (right vertical axis) as a function of cluster size over which the probe pulse has a negligible effect on the average ion charge state is shown as the (blue) x’s.[]{data-label="sat_intensity"}](figure2.png) The range of intensities over which the cluster is effectively transparent ($I_{\rm high} - I_{\rm low}$) to the probe pulse also increases with the size of the cluster (shown as the blue x’s in Fig. \[sat\_intensity\]). This indicates that the effective saturation is more pronounced in all measures in larger clusters. It was further found that AICS $\approx \alpha \ln(\beta N)$, where $\alpha =0.227$, $\beta=35457.1$ and $N$ is the cluster size less than 2057 for argon [@Rishi_coming]. The cluster size distribution will change the AICS but not by much due to the logarithmic relationship between AICS and $N$. Thus, experiments with the cluster size peaked at a few hundred atoms would have their signals enhanced by the cluster beam’s size distribution. Thus far the results have been for the spatial peak of the laser pulse(s) and may be achievable if the beam is masked to reduce the wings of the pulse as in Ref. [@Hoener2008]. Otherwise, we now consider what an experiment would detect due to the spatial distribution of the pulse. While a small subset of clusters will be irradiated by both pulses at the peak intensities, many clusters spatially located in the wings of the pulse will be irradiated by a pump pulse of insufficient intensity to saturate the ionization channel. The probe pulse will then increase their ionization. In the pump-probe setup, clusters were assumed to interact with the same intensity region of both pulses, i.e., the pulses were assumed to be spatially identical and focused at the same location. The resulting time-of-flight (TOF) signal for the pump pulse alone at $I=2.5 \times10^{15}$ W/cm$^2$ is shown as the dashed red line in Fig. \[fig\_tof1\]. It was calculated using the methodology from reference [@RecombAckad], where the signal is integrated over the intensities of the pulse for a single cluster size and using the TOF setup described in reference [@Thomas2009]). It shows that the signal will contain primarily singly charged ions with an almost linear decrease in the higher charge states. The signal only sees a small change when an identical probe pulse is included (shown as the blue boxes in Fig. \[fig\_tof1\]). If the probe pulse is below the saturation intensity (but with the same spatio-temporal profile), the TOF is quite close to the pump-only signal. However, increasing the probe pulse to beyond the saturation intensity results in some increase in the signal from the multiply-charged states. This is to be expected as now more clusters will fall into a spatial region where they will be saturated by the probe pulse. To illuminate this effect and show the trends an experiment would see in the absence of IP lowering, the AICS is calculated over the entire spatial distribution of the pulse. The saturation of the AICS is not observable for a single pulse (red solid curve in the inset of Fig. \[fig\_tof1\]) due to the spatial wings of the pulse. However, saturation [*is*]{} observable when the clusters are further irradiated by a probe pulse. Fixing the pump pulse’s peak intensity at $2.5\times10^{15}$ W/cm$^2$ and changing the intensity of the probe pulse shows the saturation of the AICS. As the probe pulse’s intensity increases from $10^{12}$ to about $10^{14}$ W/cm$^2$, the AICS remains constant at about 1.5 (blue dashed curve in the inset of Fig. \[fig\_tof1\]). Further increases in the intensity of the probe pulse increase the AICS as more clusters in the wings of the pulse become saturated. This result is again constant with delay times less than about 200 fs. IP lowering models would show much sharper increases in the AICS as a function of the intensity. This would result in the AICS increasing even for a low intensity probe pulse since the additionally photoionized electrons, allowed by IP lowering, would be cluster bound causing additional collisional ionization events. In conclusion, we have shown that atomic-based laser-cluster interaction models predict that it is possible to induce effective transparency in the XUV using a pump-pulse setup. This effect is insensitive to the delay between the pulses, and thus insensitive to nanoplasma density; this is in contrast to what IP lowering models would predict. Experimental verification of our results would place strict limits on the role of IP lowering mechanisms for small rare-gas clusters and would provide the field with valuable data to refine its models of photoionization in laser-cluster interactions not only in the XUV, but for any wavelength where photoionization plays a major role. E. A. would like to thank M. Müller for useful discussions. This work was supported by Air Force Office of Scientific Research under AFOSR Award No. FA9550-14-1-0247.

--- abstract: 'We derive lower bounds for the variance of the difference of energies between incongruent ground states, i.e., states with edge overlaps strictly less than one, of the Edwards-Anderson model on ${\mathbb{Z}}^d$. The bounds highlight a relation between the existence of incongruent ground states and the absence of edge disorder chaos. In particular, it suggests that the presence of disorder chaos is necessary for the variance to be of order less than the volume. In addition, a relation is established between the scale of disorder chaos and the size of critical droplets. The results imply a long-conjectured relation between the droplet theory of Fisher and Huse and the absence of incongruence.' address: - | L.-P. Arguin\ Department of Mathematics\ City University of New York, Baruch College and Graduate Center\ New York, NY 10010 - | C.M. Newman\ Courant Institute of Mathematical Sciences\ New York, NY 10012 USA\ and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai\ 3663 Zhongshan Road North, Shanghai 200062, China - | D.L. Stein\ Department of Physics and Courant Institute of Mathematical Sciences\ New York University\ New York, NY 10003, USA\ and NYU-ECNU Institutes of Physics and Mathematical Sciences at NYU Shanghai\ 3663 Zhongshan Road North\ Shanghai, 200062, China author: - 'L.-P. Arguin' - 'C.M. Newman' - 'D.L. Stein' bibliography: - 'bib\_incongruent.bib' title: 'A Relation between Disorder Chaos and Incongruent States in Spin Glasses on ${\mathbb{Z}}^d$' --- Introduction ============ The Edwards-Anderson (EA) model is a nearest-neighbor model of a realistic spin glass in finite dimensions [@EA75]. As opposed to the infinite-range version, the Sherrington-Kirkpatrick (SK) model [@SK75], the critical behavior of the EA model and in particular the existence of a phase transition and the nature of this phase transition remain elusive from both mathematical and physical perspectives. We refer to [@Mac84; @BM85; @MPV87; @FH88; @KM00; @PY00; @NS03] and references therein for more details on competing pictures for the low-temperature thermodynamic structure of the EA model. In the case of the SK model, it is known that there exist at low enough temperature states with edge overlap[^1] strictly less than one [@Parisi79; @Parisi83; @MPSTV84a; @MPSTV84b; @MPV87]. Such states are said to be [*incongruent*]{}. The question of existence of incongruent ground states at zero temperature for the EA model in finite dimensions is the main motivation of the present paper. More concretely, we relate the existence of such incongruent states to non-trivial lower bounds for the variance of the difference of ground state energies, which we relate in turn to the presence and extent of edge disorder chaos.\ Background ---------- Consider a finite subset $\Lambda\subset {\mathbb{Z}}^d$; $\Lambda$ is considered to be a cube centered at the origin with side-length $L$ so that $|\Lambda|=L^d$. The set of nearest-neighbor edges $\{x,y\}$ with $|x-y|=1$ and $x,y\in \Lambda$ is denoted by $\Lambda^*$. We denote the [*couplings*]{} on $({\mathbb{Z}}^d)^{*}$, the set of all nearest-neighbor edges of ${\mathbb{Z}}^d$, by ${J}=(J_{xy}, \{x,y\}\in ({\mathbb{Z}}^d)^{*})$. We suppose that the couplings are independent and identically distributed Gaussian random variable with mean $0$ and variance $1$. The distribution of $J$ is denoted by $\nu$. The EA Hamiltonian on $\Lambda\subset {\mathbb{Z}}^d$ for the disorder $J$ is the Ising-type Hamiltonian with random couplings $J$: $$\label{eqn: H} H_{\Lambda, J}(\eta)=\sum_{\{x,y\}\in \Lambda^*} -J_{xy}\eta_x\eta_y\ ,$$ where $\eta\in \{-1,+1\}^\Lambda$ is a [*spin configuration*]{} in $\Lambda$. \[df: GS\] A spin configuration $\sigma\in \{-1,+1\}^{{\mathbb{Z}}^d}$ is a [*ground state*]{} for the EA Hamiltonian for the couplings $J$ if for every finite subset $\mathcal B$ of ${\mathbb{Z}}^d$ the configuration $\sigma$ restricted to $\mathcal B$ minimizes $$\label{eqn: GS} H_{\mathcal B, J}(\eta)+ \sum_{\{x,y\}\in \partial \mathcal B } -J_{xy} \eta_x\sigma_y \ \text{over $\eta\in\{-1,+1\}^\mathcal B$} ,$$ where $\partial \mathcal B$ stands for the edges with one vertex $x$ in $\mathcal B$ and one vertex $y$ in $\mathcal B^c$. The minimizer of is unique $\nu$-a.s. for the boundary condition given by $\sigma$ in $\mathcal B^c$. The above definition is equivalent to the property that for any finite subset $\mathcal B$ of $ {\mathbb{Z}}^d$ $$\label{eqn: GS2} \sum_{\{x,y\}\in \partial \mathcal B} J_{xy}\sigma_x\sigma_y\geq 0\ .$$ Consider the [*edge overlap*]{} between $\sigma^1$, $\sigma^2$ in $\Lambda$: $$\label{overlap} Q_{\Lambda}(\sigma^1,\sigma^2)=\frac{1}{|\Lambda^*|} \sum_{\{x,y\}\in \Lambda^*} \sigma^1_x\sigma^1_y \sigma^2_x\sigma^2_y \ .$$ Two ground states are said to be [*incongruent*]{} if $$\label{eqn: incongruent} \limsup_{\Lambda \to {\mathbb{Z}}^d} Q_{\Lambda}(\sigma^1, \sigma^2)<1\ .$$ In other words, there is a strictly positive fraction of edges in $\Lambda$ for which $\sigma_x^1\sigma^1_y\neq \sigma_x^2\sigma^2_y$.\ We write ${\mathcal{G}}(J)\subset \{-1,+1\}^{{\mathbb{Z}}^d}$ for the set of infinite-volume ground states for the couplings $J$. In Section \[sect: metastate\], we recall the construction of certain measures on ${\mathcal{G}}(J)$ from limits of finite-volume ground states with specified boundary conditions. Such a measure will be denoted by $\kappa_J$ and referred to as a [*metastate*]{}. From these measures, it is possible to study three questions: 1. Is there more than one sub-sequential limit $\kappa_J$ along an infinite sequence of volumes? 2. How many ground states are in the support of $\kappa_J$ ? 3. Do there exist two or more [*incongruent*]{} ground states in the support of these measures? To study these questions, we consider the probability measure ${\mathbb{P}}$ on triples $(J,\sigma^1, \sigma^2)$ where $$\label{eqn: prob} {{\rm d}}{\mathbb{P}}= {{\rm d}}\nu (J) \times {{\rm d}}\kappa^{(1)}_{J}(\sigma^1)\times {{\rm d}}\kappa^{(2)}_{J}(\sigma^2)\ ,$$ where $\kappa_J^{(1)}, \kappa_J^{(2)}$ are two metastates. The measure ${\mathbb{P}}$ samples the disorder $J$ and then two ground states for that disorder according to $\kappa^{(1)}_J\times\kappa^{(2)}_J$. Questions (i), (ii), and (iii) were answered for the half-plane in [@ADNS10] (see also [@AD14] for general results on the set of ground states). This paper is mainly concerned with Question (iii) for the model on ${\mathbb{Z}}^d$. Question (iii) is narrower than (i) and (ii) in general, except when periodic boundary conditions are considered. In that particular case, ${\mathbb{P}}$ is translation-invariant and the existence of a single edge where $\sigma^1$ and $\sigma^2$ differ ensures the existence of a positive density of such edges. Main Results ------------ It is possible to modify the couplings locally under the measure ${{\rm d}}{\mathbb{P}}$ as follows. First, we redefine ${\mathbb{P}}$ to add an extra independent copy $J'$ of the couplings: $$\label{eqn: prob3} {{\rm d}}{\mathbb{P}}= {{\rm d}}\nu (J)\times {{\rm d}}\nu (J')\times {{\rm d}}\kappa^{(1)}_{J}(\sigma^1)\times {{\rm d}}\kappa^{(2)}_{J}(\sigma^2)\ .$$ We consider an interpolation $J(t)$ parametrized by $t\geq 0$ where $$\label{eq:t} J_{xy}(t)={{\rm e}}^{-t}J_{xy}+\sqrt{1-{{\rm e}}^{-2t}}J'_{xy} \qquad \text{ if $\{x,y\}\in \Lambda^*$,}$$ and $J_{xy}(t)=J_{xy}$ if $\{x,y\}\notin \Lambda^*$. For each ground state $\sigma^1$ and $\sigma^2$, we will construct in Section \[sect: metastate\] a measurable map $t\mapsto \sigma^i(t)$, $i=1,2$, which for each $t$ gives a ground state for the value of the interpolated couplings at $t$. (We slightly abuse notation here since we use $\sigma^i$ for the map as well as for the initial point $\sigma^i=\sigma^i(0)$.) It turns out that the distribution of the ground states $\sigma^i(t)$ under $\kappa^{(i)}_{J}$ is exactly the one of $\sigma^i$ under $\kappa^{(i)}_{J(t)}$, cf. Section \[sect: metastate\]. The first main result of this paper is to establish a lower bound for the variance of the difference of ground state energies in terms of local coupling modifications. \[thm: main\] For all $t>0$, $$\label{eqn: variance bound} \begin{aligned} &{\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)\Big)\geq \\ &2|\Lambda^*|\int_0^t\left\{{\mathbb{E}}\Big[1-Q_\Lambda(\sigma^1,\sigma^2)\Big]-\sum_{i=1,2} \Big(2\cdot{\mathbb{E}}\big[1-Q_\Lambda(\sigma^i,\sigma^i(s))\big]\Big)^{1/2}\right\}{{\rm e}}^{-s}{{\rm d}}s \ . \end{aligned}$$ The main interest of this bound is the explicit connection between incongruence, represented by the first expectation, and disorder chaos, or rather the absence thereof, represented by the second expectation. \[df: ADC\] We say that there is absence of disorder chaos at scale $\alpha$, $0\leq \alpha\leq 1$, for ${\mathbb{P}}$, if for any ${\varepsilon}>0$ there exist $A_{\varepsilon}$ with ${\mathbb{P}}(A_{\varepsilon})>1-{\varepsilon}$ and $C=C({\varepsilon})>0$, such that $$Q_\Lambda(\sigma^i, \sigma^i(t))>1-{\varepsilon}\text{ on $A_{\varepsilon}$, $i=1,2$,}$$ for all $t\leq C|\Lambda|^{-\alpha}$ and all $\Lambda$ large enough. In other words, there is absence of disorder chaos at scale $\alpha$ if with large probability, the fraction of edges for which $\sigma^1(t)$ is different from $\sigma^1(0)$ remains small for $t\leq C|\Lambda|^{-\alpha}$. Let $\mathcal I$ be the event that incongruent states exist, that is $$\mathcal I=\left\{(J,\sigma^1, \sigma^2): \limsup_{\Lambda \to {\mathbb{Z}}^d} Q_{\Lambda}(\sigma^1, \sigma^2)<1\right\}\ .$$ Definition \[df: ADC\] and Theorem \[thm: main\] imply: \[cor: ADC-&gt; variance\] Let ${\mathbb{P}}$ be as in Equation with ${\mathbb{P}}(\mathcal I)>0$. If there is absence of disorder chaos at scale $0\leq \alpha \leq 1$, then there exists $C>0$ independent of $\Lambda$ such that $$\label{eqn: var bound 1} {\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)\Big)\geq C |\Lambda|^{1-\alpha}\ .$$ The second main result is a relation between the size of critical droplets and the absence of disorder chaos as above. Fix an edge $b=\{x_0,y_0\}$ in a box $\Lambda$. As a function of $J_b$, the ground state is locally constant, cf. Section \[sect: metastate\]. Roughly speaking, according to , the ground state changes as $J_b$ is increased (or decreased) when the energy of the boundary (which of course passes through $b$) of some connected cluster of spins first becomes negative. This connected cluster could be infinite. We write $\mathcal D_b$ for the subset of vertices of this cluster inside $\Lambda$. We refer to $\mathcal D_b$ as the [*critical droplet of the edge $b$ in $\Lambda$*]{}. The important quantity is the size of the boundary $\partial \mathcal D_b$ containing the edges at the boundary with one vertex in $\mathcal D_b$ and one in its complement; see Figure \[fig: droplet\] below for an illustration. The next theorem relates the size of critical droplet boundaries to the absence of disorder chaos. \[thm: ADC\] Let ${\mathbb{P}}$ be as in Equation . Suppose that there exist $0\leq \gamma\leq 1$ and $C<\infty$ (independent of $\Lambda)$ such that with probability one, for all large $\Lambda$, $$\label{eqn: droplet ass} |\partial\mathcal D_b| \leq C |\Lambda|^\gamma \text{ for all $b\in |\Lambda^*|$.}$$ Then there is absence of disorder chaos for ${\mathbb{P}}$ at every scale $\alpha>2\gamma$. Assumption is a statement about the distribution of the size of the droplet. Indeed, we have by a union bound that $${\mathbb{P}}(\exists b\in \Lambda^*: |\partial \mathcal D_b|>a )\leq \sum_{b\in \Lambda^*} {\mathbb{P}}(|\partial \mathcal D_b|>a )\ .$$ Therefore, taking $a=a(\Lambda)$, the assumption would be satisfied if the tail distribution decays fast enough to ensure summability. Together with Corollary \[cor: ADC-&gt; variance\], this shows that non-trivial bounds on the variance of the difference of the ground state energies can be obtained by estimating the size of the critical droplets. Theorem \[thm: ADC\] is probably far from optimal as it only gives non-trivial variance bounds for $\gamma<1/2$. It is easy to check that $\gamma=0$ at $d=1$, and one might expect that $\gamma=0$ also in $d=2$. More precise estimates combining the geometry of the droplets and their energy are needed to improve the result – see Remark \[rem: estimate\]. As a modest first step in this direction, we get that the variance is uniformly bounded away from zero. \[cor: variance one\] Let ${\mathbb{P}}$ be as in Equation with ${\mathbb{P}}(\mathcal I)>0$. Then one has for some constant $C>0$ independent of $\Lambda$, $$\label{eqn: var bound 2} {\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)\Big)\geq C \ .$$ Relations to Other Results -------------------------- A variance lower bound for the difference of ground state energies was proved in [@ANSW16] under the assumption that the average (over the metastate) of the edge correlation function differs for $\sigma^1$ and $\sigma^2$. There the variance lower bound was obtained by an adaptation of the martingale approach of [@AW90]. The corresponding result at positive temperature was proved in [@ANSW14]. As in [@AW90], the variance bound in [@ANSW14; @ANSW16] is based on the elementary inequality $$\label{eqn: var before} {\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)\Big)\geq {\text{\rm Var}}\left({\mathbb{E}}\big[H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)\ \big|J_\Lambda\big]\right)\ .$$ A non-trivial variance lower bound can then be proved if there is an inherent asymmetry between $\sigma^1$ and $\sigma^2$ [*on average*]{} over $\kappa^{(1)}_J\times\kappa^{(2)}_J$ and over all couplings but the ones in $\Lambda$. For ferromagnetic models, such as random-field ferromagnets, this is not a problem as the plus and minus states retain such an asymmetry. In [@ANSW14; @ANSW16], the needed asymmetry arose as a consequence of the assumption of the existence of incongruence in spin glasses. Of course, this assumption might not hold in general. A novel approach used in the present paper is to obtain variance lower bounds by [*conditioning on the disorder outside $\Lambda$*]{}. In effect, we use the asymmetry between incongruent states that always exists when the couplings $J_{\Lambda^c}$ outside $\Lambda$ are fixed, and the ground states $(\sigma^1, \sigma^2)$ (always assumed to be incongruent) for this choice of couplings outside $\Lambda$ are also fixed. One can then think of the ground states in $\Lambda$ for the [*boundary condition*]{} $\sigma^1$ as a function of $J_\Lambda$: $J_\Lambda\mapsto \sigma^1(J_\Lambda)$. By conditioning on $(J_{\Lambda^c},\sigma^1, \sigma^2)$ instead of $J_\Lambda$ as in , we get that the variance is bounded below by $$\label{eqn: idea} {\mathbb{E}}\Big[{\text{\rm Var}}\big(H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)\ \big |J_{\Lambda^c},\sigma^1, \sigma^2\big)\Big]\ .$$ It turns out that the couplings $J_\Lambda$ are independent of $(J_{\Lambda^c},\sigma^1, \sigma^2)$, and thus remain Gaussian, cf. Lemma \[lem: metastate\]. Therefore, variance lower bounds can be obtained on ${\text{\rm Var}}\big(H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)\ \big |J_{\Lambda^c},\sigma^1, \sigma^2\big)$ using Gaussian methods.\ When no magnetic field is present, disorder chaos in the mathematical literature often refers simply to the overlap $Q_\Lambda(\sigma^1, \sigma^1(t))$ being close to $0$ with large probability for some positive $t$; see e.g. [@C14]. With this definition, absence of disorder chaos means that $Q_\Lambda(\sigma^1, \sigma^1(t))$ is bounded away from $0$ for $t$ small. For example, for the EA model, Chatterjee [@C14] showed absence of disorder chaos in this sense by proving the bound [^2] $${\mathbb{E}}[Q_{\Lambda}(\sigma^1, \sigma^1(t))]\geq C q{{\rm e}}^{-t/(Cq)}$$ for some constant $C>0$ and $q=1/(4d^2)$; see [@C14]. This bound is [*a priori*]{} too weak to get a good lower bound using Theorem \[thm: main\]. This is because it does not preclude that $\sigma^1(t)$ has overlap strictly smaller than one with $\sigma^1(0)$, and thus severely differs from $\sigma^1(0)$ for very small $t$. Absence of disorder chaos (in the above sense) was also proved for some range of $t$ depending on the size of the system for $p$-spin spherical spin glasses by Subag in [@Subag16]. In the physics literature, where the concept arose, the definition of disorder chaos (or the closely related temperature chaos) is slightly more nuanced, in that both occur only beyond a lengthscale related to the size of the perturbation $t$ [@BM87; @FH88; @KB05]. Our Definition \[df: ADC\] is simply a formalized version of the standard physics definition, with an emphasis on the [*scale*]{} of disorder chaos represented by the parameter $\alpha$. The central result of this paper is the connection established through Theorem \[thm: main\] and Corollary \[cor: ADC-&gt; variance\] between the scale of edge disorder chaos and the size of fluctuations in incongruent ground state energies, which in turn has a direct bearing on the possible presence or absence of incongruence in short-range spin glasses [@ANSW14; @ANSW16; @Stein16]. A further, unanticipated relation is established in Theorem \[thm: ADC\], in which the size of critical droplets is shown to set the scale of disorder chaos, creating a direct link between the stability of spin glass ground states (through the size of their critical droplets) and ground state multiplicity. Finally, the results proved in this paper shed light on predictions made by the droplet theory of spin glasses [@FH86; @FH87; @HF87; @FH88], based on scaling approaches [@Mac84; @BM85], on the absence of incongruence in short-range spin glasses. This will be taken up in Sect. \[sec:scaling\]. Structure of the Paper ---------------------- The necessary background about measures on ground states is given in Section 2. In Section 3, we prove Theorem \[thm: main\] based on standard Gaussian interpolation applied to the conditional variance . The proofs of Theorem \[thm: ADC\] and those of Corollaries \[cor: ADC-&gt; variance\] and Corollary \[cor: variance one\] appear in Section 4. Finally, Section 5 discusses the connection between the approach developed here and the droplet theory of Fisher & Huse. [**Acknowledgements.**]{} The authors thank the referee for many important suggestions that led to simplifications of some of the proofs. The authors are also grateful to Nick Read for insightful remarks and for an important correction to Proposition 2.9, as well as Aernout van Enter and Jon Machta for useful comments on the manuscript. The research of LPA is supported in part by U.S. NSF Grant DMS-1513441 and by U.S. NSF CAREER DMS-1653602. The research of CMN was supported in part by U.S. NSF Grants DMS-1207678 and DMS-1507019. The research of DLS was supported in part by U.S. NSF Grant DMS-1207678. Local Modification of Couplings {#sect: metastate} =============================== In this section, we develop the necessary framework to address the dependence of infinite-volume ground states on local modifications of couplings. This theory of local excitations is based on a previous construction of the [*excitation metastate*]{}, see [@NS01; @ADNS10; @ANSW14; @AD11]. The main results are Propositions \[prop: criterion\] and \[prop: flex diff\] which together yield sufficient conditions, in terms of the size of the critical droplet of a given edge, for a ground state to remain the same at that edge under local modification of couplings. Throughout this section and henceforth, we will sometimes use the notation $\sigma_e=\sigma_x\sigma_y$ for the spin interaction at the edge $e=\{x,y\}$. We will also fix the finite box $\Lambda\subset {\mathbb{Z}}^d$, and sometimes omit its dependence in the notation. Measures on Ground States and Local Excitations ----------------------------------------------- We first construct a measure on the set of ground states $\mathcal G(J)$ in terms of finite-volume ones. Consider a box $\mathcal B_n=[-n,n]^d$ on ${\mathbb{Z}}^d$ and the EA Hamiltonian on $\mathcal B_n$ with specified boundary condition $\xi$ $$\label{eqn: H_n} H_{\mathcal B_n, J}(\eta)= \sum_{e\in \mathcal B_n^*}-J_e\eta_e + \sum_{\{x,y\}\in \partial \mathcal B_n} -J_{xy} \eta_x \xi_y\ .$$ The ground state for this Hamiltonian is the unique $\nu$-a.s. minimizer over all $\eta\in \{-1,+1\}^{\mathcal B_n}$. Its restriction on $\Lambda$ can be determined using Definition . Equivalently, it can be determined using the difference of energies which extends more easily to infinite volume. More precisely, the restriction of the ground state to $\Lambda$ is the unique $\nu$-a.s. configuration $\eta\in\{-1,+1\}^\Lambda$ such that $$\label{eqn: GS diff} H_{\Lambda, J}(\eta)-H_{\Lambda, J}(\eta')+ E_n(\eta,\eta')<0 \ \ \forall \eta'\neq \eta\ ,$$ where $$\label{eqn: E} E_n(\eta,\eta')= \sum_{e\in \mathcal B_n^*\setminus \Lambda^*}-J_e(\sigma^\eta_e-\sigma^{\eta'}_e) + \sum_{\{x,y\}\in \partial \mathcal B_n} -J_{xy} (\sigma_x^\eta-\sigma_y^{\eta'}) \xi_y\ .$$ The variable $E_n(\eta,\eta')$ is the difference of energies [*outside $\Lambda$*]{} of the states $\sigma^\eta$, $\sigma^{\eta'}$ that minimizes $H_{\mathcal B_n, J}$ over the configurations equal to $\eta$ on $\Lambda$, and similarly for $\eta'$. The advantage of this formulation is two-fold. First, as detailed below, the random variables $E_n(\eta,\eta')$, $n\geq 1$, as a function of $J$ are tight. Second, the random variables $\vec E_n=\big(E_n(\eta,\eta');\eta, \eta'\in \{-1,+1\}^\Lambda\big)$ are independent of $J_\Lambda$. This is because the restriction to fixed $\eta$ cancels out the dependence on $J_\Lambda$. These two observations lead to: \[lem: metastate\] Fix $\Lambda\subset {\mathbb{Z}}^d$. There exists a subsequence such that the joint distribution of $(J_,\vec E_n)$ converges weakly to a probability measure on $(J, \vec{E})$ where $\vec{E}=\big(E(\eta, \eta'); \eta,\eta'\in\{-1,1\}^{\Lambda}\big)$ with the properties: - [Boundedness]{}: For every $\eta,\eta'$ $$\label{eqn: bound} E(\eta,\eta')\leq \sum_{e\in \partial \Lambda}2 |J_e|\ \ a.s.$$ - [Linear relations]{}: $E(\eta,\eta)=0$ for every $\eta$, and for every $\eta, \eta',\eta''$, $$\label{eqn: linear} E(\eta, \eta'')=E(\eta, \eta')+ E(\eta', \eta'')\ \ a.s.$$ - [Independence]{}: Write $J=(J_{\Lambda},J_{\Lambda^c})$ where $J_{\Lambda}=(J_e, e\in \Lambda^*)$. Then the pair $(J_{\Lambda^c},\vec E)$ is independent of $J_\Lambda$. The tightness of the random variables $(\vec E_n, n\in {\mathbb{N}})$ follows from the inequality $$\label{eqn: bound n} E_n(\eta,\eta')\leq \sum_{e\in \partial \Lambda}2 |J_e|\ .$$ This is because $$H_{\mathcal B_n, J}(\sigma^\eta)-H_{\mathcal B_n, J}(\sigma^{\eta'}) \leq H_{\Lambda, J}(\eta)-H_{\Lambda, J}(\eta') +\sum_{e\in \partial \Lambda}2 |J_e|\ ,$$ where the inequality is obtained by replacing $\sigma^\eta_x$ for $x\in \Lambda^c$ by $\sigma^{\eta'}_x$ in the difference of energies and , which increases the energy by definition of $\sigma^\eta$. The tightness of the pair $(J,\vec E_n)$ directly follows since the $J$’s are IID. Equation is also straightforward from Equation at finite $n$. The linear relations are satisfied for every $\vec E_n$ and therefore extend to the weak limits. The same holds for the independence with $J_\Lambda$. Since we are interested in incongruent states, only the values $\eta_e=\eta_x\eta_y$ on an edges $e$ matter. With this in mind we consider the collection $\vec{E}=\big(E(\eta, \eta'); \eta,\eta'\big)$ as indexed by elements $\eta,\eta'\in \{-1,+1\}^{\Lambda^*}$ where $\eta_e=\eta_x\eta_y$. Of course, if two spin configurations are equal up to a global spin flip, then they correspond to the same element in $\{-1,+1\}^{\Lambda^*}$. We then choose as a representative the one with [*smaller energy*]{} $E$; that is, we pick $\eta$ if $E(\eta,\eta')<0$ and $\eta'$ if $E(\eta,\eta')>0$. In the case where $E(\eta,\eta')=0$, which happens for example when periodic boundary conditions are considered, the $\eta$’s are simply identified.\ We write $\kappa_{J_{\Lambda^c}}({{\rm d}}\vec E)$ for the conditional distribution of $\vec{E}$ given $J$, highlighting the independence from $J_\Lambda$, constructed from Lemma \[lem: metastate\]. The variable $\vec E$ retains the relevant information on the boundary condition to determine the ground state in the box $\Lambda$ (up to a global spin flip). Given $\vec E$, the ground state in $\Lambda$ can be determined uniquely as a function of $J_\Lambda$ as in among all configuration in $\{-1,1\}^{\Lambda^*}$, assuming there are no non-trivial degeneracies. These degeneracies will occur, for a given $\vec E$ sampled from $\kappa_{J_{\Lambda^c}}$, on the [*critical set*]{} given by the union of hyperplanes $$\label{eqn: C} \mathcal C=\mathcal C(\vec E)=\bigcup_{\substack{\eta\neq \eta'}}\big\{J_\Lambda\in {\mathbb{R}}^{\Lambda^*}:\sum_{e\in \Lambda^*}J_e(\eta_e-\eta_e')=E(\eta, \eta')\big\}\ .$$ The union is over distinct $\eta,\eta'\in \{-1,+1\}^{\Lambda^*}$. We refer to each hyperplane defining the critical set as a [*critical hyperplane*]{}. We work out the details of the cases where $\Lambda$ contains one and two edges in Remark \[rem: examples\] below. On the complement of the critical sets, it is possible to order the spin configurations in $\Lambda$ (up to spin flips) in decreasing order of their energies. In particular, it is possible to determine the ground state. \[prop: ordering\] For a given $\vec E$ with the property and $J_\Lambda\in {\mathbb{R}}^{\Lambda^*}\setminus \mathcal C$, there is a well-defined ordering $\eta^{(1)}\prec\eta^{(2)}\prec\dots$ of the elements of $\{-1,+1\}^{\Lambda^*}$ given by $$\label{eqn: diff} \eta \prec \eta ' \Longleftrightarrow E(\eta,\eta')+H_{\Lambda,J}(\eta)-H_{\Lambda,J}(\eta')<0\ .$$ The critical set corresponds to the value of $J_\Lambda$ for which $\eta^{(i)}=\eta^{(j)}$ for some pair $i\neq j$. As a reference point, take $\eta^0\in$ with $\eta_e^0=+1$ for all $e\in \Lambda^*$. If $J_\Lambda \notin \mathcal C$, there exists a unique $\eta$ that minimizes the difference of energy $$E(\eta, \eta_0)+H_{\Lambda,J}(\eta)-H_{\Lambda,J}(\eta_0)\ .$$ Indeed, if $\eta'\neq \eta$ was also a minimizer we would have by the linearity that $E(\eta, \eta')+H_{\Lambda,J}(\eta)-H_{\Lambda,J}(\eta')=0$ contradicting the fact that $J_\Lambda$ is not in $\mathcal C$. Denote this unique minimizer by $\eta^{(1)}$. We define $\eta^{(2)}$ as the minimizer of the difference of energy $E(\eta, \eta^{(1)})+H_{\Lambda,J_\Lambda}(\eta)-H_{\Lambda,J_\Lambda}( \eta^{(1)})$ over $\eta$’s not equal to $\eta^{(1)}$. By construction this difference of energy is strictly positive. Again $\eta^{(2)}$ is uniquely defined by linearity. The whole sequence $\eta^{(j)}$ is constructed this way until $\{-1,+1\}^{\Lambda^*}$ is exhausted. The relation $\eta\prec \eta'$ is straightforward from construction. The ordering introduced above defines three important maps from ${\mathbb{R}}^{\Lambda^*}\setminus \mathcal C$ to $\{-1,1\}^{\Lambda^*}$ which allow the study of excitations as a local function of the couplings. The [*ground state map*]{} is the map $$\label{eqn: gs map} \begin{aligned} \sigma(\cdot):{\mathbb{R}}^{\Lambda^*}\setminus \mathcal C&\to \{-1,+1\}^{\Lambda^*}\\ J_\Lambda&\mapsto \sigma(J_\Lambda)=\eta^{(1)} \end{aligned}$$ where $\sigma(J_\Lambda)$ is $\eta^{(1)}$ in the ordering at $J_\Lambda$ given by Proposition \[prop: ordering\]. For a given edge $b\in\{-1,+1\}^{\Lambda^*}$, we define the [*excitation map at the edge $b$*]{} as $$\label{eqn: pm map} \begin{aligned} \sigma^{+,b}(\cdot):{\mathbb{R}}^{\Lambda^*}\setminus \mathcal C&\to \{-1,+1\}^{\Lambda^*}\\ J_\Lambda&\mapsto \sigma^{+,b}(J_\Lambda) \end{aligned}$$ where $\sigma^{+,b}(J_\Lambda)\prec \eta$ for all $\eta\neq \sigma^{+,b}(J_\Lambda)$ with $\eta_b=+1$. In words, $\sigma^{+,b}(J_\Lambda)$ is the configuration of smallest energy with the restriction that $\eta_b=+1$. The map $\sigma^{-,b}(\cdot)$ is defined similarly, but restricting to $\eta$’s with $\eta_b=-1$. Note that we evidently have $\sigma(J_\Lambda)= \sigma^{+,b}(J_\Lambda)$ or $\sigma(J_\Lambda)=\sigma^{-,b}(J_\Lambda)$.\ The precise definition of $\kappa_J({{\rm d}}\sigma)$ appearing in Equation can now be given. We use the same notation for both the measure on $\vec E$ and $\sigma$ as they are directly related. \[df: GS prob\] The probability measure $\kappa_J({{\rm d}}\sigma)$ on infinite-volume ground states restricted to $\Lambda$ is the distribution of $\sigma(J_\Lambda)$ as defined in under $\kappa_{J_{\Lambda^c}}({{\rm d}}\vec E)$. [ It is not hard to check that the definition of $\kappa_J({{\rm d}}\sigma)$ is equivalent to taking weak limits of the distribution of the ground states (up to a spin flip) of $H_{\mathcal B_n, J}$ given in as a probability measures on $\{-1,+1\}^{\Lambda^*}$. The construction in Lemma \[lem: metastate\] has the disadvantage of having the dependence on $J_\Lambda$ implicit in $\sigma$, which makes impossible to study the local modification of the couplings. The advantage of working with $\vec E$ is that the dependence on $J_\Lambda$ appears solely in the Hamiltonian in $\Lambda$ as in . This property is sometimes referred to as [*coupling covariance*]{}, see e.g. [@ANSW16]. ]{} \[rem: examples\] The simplest cases of excitation metastates where $\Lambda$ contains one and two edges were worked out in [@NS01] and [@ADNS10] respectively. We briefly recall these examples here to illustrate the general theory. [*Case of one edge*]{}. Consider $\Lambda=\{x,y\}$ where $x,y$ are nearest-neighbor vertices with $b=\{x,y\}$. We have that $\eta_b=+1$ or $-1$. The collection $\vec E$ of energies has four values $E(+,-), 0,0$ and $E(-,+)=-E(+,-)$. The critical set is defined by a single equation: $$2J_b=E(+,-)\ ,$$ and consists of the [*critical value*]{} $\mathcal C_b=E(+,-)/2$. Note that $\mathcal C_b$ is independent of $J_b$ by Lemma \[lem: metastate\]. The ground state $\sigma(J_b)$ at the edge $b$ is $+1$ for $J_b>\mathcal C_b$ and $-1$ for $J_b<\mathcal C_b$. The flexibility of the edge $b$, defined in below, is the function $F_b(J_b)$ giving the energy difference between $\sigma^{+,b}$ and $\sigma^{-,b}$ in absolute value. Here it is simply $$F_b(J_b)=|2J_b-E(+,-)|=2|J_b-\mathcal C_b|\ .$$ [*Case of two edges*]{}. Take $\Lambda=\{x,y,w,z\}$ with edges $b=\{x,y\}$ and $e=\{w,z\}$. In this case, the configuration $\eta$ takes value in $\{++;+-;-+;--\}$ where we write the configuration at $b$ first and at $e$ second. The critical set is defined by six equations $$\label{eqn: ex two} \begin{aligned} 2J_b&=E(++;-+) \qquad &2J_b=E(+-;--) \\ 2J_e&=E(++;+-) \qquad &2J_e=E(-+,--) \\ 2(J_b+J_e)&=E(++,--) \qquad &2(J_b-J_e)=E(+-;-+)\ . \end{aligned}$$ There are three possible scenarios: $E(++,-+)>E(+-;--)$, $E({\tiny ++},-+)=E(+-;--)$, $E(++,-+)<E(+-;--)$. We look at the first case. It is depicted in Figure \[fig: flex\] in the $(J_b,J_e)$-plane. Note that by linearity the inequality implies also $E(++;+-)>E(-+;--)$ by adding $E(-+,+-)$ on both sides. The equations define sixteen regions where the ordering of the $\eta$’s (in terms of the energy differences) is non-degenerate. (Not all twenty-four orderings of the four states are possible, since some are precluded by the energies.) We now focus on the degeneracy of the ground state. This happens at points $(J_b,J_e)$ where the energy difference between $\sigma^{+,b}(J_b,J_e)$ and $\sigma^{-,b}(J_b,J_e)$, or between $\sigma^{+,e}(J_b,J_e)$ and $\sigma^{-,e}(J_b,J_e)$, is zero. We treat the first case. The state $\sigma^{+,b}(J_b,J_e)$ can be either $(++)$ or $(+-)$, and $\sigma^{-,b}(J_b,J_e)$ can be either $(-+)$ or $(--)$. The energy difference between each is $$E(++;+-)-2J_e\qquad E(-+;--)-2J_e\ .$$ Both are negative for $J_e$ large enough, showing that we must have $\sigma^{+,b}(J_b,J_e)=(++)$ and $\sigma^{-,b}(J_b,J_e)=(-+)$. The same way we have $\sigma^{+,b}(J_b,J_e)=(+-)$ and $\sigma^{-,b}(J_b,J_e)=(--)$ for $J_e$ small enough. We conclude that the ground state degeneracy occurs at $J_b=E(++;-+)/2$ for $J_e$ large enough, and at $J_b=E(+-;--)/2$ for $J_e$ small enough. There is also a middle region where the degeneracy occurs between $(+-)$ and $(-+)$ at $J_b=J_e+E(+-;-+)/2$. ![An illustration of the critical set for two edges $b$ and $e$ in the $(J_b,J_e)$-plane. The dotted lines are the lines where the energy difference between two states is zero. The bold lines represent a degeneracy of the ground state. They delimit four regions where the ground state is non-degenerate. []{data-label="fig: flex"}](2edge_flex.pdf){height="7cm"} Critical Droplets and Flexibilities ----------------------------------- Now that we can control the ground state as a function of $J_\Lambda$, we can study how the ground state at given edge $b$ depends on the couplings in $\Lambda$. The ground state configuration at $b$ is $+1$ if $\sigma^{+,b}(J_\Lambda)$ is the ground state and $-1$ if $\sigma^{-,b}(J_\Lambda)$ is the ground state. The difference of energies between the two determines the correct value. Changes in the ground state occur when this energy difference is zero. With this in mind, we consider the absolute value of the difference of energies or [*flexibility of the edge $b$*]{} introduced in [@NS2000]: $$\label{eqn: flex} \begin{aligned} F_b(J_\Lambda)&=\left|-\sum_{e\in \Lambda^*} J_e \big(\sigma^{+,b}_e(J_\Lambda)-\sigma^{-,b}_e(J_\Lambda)\big)+ E(\sigma^{+,b}(J_\Lambda), \sigma^{-,b}(J_\Lambda))\right| \ . \end{aligned}$$ The flexibility $F_b$ is a map that measures the sensitivity of the ground state at the edge $b$ as a function of the couplings, as highlighted in Proposition \[prop: criterion\]. The terms in the first sum are only non-zero on the edges of the [*boundary of the critical droplet of the edge $b$ in $\Lambda$*]{} at $J_\Lambda\in {\mathbb{R}}^{\Lambda^*}\setminus \mathcal C$, defined to be the set $$\label{eqn: droplet} \partial\mathcal D_b(J_\Lambda)=\{e\in \Lambda^*: \sigma_e^{+,b}(J_\Lambda)\neq \sigma_e^{-,b}(J_\Lambda)\}\ .$$ ![An illustration of the critical droplet of an edge $b$ (in gray) and its boundary in $\Lambda$. The vertices in the box $\Lambda$ are black. The edges in $\partial\mathcal D_b(J_\Lambda)$ are the ones in $\Lambda^*$ that cross the boundary of the droplet.[]{data-label="fig: droplet"}](droplet.pdf){height="6cm"} The following lemma is important to control the stability of the ground states as couplings are modified. It shows that the flexibility uniquely extends to a continuous map on ${\mathbb{R}}^{\Lambda^*}$. \[lem: continuity\] For every edge $b\in \Lambda^*$, the map $J_\Lambda\mapsto F_b(J_\Lambda)$ on $\mathcal {\mathbb{R}}^{\Lambda^*}\setminus \mathcal C$ is a piecewise affine function with $$\frac{\partial F_b}{\partial J_e} (J_\Lambda)= \begin{cases} 2\sigma_e(J_\Lambda) & \text{ if $e\in \partial\mathcal D_b(J_\Lambda)$}\\ 0 & \text{ otherwise.} \end{cases}$$ Furthermore, the map extends uniquely to a continuous function on ${\mathbb{R}}^{\Lambda^*}$. Consider $J_\Lambda \in {\mathbb{R}}^{\Lambda^*}\setminus \mathcal C$. By the definition of $\sigma(J_\Lambda)$ and the fact that ${\mathbb{R}}^{\Lambda^*}\setminus \mathcal C$ is open, we have that the map $\sigma(\cdot)$ is constant in a neighborhood $V=V(J_\Lambda)$ of $J_\Lambda$, and so are $\sigma^{+,b}(\cdot)$ and $\sigma^{+,b}(\cdot)$. Write $\sigma$, $\sigma^{+}$, $\sigma^-$ for the respective values in $V$. Suppose without loss of generality, that $\sigma=\sigma^{+}$. The critical droplet boundary $\partial\mathcal D_b$ is also constant on $V$. The flexibility of the edge $b$ on $V$ takes the form $$F_b(y)=\sum_{e\in \partial\mathcal D_b} 2y_e\sigma_e - E(\sigma^{+}, \sigma^{-})\ , \ y \in V\ .$$ Therefore, the derivative in the $y_e$-direction equals $2\sigma_e$ if $e\in \partial\mathcal D_b$ and is $0$ otherwise as claimed. The fact that $F_b$ is a piecewise affine function on $\mathcal {\mathbb{R}}^{\Lambda^*}\setminus \mathcal C$ follows from the form of the derivatives and the fact that $\sigma$ is piecewise constant. It remains to prove the extension to a unique continuous function. Take $J_\Lambda\in \mathcal C$. By the same reasoning as above, the function $F_b$ is well-defined and continuous at $J_\Lambda$ unless there are degeneracies in the definition of $\sigma^{+,b}(J_\Lambda)$ or $\sigma^{-,b}(J_\Lambda)$. This happens if there are more than one minimizer for the difference of energy among the configurations with $+1$ at the edge $b$, and the ones with $-1$ at the edge $b$. Suppose there is exactly one degeneracy for the minimizer $\sigma^{+,b}$ at $J_\Lambda$. This means that at $J_\Lambda$ there are configurations $\eta^+$ and $\tilde \eta^+$ such that $$\label{eqn: equality} \sum_{e\in \Lambda^*}J_e(\eta^+_e- \tilde\eta^+_e) =E(\eta^+, \tilde \eta^+) \ .$$ In particular, this means that $J_\Lambda$ sits on the hyperplane defined by $\eta^+,\tilde \eta^+$. Note that on this hyperplane the expressions for the flexibility in for $\eta^+$ and $\tilde \eta^+$ agree since by and by $$\label{eqn: flex equal} \begin{aligned} &-\sum_{e\in \Lambda^*}J_e(\eta^+_e- \eta^-_e) +E(\eta^+, \eta^-) \\ &=-\sum_{e\in \Lambda^*}J_e(\tilde\eta^+_e- \eta^-_e) +E(\tilde\eta^+, \eta^-) -\sum_{e\in \Lambda^*}J_e(\eta^+_e- \tilde\eta^+_e) +E(\eta^+, \tilde \eta^+)\\ &=-\sum_{e\in \Lambda^*}J_e(\tilde\eta^+_e- \eta^-_e) +E(\tilde\eta^+, \eta^-)\ . \end{aligned}$$ This implies that the choice of representative for $\sigma^{+,b}$ on the hyperplane is irrelevant as far as the flexibility is concerned. Therefore the flexibility extends continuously on the hyperplane. Now suppose that there is more than one degeneracy for $\sigma^{+,b}$ or for $\sigma^{-,b}$ at $J_\Lambda$. Without loss of generality suppose that there are $m$ configurations $\eta^1,\dots, \eta^m$ with $+1$ at the edge $b$ with the same energy difference. (The reasoning for degeneracies for the $-1$ excitation is the same.) Then by definition this is the same as having the relations $$\label{eqn: equiv} \sum_{e\in \Lambda^*}J_e(\eta^i_e- \eta^j_e) =E(\eta^i, \eta^j) \qquad \text{ for all $i,j\leq m$.}$$ In other words, $J_\Lambda$ lies at the intersection of the hyperplanes defined by the $\eta^i$’s. On each hyperplane, the flexibility is well-defined and continuous as shown above. Moreover, the same reasoning as in shows that these definitions must agree on the intersection by the relations . This concludes the proof of the lemma. We now study the stability of ground states as couplings in $\Lambda$ are varied. For this, we fix $J_\Lambda, J_{\Lambda}'\in {\mathbb{R}}^{\Lambda^*}$ and consider the curve given by the non-linear interpolation $$\label{eqn: line} J_{\Lambda}(t)={{\rm e}}^{-t}J_\Lambda+\sqrt{1-{{\rm e}}^{-2t}}J'_\Lambda,\ \ t\geq0\ .$$ \[lem: number of planes\] Consider the curve $J_\Lambda(t)$, $t\geq 0$ defined in . The number of $t$’s such that $J_\Lambda(t)$ is in the critical set is smaller than $4^{|\Lambda^*|}$. A given critical hyperplane is determined by a point $y=(y_e,e\in\Lambda^*)$ on the hyperplane and a vector $v=(v_e,e\in\Lambda^*)$ orthogonal to it. If $J_\Lambda(t)$ intersects the hyperplane at $t$, then $t$ must satisfy the equation $$\sum_{e\in\Lambda^*}v_eJ_e(t)=v_ey_e\ .$$ By writing the expression for $J_e(t)$, this yields an equation of the following form for $t$: $$a{{\rm e}}^{-t}+b\sqrt{1-{{\rm e}}^{-2t}}=c\ ,$$ where $a,b,c$ depend on $J_\Lambda, J_\Lambda', v, y$. This equation has at most two solutions. Since there are at most $2^{\Lambda^*}\cdot 2^{\Lambda^*-1}$ hyperplanes, we obtain the claimed bound. For given endpoints $(J_\Lambda, J_\Lambda')$ for the curve , we write $F_b(t)=F_b(J_\Lambda(t))$, $\sigma^{\pm, b}(J_\Lambda(t))=\sigma^{\pm, b}(t)$, and $\sigma(J_\Lambda(t))=\sigma(t)$ for simplicity. The following result gives a criterion for the stability of the ground state at an edge in terms of its flexibility. In short, the ground state remains the same as the couplings in $\Lambda$ are varied as long as the flexibility is not $0$. \[prop: criterion\] Consider $b\in \Lambda^*$ and the curve $t\mapsto J_\Lambda(t)$ defined in . For $\nu$-almost all $(J_{\Lambda}, J_\Lambda')$, we have the following implication: $$\text{ if $F_b(s)> 0\ \forall 0\leq s\leq t$, then $\sigma_b(s)=\sigma_b(0),\ \forall 0\leq s\leq t\ .$}$$ First, observe that since the curve $J_\Lambda(t)$ intersects $\mathcal C$ finitely many times by Lemma \[lem: number of planes\], the limits $\lim_{t\downarrow t_0} \sigma(t)$ and $\lim_{t\uparrow t_0} \sigma(t)$ must be well-defined. Suppose there exists $t_0>0$ such that $\lim_{t \downarrow t_0}\sigma_b(t)=+1$ and $\lim_{t \uparrow t_0}\sigma_b(t)=-1$ (or vice-versa). Then $t_0$ must belong to $\mathcal C$. Denote the two limits $\lim_{t\downarrow t_0} \sigma(t)$ and $\lim_{t\uparrow t_0} \sigma(t)$ by $\sigma^+$ and $\sigma^-$ respectively. The excitations $\sigma^{+,b}$ and $\sigma^{-,b}$ might be degenerate at $t_0$. But by the continuity proved in Lemma \[lem: continuity\], the flexibility is independent of the choice of the representatives for $\sigma^{+,b}$ and $\sigma^{-,b}$. We pick $\sigma^+$ and $\sigma^-$ for representatives. This means that the flexibility at $t_0$ can be written in two ways using $\sigma^+$ and $\sigma^-$: $$\begin{aligned} E(\sigma^{+},\sigma^{-})-\sum_{e}J_e(t_0)(\sigma^{+}_e-\sigma^{-}_e)=\lim_{t\uparrow t_0}F_b(t)= \lim_{t\downarrow t_0}F_b(t)&= E(\sigma^{-},\sigma^{+})-\sum_{e}J_e(t_0)(\sigma^{-}_e-\sigma^{+}_e)\ . \end{aligned}$$ Since one is the negative of the other (note that $E(\eta,\eta')=-E(\eta',\eta)$ by ), we conclude that $F_b(t_0)=0$ as claimed. \[prop: flex diff\] Consider $b\in \Lambda^*$ and the curve $s\mapsto J_\Lambda(s)$ defined in . We have for all $0\leq t\leq 1$ that $$\big|F_b(t)-F_b(0)\big| \leq 6 \sqrt{t} \cdot \max_{e\in \Lambda^*}(|J_e|\vee |J_e'|)\cdot \max_{s\leq t}|\partial D_b(s)|\ .$$ Let $K$ be the number of critical hyperplanes crossed by $J_\Lambda(s)$ before time $t$. By Lemma \[lem: number of planes\], this number is less than $4^{|\Lambda^*|}$. Moreover, if we denote by $t_k$, $k\leq K$, the values at which the curve intersects $\mathcal C$, we must have that it intersects exactly one hyperplane almost surely by the same lemma. This means that the maps $s\mapsto\sigma(s)$ and $s\mapsto\sigma^{\pm,b}(s)$ (and in particular the critical droplet $\mathcal D_b(s)$) are well-defined and constant on each interval $(t_k,t_{k+1})$. By the continuity of the flexibility in Lemma \[lem: continuity\], it is therefore possible to expand $F_b(t)$ as follows $$\label{eqn: decomp} \begin{aligned} F_b(t)-F_b(0) &=\sum_{k: t_k<t}\int_{t_k}^{t_{k+1}\wedge t} \nabla F_b(s) \cdot \frac{{{\rm d}}J_\Lambda}{{{\rm d}}s} (s) {{\rm d}}s\\ &=\sum_{k: t_k<t}\sum_{e\in\partial\mathcal D_b(k)}2\sigma_e(k) \{J_e(t_{k+1}\wedge t)- J_e(t_{k})\}\ , \end{aligned}$$ where we used the gradient in Lemma \[lem: continuity\]. The notation $\partial\mathcal D_b(k)$ stands for $\partial\mathcal D_b(s)$ when $s\in (t_k,t_{k+1})$, and similarly for $\sigma_e(k)$. Note that $$\begin{aligned} |J_e(t_{k+1})-J_e(t_{k})| &=|J_e|({{\rm e}}^{-t_{k}}-{{\rm e}}^{-t_{k+1}})+|J'_e|(\sqrt{1-{{\rm e}}^{-2t_{k+1}}}-\sqrt{1-{{\rm e}}^{-2t_{k}}})\\ &\leq \max_{e\in \Lambda^*}(|J_e|\vee |J_e'|) \cdot \Big({{\rm e}}^{-t_{k}}-{{\rm e}}^{-t_{k+1}}+\sqrt{1-{{\rm e}}^{-2t_{k+1}}}-\sqrt{1-{{\rm e}}^{-2t_{k}}}\Big)\ . \end{aligned}$$ Putting this estimate back in yields $$|F_b(t)-F_b(0)| \leq 2\max_{e\in \Lambda^*}(|J_e|\vee |J_e'|)\cdot \max_{s\leq t}|\partial D_b(s)|\cdot(1-{{\rm e}}^{-t} + \sqrt{1-{{\rm e}}^{-2t}})\ .$$ The final estimate follows from the fact that $1-{{\rm e}}^{-x}\leq x$ for $x\geq 0$, and $t+\sqrt{2t}\leq 3\sqrt{t}$ for $0\leq t\leq1$. A Variance Bound for Gaussian Couplings {#sect: lemma} ======================================= In this section, we prove variance bounds using the local modification of couplings described in Section \[sect: metastate\]. The main result is the proof of Theorem \[thm: main\] relating the existence of incongruent states and disorder chaos. The following result is standard, see e.g. [@AT07; @C14]. We prove it for completeness. \[lem: gaussian variance\] Let $Y=(Y_i,i\leq n)$ and $Y'=(Y'_i,i\leq n)$ be two independent copies of a $n$-dimensional Gaussian vector. Consider $h: {\mathbb{R}}^n\to {\mathbb{R}}$ in $\mathcal C^2({\mathbb{R}}^n)$ with bounded derivatives. We have $${\text{\rm Var}}(h(Y))=\int_0^{\infty} \sum_{i\leq n}{\mathbb{E}}\left[\partial_ih(Y)\cdot\partial_ih(Y(s))\right] {{\rm e}}^{-s}{{\rm d}}s\ ,$$ where $Y(s)={{\rm e}}^{-s}Y+\sqrt{1-{{\rm e}}^{-2s}}Y'$. In particular, for any $t\geq 0$, $$\label{eqn: bound pos} {\text{\rm Var}}(h(X))\geq \int_0^{t} \sum_{i\leq n}{\mathbb{E}}\left[\partial_ih(Y)\cdot\partial_ih(Y(s))\right] {{\rm e}}^{-s}{{\rm d}}s\ .$$ Consider the $(2n)$-dimensional Gaussian vector $X(t)={{\rm e}}^{-t}(Y,Y)+\sqrt{1-{{\rm e}}^{-2t}}(Y',Y'')$ where $Y''$ is yet another independent copy of $Y$. Write $X_A=X_A(s)={{\rm e}}^{-s}Y+\sqrt{1-{{\rm e}}^{-2s}}Y'$ for the first $n$ component of $X(t)$, and $X_B=X_B(s)={{\rm e}}^{-s}Y+\sqrt{1-{{\rm e}}^{-2s}}Y''$ for the $n$ last. It is clear that $$\label{eqn: var1} {\text{\rm Var}}(h(X))=\int_0^{\infty} -\frac{{{\rm d}}}{{{\rm d}}s}{\mathbb{E}}[h(X_A)h(X_B)] {{\rm d}}s .$$ Gaussian integration by parts implies that for a function $g: {\mathbb{R}}^{2n}\to {\mathbb{R}}$ of moderate growth and two independent, but not identically distributed, $2n$-dimensional vectors $Z$ and $Z'$, $$\frac{{{\rm d}}}{{{\rm d}}u}{\mathbb{E}}[g(Z(u))]=\frac{1}{2} \sum_{i,j=1}^{2n} \left({\mathbb{E}}[Z_iZ_j]-{\mathbb{E}}[Z'_iZ'_j]\right){\mathbb{E}}[\partial_i\partial_j g(Z(u))]\ ,$$ for $Z(u)=\sqrt{u}Z+\sqrt{1-u} Z'$, see e.g. [@AT07]. We apply this with $Z=(Y,Y)$, $Z'=(Y',Y'')$ and $g(Z(u))=h(\sqrt{u}Y+\sqrt{1-u} Y')\cdot h(\sqrt{u}Y+\sqrt{1-u} Y'')$. In this instance, by independence, we have ${\mathbb{E}}[Z_iZ_j]={\mathbb{E}}[Z'_iZ'_j]=0$ unless $i=j$, $i=j+n$ or $j=i+n$. The case $i=j$ gives ${\mathbb{E}}[Z_iZ_j]-{\mathbb{E}}[Z'_iZ'_j]=0$ so only the two others gives a non-zero contribution with ${\mathbb{E}}[Z_iZ_j]-{\mathbb{E}}[Z'_iZ'_j]={\mathbb{E}}[Z_iZ_j]=1$. The derivatives in both cases $i=j+n$ and $j=i+n$ are $${\mathbb{E}}[\partial_i\partial_j g(Z(u))]={\mathbb{E}}[\partial_ih(X_A)\cdot \partial_j h(X_B)]\ .$$ Putting this back in with $u={{\rm e}}^{-2s}$ yields $${\text{\rm Var}}(h(X))=\int_0^{\infty} \sum_{i\leq n}{\mathbb{E}}[\partial_ih(X_A)\cdot \partial_ih(X_B)] \ 2{{\rm e}}^{-2s}{{\rm d}}s$$ since $\frac{{{\rm d}}}{{{\rm d}}u}=-2 {{\rm e}}^{-2s}\frac{{{\rm d}}}{{{\rm d}}s}$. Observe that the joint distribution of $(X_A,X_B)$ is the same as $(Y,Y(t))$. The first claim then follows by the change of variable $s\to 2s$. The second claim is straightforward from the fact that the term ${\mathbb{E}}[\partial_ih(X_A)\cdot \partial_i h(X_B)]$ is non-negative as can be seen by conditioning on $Y$. Recall the definition of the ground state map . As given in Definition \[df: GS prob\], the variance of $H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)$ under ${{\rm d}}{\mathbb{P}}= {{\rm d}}\nu (J)\times {{\rm d}}\nu (J') \times {{\rm d}}\kappa^1_{J}(\sigma^1)\times {{\rm d}}\kappa^2_{J}(\sigma^2)$ is equal to the variance of $H_{\Lambda,{J}}(\sigma^1(J_\Lambda))-H_{\Lambda,{J}}(\sigma^2(J_\Lambda))$ under the measure $$\label{eqn: prob2} {{\rm d}}{\mathbb{P}}= {{\rm d}}\nu (J)\times {{\rm d}}\nu (J') \times {{\rm d}}\kappa^1_{J_{\Lambda^c}}(\vec E^1)\times {{\rm d}}\kappa^2_{J_{\Lambda^c}}(\vec E^2)$$ We consider $J_\Lambda(t)$ as in Equation , and $\sigma(J_{\Lambda}(t))=\sigma(t)$ for short. \[lem: variance\] Consider $\Lambda \subset {\mathbb{Z}}^d$ finite. We have for every $t\geq 0$, $$\label{eqn: variance} {\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1(0))-H_{\Lambda,{J}}(\sigma^2(0) )\Big) \geq \int_0^t \sum_{b\in \Lambda^*}{\mathbb{E}}\Big[\big(\sigma^1_b(s)-\sigma^2_b(s)\big)\cdot\big(\sigma^1_b(0)-\sigma^2_b(0)\big) \Big]{{\rm e}}^{-s}{{\rm d}}s\ .$$ By conditioning on $(J_{\Lambda_c}, \vec E)$ we get by the conditional variance formula $${\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1(J_\Lambda))-H_{\Lambda,{J}}(\sigma^2(J_\Lambda)) \Big) \geq {\mathbb{E}}\left[{\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1(J_\Lambda))-H_{\Lambda,{J}}(\sigma^2(J_\Lambda)) \Big| J_{\Lambda^c}, \vec E^1,\vec E^2\Big)\right]\ .$$ The distribution of $J_{\Lambda}$ conditioned on $(J_{\Lambda_c}, \vec E^1,\vec E^2)$ remains IID Gaussian by the independence in Lemma \[lem: metastate\]. We apply Lemma \[lem: gaussian variance\] with $Y=J_\Lambda$ and $Y(t)=J_\Lambda(t)$. To compute the derivatives, we used Proposition \[prop: ordering\] and the definition of the ground state map . Since the ground state $\sigma(J_\Lambda)$ is constant and well-defined on a set of full measure, the derivative $\partial_{J_b} \sigma^1_e(J_\Lambda)$ is $0$ $\nu$-a.s for every edge $e$. Therefore we have $$\frac{\partial}{\partial J_b}\{H_{\Lambda,{J}}(\sigma^1(J_\Lambda))-H_{\Lambda,{J}}(\sigma^2(J_\Lambda))\} =-(\sigma_b^1(J_{\Lambda})-\sigma_b^2(J_{\Lambda}))\ \ \nu-a.s.$$ We conclude that $$\begin{aligned} &{\mathbb{E}}\left[{\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1(J_\Lambda))-H_{\Lambda,{J}}(\sigma^2(J_\Lambda)) \Big| J_{\Lambda^c}, \vec E^1, \vec E^2\Big)\right]\\ &= \sum_{b\in \Lambda^*}\int_0^\infty {\mathbb{E}}\left[(\sigma_b^1(J_\Lambda)-\sigma_b^2(J_\Lambda))(\sigma_b^1(J_\Lambda(s))-\sigma_b^2(J_\Lambda(s))\right] e^{-s}{{\rm d}}s\ . \end{aligned}$$ The lower bound restricted $t\geq 0$ follows from . The restriction to one edge $b$ holds for the same reason since the integrand is positive. The theorem is an elementary consequence of Lemma \[lem: variance\]. First observe that the quantity $$\left(2-2Q_\Lambda(\sigma,\sigma')\right)^{1/2}=\frac{1}{|\Lambda^*|^{1/2}}\left(\sum_{b\in \Lambda^*} (\sigma_b- \sigma_b')^2\right)^{1/2} =:\|\sigma-\sigma'\|$$ satisfies the triangle inequality. In particular, we have $$\|\sigma-\sigma'\|\geq \Big|\|\sigma-\sigma''\|-\|\sigma'-\sigma''\|\Big|\ .$$ This inequality implies $$\begin{aligned} Q_\Lambda(\sigma,\sigma')-Q_\Lambda(\sigma',\sigma'') &=1-Q_\Lambda(\sigma',\sigma'')-1+Q_\Lambda(\sigma,\sigma')\\ &\geq \frac{1}{2}\left(\Big|\|\sigma-\sigma'\|-\|\sigma-\sigma''\|\Big|\right)^2-\frac{1}{2} \|\sigma-\sigma'\|^2\\ &=\frac{1}{2}\|\sigma-\sigma''\|^2-\|\sigma-\sigma'\|\|\sigma-\sigma''\|\\ &\geq \frac{1}{2}\|\sigma-\sigma''\|^2-2\|\sigma-\sigma'\|\ , \end{aligned}$$ since $\|\sigma-\sigma''\|\leq 2$. We apply this inequality to $\sigma=\sigma^1(0)$, $\sigma'=\sigma^1(s)$, $\sigma''=\sigma^2(0)$ (and again with $1$ replaced by $2$) to rewrite the integrand in as $$|\Lambda^*|\cdot {\mathbb{E}}\Big[\|\sigma^1(0)-\sigma^2(0)\|^2-2\sum_{i=1,2}\|\sigma^i(0)-\sigma^i(s)\|\Big]\ .$$ By putting this back in , we have $$\begin{aligned} &\frac{1}{|\Lambda^*|}{\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1(J_\Lambda))-H_{\Lambda,{J}}(\sigma^2(J_\Lambda)) \Big)\\ &\geq \int_0^t\left\{2{\mathbb{E}}\Big[1-Q_\Lambda(\sigma^1(0),\sigma^2(0))\Big]-2\sqrt{2}\sum_{i=1,2} {\mathbb{E}}\Big[\Big(1-Q_\Lambda(\sigma^i(0),\sigma^i(s))\Big)^{1/2}\Big]\right\}{{\rm e}}^{-s}{{\rm d}}s\ . \end{aligned}$$ The claim follows by applying Jensen’s inequality to the second term. Disorder Chaos and Critical Droplets ==================================== We start by establishing Corollary \[cor: ADC-&gt; variance\] as an elementary consequence of Theorem \[thm: main\]. On one hand, by definition of disorder chaos at scale $\alpha$, we have that for every ${\varepsilon}>0$ there is $C({\varepsilon})>0$ and $A_{\varepsilon}$ with ${\mathbb{P}}(A_{\varepsilon})>1-{\varepsilon}$ such that for every $t\leq C|\Lambda|^{-\alpha}$ and $\Lambda$ large enough $$\label{eqn: overlap estimate} {\mathbb{E}}\big[1-Q_\Lambda(\sigma^1, \sigma^1(t))\big] = {\mathbb{E}}\big[1-Q_\Lambda(\sigma^1, \sigma^1(t));A_{\varepsilon}\big]+ {\mathbb{E}}\big[1-Q_\Lambda(\sigma^1, \sigma^1(t));A_{\varepsilon}^c\big] \leq {\varepsilon}(1-{\varepsilon})+2{\varepsilon}\ .$$ On the other hand, if there exist incongruent states with positive ${\mathbb{P}}$-probability, we must have by Fatou’s lemma $$\label{eqn: fatou} \liminf_{\Lambda\to{\mathbb{Z}}^d}{\mathbb{E}}\Big[1-Q_\Lambda(\sigma^1(0),\sigma^2(0))\Big] \geq 1-{\mathbb{E}}\Big[\limsup_{\Lambda\to {\mathbb{Z}}^d}Q_\Lambda(\sigma^1(0),\sigma^2(0))\Big]>0\ .$$ The result follows from Theorem \[thm: main\] by taking ${\varepsilon}$ small enough and $\Lambda$ large enough so that the right-hand side of Equation is strictly greater than $0$ uniformly for $s\leq C|\Lambda|^{-\alpha}$. To prove Theorem \[thm: ADC\], we need the existence of many edges on which a given ground state is not too sensitive. Since the statements of Theorem \[thm: ADC\] involve only one replica $\sigma^1$, we set for the rest of this section $${{\rm d}}{\mathbb{P}}= {{\rm d}}\nu (J)\times {{\rm d}}\nu (J') \times {{\rm d}}\kappa_{J_{\Lambda^c}}(\vec E)\ .$$ \[lem: flex\] For any ${\varepsilon}>0$, there exists $\delta=\delta({\varepsilon})$ (independent of $\Lambda$) and a subset $B_{\varepsilon}$ of $(J, \vec E)$ with ${\mathbb{P}}(B_{\varepsilon})>1-{\varepsilon}$ such that on $B_{\varepsilon}$ $$\#\{b\in \Lambda^*: |F_b(J_\Lambda)|>\delta\}> (1-{\varepsilon})|\Lambda^*|\ .$$ Since $\#\{b\in \Lambda^*: |F_b(J_\Lambda)|>\delta\}=|\Lambda^*|-\#\{b\in \Lambda^*: |F_b(J_\Lambda)|\leq \delta\}$, it suffices to show that for given ${\varepsilon}>0$ there is a $\delta$ small enough such that $${\mathbb{P}}\left(\#\{b\in \Lambda^*: |F_b(J_\Lambda)|\leq \delta\}> {\varepsilon}|\Lambda^*|\right)<{\varepsilon}\ .$$ Markov’s inequality implies that $$\begin{aligned} {\mathbb{P}}\left(\#\{b\in \Lambda^*: |F_b(J_\Lambda)|\leq \delta\}> {\varepsilon}|\Lambda^*|\right) &\leq \frac{1}{{\varepsilon}|\Lambda^*|} {\mathbb{E}}[\#\{b\in \Lambda^*: |F_b(J_\Lambda)|\leq \delta\}]\\ &= \frac{1}{{\varepsilon}|\Lambda^*|} \sum_{b\in \Lambda^*}{\mathbb{P}}(|F_b(J_\Lambda)|\leq \delta)\ . \end{aligned}$$ We show that ${\mathbb{P}}(|F_b(J_\Lambda)|\leq \delta\})<c\delta$ (uniformly on the edges $b$) for some $c>0$. The claim then follows by taking $\delta={\varepsilon}^2/c$. The key observation is that conditioned on $(J_{\Lambda^c}, \vec E)$, the states $\sigma^{\pm,b}(J_\Lambda)$ are independent of $J_b$. This is because the contribution of $J_b$ in the difference of energies on the right side of cancels when we restrict to $\eta$’s with $\eta_b=+1$ (or $\eta_b=-1$). In particular, this means that we can write the flexibility as $$F_b(J_\Lambda)=2|J_b- \mathcal C_b|\ ,$$ where $\mathcal C_b$ is a measurable function that only depends on $\vec E$ and $(J_e; e\in \Lambda^*, e\neq b)$, see also Remark \[rem: examples\]. Therefore, the variable $J_b$ is independent of $\mathcal C_b$ under ${\mathbb{P}}$ (by independence in Lemma \[lem: metastate\]), and has the standard Gaussian distribution. This implies $$\label{eqn: flex estimate} {\mathbb{P}}(|F_b(J_\Lambda)|\leq \delta\})={\mathbb{P}}(|J_b- \mathcal C_b|\leq \delta)\leq {\mathbb{P}}(|J_b|\leq \delta)\ .$$ It remains to observe that $\nu\{|J_b|\leq \delta\}\leq 2\delta/\sqrt{2\pi}$ to finish the proof. We now have all the ingredients to prove Theorem \[thm: ADC\]. Fix ${\varepsilon}>0$. From the definition \[df: ADC\], we need to find $C=C({\varepsilon})$ and a subset $A_{\varepsilon}$ of $(J, J',\vec E)$ with ${\mathbb{P}}(A_{\varepsilon})>1-{\varepsilon}$ on which $$\#\{b\in \Lambda^*: \sigma_b(t)=\sigma_b(0),\ \forall t\leq C|\Lambda|^{-\alpha}\}>(1-{\varepsilon})|\Lambda^*|\ .$$ By Proposition \[prop: criterion\], this would follow if we find $C$ and $A_{\varepsilon}$ on which $$\#\{b\in \Lambda^*: F_b(t)>0,\ \forall t\leq C|\Lambda|^{-\alpha}\}>(1-{\varepsilon})|\Lambda^*|\ .$$ We write $B_{\varepsilon}$ for the event in Lemma \[lem: flex\]. Consider the event $\widetilde B_{\varepsilon}=\{\max_{e\in \Lambda^*} (|J_e|\vee |J'_e|)<\widetilde C\sqrt{\log |\Lambda|}\}$. A standard argument using Gaussian estimates shows that there exists $\widetilde C=\widetilde C({\varepsilon})$ large enough such that ${\mathbb{P}}(\widetilde B_{\varepsilon})>1-{\varepsilon}$. We take $A_{\varepsilon}= B_{\varepsilon}\cap \widetilde B_{\varepsilon}$. We have by construction ${\mathbb{P}}(A_{\varepsilon})>1-2{\varepsilon}$. From Proposition \[prop: flex diff\] and Equation , it follows that on $(1-{\varepsilon})|\Lambda^*|$ edges $$\label{eqn: flex ineq} \begin{aligned} F_b(t)\geq F_b(0)- 6 \sqrt{t} \cdot \max_{e\in \Lambda^*}(|J_e|\vee |J_e'|)\cdot \max_{s\leq t}|\partial D_b(s)| &\geq \delta - 6 \widetilde C \sqrt{t}\cdot \sqrt{\log |\Lambda|} \cdot C |\Lambda|^\gamma \ . \end{aligned}$$ Taking $\alpha >2\gamma$, we conclude that $F_b(t)>\delta/2$ for $t\leq (6 C \widetilde C\delta)^{-2} |\Lambda|^{-\alpha}$ and $\Lambda$ large enough. This completes the proof of the theorem. \[rem: estimate\] [ The inequality is far from optimal in general as it does not take into account the dependence between the droplet $\mathcal D_b$ and the couplings $J_\Lambda$, $J_\Lambda'$. The droplet $\mathcal D_b$ is special as it optimizes the energy on its boundary. Here we bounded the value of the couplings on the boundary in an elementary way by the size of the boundary times the maximal value of the couplings in the whole box. The factor $\log |\Lambda|$ we get from this procedure is one of the reason why we cannot handle the case $\alpha=\gamma$. To improve the result, one would have to develop a better understanding of the delicate connection between the geometry of the underlying lattice and the extreme statistics of the couplings. ]{} Similar ideas gives weaker uniform bounds for the variance. Note first that the assumption ${\mathbb{P}}(\mathcal I)>0$ implies that there exist an edge $b\in \Lambda^*$ and $c>0$ (both independent of $\Lambda$) such that for $\Lambda$ large enough ${\mathbb{E}}[1-\sigma_b^1\sigma_b^2]>c$. This is because Equation implies $$\liminf_{\Lambda\to {\mathbb{Z}}^d} \frac{1}{|\Lambda^*|}\sum_{b\in \Lambda^*} {\mathbb{E}}[1-\sigma_b^1\sigma_b^2]>0\ .$$ In particular, for $\Lambda$ large enough we must have $\sum_{b\in \Lambda^*} {\mathbb{E}}[1-\sigma_b^1\sigma_b^2]>c |\Lambda^*|$ for some $c>0$. This implies the claim. Fix such an edge. We consider the interpolation on this single edge $b$, that is, we take $J_\Lambda(t)$ as $J_e(t)={{\rm e}}^{-t}J_e+\sqrt{1-{{\rm e}}^{-2t}}J_e'$ for $e=b$ and $J_e(t)=J_e$ for $e\neq b$. In this setting, Lemma \[lem: variance\] and the same reasoning as in the proof of Theorem \[thm: main\] with $Q_\Lambda(\sigma^1,\sigma^2)$ replaced by $\sigma^1_b\sigma^2_b$ and $\|\sigma -\sigma'\|$ by $(2-2\sigma^2_b\sigma^2_b)^{1/2}$ gives the bound $$\begin{aligned} &{\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1(0))-H_{\Lambda,{J}}(\sigma^2(0)) \Big)\\ &\geq 2\int_0^t\left\{{\mathbb{E}}\Big[1-\sigma^1_b(0)\sigma^2_b(0)\Big]-\sqrt{2}\sum_{i=1,2}\Big( {\mathbb{E}}\Big[1-\sigma^i_b(0)\sigma^i_b(s)\Big]\Big)^{1/2}\right\}{{\rm e}}^{-s}{{\rm d}}s\ . \end{aligned}$$ Proceeding as in (since the first term in the bracket is greater than $c$ independently of $\Lambda$) it remains to find for any ${\varepsilon}>0$ an event $A_{\varepsilon}$ and $C=C({\varepsilon})>0$ such that ${\mathbb{P}}(A_{\varepsilon})>1-{\varepsilon}$, and on $A_{\varepsilon}$ $$\sigma_b^1(t)=\sigma_b^1(0) \qquad \forall t\leq C\ .$$ By Proposition \[prop: criterion\], this holds if $F_b(t)>0$ for $ t\leq C$. We take $A_{\varepsilon}=B_{\varepsilon}\cap \widetilde B_{\varepsilon}$ for the events $B_{\varepsilon}=\{F_b(0)>\delta\}$ and $\widetilde B_{\varepsilon}=\{ (|J_b|\vee |J'_b|)<\widetilde C\}$. Recall from Remark \[rem: examples\] that $F_b(0)=2|J_b-\mathcal C_b|$ and that $J_b$ is independent of $\mathcal C_b$. In particular, we get as in that ${\mathbb{P}}(B_{\varepsilon})>1-{\varepsilon}$ by picking $\delta$ small enough. Moreover, $\widetilde C$ can be taken large enough so that ${\mathbb{P}}(\widetilde B_{\varepsilon})>1-{\varepsilon}$. This implies ${\mathbb{P}}(A_{\varepsilon})>1-2{\varepsilon}$. On this event, we get the same way as in Proposition \[prop: flex diff\] that $$F_b(t)\geq F_b(0)- 6\sqrt{t} \cdot (|J_b|\vee |J'_b|) \geq \delta - 6\widetilde C\sqrt{t}\ .$$ It remains to take $C=(12\widetilde C\delta)^{-2}$ to ensure that $F_b(t)\geq \delta/2$ for $t\leq C$ thereby proving the corollary. One might expect the proof to hold by simply dropping all but a single edge in . However, all couplings in $\Lambda$ would then be perturbed leading to a worse estimate of the flexibility in Proposition \[prop: flex diff\]. Relations to Scaling Theories {#sec:scaling} ============================= Some of the above results have interesting consequences when combined with non-rigorous scaling theories of the spin glass phase that have been proposed in the theoretical physics literature [@Mac84; @BM85; @FH86; @FH88]. The scaling-droplet picture is one of several competing theories attempting to describe the low-temperature thermodynamic properties of the spin glass phase, and the results presented elsewhere in this paper by themselves neither favor nor disfavor any of these. However, they do shed additional light on the consequences of some of the assumptions made in the scaling-droplet picture, and these will be discussed in this section. Because scaling theories represent a non-rigorous approach (so far) to the study of the spin glass phase, no attempt will be made at mathematical rigor in this section (although the conjectures and results will be stated precisely); our goal is simply to explore what our rigorous results imply for one approach to understanding finite-dimensional spin glasses. Before turning to scaling theories, we present a simple bound on the parameter $\alpha$ introduced in Definition \[df: ADC\] that provides a necessary condition for the presence of incongruence. This relies on an [*upper*]{} bound on fluctuations of (free) energy differences derived elsewhere but never published [@AFunpub; @NSunpub] (however, a statement and proof of the bound can be found in [@Stein16]). The statement of the corresponding theorem is as follows: \[thm:upperbound\] (Aizenman-Fisher-Newman-Stein) Let $F_P$ be the free energy of the finite-volume Gibbs state generated by Hamiltonian (\[eqn: H\]) in a box $\Lambda$ of volume $L^d$ using periodic boundary conditions, and let $F_{AP}$ be that generated using antiperiodic boundary conditions. Let $X_\Lambda = F_P - F_{AP}$. Then ${\rm Var}(X_\Lambda)\le{\rm const.}\times L^{d-1}$, where ${\rm Var}(\cdot)$ denotes the variance over all of the couplings inside the box. Although stated for periodic-antiperiodic boundary conditions, the theorem applies to any pair of gauge-related boundary conditions, such as two fixed BC’s. Theorem \[thm:upperbound\] has been proved only for finite volumes, but it is reasonable to expect that it applies equally well to free energy fluctuations in finite-volume restrictions of infinite-volume pure or ground states; i.e., for two pairs of boundary conditions arising from two putative ground or pure states drawn from the metastate. We therefore propose the following conjecture: \[conj:upperbound\] The variance bound of Theorem \[thm:upperbound\] extends to ${\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)\Big)$ of Theorem \[thm: main\]; i.e., for any $\sigma^1$ and $\sigma^2$ chosen as in Theorem \[thm: main\], $$\label{eq:upperbound} {\text{\rm Var}}\Big(H_{\Lambda,{J}}(\sigma^1)-H_{\Lambda,{J}}(\sigma^2)\Big)\le A L^{d-1}\, ,$$ where $A>0$ is a constant and $|\Lambda|=L^d$. Because scaling relations are typically expressed in terms of $L$ rather than $|\Lambda|$, the relation $|\Lambda|=L^d$ will be assumed for the remainder of this section. Corollary \[cor: ADC-&gt; variance\] and Conjecture \[conj:upperbound\] when combined lead immediately to our first result, which because it relies on a conjecture will be stated as a claim rather than as a corollary: \[claim:alphad\] In order for incongruent states to appear in the zero-temperature metastate, it is necessary that $\alpha\ge 1/d$. Consequently, incongruent states are ruled out if $\alpha<1/d$. Implications for Droplet-Scaling Theories {#subsubsec:ds} ----------------------------------------- Droplet-scaling theories remain one of the main contenders for the correct description of the spin glass phase in finite dimensions. The primary assumption [@FH86; @FH88] of the droplet picture of Ising spin glasses is well-known (in what follows, we restrict the discussion to zero temperature): in any dimension in which the spin glass phase exists, the minimal excitation above the ground state on length scale $L$ about a fixed point (call it the origin) is a compact droplet of order $L^d$ coherently flipped spins with an energy cost of $L^\theta$. The (dimension-dependent) exponent $\theta$ originally arose from scaling theories that examined the properties of a disordered zero-temperature fixed point; for any dimension in which a low-temperature spin glass phase exists, $\theta>0$. Fisher and Huse (FH) moreover argued that in any dimension, $\theta\le (d-1)/2$. There are several possible versions of the droplet-scaling approach. In what follows, we will assume the simplest possible version — what can justifiably be called a “minimal” droplet picture. To begin, consider all compact, connected clusters of $N$ spins containing the origin and with $L^d\le N\le (2L)^d$. Then the droplet theory (at zero temperature) makes the following assumptions [@FH86; @FH88]: 1. The distribution $\rho_L(E_L)$ of minimal droplet energies has the scaling form $$\label{eq:scaling} \rho_L(E_L)\approx\frac{1}{\Upsilon L^{\theta_{\rm e}}}{\tilde\rho}\left[\frac{E_L}{\Upsilon L^{\theta_{\rm e}}}\right]\nonumber\, ,$$ where $\Upsilon$ is constant and of order the standard deviation of the coupling distribution and $\tilde\rho(0)>0$. In other words, the typical minimal droplet energy is order $L^{\theta_{\rm e}}$, but there is a probability falling off as $L^{-\theta_{\rm e}}$ that the minimal droplet energy is of order one. (The notation $\theta_{\rm e}$ — “${\rm e}$" for excitation — rather than simply $\theta$ is ours and not FH’s; the reason for this notation will be discussed momentarily.) 2. The surface area of the droplet boundary scales as $L^{d_s}$, where $d-1<d_s<d$. [^3] (Recent simulations of a related quantity in [@WMK18], namely the fractal dimension of the interface induced by switching from periodic to antiperiodic boundary conditions, find that $d_s<d$ for $d<6$, and seems to approach $d$ at $d=6$.) 3. Energy difference fluctuations are governed by a (dimension-dependent) “stiffness exponent” $\theta_{\rm s}$ (again, our notation, not FH’s), which governs the size of the free energy fluctuations when one switches from, say, periodic to antiperiodic boundary conditions in a volume $\Lambda_L$. That is, using the notation of Theorem \[thm:upperbound\], $$\label{eq:stiffness} aL^{2\theta_{\rm s}}\le {\rm Var}(X_\Lambda)\le bL^{2\theta_{\rm s}}\nonumber\, ,$$ where $0<a<b<\infty$ are constants. In order for a stable spin glass phase to exist in dimension $d$, it is necessary that $\theta_{\rm s}>0$. 4. Droplet excitation energies scale in the same way as ground state interface energies; i.e., $\theta_{\rm s}=\theta_{\rm e}$. This last assumption leads to what we referred to earlier as a “minimal” droplet-scaling theory, and has been the subject of some debate (see, for example, [@KP04; @KB05; @A90]). Additional exponents have been proposed in various places, and in non-scaling theories there are multiple types of excitations and interfaces with different exponents [@KP04]. However, the version of droplet theory with $\theta_{\rm s}=\theta_{\rm e}$ is the simplest and cleanest, has been shown to hold in some special cases [@BKM03], and corresponds to the original theory as proposed by FH. We therefore adopt it in what follows, and hereafter set $\theta_{\rm e}=\theta_{\rm s}=\theta$. 5. $\theta\le (d-1)/2$. At least insofar as this refers to the stiffness exponent, this has been rigorously proved, as noted above. After the scaling theories had been introduced, it was quickly noted that they implied what came to be known as “temperature chaos”, namely a rearrangement on all sufficiently large lengthscales of the pure state correlations upon an infinitesimal change of temperature [@BM87; @FH88]. This is believed to be closely related to disorder chaos, and the behavior of the two is expected to be similar; in particular, the exponent governing the lengthscale beyond which edge and spin overlaps fall to zero is the same in all treatments of both kinds of spin glass “chaos”. In analyzing disorder chaos, one begins by considering (see, for example, [@KB05]) a small perturbation of the couplings of the form $$\label{eq:pert} J_{xy}\to J'_{xy}=\frac{J_{xy}+\eta_{xy}\Delta J}{\sqrt{1+(\Delta J)^2}}$$ where $\eta_{xy}$ is a normally distributed random variable with zero mean and unit variance. Scaling theory then predicts [@BM87; @FH88; @KB05] that a new ground state will appear outside of a characteristic length $\ell_c$ that is governed by the various exponents introduced above. We therefore add this to the above assumptions: \(vi) Upon changing the couplings in the manner (\[eq:pert\]) above, the ground state rearranges beyond a lengthscale $\ell_c$ governed by $$\label{eq:lc} \ell_c(\Delta J)=\Delta J^{-1/\xi}$$ where the new exponent $\xi=d_s/2-\theta$. For a system of linear size $L$, the spin overlap $q$ obeys the scaling law [@KB05] $$\label{eq:spin} \langle q(\Delta J,L)\rangle=F(L/\ell_c)=F(\Delta J^{1/\xi}L)$$ where $F(x)\approx 1-ax^\xi$ for $x\ll 1$ and $F(x)\approx bx^{-d/2}$ for $x\gg 1$. The edge overlap behaves similarly. To express this using our notation, we relate $\Delta J$ in (\[eq:pert\]) and $t$ in (\[eq:t\]). For $\sqrt{t}<\epsilon\ll 1$ and $\Delta J<\epsilon\ll 1$, we have $\Delta J=(2t)^{1/2}+O(\epsilon^2)$. Therefore to order $\epsilon^2$, $$\label{eq:t1} \ell_c= t^{-1/2\xi}\, .$$ In (\[eq:t1\]) we rescaled $\ell_c$ by a factor of order one to eliminate a multiplicative constant. This has no effect on the analysis to follow. It should be emphasized that Eq. (\[eq:lc\]) of (vi) is not a separate assumption: it follows directly from (i) (at least for disorder chaos). For ease of presentation and future reference, it will be listed along with the assumptions above, but it should be kept in mind that it is a prediction of scaling theory, not an assumption. We now turn to a discussion of what the results proved in this paper imply about the minimal scaling theory described above. One of the central conclusions of the droplet picture is that the ordered spin glass phase consists of a single pair of spin-reversed pure states (at $T>0$) or ground states (at $T=0$) in all dimensions in which a spin glass phase occurs [@FH87]. The argument against the presence of incongruent states relied first on the inequality $\theta\le (d-1)/2$, which (at least as far as the spin glass stiffness is concerned) is no longer in dispute. The second part of the argument relied on a conjecture that the standard deviation of the energy fluctuations arising from the presence of incongruent states would scale at least as fast as the square root of the volume, leading to a contradiction. However, at the time this argument was put forward, there was little firm support for any sort of lower bound on (free) energy fluctuations arising from incongruence. [^4] As a consequence, while it was generally accepted that the droplet theory leads to a two-state picture, the conclusion has remained mostly conjectural. (Some authors assert that the droplet picture simply assumes at the outset that there is only a single pair of ground states [@KP04].) However, we are now in a position to solidify the argument that the droplet theory indeed leads to a two-state picture, at least insofar as incongruence is concerned. The claim we make is the following: \[claim:twostate\] If the scaling-droplet theory as defined by Assumptions (i)-(vi) above is correct, then in any finite dimension the zero-temperature metastate, generated using coupling-independent boundary conditions, is supported on a single pair of ground states. By Definition \[df: ADC\], the absence of disorder chaos on scale $\alpha$ means that $$\label{eq:Q} Q_\Lambda(\sigma^i, \sigma^i(t))>1-{\varepsilon}\text{ on $A_{\varepsilon}$, $i=1,2$,}$$ for all $t\leq C|\Lambda|^{-\alpha}$ and all $\Lambda$ large enough. Using Eq. (\[eq:t1\]), this leads to the identification $$\label{eq:alphaxi} \alpha=2\xi/d=d_s/d-2\theta/d\, .$$ In a dimension with a spin glass phase $0<\theta<d_s/2$, so $0<\alpha<1$. Eq. (\[eq:alphaxi\]) says that the minimal droplet excitation in a volume of linear dimension $L$ sets the scale for the absence of disorder chaos. Moreover, Corollary \[cor: ADC-&gt; variance\] says that, if there is ADC at scale $\alpha$, and ${\mathbb\sigma}^1$ and ${\mathbb\sigma}^2$ are incongruent spin configurations in $\Lambda$, then $$\label{eq:corollary1} {\rm Var}\left(H_{\Lambda,J}({\bf\sigma}^1)-H_{\Lambda,J}({\bf\sigma}^2)\right)\ge C|\Lambda|^{1-\alpha}\, .$$ When combined with (\[eq:alphaxi\]), this becomes $$\label{eq:corollary2} {\rm Var}\left(H_{\Lambda,J}({\bf\sigma}^1)-H_{\Lambda,J}({\bf\sigma}^2)\right)\ge C L^{d-d_s+2\theta}\ge CL^{2\theta+\delta}\ ,$$ where $\delta(d)\equiv d-d_s>0$ using assumption (ii) of the droplet theory. But by Assumptions (iii) and (iv) of the droplet theory, $$\label{eq:corollary3} {\rm Var}\left(H_{\Lambda,J}({\bf\sigma}^1)-H_{\Lambda,J}({\bf\sigma}^2)\right)\le bL^{2\theta}\, ,$$ leading to a contradiction for sufficiently large $L$ and demonstrating the above claim that the minimal droplet theory is indeed a two-state theory. It is interesting to note that while the original two-state argument relied on the inequality $\theta\le (d-1)/2$, this is nowhere used in the above argument; indeed, at least for the purposes of the argument, $\theta$ can be anything at all. The other assumptions of the droplet theory were all necessary, however. Of particular interest is that the droplet [*geometry*]{} plays a crucial role in setting the scale of ground state energy difference fluctuations. Further Relations {#subsubsec:further} ----------------- We conclude this discussion with an argument that uses no scaling assumptions and arrives at another relation connecting droplet geometries and energies to the presence or absence of incongruence. In this case, however, the droplets under consideration are not low-energy excitations above the ground state but rather the “critical droplets”, introduced above Theorem \[thm: ADC\], that measure the stability of a given ground state pair. Consider the critical droplet boundary $\partial\mathcal D_b$ (of energy order one). This naturally leads to a new exponent $d_f$, defined as $|\partial\mathcal D_b|={\rm const.}\ L^{d_f}$. Then Theorem \[thm: ADC\] provides the relation $\alpha=2d_f/d$, and Corollary \[cor: ADC-&gt; variance\] gives the bound $$\label{eq:newbound} {\rm Var}\left(H_{\Lambda,J}({\bf\sigma}^1)-H_{\Lambda,J}({\bf\sigma}^2)\right)\ge {\rm const.}\ L^{d(1-\alpha)}={\rm const.}\ L^{d-2d_f}\, .$$ Combining this with (\[eq:corollary3\]) then implies that if $d_f<(d-2\theta)/2$, there cannot be incongruent ground states. This result bypasses the issue of whether  $\theta_{\rm e}=\theta_{\rm s}$; only the “stiffness” $\theta_{\rm s}$ enters. [^1]: In the SK model, unlike in the EA model, edge and spin overlaps are trivially related. [^2]: The bound is proved in finite volume for fixed boundary conditions, but also holds when the boundary conditions are sampled from a metastate. [^3]: At first glance it might appear that the condition $d_s < d$ is already incompatible with the existence of incongruent states. However, it is neither a necessary nor sufficient condition for incongruence to be absent. [^4]: However, as mentioned in the introduction, recent work by the authors in collaboration with J. Wehr [@ANSW14; @ANSW16] has proved a lower bound for the variance scaling with the volume for at least certain pairs of incongruent states.

--- abstract: | We develop a nonlinear semi-parametric Gaussian process model to estimate periods of Miras with sparsely sampled light curves. The model uses a sinusoidal basis for the periodic variation and a Gaussian process for the stochastic changes. We use maximum likelihood to estimate the period and the parameters of the Gaussian process, while integrating out the effects of other nuisance parameters in the model with respect to a suitable prior distribution obtained from earlier studies. Since the likelihood is highly multimodal for period, we implement a hybrid method that applies the quasi-Newton algorithm for Gaussian process parameters and search the period/frequency parameter space over a dense grid. A large-scale, high-fidelity simulation is conducted to mimic the sampling quality of Mira light curves obtained by the M33 Synoptic Stellar Survey. The simulated data set is publicly available and can serve as a testbed for future evaluation of different period estimation methods. The semi-parametric model outperforms an existing algorithm on this simulated test data set as measured by period recovery rate and quality of the resulting Period-Luminosity relations. author: - 'Shiyuan He, Wenlong Yuan, Jianhua Z. Huang, James Long & Lucas M. Macri' bibliography: - 'm33gp.bib' title: | Period estimation for sparsely sampled quasi-periodic\ light curves applied to Miras --- Introduction ============ The determination of reliable periods for variable stars has been an area of interest in astronomy for at least four centuries, since the discovery of the variability of Mira ($o$ Ceti) by Fabricius in 1596 and the first attempts to determine its period by Holwarda & Bouillaud in the mid-1600s. The availability of electronic computers for astronomical research half a century ago enabled the development of many algorithms to estimate periods quickly and reliably, such as @Lafler1965 [@Lomb1976; @Scargle1982]. The aforementioned algorithms work best in the case of periodic variations with constant amplitude and Mira variables present several challenges in this regard. While their periods of pulsation are stable except for a few intriguing cases [@Templeton2005], Mira light curves can exhibit widely varying amplitudes from cycle to cycle [see, for example, the historical light curve of Mira compiled by @Templeton2009]. In the case of C-rich Miras, the stochastic changes in mean magnitude across cycles [e.g., @Marsakova1999] only complicate the problem further. The wide variety of light curves for long-period variables, already recognized by @Campbell1925 and @Ludendorff1928, may complicate the identification of Miras among other stars. Lastly, from a purely practical standpoint, it is simpler to obtain light curves spanning several cycles for RR Lyraes or Cepheids (with periods ranging from $\sim 0.5$ to $\sim 100$ d) than for Miras (with periods ranging from $\sim 100$ to $\sim 1500$ d). Despite these challenges, the identification and determination of robust periods for Miras — especially in the regime of sparsely sampled, low signal-to-noise light curves — would be very beneficial for the determination of distances to galaxies of any type. Thanks to the unprecedented temporal coverage of the Large Magellanic Cloud (LMC) by microlensing surveys, the availability of large samples of extremely well-observed Miras has led to a thorough characterization of their period-luminosity relations at various wavelengths [@Wood1999; @Ita2004; @Soszynski2007]. The dispersion of the $K$-band period-luminosity relation [@Glass2003 $\sigma=0.13$ mag], is quite comparable to that of Cepheids at the same wavelength [@Macri2015 $\sigma=0.09$ mag] and makes them competitive distance indicators. The third phase of the OGLE survey [@Udalski2008] imaged most of the LMC with little interruption over 7.5 years and resulted in the discovery of 1663 Miras [@Soszynski2009] with a median of 466 photometric measurements per object. The temporal sampling of these light curves and their photometric precision are exceptional relative to typical astronomical surveys and make period estimation relatively easy. In comparison, a similar span of observations of M33 by the DIRECT [@Macri2001] and M33SSS projects [@Pellerin2011] in the $I$-band consists of a median number of 44 somewhat noisy measurements, heavily concentrated in a few observing seasons. Representative Mira light curves from the OGLE & DIRECT/M33SSS surveys are shown in Fig. \[fig:example.mira.lc\]. There are several reasons for the striking difference in quality between these two data sets. The LMC Miras are among the brightest objects in the OGLE fields, whereas their M33 counterparts are among the faintest in the aforementioned surveys of this galaxy. While the effective exposure times of all these surveys are quite comparable, after taking into account differences in collecting area of their respective telescopes, M33 lies approximately 6.2 mag farther in terms of its $I$-band apparent distance modulus. Furthermore, the main goal of the OGLE project (detection of microlensing events) requires a very dense temporal sampling of the survey fields; this is achieved by using a dedicated telescope and is helped by the fact that the LMC is observable nearly all year long from the site. In contrast, the observations of M33 were carried out using shared facilities (available only a few nights per month) with the primary purpose of studying Cepheids and eclipsing binaries (which do not require exceptionally dense temporal sampling), and the galaxy is only observable all night long for $\sim 1/3$ of the year. Standard period estimation algorithms, which work well for high signal-to-noise, well sampled light curves such as those obtained by OGLE, will fail on more typical data sets represented by the M33 observations. The purpose of this work is to develop and test a methodology for estimating periods for sparsely sampled, noisy, quasi-periodic light curves such as those of Miras observed in M33 by the aforementioned projects. \[fig:example.mira.lc\] {width="49.00000%"} {width="49.00000%"} The rest of the paper is organized as follows. In §\[sec.background\] we review several existing period estimation methods. In §\[sec.model\] we introduce a new semi-parametric (SP) model for Mira variables which uses a Gaussian process to account for deviations from strict periodicity. We use maximum likelihood to estimate the period and the parameters of the Gaussian process, while other nuisance parameters in the model are integrated out with respect to some prior distributions using earlier studies. Since the likelihood is highly multimodal for the period/frequency parameter, we implement a hybrid method that applies the quasi-Newton algorithm for Gaussian process parameters and a grid search for the period/frequency parameter. In order to assess the effectiveness of the SP model, in §\[sec.construct.test\] we carefully construct a simulated data set by fitting smooth functions to the light curves of well-observed OGLE LMC Miras and resampling them at the cadence, noise level, and completeness limits of the aforementioned M33 observations. Using the simulated data, in §\[sec.evaluation\] we compare the performance of existing period estimation methods to our SP model. We find that our proposed model shows an improvement over the generalized Lomb-Scargle (GLS) model under various metrics. In §\[sec.discussion\], we conclude and discuss some future applications. Simulated light curves for reproducing the results in the paper and performance benchmarking are made publicly available as supplementary material. Period estimation techniques {#sec.background} ============================ Let $y_i$ be the magnitude of a variable star observed at time $t_i$ (in units of days) with uncertainty $\sigma_i$. The data set for this object, obtained as part of a time-series survey with $n$ epochs is $\{(t_i,y_i,\sigma_i)\}_{i=1}^n$. One common approach to estimate the primary frequency of such an object is to assume some parametric model for brightness variation and then use maximum likelihood to estimate parameters. @Zechmeister2009 define the GLS model as $$\label{eq:gls} y_i = m + a\sin(2\pi f t_i + \phi) + \sigma_i\epsilon_i,$$ where $\epsilon_i \sim \mathcal{N}(0,1)$, $m$ is the mean magnitude, $a$ is the amplitude, $\phi \in [-\pi,\pi]$ is the phase, and $f$ is the frequency [see @Reimann1994 for early work in this model]. Using the sine angle addition formula and letting $\beta_1 = a\cos(\phi)$ and $\beta_2 = a\sin(\phi)$ one obtains $$\label{eq:gls2} y_i = m + \beta_1\sin(2\pi f t_i) + \beta_2\cos(2\pi f t_i) + \sigma_i\epsilon_i.$$ The likelihood function of this model is highly multimodal in $f$. However at a fixed $f$ the model is linear in the parameters $(m,\beta_1,\beta_2)$. These two facts motivate the computation strategy of performing a grid search across frequency and minimizing a weighted least squares $$\begin{split} & (\widehat{m}(f),\widehat{\beta}_1(f),\widehat{\beta}_2(f) ) \\ & \qquad = \operatorname*{arg\,min}_{m,\beta_1,\beta_2} \sum_{i=1}^n \frac{1}{\sigma_i^2} \left\{y_i - m \right.\\ & \qquad \qquad \left. -\beta_1\sin(2\pi f t_i) - \beta_2\cos(2\pi f t_i)\right\}^2, \end{split}$$ at every frequency $f$ on the grid. Under the normality assumption, the weighted least squares minimization is equivalent to maximizing the likelihood. Since the model is linear, computation of $\widehat{m}(f),\widehat{\beta}_1(f),\widehat{\beta}_2(f)$ is straightforward. The residual sums of squares at $f$ is $$\begin{split} {\rm RSS}(f) & = \sum_{i=1}^n \frac{1}{\sigma_i^2} \{y_i - \widehat{m}(f) \\ & \quad - \widehat{\beta}_1(f)\sin(2\pi f t_i) - \widehat{\beta}_2(f)\cos(2\pi f t_i)\}^2, \end{split}$$ and the maximum likelihood estimator for $f$ is $$\widehat{f} = \operatorname*{arg\,min}_{f} {\rm RSS}(f).$$ Define ${\rm RSS}_0$ as the (weighted) sum of squared residuals when fitting a model with only an intercept term $m$. The periodogram is defined as $$\label{eq:glsP} S_{\rm LS}(f) = \frac{(n-3)({\rm RSS}_0 - {\rm RSS}(f))}{2{\rm RSS}(f)}.$$ The periodogram has the property that if the light curve of the star is white noise (i.e., $y_i = m + \epsilon_i$), $S_{\rm LS}(f)$ has an $F_{2,n-3}$ distribution. Thus the periodogram may be used for controlling the “false alarm probability,” the potential that a peak in the periodogram is due to noise [@Schwarzenberg1996]. A large number of period estimation algorithms in astronomy are closely related to GLS. The LS method is identical to GLS but first normalizes magnitudes to mean $0$ and does not fit the $m$ term [@Lomb1976; @Scargle1982]. The “harmonic analysis of variance” includes an arbitrary number of harmonics in Equation [@Quinn1991; @Schwarzenberg1996]. @Bretthorst2013 incorporates Bayesian priors on the parameters $\beta_1$ and $\beta_2$. The method is similar to performing a discrete Fourier transform and selecting the frequency which maximizes the @Deeming1975 periodogram. However, @Reimann1994 showed that GLS has better consistency properties than the Deeming periodogram. ![Light curve of a Mira in the LMC observed by OGLE (black points), decomposed following Eqn. \[eqn:basicDecomposition\]. Top panel: fitted light curve; middle panel: periodic signal, $m+q(t)$; bottom panel: stochastic variations, $m+h(t)$.[]{data-label="fig:onemira"}](fig02.eps){width="49.00000%"} It is also possible to use non-sinusoidal models but compute and minimize the residual sum of squares as above. For example, @Hall2000 consider the Nadaraya-Watson estimator and @Reimann1994 uses the Supersmoother algorithm. @Wang2012 used Gaussian processes with a periodic kernel and found the period with maximum likelihood or minimum leave-one-out cross-validation error. None of the above methods account for the non-periodic variation present in Miras. While these methods are adequate for densely sampled Mira light curves (where the quantity of data overwhelms model inadequacy), their performance deteriorates in the sparsely sampled regime. In Section \[sec.evaluation\], we compare our proposed model with the LS method. The SP model {#sec.model} ============ Suppose the data $\{(t_i,y_i,\sigma_i)\}_{i=1}^n$ are modeled by $$y_i = g(t_i) + \sigma_i\epsilon_i\, ,$$ where $g(t_i)$ is the light curve signal and the $\epsilon_i\sim N(0,1)$ is independent of other $\epsilon_j$s. The signal of the light curve is further decomposed into three parts, $$\begin{split} g(t) & = m + q(t) + h(t)\\ & = m + \beta_1\cos(2\pi ft) + \beta_2\sin(2\pi ft) + h(t)\, , \label{eqn:basicDecomposition} \end{split}$$ where $m$ is the long-run average magnitude, $q(t) = \beta_1\cos(2\pi ft) + \beta_2\sin(2\pi ft)$ with frequency $f$ is the exactly periodic signal, and $h(t)$ is the stochastic deviation from a constant mean magnitude, caused by the formation and destruction of dust in the cool atmospheres of Miras. Fig. \[fig:onemira\] provides an example of the decomposition for a Mira light curve. The first two terms $m + q(t)$ in Eqn. \[eqn:basicDecomposition\] are exactly the same as the GLS model of Eqn. \[eq:gls2\]. To simplify notation, we define ${\mathbf{b}}_f(t) =(\cos(2\pi ft),\sin(2\pi ft))^T$, so that $q(t) = {\mathbf{b}}_f(t)^T{\boldsymbol{\beta}}$. The subscript in ${\mathbf{b}}_f(t)$ emphasizes that the basis is parameterized by the frequency $f$. An SP statistical model is constructed in Eqn. \[eqn:basicDecomposition\] if we assume $h(t)$ is a smooth function that belongs to a reproducing kernel Hilbert space $\mathcal{H}$ with norm $\Vert\cdot\Vert_{\mathcal{H}}$ and a reproducing kernel $K(\cdot,\cdot)$. For this model, if the frequency $f$ is known, we obtain a least squares kernel machine considered in @Liu2007. Because the frequency is unknown, the response function is nonlinear in $f$. This nonlinearity and the multimodality in $f$ of the residual sum of squares provide additional challenges that require a novel solution. Besides the additive formulation in Eqn. \[eqn:basicDecomposition\], another possible solution to account for the quasi-periodicity is a multiplicative model such as $g(t) = m + h(t) q(t)$, where the amplitude of the strictly periodic term $q(t)$ is modified by a smooth function $h(t)$. However, the multiplicative model is more computationally intensive in nature and requires imposing a positive constraint on $h(t)$. As we will show in the following subsections, the $h(t)$ term in the additive model can be easily absorbed into the likelihood function. Nevertheless, the multiplicative approach is an interesting alternative approach to model formulation and is open to future study. Equivalent formulations ----------------------- Following §5.2 of @Rasmussen2005, for fixed $f$, the parameters $m,\beta_1,\beta_2$ and $h(t)$ in Eqn. \[eqn:basicDecomposition\] are jointly estimated by minimizing $$\begin{split} & \sum_{i=1}^n \frac{1}{\sigma_i^2} [ y_i - m - \beta_1\cos(2\pi ft_i) \\ & \qquad - \beta_2\sin(2\pi ft_i) - h(t_i) ]^2 + \lambda \Vert h(\cdot)\Vert_{\mathcal{H}}^2, \label{eqn:generalFormulation} \end{split}$$ where $\lambda$ is a regularization parameter. A smoothing/penalized spline model for $h(t)$ is a special case of the general formulation of Eqn. \[eqn:generalFormulation\] with a specifically defined kernel; see §6.3 of @Rasmussen2005. For fixed $\lambda$, the solution of $h(t)$ is a linear combination of $n$ basis functions $K(t_i,t)$, $i=1,2,\cdots,n$, by the representer theorem [@Kimeldorf1971; @OSullivan1986]. It is still left for us to choose the regularization parameter $\lambda$ to balance data fitting and the smoothness of the function $h(t)$. An equivalent point of view to the above regularization approach is to impose a Gaussian process prior on the function $h(t)$; see §5.2.3 of @Rasmussen2005. The benefit of this view is that it provides an automatic method for selecting the regularization parameter $\lambda$. In particular, we can absorb $\lambda$ into the definition of the norm $\Vert \cdot\Vert_{\mathcal{H}}$ and assume the term $h(t)$ in Eqn. \[eqn:basicDecomposition\] follows a Gaussian process, $h(t) \sim {\mathcal{GP}}(0,k_{{\boldsymbol{\theta}}}(t,t'))$, with the squared exponential kernel $k_{{\boldsymbol{\theta}}} (t,t') = \theta_1^2\exp \left(-\frac{(t-t')^2}{2\theta_2^2}\right),$ and parameters ${\boldsymbol{\theta}}= (\theta_1,\theta_2)$. The Gaussian process assumption implies that at any finite number of time points $t_1,t_2,\cdots,t_s$, the vector $(h(t_1),\cdots,h(t_s))$ is multivariate normally distributed, with zero mean and covariance matrix $\mathbf{K} = (k(t_i,t_j))$. This imposes a prior on the function space of $h(t)$. We also impose priors on $m$ and ${\boldsymbol{\beta}}$ in Eqn. \[eqn:basicDecomposition\]. In particular, we assume $m \sim {\mathcal{N}}(m_0, \sigma_m^2)$ and $ {\boldsymbol{\beta}}\sim {\mathcal{N}}({\mathbf{0}},\sigma_b^2{\mathbf{I}})$. The prior mean $m_0$ can be interpreted as the average magnitude of Miras in a certain galaxy, and $\sigma_m^2$ is the variance of Miras in that galaxy; the prior variance $\sigma_b^2$ is the variance of the light curve amplitude. These prior parameters can be determined using previous studies. For example, in §\[sec.evaluation\], we use well-sampled light curves of LMC Miras [@Soszynski2009] to obtain values of these parameters. It is advisable to check the sensitivity of these prior specifications. The benefit of using priors on $m$ and ${\boldsymbol{\beta}}$ is three-fold: first, they introduce regularization by using information from early studies; second, they provide a natural device for separating the estimation of frequency and the light curve signal component using Bayesian integration when the parameter of interest is the frequency; lastly, the regularization parameter ${\boldsymbol{\theta}}$ of the non-parametric function is allowed to be chosen by the maximum likelihood, without resorting to the computationally expensive cross-validation method. In summary, we have built the following hierarchical model for a Mira light curve: $$\label{eqn:hier} \begin{split} & y_i | m,{\boldsymbol{\beta}}, g(t_i) \sim {\mathcal{N}}(g(t_i), \sigma_i^2), \\ & g(t) = m + {\mathbf{b}}_f(t)^T{\boldsymbol{\beta}}+ h(t), \\ & m \sim {\mathcal{N}}(m_0, \sigma_m^2), {\boldsymbol{\beta}}\sim {\mathcal{N}}({\mathbf{0}},\sigma_b^2{\mathbf{I}}),\\ & h(t)| {\boldsymbol{\theta}}\sim {\mathcal{GP}}(0,k_{{\boldsymbol{\theta}}} (t,t')), \end{split}$$ where ${\boldsymbol{\theta}}$ and $f$ are fixed parameters. In this model, the frequency parameter $f$ is of key interest to our study. We do not perform a fully Bayesian inference by imposing a prior distribution on $f$ because the likelihood function of $f$ is highly irregular, with numerous local maxima, and Monte Carlo computation of the posterior is expensive and intractable for large astronomical surveys. Previously, @Baluev2013 applied a Gaussian process model to study the impact of red noise in radial velocity planet searches. While his maximum likelihood method is a classical frequentist approach in statistics, our approach can be considered as a hybrid of Bayesian and frequentist approaches. We treat the parameter of interest $f$, and the parameters for the kernel ${\boldsymbol{\theta}}$ of the Gaussian process as fixed, and impose a prior distribution on other parameters. This is similar to the type-II maximum likelihood estimation of parameters of a Gaussian process or regularization parameters in function estimation; see §5.2 of @Rasmussen2005. From the Bayesian point of view, ${\boldsymbol{\theta}}$ and $f$ are treated as hyper-parameters that in turn are estimated by the empirical Bayes method. Because the Gaussian process plays a critical role in modeling departure of light curves from periodicity, we may also refer to our model more precisely as the nonlinear SP Gaussian process model. Estimation of the frequency and the periodogram ----------------------------------------------- Let ${\mathbf{y}}= (y_1,y_2,\cdots, y_n)$ be the observation vector of the magnitudes of a light curve. By integrating out $m,{\boldsymbol{\beta}}$ and ${\mathbf{h}}$ from the joint distribution given by Eqn. \[eqn:hier\], we get the marginal distribution of ${\mathbf{y}}$, $p({\mathbf{y}}|{\boldsymbol{\theta}}, f)$, which is a multivariate normal with mean ${\boldsymbol{\mu}}= m_0{\mathbf{1}}$ and covariance matrix $${\mathbf{K}}_y = \left( \sigma_m^2 + \sigma_b^2 {\mathbf{b}}_f(t_i)^T {\mathbf{b}}_f(t_j) + k_{{\boldsymbol{\theta}}}(t_i, t_j) +\sigma^2_i\delta_{ij} \right)_{n\times n},$$ where $\delta_{ij} = 1$ if $i=j$ and $\delta_{ij} = 0$ if $i\neq j$. Therefore, the log likelihood of ${\boldsymbol{\theta}}$ and $f$ is $$\begin{split} Q({\boldsymbol{\theta}},f) = &\log(p({\mathbf{y}}| {\boldsymbol{\theta}}, f))\\ = &-\frac{1}{2} ({\mathbf{y}}-m_0{\mathbf{1}})^T{\mathbf{K}}_y^{-1}({\mathbf{y}}-m_0{\mathbf{1}}) \\ &\qquad -\frac{1}{2}\log\det {\mathbf{K}}_y -\frac{n}{2}\log(2\pi) \label{eqn:mainObj}. \end{split}$$ The maximum likelihood estimator of ${\boldsymbol{\theta}}$ and $f$ is obtained by maximizing $Q({\boldsymbol{\theta}},f)$. Since the likelihood function is differentiable with respect to ${\boldsymbol{\theta}}$ but highly multimodal in the parameter $f$, standard optimization methods cannot be directly used to jointly maximize over ${\boldsymbol{\theta}}$ and $f$. We adopt a profile likelihood method as follows. For each frequency $f$ over a dense grid, we compute the maximum likelihood estimator $\widehat{{\boldsymbol{\theta}}}_f = \operatorname*{arg\,max}_{{\boldsymbol{\theta}}} Q({\boldsymbol{\theta}},f)$. This can be done using the quasi-Newton method. Then we perform a grid search to find the maximum profile likelihood estimator of $f$, i.e., $$\label{eqn:fhat} \hat{f} = \operatorname*{arg\,max}_f Q(\widehat{{\boldsymbol{\theta}}}_f, f)\, ,$$ the estimated period is $\hat{P}=1/\hat{f}$. The details of the algorithm are given in §\[sec.quasi\]. The profile log-likelihood as a function of the frequency $f$ is adopted as the *periodogram* for our model, $$S_{SP}(f) = Q(\widehat{{\boldsymbol{\theta}}}_f, f)\, . \label{eqn:periodogram}$$ It contains the spectral information of the signal. The frequency of the dominant harmonic component is expected to be the location of the peak of this profile likelihood. Computation of the periodogram {#sec.quasi} ------------------------------ Now we present the details of computing the profile likelihood. Because $Q({\boldsymbol{\theta}},f)$ is highly multimodal in the frequency parameter $f$, we follow the commonly used strategy of optimization through grid search. On the other hand, since $Q({\boldsymbol{\theta}},f)$ is differentiable in parameter ${\boldsymbol{\theta}}$, the quasi-Newton method can be employed to optimize over ${\boldsymbol{\theta}}$ for fixed $f$, and obtain the profile likelihood (Eqn. \[eqn:periodogram\]). The gradient of the log likelihood (Eqn. \[eqn:mainObj\]) with respect to $\theta_j (j=1,2)$ is $$\frac{\partial}{\partial \theta_j} Q({\boldsymbol{\theta}},f)\!=\! \frac{1}{2}\mathrm{tr}\left(({\boldsymbol{\alpha}}{\boldsymbol{\alpha}}^T\!-\!{\mathbf{K}}_y^{-1})\frac{\partial {\mathbf{K}}_y}{\partial \theta_j}\right)$$ where ${\boldsymbol{\alpha}}\!=\!{\mathbf{K}}^{-1}_y({\mathbf{y}}-m_0{\mathbf{1}})$. In general, the objective function for the Gaussian process model is not convex in its kernel parameters ${\boldsymbol{\theta}}$ and global optimization cannot be guaranteed. Fig. \[fig:lc.surface\] shows a surface plot of $Q({\boldsymbol{\theta}},f)$ as a function of ${\boldsymbol{\theta}}$ for one simulated light curve, with $f$ fixed at the true frequency. The surface exhibits unimodality in this case, although it is not convex. The computation involved in calculating the profile likelihood through the quasi-Newton method can be intensive. Since the objective function (Eqn. \[eqn:mainObj\]) is non-convex in ${\boldsymbol{\theta}}$, generally multiple starting points should be attempted to find the global optimizer when applying the quasi-Newton method. In addition, evaluating the objective function and the gradient function requires inversion of the covariance matrix whose computation cost is of the order $O(n^3)$. During each quasi-Newton iteration, these evaluations could be repeated several times because multiple step size might be attempted. To make the computation more challenging, all of the above needs to be repeated at hundreds or even thousands of densely gridded $f$s per light curve. Furthermore, the method may need to be applied to hundreds of thousands or millions of light curves from large astronomical surveys. In order to speed up computation over the dense grid of frequency values, we use the result of applying the quasi-Newton method at one frequency value as a warm start for the subsequent frequency value. Specifically, the optimizer $\widehat{{\boldsymbol{\theta}}}_f$ and its approximate inverse Hessian matrix are provided as quantities to start the quasi-Newton iterations for the next frequency value on the dense grid. When the initial point is near the local minimizer and the inverse Hessian matrix is a good approximation to the true Hessian matrix, the quasi-Newton algorithm will converge at superlinear rate; the step size of $\alpha=1$ will be accepted by the Wolfe descent condition, avoiding evaluation of the objective function multiple times to determine the appropriate step size during each iteration [see Ch. 6 of @Nocedal2006 for a more rigorous mathematical discussion]. We find that a warm start can speed up the computation significantly but sometimes we need to restart with random initial values to ensure convergence to the global optimum. The pseudocode provided in the Appendix describes our algorithm. Estimation of the signal and its components ------------------------------------------- After the parameters $f$ and ${\boldsymbol{\theta}}$ are fixed at their maximum likelihood estimates $\widehat{f}$ and $\widehat{{\boldsymbol{\theta}}}_{\hat{f}}$, we can perform the inference of the light curve signal $g(t)$ and its components in the standard Bayesian framework. Interested readers may consult Ch. 2 of @Rasmussen2005 for a detailed discussion of this topic. Firstly, we could obtain the posterior distribution of ${\boldsymbol{\gamma}}= (m,{\boldsymbol{\beta}}^T)$, the parameters for the long run average magnitude and the exactly periodic term. The prior of ${\boldsymbol{\gamma}}$ is ${\mathcal{N}}({\boldsymbol{\gamma}}_0,{\boldsymbol{\Sigma}}_\gamma)$ with ${\boldsymbol{\gamma}}_0=(m_0,0,0)^T$ and ${\boldsymbol{\Sigma}}_\gamma = \mathrm{diag}(\sigma_m^2,\sigma_b^2,\sigma_b^2)$. Its posterior distribution is ${\boldsymbol{\gamma}}| {\mathbf{y}}\sim {\mathcal{N}}(\bar{{\boldsymbol{\gamma}}}, \bar{{\boldsymbol{\Sigma}}}_\gamma)$ with $$\label{equ:gammapost} \begin{split} \bar{{\boldsymbol{\gamma}}} = & \left({\mathbf{H}}^T{\mathbf{K}}_c^{-1}{\mathbf{H}}+{\boldsymbol{\Sigma}}_\gamma^{-1}\right)^{-1} \\ & \left({\boldsymbol{\Sigma}}_\gamma^{-1}{\boldsymbol{\gamma}}_0+{\mathbf{H}}^T{\mathbf{K}}_c^{-1}{\mathbf{y}}\right)\, , \\ \bar{{\boldsymbol{\Sigma}}}_\gamma = & \left({\mathbf{H}}^T{\mathbf{K}}_c^{-1}{\mathbf{H}}+ {\boldsymbol{\Sigma}}_\gamma^{-1}\right)^{-1}, \end{split}$$ [where]{} $${\mathbf{h}}(t)\!=\!(1,{\mathbf{b}}_{\hat{f}}(t)^T)^T, {\mathbf{H}}\!=\!({\mathbf{h}}(t_1),{\mathbf{h}}(t_2),\cdots,{\mathbf{h}}(t_n))^T,$$ and$\,{\mathbf{K}}_c\!=\!\big(k_{\widehat{{\boldsymbol{\theta}}}_{\hat{f}}}(t_i,\!t_j)\!+\!\sigma_i^2\delta_{ij}\!\big)\!_{{\tiny\textit{n}}\times\!{\tiny\textit{n}}}\,$with$\,\widehat{f}\,$and$\,\widehat{{\boldsymbol{\theta}}}_{\hat{f}}\,$plugged in. Consider the prediction of light curve magnitude at a specific time point $t^*$. Define the vector ${\mathbf{k}}^* = (k_{\widehat{{\boldsymbol{\theta}}}}(t^*,t_1),\ \cdots,\ k_{\widehat{{\boldsymbol{\theta}}}}(t^*,t_n))^T$. Conditional on $({\mathbf{y}}, {\boldsymbol{\gamma}})$, the distribution of $g(t^*)| {\mathbf{y}}, {\boldsymbol{\gamma}}$ is a multivariate normal with mean $ {\mathbf{h}}(t^*)^T \gamma + {\mathbf{k}}_{{\boldsymbol{\theta}}}(t^*,{\mathbf{t}}) {\mathbf{K}}_c^{-1} ({\mathbf{y}}- {\mathbf{H}}\gamma)$ and variance $k_{\widehat{{\boldsymbol{\theta}}}}(t^*,t^*) - ({\mathbf{k}}^*)^T {\mathbf{K}}_c^{-1} {\mathbf{k}}^*$. With the posterior distribution of ${\boldsymbol{\gamma}}$ given in Eqn. \[equ:gammapost\], we are able the remove ${\boldsymbol{\gamma}}$ from the above conditional distribution of $g(t^*)$. Finally, we get the posterior distribution of the signal at $t^*$ as $g(t^*)| {\mathbf{y}}\sim {\mathcal{N}}(\bar{g}^*, \bar{\sigma}_{g^*}^2)$ with $$\begin{split} \bar{g}^* = & {\mathbf{h}}(t^*)^T \bar{\gamma} + {\mathbf{k}}(t^*,{\mathbf{t}}) {\mathbf{K}}_c^{-1} ({\mathbf{y}}- {\mathbf{H}}\bar{\gamma})\, ,\\ \bar{\sigma}^2_{g^*} = & k_{\widehat{{\boldsymbol{\theta}}}}(t^*,t^*) - ({\mathbf{k}}^*)^T {\mathbf{K}}_c^{-1} {\mathbf{k}}^*+ {\mathbf{r}}^T\bar{{\boldsymbol{\Sigma}}}_\gamma{\mathbf{r}}\, , \end{split} \label{equ:gpredict}$$ where ${\mathbf{r}}={\mathbf{h}}(t^*)-{\mathbf{H}}^T{\mathbf{K}}_c^{-1}{\mathbf{k}}^*$. Simulation of M33 light curves {#sec.construct.test} ============================== It is not possible to evaluate the period estimation accuracy of our method directly on the M33 data because the “ground truth” is unknown. Instead, we construct a test data set by smoothing the well-sampled OGLE light curves to infer continuous functions, then resample these functions to match the observational patterns of the M33 data, and at last add noise to the light curves. This data set can serve as a testbed for future studies of comparing different period estimation methods. We will now describe the M33 observations and the construction of the test data set. As the whole simulation procedure is a complicated process, we will discuss its components in detail from §4.1 to §4.4. The whole simulation procedure will be summarized in §4.5. Characteristics of the M33 observations --------------------------------------- Most of the disk of M33 was observed by the DIRECT [@Macri2001] and M33SSS [@Pellerin2011] projects in the $BVI$ bands, with a combined baseline of $7-9$ years and a sampling pattern that depends on the exact location within the disk (see Fig. \[fig:obs.gaps\]). The large area of coverage and long baseline of these observations make them suitable for Mira searches. We use the $I$-band observations to carry out the simulations, as this is the wavelength range where Miras are brightest (out of the three bands used by these projects). Detailed descriptions of the M33 observations can be found in the above referenced papers. We use the data products from a new reduction that will be presented in a companion paper (W. Yuan et al. 2016, in prep.). $I$-band light curves are available for $\sim 2.5\times 10^5$ stars, with a median of 44 measurements and a maximum of 170. . We model the relation between a magnitude measurement $m$ and its uncertainty $\sigma$ as $$\sigma = a(t_i',F)^{[m-b(t_i',F)]} + c(t_i',F)\, , \label{equ.sigma.mag}$$ for each observation field $F$ and each observation night $t_i'$, where $a(t_i',F)$, $b(t_i',F)$ and $c(t_i',F)$ are field- and night-specific constants. There are 31 different fields in total, $F=0,1,\cdots, 9, a,b,\cdots, u$. The parameters are determined via least-squares fitting using all the measurements for the specific field $F$ and night $t_i'$. Fig. \[fig:simu.sigmag\] shows the $m-\sigma$ relation for a typical field. ![$m-\sigma$ relation for a given night and field within M33. The solid red line is the best-fit relation using the empirical function $\sigma = a^{(m-b)} + c$, with $a=2.666,\,b=23.117,\,c=0.008$.[]{data-label="fig:simu.sigmag"}](fig05.eps){width="49.00000%"} In order to test the SP periodogram we need sparsely sampled, moderately noisy Mira light curves with known periods. Thus, we characterize the sampling patterns and noise levels of the M33 observations and simulated Mira light curves of known periods using the OGLE observations of these objects in the LMC. Matching the M33 observation pattern ------------------------------------ The first step in simulating a Mira light curve is to randomly select a sampling pattern based on the light curve of an actual star in some field $F$, $\{t_i'\}_{i=1}^n$ with $n\in [10,170]$. A random time shift $s$ is added, $t_i = t_i' + s$ for $i=1,2,\cdots, n$. The random shift $s$ follows a uniform distribution over the interval $[0, P_0]$, where $P_0$ is the true period of the LMC Mira selected during the artificial light curve generation process. This helps to simulate a large number of unique light curves sampled at random phases using the limited number of template light curves. The Mira template light curves ------------------------------ The template Mira light curves are obtained by using our SP model to fit the Mira light curves in the LMC, collected by the OGLE project [@Soszynski2009]. A total number of 1663 Miras have been observed in $I$ with very high accuracy, excellent phase coverage, and a long baseline (the median and mean number of observations are 466 and 602, respectively, with a baseline of $\sim 7.5$ years for most fields). Because the LMC light curves are densely sampled with high quality, we can adopt a more complicated model to provide a higher fidelity fit. Following §5.4.3 of @Rasmussen2005, instead of Eqn. \[eqn:basicDecomposition\], the signal light curve $g(t)$ is decomposed into $$\label{eqn:fullgpmodel} g(t) = m + l(t) + q(t) + h(t),$$ where $m$ is the long run average magnitude, $l(t)$ is the long-term (low-frequency) trend across different cycles, $q(t)$ is the periodic term, and $h(t)$ is small-scale (high-frequency) variability within each cycle. The latter three terms are modeled by the Gaussian process with different kernels. In particular, we use the squared exponential kernel $k_l(t_1,t_2) = \theta_1^2 \exp(-\frac{1}{2}\frac{(t_1-t_2)^2}{\theta_2^2})$ for $l(t)$, another squared exponential kernel $k_h(t_1,t_2) = \theta_6^2 \exp(-\frac{1}{2}\frac{(t_1-t_2)^2}{\theta_7^2})$ for $h(t)$, and lastly a periodic kernel $$\begin{split} k_q(t_1,t_2) = \theta_3^2 \exp \bigg ( -\frac{1}{2}&\frac{(t_1-t_2)^2}{\theta_4^2} \\ &- \frac{2\sin^2(2\pi f(t_1-t_2))}{\theta_5^2} \bigg ) \end{split}$$ for $q(t)$. Note the periodic kernel allows the light curve amplitude to change across cycles. The maximum likelihood method is applied to fit each LMC light curve, fixing $f$ to the OGLE value and solving for the unknown parameters $(\theta_1,\theta_2, \cdots, \theta_7)$. Fig. \[fig:complexDecomposition\] is an illustration of the model fitting result using Eqn \[eqn:fullgpmodel\] based on the same light curve as in Fig. \[fig:onemira\]. Notice that the more complex model in Fig. \[fig:complexDecomposition\] is only suitable for a densely sampled light curve. Once the sampling pattern is chosen, one of the template light curves will be selected according to the luminosity function described in the next subsection. With the selected template, the magnitude of the simulated light curve signal at $t_i'$ with shift $s$ is $g(t_i'+s)$, which is computed with Eqn. \[eqn:fullgpmodel\] in a similar way as Eqn. \[equ:gpredict\]. ![Light curve of a Mira in the LMC observed by OGLE (black points), decomposed following Eqn \[eqn:fullgpmodel\]. Top panel: the fitted light curve; second panel: long-term signal, $m+l(t)$; third panel: periodic term, $m + q(t)$; bottom panel: stochastic variations, $m+h(t)$.[]{data-label="fig:complexDecomposition"}](fig06.eps){width="49.00000%"} Matching the luminosity function\ to the M33 observations --------------------------------- While the OGLE observations of LMC Miras are deep enough to detect these objects over their entire range of luminosities, the M33 observations become progressively more incomplete for fainter and redder objects. We derived an empirical completeness function for the M33 observations as follows. We fitted the observed luminosity function $\mathcal{F}_0(I)$ using an exponential for $I \in [18.5,20]$ mag and extrapolated to fainter magnitudes, obtaining $\mathcal{F}_1(I)$. The empirical completeness function is then $\mathcal{C}(I) = \mathcal{F}_1(I) / \mathcal{F}_0(I)$. We randomly picked $\{t_i'\}_{i=1}^n$ from the M33 light curves. For each $\{t_i'\}_{i=1}^n$, we selected a (LMC-based) template using $\mathcal{C}(I+6.2)$ as the probability distribution. The value of $+6.2$ mag accounts for the approximate difference in distance modulus between the LMC and M33. In this way the resulting luminosity function of the simulated light curves is statistically the same as that of the real M33 observations. The simulation procedure ------------------------ With all the components discussed above, we are able to present the whole simulation procedure here. In order to generate one simulated Mira light curve matching the sampling characteristics of the M33 observations, the first step is to randomly select a sampling pattern $\{t_i'\}_{i=1}^n$, and then add a random shift $s$, $t_i = t_i' + s$, $i=1,2,\cdots, n$. The second step is to randomly select a template light curve according to the luminosity function, then compute the light curve signal $g(t_i'+s)$ for the selected sampling pattern $\{t_i'\}_{i=1}^n$. The third step is to use the best-fit relations (Eqn. \[equ.sigma.mag\]) to add photometric noise via $$y_i = g(t_i' + s)+6.2+ \sigma_i\epsilon_i\, ,$$ where $+6.2$ mag is the approximate relative distance modulus, $\epsilon_i$ is drawn from $\mathcal{N}(0,1)$, and $\sigma_i$ is computed from $$\sigma_i = a(t_i',F)^{[g(t_i)+6.2-b(t_i',F)]} + c(t_i',F).$$ for the selected observation pattern $t_i'$ and field $F$. Following this procedure, we generate one simulated light curve $\{t_i',y_i,\sigma_i\}_{i=1}^n$. The procedure is repeated until $10^5$ suitable light curves are generated, excluding any with $<10$ data points or sampling on $<7$ nights. Performance evaluation {#sec.evaluation} ====================== Having generated the test data set, we evaluate the performance of the SP model and compare it with the GLS model. We choose prior parameters for the SP model of $m_0 = 15.62 + 6.2$, $\sigma_m = 10$ and $\sigma_b =1$. The adopted value of $m_0$ is the average $I$ magnitude of Miras in the LMC and once again $+6.2$ is the approximate relative distance modulus between M33 and the LMC. The values of $\sigma_m$ and $\sigma_b$ are larger than those derived from the LMC samples in order to make those priors non-informative. Although fitting the SP model is computationally slower than the LS model, we find that our model gives an overall improvement in various metrics. For both methods, the periodograms are computed on a dense frequency grid from $1/2000$ to $1/100$ with a spacing of the order of $10^{-5}$. For the GLS method, we chose a spacing of (0.05/time span) or $\sim2.5\times 10^{-5}$, which results in optimal performance for this simulation. For our SP method, we chose a slightly smaller value of $10^{-5}$ to facilitate the warm start mechanism in our algorithm (see Appendix) given that small changes in frequency result in tiny changes of the objective function. The aliasing effect ------------------- We fit the entire simulated data set using the SP model. Fig. \[fig:lc.and.spec\] gives an example of a simulated light curve and its SP periodogram (Eqn. \[eqn:periodogram\]). In this example, the true frequency (labeled by the blue dotted line) is successfully recovered. Aliasing frequencies at $f\pm 1/365$ d affect most periodograms when dealing with sparsely observed astronomical data. The red dashed line in Fig. \[fig:lc.and.spec\] indicates the aliasing frequency at $f+1/365$ where a strong peak exists. This is not a rare case, and for some light curves the one-year beat aliasing frequencies have higher log likelihoods than the true frequencies. Fig. \[fig:fvsf\] compares the recovered and true frequencies for all simulated light curves. Two secondary strips parallel to the main one and offset by $\pm0.00274$ represent $\hat{f} = f \pm 1/365$, respectively. Other aliasing frequencies, such as $2f$, $3f$, $0.5f$, etc., are also noticeable. Lastly, due to the sampling pattern of some light curves, the side lobes of the main peak can be higher than the central value. These manifest as close parallel strips to the aforementioned features. Accuracy assessment ------------------- The estimated frequency is considered as correct if $\Delta f=|\hat{f}-f_0|<C_f$ for each light curve. The estimation accuracies for the two methods are summarized in Table \[tbl:accuracy\] for several different values of $C_f$. We choose $C_f = 2.7\times 10^{-4}$ to stringently bind the one-to-one strip in Fig. \[fig:fvsf\]. Overall, the SP correctly estimates the period for 69.4% of the light curves, while the LS model has an accuracy of 63.6%. The improvement of SP over LS is more evident for C-rich Miras, with about 10% higher accuracy, while the improvement for O-rich Miras is smaller, with about 3% higher accuracy. The difference in performance arises because C-rich Miras often exhibit larger stochastic deviations that can be better captured by the SP model, while O-rich Miras have more stable light curves that can be modeled reasonably well with the LS method. We also compute the estimation accuracy of each method by grouping the light curves according to the number of observations, as shown in the left panels of Fig. \[fig:accuracy\]. The top and bottom rows show results for C- and O-rich Miras, respectively. The performance difference is once again more evident in the C-rich category. Note that accuracy is not a monotonic function of the number of observations, implying this is not a good indicator [*per se*]{} of the information content of the light curves for frequency (period) estimation. Thus, we define another metric, called [*phase coverage*]{}. Recall that the times of observation for a given light curve are $t_1,t_2,\cdots,t_n$. Given a period of $P$, these are converted into corresponding phases by $s_i = (t_i\ \mathrm{ mod }\ P)/P,\ i=1,2,\cdots, n\,$ in the closed interval $[0,1]$. Now, define $$J = \Big(\bigcup_i (s_i-l,s_i+l)\Big)\cap [0,1]\, ,$$ for a specific $l>0$, the phase coverage can be measured by $\lambda(J)$ where $\lambda(\cdot)$ is the Lebesgue measure (we choose $l=0.02$). $\lambda(J)$ describes the “length” of the union of the intervals $J$. For example, $\lambda(J) = 0.1$ for $J = (0.1,0.2)$, and $\lambda(J) = 0.2$ for $J = (0.1,0.2)\cup (0.5, 0.6)$. We divide the light curves into 100 groups such that their $\lambda(J)$ is in one of the intervals $(k/100,(k+1)/100]$ for $k=0,1,\cdots,99$ and compute the estimation accuracy for each subset. The results for the two models are plotted in the middle column of Fig. \[fig:accuracy\]. Now the estimation accuracy is monotonically increasing as a function of phase coverage. The accuracy improvement of our method is highest when the phase coverage is around 0.5 for C-rich Miras. As the phase coverage approaches the extremes (0 or 1), the performance difference between the two methods diminishes. At $\lambda(J) \approx 0$, both methods will fail because this is a hopeless situation. At the other extreme, when $\lambda(J)\approx 1$ and abundant information is available for frequency estimation, both methods have an accuracy close to 1. The periodogram $S_{\rm SP}(f)$ of our model defined in Eqn. \[eqn:periodogram\] provides more information than just the optimal frequency. Suppose $f_1$ is the largest local maximal (global maximum) of $S_{\rm SP}(f)$, and $f_2$ is the second largest local maximal of $S_{\rm SP}(f)$. Now define [conf]{} $=S_{\rm SP}(f_1)-S_{\rm SP}(f_2)\ge 0$. The value of [conf]{} serves as a confidence measurement of the global optimal estimate in Eqn. \[eqn:fhat\]. Larger values of [conf]{} indicate smaller uncertainty in our estimate, and thereby the estimate is more reliable. Now, let $c_0$ be the smallest value, and let $c_1, \dots, c_{100}$ be the $1$st–$100$th percentiles of all the [conf]{} values computed for all the light curves. Each light curve can be assigned to a percentile group if its [conf]{} is in $(c_{k-1},c_{k}]$ for some $k\in\{1,2,\cdots,100\}$. After assigning all light curves by [conf]{} to their corresponding percentile groups, the estimation accuracy in each group can be computed. The same procedure is applied to the GLS model, with the $p$-value of the F-statistics given in Eqn. \[eq:glsP\] for the top peak being used as its [conf]{} measurement. The result is plotted in the right column of Fig. \[fig:accuracy\]. The accuracy of our SP method is much higher than the LS model in the top 40 groups. In particular, the accuracy of our method is higher than 90% in the top 20 groups for both C- and O-rich Miras. Light curves with high values of [conf]{} are particularly reliable for constructing Period-Luminosity relations (hereafter, PLRs) based on the “Wesenheit” function [@Madore1982]. This function enables a simultaneous correction for the effects of dust attenuation and finite width of the instability strip by defining a new magnitude $W_I\!=\!I\!-\!1.55(V\!-\!I)$, where $V$ and $I$ are the mean magnitudes in those filters. Figure \[fig:pl.compare\] compares PLRs based on $W_I$ magnitudes and periods determined by OGLE and estimated with each of the two models. The top and bottom rows display the PLRs for C- and [ccrrr]{} & SP & 58.1 & 55.3 & 56.5\ & LS & 49.4 & 51.6 & 50.6\ & SP & 69.6 & 63.8 & 66.3\ & LS & 60.1 & 60.6 & 60.4\ & SP & 73.5 & 66.2 & 69.4\ & LS & 63.7 & 63.5 & 63.6\ & 43,116 & 56,884 & [O-rich Miras, respectively. The leftmost column shows the PLRs based on the actual OGLE periods, while the next two sets of columns show the corresponding relations based on SP or LS periods for the simulated light curves with the top 10% and 40% values of [conf]{}.]{} In order to provide a quantitative comparison of the improvement obtained with our SP method, we calculated the dispersion of the actual $W_I$ PLRs and their recovered counterparts as a function of [conf]{} value as follows, separately for C- and O-rich Miras. First, we selected all objects of a given class with $2\!<\!\log P\!<\!3$. If the @Soszynski2009 catalog did not provide a $V$ measurement for a given variable, the missing value was estimated through linear interpolation of the $(I,V\!-\!I)$ relation for objects of the same class within $|\Delta\log P|<0.05$ dex. We fitted a quadratic PLR $$m=a+b(\log P-2.3)+c(\log P-2.3)^2$$ with iterative $3\sigma$ clipping (removing $\sim 5$% of the data). We then computed the dispersion of the initially selected OGLE sample about the best-fit relation, including outliers. This yielded “benchmark” dispersions of 0.45 & 0.54 mag for C- & O-rich variables, respectively. Keeping the best-fit relation fixed, we computed the dispersion of recovered PLRs using all artificial light curves within a certain range of [conf]{} (top 10%, top 20%, $\dots$), using the periods and [conf]{} values derived by the SP or the LS method. As in the case of the OGLE samples, we only considered objects with $2\!<\!\log P\!<\!3$. The results are plotted in Fig. \[fig:pl.sig\]. The SP subsamples exhibit lower [llrrrr]{} 00082 & O & 14.241 & 16.509 & 164.84 &\ 00094 & C & 15.120 & 18.885 & 332.30 &\ 00098 & C & 15.159 & 17.921 & 323.10 &\ 00115 & C & 14.932 & 16.947 & 176.13 &\ 00355 & O & 14.199 & 16.219 & &\ [(or at worst, equal) dispersions than their LS counterparts for all percentiles and for both subtypes. As discussed previously, the improvement provided by our method is strongest for C-rich Miras and diminishes in significance as one includes light curves with progressively lower confidence values.]{} Summary {#sec.discussion} ======= In this paper, we developed a nonlinear SP Gaussian process model for estimating the periods of sparsely sampled quasi-periodic light curves, motivated by the desire to detect Miras in an existing set of observations of M33. We conducted a large-scale high-fidelity simulation of Mira light curves as observed by the DIRECT/M33SSS surveys to compare our model with the GLS method. Our model shows improved accuracy under various metrics. The simulation data set is provided as a testbed for future comparison with other methods. The SP model will be used in a companion paper to search for Miras in M33, estimate their periods, and study the resulting PLRs.   SH was partially supported by Texas A&M University-NSFC Joint Research Program. WY & LMM acknowledge financial support from the NSF through AST grant \#1211603 and from the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University. JZH was partially supported by NSF grant DMS-1208952. The authors acknowledge the Texas A&M University Brazos HPC cluster that contributed to the research reported here. [[*Simulated light curves:*]{} A tarfile, containing $10^5$ simulated light curves are generated following the procedure of §\[sec.construct.test\]. Each light curve is stored in one file with three columns: MJD, $I$ magnitude, and uncertainty. The file name, e.g., lc006788.dat is generated sequentially and is only meant for bookkeeping purposes. A mapping between simulated light curve ID and the original OGLE object is given in the file “lc.dat”, which can also be found in the tarfile.]{} [[*Mira variables:*]{} Table\[tbl:oglemira\] summarizes the relevant properties of OGLE LMC Miras from @Soszynski2009 that were used to simulate the light curves: OGLE ID, main period and mean $I$ & $V$ magnitudes. It includes some extrapolated values of $V$ for objects with missing data (suitably identified with a “\*”). This table can be used to compare true versus derived periods and to generate Period-Luminosity relations.]{} [[*Software:*]{} The related software package, `varStar`, has been released under a GPL3 license [@He2016]. The active software development repository can be found at [github.com/shiyuanhe/varStar](github.com/shiyuanhe/varStar).]{} * *Quasi-Newton’s Method with Grid Search**\ **Input:** Maximal and minimal trial frequencies $f_M\!>\!f_m\!>\!0$; frequency step $\Delta f$; $n$ observations $\{t_i,y_i,\sigma_i\}$.\ **Output:** Periodogram $S(f)$ evaluated at the trial frequencies.\ Initialize ${\boldsymbol{\theta}}^{(0)}$ and $\mathbf{H}^{(0)}$, and $f\gets f_m$; $p \gets 0$; $\mathbf{t}_p \gets -\mathbf{H}^{(p)} \frac{\partial}{\partial {\boldsymbol{\theta}}} Q({\boldsymbol{\theta}}^{(p)}, f)$; ${\boldsymbol{\theta}}^{(p+1)}\gets {\boldsymbol{\theta}}^{(p)} +\alpha_p \mathbf{t}_p$ and the step size $\alpha_p$ satisfying the Wolfe condition; $\mathbf{d}_p \gets {\boldsymbol{\theta}}^{(p+1)}-{\boldsymbol{\theta}}^{(p)}$, $\mathbf{e}_p \gets \frac{\partial}{\partial {\boldsymbol{\theta}}} Q({\boldsymbol{\theta}}^{(p+1)}, f)- \frac{\partial}{\partial {\boldsymbol{\theta}}} Q({\boldsymbol{\theta}}^{(p)}, f)$; $\rho_p= 1/\mathbf{d}_p^T\mathbf{e}_p$; $\mathbf{H}^{(p+1)}\gets (\mathbf{I}-\rho_p \mathbf{d}_p\mathbf{e}_p^T) \mathbf{H}^{(p)} (\mathbf{I}-\rho_p \mathbf{e}_p\mathbf{d}_p^T) +\rho_p\mathbf{d}_p\mathbf{d}_p^T$; $p \gets p + 1$; $\widehat{{\boldsymbol{\theta}}}_{f} \gets {\boldsymbol{\theta}}^{(p)}$, and $S(f)\gets Q(\widehat{{\boldsymbol{\theta}}}_{f},f)$; $\mathbf{H}^{(0)} \gets \mathbf{H}^{(p)}$ and ${\boldsymbol{\theta}}^{(0)}\gets {\boldsymbol{\theta}}^{(p)}$;

\ \ Feza Gürsey Institute P.O.Box 6 Çengelköy, Istanbul 81220 Turkey\ May 7, 2001 PACS numbers: 11.10.Ef 02.30.Wd 02.30.Jr 03.40.Gc [**Abstract**]{} We propose a general scheme to construct multiple Lagrangians for completely integrable non-linear evolution equations that admit multi-Hamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a conserved quantity higher/lower than the Hamiltonian in the potential part of the new Lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map. This leads to some remarkable new developments. We show that nonlinear evolutionary systems that admit $N$-fold first order local Hamiltonian structure can be cast into variational form with $2N-1$ Lagrangians which will be local functionals of Clebsch potentials. This number increases to $3 N -2$ when the Miura transformation is invertible. Furthermore we construct a new Lagrangian for polytropic gas dynamics in $1+1$ dimensions which is a [*local*]{} functional of the physical field variables, namely density and velocity, thus dispensing with the necessity of introducing Clebsch potentials entirely. This is a consequence of bi-Hamiltonian structure with a compatible pair of first and third order Hamiltonian operators derived from Sheftel’s recursion operator. Introduction ============ In this paper we shall point out a general technique for the construction of inequivalent solutions to the inverse problem in the calculus of variations. We shall show that completely integrable partial differential equations in $1+1$ dimensions that admit multi-Hamiltonian structure can be cast into variational form with multiple Lagrangians. It is remarkable that all these new Lagrangians can be obtained directly from our present knowledge of complete integrability of the evolutionary system without doing any new calculations! One of the important properties we expect from a completely integrable system is multi-Hamiltonian structure. A vector evolutionary system can then be cast into Hamiltonian form in more than one way $$u^i_{t_{\aleph+\alpha-1}} = \{u^i, H_\alpha \}_\aleph = J^{ik}_{\aleph} \delta_k H_\alpha \qquad \left\{ \begin{array}{l} i=1,2,...,n \\ \aleph=1,2,... N. \\ \alpha=-1,0,1,..., \infty \end{array} \right. \label{hameq}$$ where the variational derivative is denoted by $\delta_k\equiv \delta /\delta u^k$ and $J$ is a matrix of differential operators satisfying the properties of a Poisson tensor, namely skew-symmetry and Jacobi identity. For integrable systems there exists more than one such Hamiltonian operator and Hamiltonian function as the respective Hebrew and Greek indices indicate. Then, by the theorem of Magri [@magri] completely integrable systems admit infinitely many conserved Hamiltonian functions which are in involution with respect to Poisson brackets defined by compatible Hamiltonian operators. The essential element in the multi-Hamiltonian approach to integrability is the construction of the Hamiltonian operators themselves. Fortunately this is a rich subject [@dorfman] that can be put to good use. We shall be interested in the consequences of multi-Hamiltonian structure on the Lagrangian formulation of completely integrable evolutionary equations. We shall work in the opposite direction to the traditional approach of deriving Hamiltonian structure from a Lagrangian. The crucial fact that we shall exploit is the relationship between Hamiltonian operators and Dirac brackets [@dirac] for degenerate Lagrangian systems which was first pointed out by Macfarlane [@mac]. In the case of completely integrable systems we have much more information on Hamiltonian structure than Lagrangian and it became clear only recently [@pavlov], [@nhepth], [@pavlov2] how we can construct multiple Lagrangians for systems that admit multi-Hamiltonian structure. We shall now present the general and most simple technique for generating these new Lagrangians. Multi-Lagrangians {#sec-main} ================= Evolutionary systems (\[hameq\]) cannot be cast into variational form with a local expression for the Lagrangian using the velocity fields $u^i$ alone but require the introduction of Clebsch potentials. In $1+1$-dimensions the general expression for Clebsch potentials is given by $$u^i = \phi^i_x \label{clebsch}$$ and in this paper we shall only consider Lagrangians that are local functionals of these potentials. In the time-honored way we shall split the Lagrangian density for eqs.(\[hameq\]) into two $${\cal L} = {\cal T} - {\cal V} \label{lagtot}$$ that consist of the kinetic and potential pieces respectively. For the first Lagrangian density, an enumeration which will become clear presently, the kinetic term is always given by $${\cal T}_{1} = g_{ik} \; \phi^i_t \, \phi^k_x \label{kinetic}$$ where $g_{ik}$ are constants with $\det g_{ik} \ne 0$ and $${\cal V}_{1} = 2 {\cal H}_1 \label{potential}$$ is the Hamiltonian density. We note that the Hamiltonian function that appears in (\[hameq\]) is the space integral of the density. We shall number the conserved Hamiltonians by reserving the subscript $1$ to the “usual" Hamiltonian function but of course there exists conserved quantities such as Casimirs and the momentum which are of lower order. In fact, for complete integrability, an $n$-component vector evolutionary system (\[hameq\]) must admit $n$ infinite series of conserved Hamiltonians. We shall denote their densities by $${\cal H}_{\alpha ; [i]} \qquad \qquad i=1,2,..n; \quad \alpha=-1,...,\infty \label{series}$$ and recall that each series starts with a Casimir $${\cal H}_{-1 ; [i]} = g_{ik} u^k \label{casimir}$$ which will carry the label minus one. One of these series is distinguished in that it contains the “usual" Hamiltonian function which is the one that appears in eq.(\[hameq\]). For the $2$-component systems that we shall discuss in this paper these are the Eulerian and Lagrangian series. We note also that the two series may coincide up to a relabelling dictated by the recursion operator. This is in fact the case for the $\gamma=2$ case of gas dynamics and in most examples of completely integrable dispersive equations except the Boussinesq equation. The potential part of the Lagrangian does not depend on the velocities and from eq.(\[kinetic\]) it follows that the Hessian $$\det \left| \frac{\partial^2 {\cal L}_1}{\partial \phi^i_t \, \partial \phi^k_t} \right| = 0$$ vanishes identically. We have therefore a degenerate Lagrangian system and in order to cast it into Hamiltonian form we must use Dirac’s theory of constraints [@dirac], or the covariant Witten-Zuckerman theory [@witten; @zuck] of symplectic structure. In particular, the first Hamiltonian operator obtained from the first Lagrangian is given by $$J^{ik}_{1} = g^{ik} \, D \qquad D \equiv \frac{d}{d x} \label{j0}$$ where $g^{ik}$ is the inverse of the coefficients in the kinetic part of the first Lagrangian (\[kinetic\]) which is non-degenerate. The construction of multiple Lagrangians relies on the use of the Lenard recursion relation which is implicit in eqs.(\[hameq\]) that in the Greek and Hebrew indices we have a symmetric matrix $$J^{ik}_{[\aleph} \delta_{|k|} H_{\alpha]} =0 \label{lenard}$$ where square brackets denote complete skew-symmetrization and bars enclose indices which are excluded in this process. Provided we can invert these Hamiltonian operators, we can construct recursion operators $$R_{\;\aleph_2 \; \;k}^{\aleph_1 \;\; i} = J_{\aleph_2}^{im} (J^{mk}_{\aleph_1})^{-1} \label{recop}$$ that map gradients of conserved Hamiltonians into each other (\[lenard\]). For the construction of Lagrangians we start with the crucial observation that the first Lagrangian is of the form $${\cal L}_1 = {\cal H}_{-1 [i]} \, \phi^i_t - 2 {\cal H}_1 \label{observe}$$ which is manifest from (\[kinetic\]). The original fields that enter into the evolutionary system (\[hameq\]) are Casimirs which is evident from the subscript minus one. The second Lagrangian will be of the same general structure as (\[observe\]) if we further suppose that eqs.(\[hameq\]) can be written in bi-Hamiltonian form. Thus there will exist $H_2$ which is the next conserved Hamiltonian function in the hierarchy and the momentum $H_0$ which comes after Casimirs. The higher Lagrangian should simply be $${\cal L}_2 = {\cal H}_{0 [i]} \, \phi^i_t - 2 {\cal H}_2$$ but there is an important refinement that we need to insert here. It is not the conserved density but rather the momentum map that enters into the kinetic part of the Lagrangian. The two differ only by total derivatives which is irrelevant in the context of conservation laws and therefore generally skipped over. However, these divergence terms are of crucial interest as the momentum map in the theory of symplectic structure. We shall show that given $\alpha^{th}$ local Hamiltonian structure, the full new Lagrangian is simply given by $${\cal L}_\alpha = \{ {\cal H}_{\alpha-2 [i]} + ({\cal G}_{\alpha -2 [i]} )_x \} \phi^i_t - 2 {\cal H}_{\alpha} \label{genexp}$$ where ${\cal G}_{\alpha [i]}$ is a functional of the potentials. The coefficient of $\phi^i_t$ above is the momentum map and this is the only calculation necessary to find the new Lagrangian. The fact that it is the momentum map rather than the conserved density that plays an important role in the Lagrangian can be seen at the level of the first Lagrangian. Now the Casimirs play the role of the momentum map and they are used to construct the next higher conserved quantity according to the construction of the canonical energy-momentum tensor $${\cal H}_0 = \frac{\partial {\cal L}_1}{\partial \phi^i_t} \phi^i_x = {\cal H}_{-1 [i]} u^i = \frac{1}{2} g^{ik} {\cal H}_{-1 [i]}{\cal H}_{-1 [k]}$$ which is the momentum. This classical result for Lagrangians linear in the velocity can be generalized at each level we have a higher Lagrangian. We have $$2 {\cal H}_{\alpha-1} = g^{ik} [ {\cal H}_{\alpha-2 [i]} + ({\cal G}_{\alpha-2 [i] } )_x ] {\cal H}_{-1 [k]} \label{check}$$ ending at the level where a local Lagrangian is no longer possible. In fact the validity of this equation is directly related to the existence of the Lagrangian. If a check of (\[check\]) fails for some $\alpha$, then there exists no local Lagrangian at $\alpha^{th}$ level. Now we come to an important reservation that our new Lagrangians will necessarily carry. The Euler equations that follow from the variation of the action with the second Lagrangian will be $$R_{\;2 \; \;k}^{1 \;\; i} \, \left[ u^k_{t} - J^{km}_{1} \delta_m H_1 \right]=0 \label{crucial}$$ so that the first variation of the second action will certainly be an extremum for the original equations of motion (\[hameq\]) but the Euler equations (\[crucial\]) require something weaker, namely linear combinations of functionals in the kernel of the recursion operator can be added to the right hand side of the equations of motion and the new action will still be an extremum. From this construction it is manifest that for every Hamiltonian function in the infinite hierarchy of conserved Hamiltonians that we have for completely integrable systems, there exists a degenerate Lagrangian (\[genexp\]) that yields the equations of motion as its Euler equation up to functionals in the kernel of the recursion operator. The number of Lagrangians that can be constructed in this way is therefore infinite in number. Given bi-Hamiltonian structure we have two local Hamiltonian operators but the Lenard recursion operator (\[recop\]) is non-local. However, the special form of the first Hamiltonian operator (\[j0\]) leads to a local expression for the second Lagrangian in terms of Clebsch potentials. But it is clear that the repeated application of the recursion operator will require the introduction of non-local terms in higher Lagrangians. Strictly speaking, this is not a problem because the original Lagrangian is itself non-local in terms of the velocity fields $u^i$ which are the original variables. We swept this problem under the rug by introducing Clebsch potentials. Higher Lagrangians for evolutionary equations (\[hameq\]) can be written in local form by introducing potentials for the Clebsch potentials themselves! If, however, the equations of motion admit $N$ local Hamiltonian operators, then our construction guarantees the existence of $N$ Lagrangians which are local functionals of the Clebsch potentials. Thus we have [**Theorem 1**]{} [*A completely integrable system that admits $N$-fold local first order Hamiltonian structure can be given $N$ different variational formulations with degenerate Lagrangians that are local functionals of the Clebsch potentials*]{}. By a convenient abuse of language we claim that we have a Lagrangian for an equation that involves fields when the Lagrangian is in fact only a functional of the Clebsch potentials for these fields. Then we have the audacity to put in by hand the expression for the fields in terms of potentials after the variation! This can be at best only a shorthand for the real variational principle where we must impose the relationship between the fields and their potentials through Lagrange multipliers. So far we have been guilty of this abuse ourselves. But now we must say that the first Lagrangian is actually $${\cal L}_{\ge 1}^{full} = {\cal L}_1(\phi^i, \phi^i_x, \phi^i_{xx}...) + \lambda_i ( u^i - \phi^i_x ) \label{reallag}$$ so that upon variation with respect to all the variables $\phi^i, u^i, \lambda_i$ we get (\[clebsch\]), $\lambda_i=0$ and we arrive at the equations of motion (\[hameq\]) expressed in terms of the original fields $u^i$ without fudging. Now this obvious observation may seem correct but naive, however, we shall now find that it dramatically increases the number of new Lagrangians we can construct for integrable systems. For every evolutionary equation that admits, say for simplicity, bi-Hamiltonian structure there exists a differential substitution $$u^i=M^i(r^k, r^k_x, ...) \label{m}$$ that brings the second Hamiltonian operator to the canonical form (\[j0\]) of Darboux. This differential substitution is a Miura transformation. Strictly speaking the theorem of Darboux remains unproved in field theory where the number of degrees of freedom is infinite but we shall assume it. Miura transformation works in a direction opposite to the usual action of the recursion operator. It leads to Hamiltonian equations $$r^i_{t} = \{r^i, H_0\}_1 = g^{ik} D \, \delta_{r^k} H_0\Big|_{u^m=M^m(r^n)} \label{hameqmod}$$ where $H_0$ is the momentum for eqs.(\[hameq\]) expressed through (\[m\]). These are modified equations, different from the original equations, but the two sets are related by $$u^i_{t} - J_2^{ik} \delta_{u^k} H_0 (u) = {\cal O}^i_j (r^l) \, \Big[ r^j_{t} - J_1^{jk} \delta_{r^k} H_0 \Big|_{u^m=M^m(r^n)} \Big] \label{mm3}$$ up to functions in the kernel of some matrix differential operator ${\cal O}^i_j$. A comparison of eqs.(\[crucial\]) and Miura’s relation (\[mm3\]) shows us that using the differential substitution of Miura we can obtain new Lagrangians for nonlinear evolution equations that admit multi-Hamiltonian structure in the opposite direction to our earlier construction. Transforming to the variables $r^i$ and using Clebsch potentials $$r^i = \psi_x^i \label{newclebsch}$$ we can write the classical Lagrangian for the modified system (\[hameqmod\]) $${\cal L}_1^{modified} = g_{ik} \psi_x^i \psi_t^k - 2 {\cal H}_0 \Big|_{u^m=M^m(\psi_x,\psi_{xx}...) } \label{lagmod}$$ where the labelling of ${\cal H}_0$ refers to its expression in the original variables $u^i$ but these need to be substituted for in terms of $r^i$ according to (\[m\]) and expressed through the potentials (\[newclebsch\]). We note that the Casimirs $ r^i = \psi^i_x$ for the modified system are absent in the polynomial ${\cal H}_\alpha(u)$ hierarchy. We would expect naively that the Euler equations resulting from the first variation of the action with the Lagrangian (\[lagmod\]) would result in the modified equations (\[hameqmod\]). This would indeed be the case if we were to impose the constraint between the fields $r^i$ and their potentials $\psi^i$ as in (\[reallag\]) but now using (\[newclebsch\]). However, by imposing the constraint through Miura’s differential substitution $${\cal L}_0^{full} = {\cal L}_1^{modified}( \psi^i_x, \psi^i_{xx}...) + \lambda_i [ u^i - M^i(\psi^i_x,\psi^i_{xx}...) ] \label{reallag2}$$ we obtain a new Lagrangian for the original equations (\[hameq\]) in the original variables $u^i$. We shall use this construction to derive new Lagrangians, in particular for KdV in section \[sec-kdv\]. It is evident that this construction can be extended when there exists multi-Hamiltonian structure but, as we shall find in the example of KdV, sometimes it is possible to arrive at local Lagrangians using non-local Hamiltonian operators as well. Now we conclude [**Theorem 2**]{} [*The first Lagrangian of every modified equation obtained through a Miura transformation will serve as a new zeroth Lagrangian for the original equations of motion provided the constraint between the fields and their potentials is imposed through a Miura-type differential substitution. For $N$-fold Hamiltonian structure there exists $N-1$ such new Lagrangians*]{}. Miura transformation is in general not invertible because it is a differential substitution. But there exists interesting examples where it reduces to a point transformation which is invertible. In that case we can construct $N-1$ further Lagrangians. We conclude that for an evolutionary system that admits $N$ fold first order Hamiltonian structure, the number of different variational principles where the first variation will be an extremum by virtue of the original equations of motion is $2N-1$ and in the case Miura transformation is invertible $3N-2$. We illustrate this situation for the case of bi-Hamiltonian structure in tables \[table1a\] and \[table1b\]. The general situation is much more complicated than what these tables would lead us to expect. Starting with tri-Hamiltonian structure the individual entries in each one of these tables will need to be table by itself because there are inequivalent Hamiltonian operators that yield the same equations of motion with the same Hamiltonian function. We shall discuss this interesting situation in a future publication on the Chaplygin-Born-Infeld equation. [c||c|c|c]{} equations of motion & $u_t = J_2 \delta H_0 $ & $u_t = J_1 \delta H_1$ & $ J_2 J_1^{-1} u_t = - \delta_\phi H_2 $\ \ local Hamiltonian op. & $J_2$ & $J_1$ & no\ \ local Lagrangian & no & ${\cal L}_1$ & ${\cal L}_2$\ \ modified equations & $r_t = \tilde{J}_2 \delta H_0$ & $r_t = \tilde{J}_1 \delta H_1$ &\ \ \[-4.5mm\] local Hamiltonian op. & $\tilde{J}_2 = J_1$ & $\tilde{J}_1$ &\ \ \[-4.5mm\] local Lagrangian & ${\cal L}_0$ & no & [c||c|c|c]{} equations of motion & & $u_t = J_2 \delta H_0 $ & ...\ \ local Hamiltonian op. & & $J_2$ & ...\ \ local Lagrangian & & no & ...\ \ modified equations & $\tilde{J}_1 \tilde{J}_2^{-1} r_t = - \delta_\psi H_{-1} $& $r_t = \tilde{J}_2 \delta H_0$ & ...\ \ \[-4.5mm\] local Hamiltonian op. & no & $\tilde{J}_2 = J_1$ & ...\ \ \[-4.5mm\] local Lagrangian & ${\cal L}_{-1}$ & ${\cal L}_0$ & ... KdV {#sec-kdv} === KdV stands as the symbol of completely integrable systems. We think we know it, but it turns out to be so rich that there is still new information to be learned about it. We recall that KdV $$u_t + 6 u \, u_x - u_{xxx} =0 \label{kdv}$$ admits the Kruskal sequence of conserved Hamiltonian densities $$\begin{aligned} {\cal H}_{-1}^{KdV} &=& u \label{kdvcasimir1} \\ {\cal H}_{0}^{KdV} &=& \frac{1}{2} u^2 \label{kdvcasimir2} \\ {\cal H}_{1}^{KdV} &=& u^3 + \frac{1}{2} u_x^{2} \label{kdvh1} \\ {\cal H}_{2}^{KdV} &=& \frac{5}{2} \, u^{4} + 5 \, u \, u_{x}^{2} + \frac{1}{2} \, u_{xx}^{\;\;2} \label{kdvh3}\\ & ... & \nonumber\end{aligned}$$ which are in involution with respect to Poisson brackets defined by two Hamiltonian operators $$J_1 = D, \qquad J_2 = - D^3 + 2 u D + 2 D u \label{kdvops}$$ that form a Poisson pencil. By introducing the potential $$u = \phi_x$$ KdV can be cast into variational form with two Lagrangians $$\begin{aligned} {\cal L}_1^{KdV} & = & {\cal H}_{-1}^{KdV} \, \phi_t - 2 {\cal H}_1^{KdV} \label{lagkdv}\\ {\cal L}_2^{KdV} & = & ( {\cal H}_{0}^{KdV} + \phi_{xx} ) \phi_t - 2 {\cal H}_2^{KdV} \label{pavlov}\end{aligned}$$ which consist of the classical Lagrangian and the second Lagrangian [@pavlov] respectively.[^1] Here we observe that both (\[lagkdv\]) and (\[pavlov\]) are examples of our general expression (\[genexp\]) for higher Lagrangians. The second application of Lenard’s recursion operator to $J_1$ results in a third Hamiltonian operator which is non-local so we cannot continue to generate higher Lagrangians. But we can use Theorem 2 to generate new lower Lagrangians for KdV. For this purpose we note that in both Lagrangians (\[lagkdv\]) and (\[pavlov\]) we should have added the constraint $ \lambda ( u - \phi_x )$ and written the full Lagrangian. But following the convenient abuse of language we did not do so because it was manifest. It is, however, necessary to write the full Lagrangian in the case of lower Lagrangians. According to our general construction of lower Lagrangians we first recall the original Miura transformation $$u = r^2 + r_{x} \label{miura}$$ that brings $J_2$ to the canonical form of $J_1$ in the variable $r$. The equation of motion for $r$ is mKdV which is different from (\[kdv\]) but under the substitution (\[miura\]) we have Miura’s result $$u_t + 6 u \, u_x - u_{xxx} = (D + 2 r ) \left( r_t + 6 r^2 r_x - r_{xxx} \right) =0 \label{miura2}$$ so that, on shell, if mKdV is satisfied then so is KdV. Now we can introduce the Clebsch potential for the modified field variable $$r = \psi_x$$ and write the first Lagrangian for mKdV $${\cal L}_1^{mKdV} = \psi_x \psi_t + {\cal H}_0^{KdV}\Big|_{u=\psi_x^2+\psi_{xx} } \label{mkdvlag}$$ in a straight-forward manner. But now enforcing the constraint in the full Lagrangian through the Miura transformation $${\cal L}_{0}^{KdV \; full} = {\cal L}_1^{mKdV} + \lambda ( u - \psi_x^2 - \psi_{xx} ) \label{kdv0}$$ we shall arrive at a new Lagrangian for KdV because the Euler equation that comes from the first variation of this action will be satisfied by virtue of (\[miura2\]). Unlike (\[pavlov\]) which is a higher Lagrangian, (\[kdv0\]) is a lower Lagrangian in the sense of the action of the recursion operator on the equations of motion in the resulting Euler equation. And the saga of KdV continues! We consider the third Hamiltonian operator for KdV $$J_3 = R^2 J_1 \label{j3kdv}$$ which is nonlocal but the relationship between differential substitutions and Hamiltonian structures of KdV [@max7] enables us to construct another new local Lagrangian for KdV. For this purpose we recall that the differential substitution $$r = \alpha q + \frac{\varepsilon}{q} + \frac{q_x}{2 q}$$ which transforms mKdV into twice modified KdV $$q_t = \left( q_{xx} - \frac{3 q_x^2}{2 q} + \frac{6 \varepsilon^2}{q} - 2 \alpha^2 q^3 \right)_x$$ is a Miura transformation for (\[j3kdv\]). This can best be seen by the expression $$J_3 = \frac{1}{2}( q^2 D + D q^2 ) - q_x D^{-1} q_x$$ for the third non-local Hamiltonian operator for KdV in terms of twice modified variable $q$. We recall that $J_3$ is fifth order in $u$. We have the Miura relation $$\begin{aligned} r_t + 6 r^2 r_x - r_{xxx} & = & \left( \alpha - \frac{\varepsilon}{q^2} - \frac{q_x}{2 q^2} + \frac{1}{q} D \right) \nonumber \\ && \left[ q_t - \left( q_{xx} - \frac{3 q_x^2}{2 q} + \frac{6 \varepsilon^2}{q} - 2 \alpha^2 q^3 \right)_x \right] \label{miura3}\\ &=& 0 \nonumber\end{aligned}$$ between modified and twice modified KdV’s. Introducing the potential for the twice modified variable $q=\chi_x$ we have $$u = \Phi(\chi_x,\chi_{xx},\chi_{xxx}) \equiv \frac{\chi_{xxx}}{\chi_x} - \frac{\chi_{xx}^{\;2}}{\chi_x^2} + 2 \alpha \chi_{xx} + \alpha^2 \chi_{x}^2 + 2 \alpha \varepsilon + \frac{\varepsilon^{2}}{\chi_x^2} \label{uq}$$ in terms of the original field $u$. The first Lagrangian for twice modified KdV is simply $${\cal L}_{1}^{m_2KdV} = \chi_{x} \, \chi_{t} + {\cal H}_{-1}^{KdV} \Big|_{ u = \Phi(\chi_{x}, \chi_{xx},\chi_{xxx} ) } \label{mmkdvlag}$$ and therefore the second lower Lagrangian for KdV is given by $${\cal L}_{-1}^{KdV \; full} = {\cal L}_1^{m_2KdV} + \lambda [ u - \Phi(\chi_{x}, \chi_{xx}, \chi_{xxx} ) ] \label{kdvlagm2}$$ which provides another illustration of (\[reallag2\]). This process can be continued. We note that an alternative to the Clebsch potential for KdV is the Schwartzian which was pointed out by Schiff [@sch]. We shall postpone consideration of Schwartzian potentials to future work. Polytropic gas dynamics {#sec-gas} ======================= The simplest examples for applying our construction of multi-Lagrangians consist of quasi-linear second order hyperbolic equations that Dubrovin and Novikov [@dn] have called equations of hydrodynamic type. The distinguished example in this set consists of the Eulerian equations of polytropic gas dynamics in $1+1$ dimensions $$\begin{aligned} \rho _{t} + u \, \rho _{x} + \rho \, u_{x} &=&0 \label{GD1} \\ u_{t}+ u \, u_{x}+ \rho^{\gamma-2} \rho _{x} &=&0 \nonumber\end{aligned}$$ and in particular for $\gamma=-1$ we have the case of Chaplygin gas, or Born-Infeld equation that was recently shown to have a string theory antecedent [@jackiw2]. This system can be cast into quadri-Hamiltonian form [@gn1]. For the Chaplygin-Born-Infeld case the complete Hamiltonian structure can be found in [@annov] and its symmetries were given in [@hor]. In the following we shall use the labelling $u^1 = \rho$ and $u^2=u$. First we have three local Hamiltonian structures of first order [@n1] $$J_1 = \left( \begin{array}{cc} 0 & D \\ D & 0 \end{array} \right) = \sigma^1 D, \label{j1}$$ $$J_2 = \left( \begin{array}{cc} \rho \, D + D \, \rho & (\gamma-2)\, D \, u + u \, D \\ D \, u + (\gamma-2)\, u \, D & \rho^{\gamma-2} D + D \, \rho^{\gamma-2} \end{array} \right), \label{j2}$$ $$J_3 = \left( \begin{array}{cc} u \, \rho \, D + D \, u \, \rho & \begin{array}{c} D \left[ \frac{1}{2} (\gamma-2) u^2 + \frac{1}{\gamma-1} \rho^{\gamma-1} \right] \\ + \left[ \frac{1}{2} u^2 + \frac{1}{\gamma-1} \rho^{\gamma-1} \right] D \end{array} \\ \begin{array}{c} D \left[ \frac{1}{2} u^2 + \frac{1}{\gamma-1} \rho^{\gamma-1} \right] \\ + \left[ \frac{1}{2} (\gamma-2) u^2 + \frac{1}{\gamma-1} \rho^{\gamma-1} \right] D \end{array} & u \, \rho^{\gamma-2} \, D + D \, u \, \rho^{\gamma-2} \end{array} \right) \label{j3}$$ which form a Poisson pencil ${\cal J} = J_1 + c_1 J_2 + c_2 J_3$ with $c_1, c_2$ constants, [*i.e.*]{} these Hamiltonian operators are compatible. In eq.(\[j1\]) $\sigma^1$ is the Pauli matrix and this is the canonical Darboux form of first order Hamiltonian operators. The equations of polytropic gas dynamics admit two infinite hierarchies of conserved Hamiltonians which are in involution with respect to Poisson brackets defined by all three of these Hamiltonian operators. In the first set, which is called Eulerian [@gn1], the Hamiltonian densities are given by $$\begin{aligned} {\cal H}_{-1}^E &=& \rho \label{casimir1} \\ {\cal H}_{0}^E &=& u\, \rho \label{momentum} \\ {\cal H}_{1}^E &=& \frac{1}{2} u^{2} \rho +\frac{1}{\gamma (\gamma -1)} \rho^{\gamma} \label{hamiltonian} \\ {\cal H}_{2}^E &=& \frac{1}{6} u^{3} \rho + \frac{1}{\gamma (\gamma-1)} u\, \rho^{\gamma} \label{h5}\\ {\cal H}_{3}^E &=& \frac{1}{24} u^{4} \rho +\frac{1}{2 \gamma (\gamma -1)} u^{2} \rho^{\gamma } +\frac{1}{2 \gamma (\gamma -1)^{2} (2\gamma -1)} \rho^{2\gamma -1} \label{h6}\\ & ... & \nonumber\end{aligned}$$ where (\[momentum\]) is the momentum, (\[hamiltonian\]) is the familiar Hamiltonian function, the Casimir is in (\[casimir1\]) and the rest consist of higher Hamiltonians. Therefore, the Euler series is the distinguished one in the terminology of section \[sec-main\]. The second series $$\begin{aligned} {\cal H}_{-1}^L &=& u \label{casimir2} \\ {\cal H}_{0}^L &=& \frac{1}{2} (\gamma-2) u^{2} +\frac{1}{\gamma -1} \rho^{\gamma-1} \label{h1l} \\ {\cal H}_{1}^L &=& \frac{1}{6} (\gamma-2) u^{3} + \frac{1}{\gamma-1} u\, \rho ^{\gamma-1} \label{h2l}\\ {\cal H}_{2}^L &=& \frac{1}{24} (\gamma-2) u^{4} +\frac{1}{4 (\gamma -1)} u^2 \rho^{\gamma -1} +\frac{1}{2 (\gamma -1)^2 (2 \gamma - 3) } \rho^{2 (\gamma -1)} \label{h3l}\\ & ... & \nonumber\end{aligned}$$ is the Lagrangian series which starts with the Casimir (\[casimir2\]). Note that for $\gamma=2$ this series is no longer polynomial as logarithms will enter and the same remark holds for integer and half-integer values of $\gamma$ in both series. Finally, we note that the recursion operator $R_{2}^{\;1} = J_2 J_1^{-1}$ can be used to write infinitely many Hamiltonian operators by letting it to act $n$ times on $ J_1 $. However, in general none of these operators will be local. In particular we note that $$R_{3}^{\;1} = J_3 \, (J_1)^{-1} \ne ( R_{2}^{\;1})^2, \qquad J_3 \ne J_2 J_1^{-1} J_2$$ except in the case of shallow water waves where $\gamma=2$ which admits extension to integrable dispersive equations. Next, there is a third order Hamiltonian operator [@on] which was obtained from Sheftel’s remarkable recursion operator [@sheftel] $$J_4 = D U_x^{-1} \, D U_x^{-1} \, \sigma^1 D \label{sheftel}$$ where $$U = \left( \begin{array}{cc} u & \rho \\ \frac{1}{\gamma-2} \rho^{\gamma-2} & u \end{array} \right) \label{sheftelu}$$ which is only compatible with $J_0$. Higher conserved Hamiltonians start with the density [@sheftel], [@verosky] $$\hat {\cal H}^{SV(E)}_{-1} = \frac{\rho_x}{ u_x^{\;2} - \rho^{\gamma - 3} \rho_x^{\;2}} \label{verosky}$$ which is part of the Eulerian series. There is also a Lagrangian series starting with $$\hat {\cal H}^{SV(L)}_{-1} = - \frac{u_x}{ u_x^{\;2} - \rho^{\gamma - 3} \rho_x^{\;2}} \label{verosky2}$$ and both form new infinite hierarchies of conservation laws. We will be interested in the Lagrangian formulation of the equations of polytropic gas dynamics (\[GD1\]) that correspond to all these Hamiltonian structures. Introducing the Clebsch potentials [@n3] $$u=\varphi_{x}, \qquad \rho =\psi_{x} \label{potentials}$$ we have the first Lagrangian representation for this system $${\cal L}_1^{\gamma} = {\cal H}_{-1}^L \psi_{t} + {\cal H}_{-1}^E \varphi_{t} - 2 {\cal H}_{1}^E (\varphi _{x},\psi_{x}) \label{lag1}$$ but using the recursion operators $J_2 \, J_1^{-1}$ and $J_3 \, J_1^{-1}$ we find two further Lagrangians $$\begin{aligned} {\cal L}_{2}^{\gamma} & = & {\cal H}_{0}^L \psi _{t} + {\cal H}_{0}^E \varphi_{t} - 2 {\cal H}_{2}^E (\varphi _{x},\psi_{x}) \label{lag2} \\ {\cal L}_{3}^{\gamma} & = & {\cal H}_{1}^L \psi _{t} + {\cal H}_{1}^E \varphi_{t} - 2 {\cal H}_{3}^E (\varphi _{x},\psi_{x}) \label{lag3}\end{aligned}$$ which are local functionals of the Clebsch potentials. The Lagrangian obtained through the action of the recursion operator $J_4 \, J_1^{-1}$ is the most interesting one. Because $J_4$ is a third order operator, the fourth Lagrangian $$\begin{aligned} {\cal L}_{4}^{\gamma} & = & {\cal H}^{SV(E)}_{-1} u_{t} + {\cal H}^{SV(L)}_{-1} \rho_{t} - 2 {\cal H}_{-1}^E (\varphi _{x},\psi_{x}) \label{lag4} \\ {\cal L}_{4}^{\gamma}& = & \frac{\rho_{x} u_{t} - u_{x} \rho _{t}} {u_{x}^{2}-\rho ^{\gamma -3}\rho_{x}^{2}}- 2 \, \rho \label{p}\end{aligned}$$ is [*local in the velocity fields*]{}. This is a general property of bi-Hamiltonian structure with a pair of first and third order Hamiltonian operators. Here we find a remarkable situation in that the number of Lagrangians that we can construct by repeated application of Sheftel’s recursion operator $J_4 J_1^{-1}$ is [*infinite*]{} in number. All of these Lagrangians will be [*local*]{} in the original field variables $\rho$ and $u$. Now we come to lower Lagrangians that will arise from Miura transformations. The Miura transformations that bring the Hamiltonian operators (\[j2\]) and (\[j3\]) to the Darboux form of (\[j1\]) are point transformations for equations of hydrodynamic type. Dubrovin and Novikov had pointed out that first order Hamiltonian operators for equations of hydrodynamic type are given by $$J^{ik} = g^{ik} \, D - g^{im} \, \Gamma^k_{mn} u^n_x \label{dnop}$$ where $g_{ik}$ are the components of a Riemannian metric which is flat by virtue of the Jacobi identities. The Miura transformation provides manifestly flat coordinates for this metric. For example from (\[j2\]) we find the flat metric $$d s_2^2 = \frac{2}{4 \rho^{\gamma-1} - (\gamma-1)^2 u^2} \left[ \rho^{\gamma-2} d \rho^2 - (\gamma-1) u \, d \rho \, d u + \rho \, d u^2 \right] \label{metrics}$$ and it can be verified that the Miura transformation $$\rho = r \, p \qquad u = \frac{1}{\gamma-1} \left( r^{\gamma-1} + p^{\gamma-1} \right)$$ brings it into the manifestly flat form $ 2 d r \, d p$. In these variables we find the first modified equations of gas dynamics $$\begin{aligned} r_t + \frac{\gamma}{\gamma-1} (r^{\gamma-1} + p^{\gamma-1}) r_x + \gamma r \, p^{\gamma-2} p_x =0 \label{modgas1} \\ p_t + \gamma p \, r^{\gamma-2} r_x+ \frac{\gamma}{\gamma-1} (r^{\gamma-1} + p^{\gamma-1}) p_x =0 \nonumber\end{aligned}$$ and linear combinations of these equations with variable coefficients give eqs.(\[GD1\]) of gas dynamics. Introducing the potentials $$r = \chi_x , \qquad p = \upsilon_x$$ we have the Lagrangian $${\cal L}_{0}^{\gamma full} = \chi_{x} \upsilon_{t} +\upsilon_{x} \chi_{t} - 2 {\cal H}_0^E + \lambda \left( u - \frac{ \chi_x^{\gamma-1} + \upsilon_x^{\gamma-1}}{\gamma-1} \right) + \sigma \left( \rho - \chi_x \upsilon_x \right) \label{lag0gd}$$ where ${\cal H}_0^E$ is the momentum (\[momentum\]) expressed in terms of the potentials $\chi$ and $\upsilon$. Transforming to the first modified variables $r, p$ we get $\tilde{J}_1, \tilde{J}_2= J_1$ and $\tilde{J}_3$ defining the tri-Hamiltonian structure of eqs.(\[modgas1\]). Now there is a new lower Lagrangian that we can construct from the recursion operator $\tilde{J}_1 \tilde{J}_2^{-1}$. We find $$\tilde{J}_1 = \left( \begin{array}{cc} (1-\gamma) \left[ r p^{\gamma-2} \Delta D + D r p^{\gamma-2} \Delta \right] & \begin{array}{c} \left[ (\gamma-2) r^{\gamma-1} + p^{\gamma-1} \right] \Delta D \\ + D \left[ r^{\gamma-1} + (\gamma-2) p^{\gamma-1} \right] \Delta \end{array} \\ \begin{array}{c} \left[ r^{\gamma-1} + (\gamma-2) p^{\gamma-1} \right] \Delta D\\ + D \left[ (\gamma-2) r^{\gamma-1} + p^{\gamma-1} \right] \Delta \end{array} & (1 - \gamma ) \left[ p r^{\gamma-2} \Delta D + D p r^{\gamma-2} \Delta\right] \end{array} \right), \label{jr}$$ $$\Delta \equiv \frac{1}{(\gamma-1) (r^{\gamma-1} - p^{\gamma-1})^{2} }$$ where the labelling of the variables is in the order $r$ and $p$. The new Lagrangian is given by $${\cal L}_{-1}^{\gamma full} = \frac{\chi_x \upsilon_{t} - \upsilon_{x} \chi_t}{ \chi_x^{\gamma-1} - \upsilon_x^{\gamma -1}} - {\cal H}_{-1}^E + \lambda \left( u - \frac{ \chi_x^{\gamma-1} + \upsilon_x^{\gamma-1}}{\gamma-1} \right) + \sigma \left( \rho - \chi_x \upsilon_x \right) \label{lowlaggas}$$ where the momenta do not belong to the polynomial series of conserved Hamiltonians. However, we can identify the lower momenta from this Lagrangian $$\begin{aligned} {\cal H}_{-2}^{\gamma \, \pm} = \xi_{\pm}^{\frac{3-\gamma}{\gamma-1}} \; ( \xi_+ \xi_- )^{-1/2} , \nonumber \\[2mm] \xi^2+ (\gamma-1) u \, \xi + \rho^{\gamma-1} = 0 \nonumber\end{aligned}$$ where $\pm$ refers to Eulerian and Lagrangian series as well as the roots of the quadratic equation. We now turn to the third Hamiltonian structure (\[j3\]) defined by the flat metric $$\begin{aligned} d s_3^2 & = & - \frac{8 (\gamma-1)^2 }{[(\gamma-1)^2 u^{2} -4 \rho ^{\gamma -1}]^{2}} \Big\{ u \rho^{\gamma-2} d \rho ^{2} \nonumber \\ && - \frac{1}{2 (\gamma -1) } \left[ (\gamma-1)^2 u^{2}+ 4 \rho ^{\gamma -1} \right] d \rho \, d u + u \rho \, du^{2} \Big\}\\ &=& 2 d q \, d w \nonumber\end{aligned}$$ and the coordinate transformation that brings it to the manifestly flat form is given by $$\begin{aligned} q & = & \left[ (\gamma-1)^2 u^2 - 4 \rho^{\gamma -1} \right]^{\frac{\gamma -3}{2(1-\gamma)}} \label{c1} \\ w&=& \int^z \frac{1}{\sqrt{1+\xi^2} } \, \xi^{\frac{\gamma -3}{1-\gamma}} \; d \xi \label{inte} \\ z&=& \sinh \left\{ \frac{1}{2} \ln \frac{ (\gamma -1) u + 2 \rho^{(\gamma-1)/2} }{ (\gamma -1) u - 2 \rho^{(\gamma-1)/2}} \right\} \nonumber\end{aligned}$$ where, in general, the last integral cannot be done in closed form. For some specific values of $\gamma$ the integral (\[inte\]) is elementary as in the notable case of Chaplygin-Born-Infeld. But this paper is devoted to the general case of polytropic gas dynamics and we shall not consider inverting (\[c1\]), (\[inte\]) to obtain $u, \rho$ as functions of $q$ and $w$. We shall only remark that after this inversion we can obtain two more new Lagrangians. The Lagrangians (\[lag1\]), (\[lag2\]) and (\[lag3\]) for polytropic gas dynamics are examples illustrating the general expression (\[genexp\]) for higher Lagrangians. For equations of hydrodynamic type there is no dispersion and hence ${\cal G}$ vanishes identically. We have given only two (\[lag0gd\]), (\[lowlaggas\]) of the four lower Lagrangians because the integral (\[inte\]) must be carried out before we arrive at the second modified equations of gas dynamics which will lead to two further new Lagrangians. Certainly the Lagrangian (\[p\]) which is derived from bi-Hamiltonian structure with a first and third order operators according to (\[genexp\]) is the most remarkable one because this is the first time it has been possible to write a Lagrangian for polytropic gas dynamics that is local in the original field variables, namely the density and velocity. Furthermore it is only the first element in an infinite series of such Lagrangians. Kaup-Boussinesq system ====================== Gas dynamics with $\gamma = 2$ governs the behavior of long waves in shallow water. From the point of view of complete integrability it is a remarkable case, because in this case we find several completely integrable dispersive generalizations of eqs.(\[GD1\]). Most prominent among them is the well-known Kaup-Boussinesq system [@kaup1] $$u_{t} = \left(\frac{u^{2}}{2}+\rho \right)_x \qquad \rho_{t} = \left(u\rho +\varepsilon ^{2}u_{xx} \right)_x \label{kbous}$$ which admits tri-Hamiltonian structure. The first Hamiltonian structure is given by the Hamiltonian operator (\[j1\]) and $$J_{2}^{KBq} = \left( \begin{array}{cc} D & \frac{1}{2} \, D \, u \\[2mm] \frac{1}{2} \, u D & \frac{1}{2} ( \rho \, D + D \, \rho ) + \varepsilon^{2} D^{3} \end{array} \right) \label{j2kbq}$$ where $D^{-1}$ denotes the principal value integral, is the second Hamiltonian operator for the Kaup-Boussinesq system. In the limit $\varepsilon \rightarrow 0$ this Hamiltonian operator reduces to (\[j2\]) with $\gamma=2$. The recursion operator is given by $$R^{1 \; K Bq}_{2} =\left( \begin{array}{cc} \frac{1}{2}u+\frac{1}{2}u_{x} D^{-1} & 1 \\ \varepsilon ^{2} D^{2}+\rho +\frac{1}{2}\rho _{x} D^{-1} & \frac{1}{2}u \end{array} \right)$$ and there is a third local Hamiltonian operator obtained by the action of the recursion operator $J_{2}^{KBq}= (R^{1 \; K Bq}_{2})^2 J_0$ as in the $\gamma=2$ case of gas dynamics. The conserved Hamiltonians in the Eulerian and Lagrangian series are $$\begin{aligned} {\cal H}_{-1}^{KBq} & = & \rho \label{hkb1} \\ {\cal H}_0^{KBq} & = & u \, \rho \label{hkb2} \\ {\cal H}_{1}^{KBq} &=& \frac{1}{2} \left( \rho u^{2}+\rho ^{2} + \varepsilon ^{2} u \, u_{xx} \right) \label{hkb3} \\ {\cal H}_{2}^{KBq} &=&\frac{1}{2}\left[ \rho u^{3}+3\rho ^{2}u-\varepsilon ^{2}(4u_{x}\rho _{x}+3uu_{x}^{2} ) \right] \label{hkb5} \\ H_{3}^{KBq}&=&\frac{1}{4}u^{4}\rho +\frac{3}{2}u^{2}\rho ^{2}+\frac{1}{2}\rho ^{3}+\varepsilon ^{4}u_{xx}^{2} \label{hkb6} \\ && -\varepsilon ^{2}(\frac{5}{2}\rho u_{x}^{2}+4uu_{x}\rho _{x}+\rho _{x}^{2}+\frac{3}{2}u^{2}u_{x}^{2}) \nonumber \\ & ... & \nonumber\end{aligned}$$ and the degeneracy in the $\gamma = 2$ case of gas dynamics is repeated in its dispersive generalization. In particular, the Lagrangian and Eulerian series coincide apart from a relabelling $$\begin{aligned} {\cal H}_{-2}^{KBq(E)} = & u & = {\cal H}_{-1}^{KBq(L)} \nonumber \\ {\cal H}_{-1}^{KBq(E)} = & \rho & = {\cal H}_{0}^{KBq(L)} \label{degen} \\ & ... & \nonumber \\ {\cal H}_{-2+n}^{KBq(E)} & =& {\cal H}_{-1+n}^{KBq(L)} \nonumber\end{aligned}$$ that is dictated by the recursion operator. With the aid of the Clebsch potentials $$u = \varphi_x, \qquad \rho = \psi_x \label{potKBq1}$$ we obtain $${\cal L}_{1}^{KBq} = {\cal H}_{-1}^{KBq} \varphi_{t} + {\cal H}_{-2}^{KBq} \psi_{t} - 2 {\cal H}_{1}^{KBq} (\varphi _{x},\psi _{x},\varphi _{xx},\psi _{xx},...) \label{lagKBq0}$$ for the first Lagrangian. Using the technique we have presented in section \[sec-main\] we shall now construct higher Lagrangians. These three local Hamiltonian structures enable us to construct two new Lagrangians $${\cal L}_{2}^{KBq} =( {\cal H}_{0}^{KBq} +\varepsilon ^{2}\varphi _{xxx})\varphi _{t}+ {\cal H}_{-1}^{KBq} \psi _{t} - 2 {\cal H}_{2}^{KBq}(\varphi _{x},\psi _{x},\varphi _{xx},\psi _{xx},...) \label{lagKBq1}$$ and $$\begin{aligned} {\cal L}_{3}^{KBq} & = & \left[ {\cal H}_{1}^{KBq} +\varepsilon ^{2} \left( 2 \psi_{xxx}+\varphi _{xx}^{2} + \varphi _{x}\varphi _{xxx} \right) \right] \varphi_{t} \nonumber \\ &&+ \left( {\cal H}_{0}^{KBq} +\varepsilon^{2}\varphi_{xxx} \right)\psi_{t} -2 {\cal H}_{3}^{KBq}(\varphi_{x},\psi _{x},...) \label{lagKBq2}\end{aligned}$$ for the Kaup-Boussinesq system. The determination of ${\cal G}_{\beta ; [i]}$ is according to eq.(\[genexp\]) with $\beta= 2, 3$ and $[2] = [1] -1$ because of the relabelling difference (\[degen\]) between the Lagrangian and Eulerian series. Note that the momentum map which is the coefficient of $\phi_t$ in (\[lagKBq1\]) is exactly the same as the momentum in front of $\psi_t$ in (\[lagKBq2\]). The reason for this goes back to the degeneration of the Eulerian and Lagrangian series into one and the fact that it is the momentum map that is the important element in the general construction (\[genexp\]). In the dispersionless limit the Lagrangians (\[lagKBq0\]), (\[lagKBq1\]), (\[lagKBq2\]) reduce to the gas dynamics Lagrangians (\[lag1\]), (\[lag2\]) and (\[lag3\]) with $\gamma=2$. Kaup-Broer System ================= There is another completely integrable dispersive version of the $\gamma=2$ case of gas dynamics which is the Kaup-Broer system [@kaup1], [@broer]. The triangular invertible differential substitution $$\rho =\eta +\varepsilon u_{x} \label{kbqtokbr}$$ transforms the Kaup-Boussinesq system (\[kbous\]) into the Kaup-Broer system $$\begin{aligned} u_{t} &= & u\, u_{x} +\eta_x +\varepsilon u_{xx} \nonumber \\ \eta _{t} &=& \left( \eta u \right)_x -\varepsilon \eta _{xx} \label{kbroer}\end{aligned}$$ which also has three local Hamiltonian structures [@kuper]. For the Kaup-Broer system the conserved Hamiltonians in the Eulerian series are given by $$\begin{aligned} {\cal H}_{0}^{KBr} &=&u\eta \\ {\cal H}_{1}^{KBr} &=& \frac{1}{2}[u^{2}\eta +\eta ^{2}-2\varepsilon \eta u_{x}]\\ {\cal H}_{2}^{KBr}&=&\frac{1}{2} [u^{3}\eta +3u\eta ^{2}+6\varepsilon \eta uu_{x}-4\varepsilon ^{2}u_{x}\eta _{x}], \\ {\cal H}_{3}^{KBr} &=&\frac{1}{4}u^{4}\eta +\frac{3}{2}u^{2}\eta ^{2}+\frac{1}{2} \eta ^{3}+\varepsilon (\frac{3}{2}\eta ^{2}u_{x}-u^{3}\eta _{x}) \\ && +\varepsilon ^{2}(2u^{2}\eta _{xx}-\eta u_{x}^{2}-\eta _{x}^{2})-2\varepsilon ^{3}\eta _{x}u_{xx} \nonumber\end{aligned}$$ which can be obtained from (\[hkb2\])-(\[hkb5\]) through the substitution (\[kbqtokbr\]). The first Hamiltonian operator for the Kaup-Broer system is given by (\[j1\]) and the second Hamiltonian operator $$J_1^{KBr} = \left( \begin{array}{cc} D & \frac{1}{2} \, D \, u + \varepsilon D^{2} \\ \frac{1}{2} \, u\, D -\varepsilon D^{2} & \frac{1}{2} ( \eta \, D + D \, \eta ) \end{array} \right) \label{j2kbr}$$ can be obtained from (\[j2kbq\]) of the Kaup-Boussinesq system using the substitution (\[kbqtokbr\]). For Kaup-Broer system we introduce the potentials $$\eta = w_{x}, \qquad \psi = w + \varepsilon \varphi_{x} \label{potKBr}$$ and arrive at the first Lagrangian $${\cal L}_{1}^{KBr} = {\cal H}_{-1}^{KBq} \varphi_{t} + {\cal H}_{-2}^{KBr} w_{t} - 2 {\cal H}_{1}^{KBr}(w_{x},\varphi _{x},w_{xx},\varphi _{xx},...) \label{lagKBr1}$$ but now we can derive two further Lagrangians using the recursion operator obtained from the Hamiltonian operators (\[j2kbr\]) and (\[j1\]). Following our procedure of section \[sec-main\] we find the second Lagrangian $${\cal L}_{2}^{KBr} = ( {\cal H}_{0}^{KBr} -2\varepsilon w_{xx})\varphi_{t} + {\cal H}_{-1}^{KBr} w_{t} -2 {\cal H}_{2}^{KBr}(w_{x},\varphi _{x},w_{xx},\varphi _{xx},...) \label{lagKBr2}$$ which is the same as the Lagrangian of Kaup-Boussinesq system (\[lagKBq2\]) subject to the differential substitution (\[kbqtokbr\]). Similarly we find $$\begin{aligned} {\cal L}_{3}^{KBr} & = & ({\cal H}_{1}^{KBr} +2\varepsilon ^{2}w_{xxx}) \varphi _{t} \label{bk} \\ && + ( {\cal H}_{0}^{KBr}+\varepsilon \varphi_{x}\varphi_{xx} +\varepsilon ^{2} \varphi_{xxx})w_{t} -2 {\cal H}_{3}^{KBr}(\varphi _{x},w_{x},...) \nonumber\end{aligned}$$ as the third Lagrangian for the Kaup-Broer equations (\[kbroer\]). As in the case of Kaup-Boussinesq, these Lagrangians reduce to $\gamma=2$ gas dynamics Lagrangians in the dispersionless limit. In the Kaup-Broer Lagrangians we find another example of the general formula (\[genexp\]) for Lagrangians. Nonlinear Shrödinger equation ============================= We shall consider the nonlinear Shrödinger equation in the $2$-component real version $$\begin{aligned} \upsilon _{t} & = & \left[\frac{\upsilon ^{2}}{2} +\eta +\varepsilon ^{2}(\frac{\eta _{xx}}{\eta } -\frac{\eta_{x}^{2}}{2\eta ^{2}}) \right]_x \nonumber \\ \eta_{t} & = & (\eta \upsilon )_{x}, \label{r2nls}\end{aligned}$$ which is a reaction-diffusion system. Again this reduces to the $\gamma=2$ case of gas dynamics in the dispersionless limit. This version of NLS can be obtained by another triangular differential substitution $$u = \upsilon +\varepsilon \eta _{x}/\eta \label{kbrtonls}$$ from the Kaup-Broer system. NLS has the same first local Hamiltonian structure (\[j1\]) as in the case of Kaup-Boussinesq or Kaup-Broer systems. Once again the second Hamiltonian operator for NLS can be found by the transformation (\[kbrtonls\]) from the second Hamiltonian operator (\[j2kbr\]) of the Kaup-Broer system. Thus for the $2$-component real version of NLS the second Hamiltonian operator is given by $$J^{NLS}_2 = \left( \begin{array}{cc} D + \varepsilon^{2} \left\{ \begin{array}{c} \eta^{-1} \, D^3 + D^3 \, \eta^{-1} \\ -\frac{1}{2} \left[ (\eta^{-1})_{xx} \, D + D \, ( \eta^{-1})_{xx} \right] \end{array} \right\} & \frac{1}{2} \, D \, \upsilon \\ \frac{1}{2} \, \upsilon D &\frac{1}{2} ( \eta \, D + D \, \eta ) \end{array} \right) \label{j2nls}$$ and the conserved Hamiltonians are $$\begin{aligned} {\cal H}_{-2}^{NLS} &=& \upsilon \\ {\cal H}_{-1}^{NLS} &=& \eta \\ {\cal H}_{0}^{NLS} &=& \upsilon \eta \\ {\cal H}_{1}^{NLS} &=& \frac{1}{2} \left( \eta \upsilon^2 + \eta^2 - \varepsilon^{2} \frac{\eta_x^2}{\eta} \right)\\ {\cal H}_{2}^{NLS} &=& \frac{1}{2} \left[ \eta \upsilon^2 + 3 \upsilon \eta^2 + \varepsilon^{2} \left( \upsilon_x \eta_x - 3 \frac{\upsilon \eta_x^2}{\eta} \right) \right] \\ {\cal H}_{3}^{NLS} &=&\frac{3}{4} \eta^2 \upsilon^2 + \frac{1}{4} \eta^3 + \frac{1}{8} \upsilon^4 \eta + \varepsilon^{4} \left( \frac{\eta_{xx}^{\;\;2}}{2 \eta} - \frac{ 5 \eta_x^{\;4} }{ 24 \eta^3 } \right) \\ & & + \varepsilon^{2} \left( \upsilon^2 \eta_{xx} - \frac{5}{4} \eta_{x}^{\;2} - \frac{3}{4} \frac{\eta_{x}^{\;2}}{\eta} \upsilon^2 - \frac{1}{2} \upsilon_{x}^{\;2} \eta \right) \nonumber \\ &...& \nonumber\end{aligned}$$ which forms an infinite sequence combining both Eulerian and Lagrangian series according to (\[degen\]). In order to construct the Lagrangians for NLS we introduce the potentials $$\begin{aligned} \upsilon & = & z_{x} \nonumber \\ z & = & \varphi -\varepsilon \ln w_{x} \label{potup}\end{aligned}$$ and the first Lagrangian $${\cal L}^{NLS}_{1}= {\cal H}^{NLS}_{-1} z_{t}+ {\cal H}^{NLS}_{-2} w_{t} - 2 {\cal H}_{1}^{NLS}(w_{x},z_{x},...)$$ is the classical result. Once again we shall use the techniques of section \[sec-main\] to construct higher Lagrangians with the recursion operator obtained from (\[j2nls\]) and (\[j1\]). We obtain two higher Lagrangians for NLS $${\cal L}^{NLS}_{2}= {\cal H}^{NLS}_{0} z_{t}+ \left[ {\cal H}^{NLS}_{-1} +\varepsilon ^{2} \left(\frac{w_{xxx}}{w_{x}} -\frac{w_{xx}^{2}}{w_{x}^{2}} \right)\right] w_{t} -2 {\cal H}_{2}^{NLS}(w_{x},z_{x},...)$$ and $$\begin{aligned} {\cal L}^{NLS}_{3}&=& \left\{{\cal H}^{NLS}_{0}+\varepsilon^{2} \left[ z_{xxx} + \left(\frac{z_{x}w_{xx}}{w_{x}} \right)_{x} \right] \right\} w_{t} \\ &&+ \left( {\cal H}^{NLS}_{1} + 2 \varepsilon ^{2} w_{xxx} \right) z_{t} -2 {\cal H}_{3}^{NLS}(z_{x},w_{x},...). \nonumber\end{aligned}$$ that are local functionals of the potentials. Here again, in the dispersionless limit we find the $\gamma=2$ gas dynamics Lagrangians. The remarkable strength of the general expression (\[genexp\]) for new Lagrangians is manifest. Boussinesq Equation =================== In order to discuss the bi-Hamiltonian structure and the Lagrangians for the Boussinesq equation in a unified framework we first turn to its dispersionless limit. For polytropic gas dynamics we had $$\rho _{t} = (\rho u)_{x}, \qquad u_{t} = \left(\frac{u^{2}}{2}+\frac{\rho ^{\gamma -1}}{\gamma -1}\right)_{x}$$ with its first nontrivial commuting flow $$\rho_{y} = u_{x}, \qquad u_{y} = \left(\frac{\rho ^{\gamma -2}}{\gamma -2}\right)_{x}$$ both of which reduce to a second order quasi-linear wave equation [@gn1]. If we express Boussinesq equation in the form $$\rho_{yy} - \left( \frac{1}{2} \rho^2 - \varepsilon^2 \rho_{xx} \right)_{xx} = 0 \label{boussinesq}$$ or $$\rho_{y}=u_{x}, \qquad u_{y}= \left(\frac{\rho ^{2}}{2} - \varepsilon^2 \rho_{xx}\right)_x \label{b1}$$ as a first order evolutionary system and compare its dispersionless limit to polytropic gas dynamics, we find that it corresponds to the commuting flow for $\gamma =4$. The completely integrable dispersive equation $$\begin{aligned} \rho _{t} & = & \left[\rho u-2\varepsilon ^{2}u_{xx} \right]_{x}, \label{commb} \\ u_{t}& = & \left[\frac{u^{2}}{2}+\frac{1}{3}\rho ^{3}-\frac{3}{2} \varepsilon ^{2}(2\rho \rho _{xx}+\rho _{x}^{2})+2\varepsilon ^{4}\rho_{xxxx}\right]_{x} \nonumber\end{aligned}$$ is the commuting flow to the Boussinesq equation. This system admits bi-Hamiltonian structure [@olver] with the Hamiltonian operators (\[j1\]) and $$J^{B}_{2} = \left( \begin{array}{cc} \rho D + D \rho - 8 \varepsilon^{2} D^{3} & 3 u \, D + 2 u_{x} \\[4mm] 3 D u - 2 u_{x} & \begin{array}{c} 8 ( \rho ^{2} D + D \rho^2 ) + 8 \varepsilon ^{4} D^{5} \\ - \varepsilon ^{2} [ 5 ( \rho \, D^{3} + D^3 \rho ) - 3 ( \rho_{xx} D + D \rho_{xx} ) ] \end{array} \end{array} \right)$$ which are compatible. The conserved Hamiltonian densities for the Boussinesq system are given by $$\begin{aligned} {\cal H}_{-1}^{E} &=& \rho, \\ {\cal H}_{0}^{E} &=& \rho u, \\ {\cal H}_{1}^{E} &=&\frac{1}{4}\left[2\rho u^{2}+\frac{1}{3}\rho ^{4}+\varepsilon^{2}(6\rho \rho _{x}^{2}+4u_{x}^{2})+4\varepsilon ^{4}\rho _{xx}^{2}\right], \nonumber \\ {\cal H}_{2}^{E} &=&\frac{1}{28}\left[\frac{14}{3}\rho u^{3}+\frac{7}{3}\rho ^{4}u+14\varepsilon ^{2}(2uu_{x}^{2}+4\rho ^{2}\rho _{x}u_{x}+3u\rho \rho _{x}^{2})\right. \label{bcomham} \\ && \left. +28\varepsilon ^{4}(u\rho _{xx}^{2}+\rho _{x}^{2}u_{xx}+4\rho \rho _{xx}u_{xx})+64\varepsilon ^{6}\rho _{xxx}u_{xxx}\right]\end{aligned}$$ in the Eulerian sequence and we have also $$\begin{aligned} {\cal H}_{-1}^{L} &=& u, \\ {\cal H}_{0}^{L} &=& u^{2}+\frac{1}{3}\rho ^{3}+\varepsilon ^{2}\rho_{x}^{2} , \\ {\cal H}_{1}^{L} &=& \frac{1}{3}u^{3}+\frac{1}{3}\rho ^{3}u-\varepsilon ^{2}u(4\rho \rho _{xx}+3\rho _{x}^{2})+\frac{16}{5}\varepsilon ^{4}u_{xx}\rho _{xx}, \label{boussham}\\ {\cal H}_{2}^{L} &=& \frac{2}{3}u^{4}+\frac{4}{3}\rho ^{3}u^{2}+\frac{4}{45}% \rho ^{6}+\varepsilon ^{2}(\frac{28}{3}\rho ^{3}\rho _{x}^{2}+4u^{2}\rho _{x}^{2}+32\rho u\rho _{x}u_{x}+8\rho ^{2}u_{x}^{2}) \nonumber \\ &&+\varepsilon ^{4}(\frac{136}{5}\rho ^{2}\rho _{xx}^{2}-\frac{248}{5}\rho _{x}^{4}+\frac{128}{5}uu_{xx}\rho _{xx}+\frac{16}{5}u_{x}^{2}\rho _{xx}+% \frac{96}{5}\rho u_{xx}^{2}) \\ &&+\varepsilon ^{6}(32\rho \rho _{xxx}^{2}-\frac{592}{15}\rho _{xx}^{3}+% \frac{64}{5}u_{xxx}^{2})+\frac{64}{5}\varepsilon ^{8} \rho_{xxxx}^{2} \nonumber\end{aligned}$$ in the Lagrangian sequence. The Hamiltonian function of Boussinesq system with the first order Hamiltonian operator in Darboux form (\[j1\]) is $\frac{1}{2}H_{0}^{L}$. We note that the system (\[b1\]) for the Boussinesq equation differs from all dispersive integrable examples we encountered earlier in that its familiar Hamiltonian function (\[boussham\]) is in the Lagrangian sequence. This is because Boussinesq equation is the family of commuting flows to the regular gas dynamics hierarchy. The first commuting higher flow for the Boussinesq system (\[commb\]) has the Hamiltonian function (\[bcomham\]) in the Eulerian series. By introducing potentials $$u = \varphi_{x}, \qquad \rho = \psi_{x}$$ we can obtain two local Lagrangian densities for the Boussinesq system. First we have the classical Lagrangian $$\begin{aligned} {\cal L}_{1}^{B(L)} & = & {\cal H}_{-1}^{L \; \gamma=4} \psi_{y}+ {\cal H}_{-1}^{E \; \gamma=4} \varphi_{y}-{\cal H}_{0}^{L \; \gamma=4} \label{bol1} \\ {\cal L}_{1}^{B(E)} &=& {\cal H}_{-1}^{L \; \gamma=4} \psi _{t}+ {\cal H}_{-1}^{L \; \gamma=4} \varphi_{t}-2 {\cal H}_{1}^{E \; \gamma=4} \label{cbol1}\end{aligned}$$ for Boussinesq system and its first nontrivial commuting flow (\[commb\]). The second Lagrangians are given by $${\cal L}_{2}^{B(L)}= ( {\cal H}_{0}^{E \; \gamma=4} -4\varepsilon ^{2}\varphi _{xxx})\varphi _{y}+ [ {\cal H}_{0}^{L \; \gamma=4} - 5 \varepsilon ^{2} ( \psi _{x}\psi_{xx})_x +4\varepsilon ^{4}\psi_{xxxxx} ] \psi _{y} - {\cal H}_{1}^{L \; \gamma=4}$$ $${\cal L}_{2}^{B(E)} = ( {\cal H}_{0}^{E \; \gamma=4} -4\varepsilon ^{2}\varphi _{xxx})\varphi _{t}+ [ {\cal H}_{0}^{L \; \gamma=4} - 5 \varepsilon ^{2} ( \psi _{x}\psi_{xx})_x +4\varepsilon ^{4}\psi_{xxxxx} ] \psi _{t} - 2 {\cal H}_{2}^{E \; \gamma=4}$$ according to the general construction of Lagrangians in (\[genexp\]). Here we see also that the Lagrangian for the commuting flow is obtained by flipping the Hamiltonian functions between the Lagrangian and Eulerian series while keeping the momenta fixed. In section \[sec-gas\] we had constructed Lagrangians for gas dynamics using the Hamiltonians from the Eulerian series in the potential part of the Lagrangian. The general formula (\[genexp\]) can readily be used to construct Lagrangians for the commuting flow (\[b1\]) by this simple flip in the potential. Conclusion ========== This is the first time it has been possible to write a Lagrangian for polytropic gas dynamics that is local in the original field variables, namely the density and velocity. It is a result of the general expression (\[genexp\]) that serves to identify immediately multi-Lagrangians for completely integrable systems. What is even more remarkable is that this is only the first element in an infinite series of such local Lagrangians for polytropic gas dynamics. It is worth emphasizing again that the scheme we have presented in section \[sec-main\] is a universal one for the construction of multi-Lagrangians appropriate to evolutionary systems. The expressions (\[genexp\]) and (\[reallag2\]) for Lagrangians of completely integrable systems has general validity. We note that (\[genexp\]) with $\alpha=1$ is true even in the case of non-integrable equations, provided the equations are presented in the form of conservation laws and the system admits one further conserved quantity, namely the Hamiltonian. We have discussed in detail the higher Lagrangians for the completely integrable non-linear evolution equations of polytropic gas dynamics, Kaup-Boussinesq, Kaup-Broer, NLS and Boussinesq equations all of which bear out the universal applicability of (\[genexp\]) in the construction of higher Lagrangians. We have also presented the lower Lagrangians (\[reallag2\]) fully for KdV and partially for gas dynamics owing to the difficulty of writing the second modified variables in closed form. The invariance group of these multi-Lagrangians and their Noether currents should prove to be of interest in discovering new hidden symmetries of fluid mechanics. We did not discuss this important issue here. Recently Jackiw and co-authors [@jackiw] have used hidden symmetries in the classical Lagrangian for fluid mechanics to construct very interesting field theory models of fluid mechanics. Multi-Lagrangians may prove to be of interest in this connection also. [99]{} Magri F 1978 J. Math. Phys. [**19**]{} 1156\ Magri F 1980 in [*Nonlinear Evolution Equations and Dynamical Systems*]{} Boiti M, Pempinelli F and Soliani G editors Lecture Notes in Phys., [**120**]{}, Springer, New York, p. 233\ Magri F, Morosi C and Tondo G 1988 Comm. Math. Phys. [**115**]{} 457 Dorfman I Ya 1993 [*Dirac Structures*]{} J. Wiley & Sons Dirac P A M 1964 [*Lectures on Quantum Mechanics*]{} Belfer Graduate School of Science Monographs series 2, New York\ Hanson A, Regge T and Teitelboim C 1976 Acad. Naz. Lincei (Rome)\ Sundermeyer K 1982 [*Constrained Dynamics*]{}, Lecture Notes in Physics Vol. 169 Springer Verlag\ Kosmann-Schwarzbach Y 1986 [*Géométrie de systémes Bihamiltoniens*]{}, Univ. of Montreal Press No.[**102**]{} 185\ Jackiw R 1994 in [*Constraint Theory and Quantization Methods*]{} Colomo F, Lusanna L and Marmo G eds. World Scientific, Singapore Macfarlane A J 1982 CERN preprint TH 3289 Pavlov M V 2000 unpublished Nutku Y “New symplectic structure for KdV“ in ”Integrable Hierarchies and Modern Physics Theories, NATO ARW-UIC 2000" Editors H. Aratyn and A. Sorin. Kluver Academic Publishers Series II, Vol. 18 (2001) hep-th/0011052 Pavlov M V 2001 “Four Lagrangian Representations for the Extended Harry-Dym Equation" to be published. Witten E 1986 Nuclear Physics [**B 276**]{} 291\ Crnković C and Witten E 1986 in [*Three Hundred Years of Gravitation*]{}, Hawking S W and Israel W editors Zuckerman G J 1986 in [*Mathematical Aspects of String Theory*]{} Yau S T editor, World Scientific, Singapore Schiff 1992 “The KdV action and Deformed Minimal Models" hep-th/9205105 Dubrovin B A and Novikov S P 1983 Sov. Math. Dokl. [**27**]{} 665 Jackiw R and Polychronakos A P 1999 Commun. Math. Phys. [**207**]{} 107 Gümral H and Nutku Y 1990 J. Math. Phys. [**31**]{} 2606 Arik M, Neyzi F, Nutku Y, Olver P J and Verosky J 1989 J. Math. Phys. [**30**]{} 1338 Hassaine M and P. Horva’thy 2000 Ann. Phys. (N.Y.) [**282**]{} 218 Nutku Y 1987 J. Math. Phys. [**28**]{} 2579 Olver P J and Nutku Y 1988 J. Math. Phys. [**29**]{} 1610 Sheftel M B 1983 Theor. Math. Phys. [**56**]{} 878\ Sheftel M B 1986 Func. Anal. Appl. [**20**]{} 227\ Sheftel M B 2001 [*Group analysis of hydrodynamic-type systems*]{} math-ph/0104014 Verosky 1984 J. Math. Phys. [**25**]{} 884 Nutku Y. 1985 J. Math. Phys. [**26**]{} 1237 Kaup D J 1975 Prog. Theor. Physics [**54**]{} 396 Broer L J F 1975 Appl. Sci. Res. [**31**]{} 377 Kuperschmidt 1985 Comm. Math. Phys. [**99**]{} 51 Pavlov M V 1998 Phys. Lett. A [**243**]{} 295 Olver P J 1986 [*Applications of Lie Groups to Differential Equations*]{}, Graduate Texts in Mathematics Vol. 107 Springer Verlag Jackiw R 2000 [*A Particle Field Theorist’s Lectures on Super-symmetric, Non-Abelian Fluid Mechanics and $d$-Branes*]{} physics/001042 and references therein. [^1]: Note that there is an error in potential term of the second Lagrangian in [@nhepth]. The results that follow are stated correctly.

--- abstract: 'Two of the defining elements of Social Networking Services are the social profile, containing information about the user, and the social graph, containing information about the connections between users. Social Networking Services are used to connect to known people as well as to discover new contacts. Current friend recommendation mechanisms typically utilize the social graph. In this paper, we argue that psychometrics, the field of measuring personality traits, can help make meaningful friend recommendations based on an extended social profile containing collected smartphone sensor data. This will support the development of highly distributed Social Networking Services without central knowledge of the social graph.' author: - title: 'Towards Psychometrics-based Friend Recommendations in Social Networking Services' --- =1 Introduction ============ Social Networking Services (SNSs) are one of the most used services on the World Wide Web [@greenwood_social_2016]. Two typical elements of a SNS are the *social profile*, containing information about a user, for example her interests, and the *social graph*, containing information about the connections between users. In our previous work, we argued that the smartphone is the optimal social networking device [@beierle_towards_2015]. It typically has only one user and, with recent developments in smartphone sensor technologies and available APIs, more and more personal data – like location traces, most frequently used apps, etc. – is available that could potentially extend existing social profiles. One of the typical applications in SNSs are friend recommendations. When recommending new connections in an SNS, typically, the social graph is utilized [@yin_unified_2010]. While doing so enables the incorporation of graph-based properties like the number of mutual friends, there are also studies that look into the similarity of attributes of neighboring nodes, thus incorporating the social profile in the recommendation process [@mohajireen_relational_2011]. The basis for the cited studies about friend recommendations is the insight that *homophily* – the tendency for people to associate themselves with people who are similar to them – is structuring any type of network [@mcpherson_birds_2001]. Looking further into the fields of psychology and social sciences, *psychometrics*, the academic field that deals with measuring psychological personality traits, seems like a promising research area providing results that could help improve friend recommendations in SNSs. Recently, the company *Cambridge Analytica* was in the media because of their alleged success in utilizing psychometrics in targeted political campaign advertisements [@blakely_data_2016], though their impact on the campaign remains somewhat unclear [@confessore_data_2017]. Although the use case is different – targeted advertising instead of friend recommendation – this shows the potential of applying psychology research results to other fields. In this paper, we argue that combining current smartphone technologies with findings from psychometrics will enable meaningful friend recommendations based on social profiles without requiring knowledge of the social graph. Our main contributions of this short paper is a thorough analysis of the theoretical background of psychometrics in relation to SNSs and mobile devices, including a proposal of how to integrate the insights into friend recommendations in SNSs. Analysis and Concept ==================== In this section, we give a detailed analysis of relevant work related to psychometrics, social networking, and smartphone usage. In Section \[sec:2:psych\], we give a literature review on how and why people actually connect with each other in (offline) social networks. We outline the concepts of homophily and personality from psychology and social sciences. In order to ensure that the same concepts hold true in SNSs, we look into existing research on SNSs and personality in Section \[sec:2:socialnetworkpersonality\]. In Section \[sec:2:smartphonepersonality\], we investigate the relationship between smartphone usage and personality. As we will deal with attributes of users rather then existing connections between users, we will look at existing definitions and components of social profiles in Section \[sec:2:definition\]. Psychology and Social Sciences {#sec:2:psych} ------------------------------ In this section, we want to investigate two concepts from psychology and social sciences: homophily and personality. Those two concepts will help us conceptualize the parameters we need in a system for psychometrics-based friend recommendations. Furthermore, it will answer the questions “When do people become friends?”, i.e., “When do people create edges in a social graph?”, which are necessary to be asked in social networking. Homophily is the concept that people tend to associate themselves with other people that are similar to them. According to McPherson et al., this principle structures network ties of every type, including friendship, work, or partnership [@mcpherson_birds_2001]. Some of the categories in which people have homophilic contacts are ethnicity, age, education, and gender. The social profile is representing a user. The personality of a person influences a multitude of aspects, e.g., job performance, satisfaction, or romantic success [@golbeck_predicting_2011] and is a “key determinant for the friendship formation process” [@burgess_school_2011]. One of the established ways to talk and research about personality is the so-called Big Five or Five-Factor model [@tupes_recurrent_1992; @mccrae_introduction_1992]. The five personality factors spell the acronym OCEAN and are *openness to experience*, *conscientiousness*, *extraversion*, *agreeableness*, and *neuroticism*. In their study, Selfhout et al.show the importance of homophily for friendship networks [@selfhout_emerging_2010]. For three of the five factors (openness to experience, extraversion, and agreeableness), they conclude that people tend to select friends with similar levels of those traits. Additionally to friendship, there are several studies finding correlations between different aspects of everyday life and the five factors. Especially interesting for social networking related questions is the correlations between the five factors and preferences or interests. [@rawlings_music_1997] and [@rentfrow_re_2003] are two of the studies that find correlations between personality and the music the persons prefer to listen to. As we will show in Section \[sec:2:definition\], music preference is a typical element for a social profile. It is the most commonly filled attribute in publicly accessible Facebook profiles after gender [@farahbakhsh_analysis_2013]. Social Networking Services and Personality {#sec:2:socialnetworkpersonality} ------------------------------------------ Several studies suggest that the findings about (offline) social networks are also valid when dealing with SNSs. Liu claims the social profile is a “performance” by the user who expresses herself by crafting the profile [@liu_social_2007]. While this might be true, various studies show that this does not imply that this “performance” distorts the personality that is expressed in the profile. For example, Back et al. conclude in their study that “Facebook Profiles Reflect Actual Personality, Not Self-Idealization,” as the title of their paper indicates [@back_facebook_2010]. In their study, Goldbeck et al. show that Facebook profiles can be used to predict personality [@golbeck_predicting_2011]. Another study comes to the same conclusion and shows “that Facebook-based personality impressions show some consensus for all Big Five dimensions” [@gosling_personality_2007]. Smartphone Usage and Personality {#sec:2:smartphonepersonality} -------------------------------- Some studies on pre-smartphone-era cell phones found correlations between personality traits and mobile phone usage. For example, these are the results of a study done on the general use of mobile phones (calls, text messages, changing ringtones and wallpapers) [@butt_personality_2008], as well as of a study about using mobile phone games [@phillips_personality_2006]. While these studies were based on self reports by users, Chittaranjan et al. conducted two user studies in which they collected usage data on Nokia N95 phones [@chittaranjan_whos_2011; @chittaranjan_mining_2013]. In those studies, the authors were looking at Bluetooth scan data, call logs, text messages, calling profiles, and application usage. At the time the study was conducted, apps were not as common as nowadays with Android and iOS. The authors state that “features derived from the App Logs were sparse due to the low frequency of usage of some of the applications” [@chittaranjan_mining_2013]. It will be interesting to compare the results from their study to a new study where the usage of apps is commonplace. The results of the cited studies indicate that “several aggregated features obtained from smartphone usage data can be indicators of the Big-Five traits” [@chittaranjan_mining_2013]. In a more recent study, Lane and Manner showed relations between the usage of apps and the five personality dimensions [@lane_influence_2012]. Apps were categorized in different application types: communication, games, multimedia, productivity, travel, and utilities. Overall, the referenced studies indicate strong correlations between smartphone usage behavior and personality traits. Social Profiles {#sec:2:definition} --------------- The social profile is one of the central elements of SNSs. In Boyd and Ellison’s definition of *Social Network Sites*, the “public or semi-public profile within a bounded system” is the first defining element, and the “backbone” of the SNS [@boyd_social_2007]. Typical elements of a social profile are “age, location, interests, and an ’about me’ section,” and a photo. In [@richter_functions_2008], the social profile is the first defining functionality of an SNS. Here, the authors call the functionality “identity management,” as the profile is a “representation of the own person.” In a survey paper about SNSs, the social profile is described as the “core” of an Online Social Network [@heidemann_online_2012]. Another recent survey describes the creation and maintenance of user profiles as the “basic functionality” of SNSs [@paul_survey_2014]. In [@gondor_sonic:_2015], several SNSs from different categories, like general (e.g. Facebook), business oriented (e.g. LinkedIn) or special purpose (e.g. Twitter), were analyzed. The social profile is an element that is present in all of those SNSs. Rohani and Hock state that the type of information included in social profiles differs between different SNSs [@rohani_social_2009]. In their analysis of publicly disclosed Facebook profile information, Farahbakhsh et al. distinguish between personal and interest-based attributes [@farahbakhsh_analysis_2013]. Personal attributes include a friend list, current city, hometown, gender, birthday, employers, college, and high school. Interest-based attributes are music, movie, book, television, games, teams, sports, athletes, activities, interests, and inspirations. Lampe et al. distinguish between three different types of information: referents, interests, and contact [@lampe_familiar_2007]. Referents include verifiable attributes: hometown, high school, residence, concentration. Contact information are also verifiable, for example website, email, address, or birthday. As the authors indicate, interests are less verifiable. Interests include an ’about me’ section, favorite music, movies, TV shows, books, quotes, and political views. As the cited studies in Section \[sec:2:smartphonepersonality\] and also social networking related studies (e.g., [@bao_recommendations_2015]) suggest, more detailed user data additional to the data typically available in a social profile can help improve recommendations in SNSs. Concept and Prototype {#sec:2:concept} ===================== Research in psychology and social sciences indicates that homophily in age, education, etc., as well as in personality traits, is a strong indicator for friendship, i.e., for the creation of an edge in a social graph. Several of the aforementioned studies conclude with findings about correlations between smartphone usage and personality traits. Combining those insights, in order to make meaningful recommendations for new connections in an SNS, we can recommend users that show a similar behavioral pattern with their smarthphones. As the existing studies suggest, this will indicate the similarity of their personality traits. Doing this, we do not necessarily need to know which behavior indicates which personality trait. Such a mechanism for friend recommendations can have several benefits: (1) By logging information about the smartphone usage behavior (a lot of which can be done unobtrusively, see [@beierle_privacy-aware_2016]), we could automatically set up or update an existing social profile in an SNS, eliminating or reducing the tedious task to keep such a profile up to date [@lampe_familiar_2007]. (2) When fully relying on social profile data for friend recommendations, the social graph is not needed in the recommendation process. This will enable the development of highly distributed SNSs, for example in device-to-device scenarios where two randomly meeting people could determine – without contacting some centralized server – how similar they are and thus are recommended to be friends. (3) By collecting highly personalized user data, further studies in the fields of psychometrics could be possible. We are implementing an Android prototype. With the *Google Awareness API*, Google offers to get the user’s current time, location, nearby places, nearby beacons, headphone state, activity, and weather.[^1] Android also enables developers to retrieve a list of the most frequently used apps. Most music players broadcast what the user is listening to, so other apps can retrieve this information. The available APIs and mechanisms allow for an unobtrusive collection of user data on Android smartphones. After collecting the mentioned data, in order to estimate their similarity, two users can share their data in a device-to-device manner by utilizing appropriate data structures and technologies like Bluetooth, Wi-Fi Direct, or Wi-Fi Aware. Related Work ============ In this section, we review related work on smartphone sensor data collection (Section \[sec:relwork:smartphonedata\]) and on link prediction in SNSs (Section \[sec:relwork:linkprediction\]). Smartphone Data Collection {#sec:relwork:smartphonedata} -------------------------- In [@pejovic_anticipatory_2015], the authors survey the state of the art of “Anticipatory Mobile Computing,” describing how the advances in mobile technology will enable predicting future contexts and acting on it. This field has some similarities to our work, especially with respect to collecting and using sensor data, but it does not focus on social networking. In [@hashemi_user_2016], the authors present a framework called “BaranC” for monitoring and analyzing digital interaction of users with their smartphones. The architecture uses cloud technologies to analyze data and thus raises privacy concerns. The goal of BaranC is not social networking but offering personalized services. In another work by the same authors, an application utilizing their framework is presented [@hashemi_next_2016], predicting the next application a user will use. Wang et al. present a system that collects a multitude of sensor data from mobile devices and queries the users with questionnaires. The collected data is then used to accurately predict the GPA of the undergraduate students who participated [@wang_smartgpa:_2015]. Xiong et al. present a system for social sciences studies that collects sensor data and enables researchers to create surveys for study participants [@xiong_sensus:_2016]. Again, in both those cases, the developed concepts did not focus on social networking. Link Prediction {#sec:relwork:linkprediction} --------------- One of the common ways to research about links in social networks is *link prediction*. The key difference to our work is that here, the social graph is used to calculate the prediction or recommendation, while our approach is also feasible in device-to-device mobile SNS scenarios where the social graph is not known. Yin et al. analyzed links in social networks based on “intuition-based” aspects: homophily (shared attributes), rarity (matching uncommon attributes), social influence (more likely to link to person that shares attributes with existing friends), common friendship (mutual friends), social closeness (being close to each other in the social graph), preferential attachment (more likely to link to a popular person) [@yin_unified_2010]. Most aspects focus on the social graph or global knowledge about attribute distribution (in the case of rarity). In the work by Mohajureen et al., the authors use the attributes of neighboring nodes in the friend recommendation process [@mohajireen_relational_2011]. For this algorithm to work, the social graph as well as the features of each user have to be available. A somewhat special case of link prediction or friend recommendation is described in *WhozThat?* [@beach_whozthat?_2008]. Here, the idea is to retrieve information about another person you just met. Via Bluetooth, user handles from an SNS are exchanged and data about the other person can be retrieved from that SNS. As described in Section \[sec:2:concept\], by following our concept, the same scenario can be realized in a distributed manner without contacting existing centralized SNS. Conclusion and Future Work ========================== In this paper, we proposed the extension of social profiles with smartphone sensor data. We showed that research results from the field of psychometrics suggest that we then can calculate relevant friend recommendations based on those profiles, without utilizing the social graph. This will enable recommendations in highly distributed SNSs. Future work includes conducting a user study with our prototype to confirm our conclusions. Furthermore, in the device-to-device SNS scenario, the issue of privacy-awareness will be further investigated so that similarity estimations between users are possible without sharing all the collected sensitive sensor data. Acknowledgment {#acknowledgment .unnumbered} ============== This work has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 645342, project reTHINK and from project DYNAMIC[^2] (grant No 01IS12056), which is funded as part of the Software Campus initiative by the German Federal Ministry of Education and Research (BMBF). [10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{} S. Greenwood, A. Perrin, and M. Duggan. (2016, Nov.) Social [[Media Update]{}]{} 2016. <http://www.pewinternet.org/2016/11/11/social-media-update-2016/>. F. Beierle, S. G[ö]{}nd[ö]{}r, and A. K[ü]{}pper, “Towards a [[Three]{}]{}-tiered [[Social Graph]{}]{} in [[Decentralized Online Social Networks]{}]{},” in *Proc. 7th [[International Workshop]{}]{} on [[Hot Topics]{}]{} in [[Planet]{}]{}-Scale [[mObile Computing]{}]{} and [[Online Social neTworking]{}]{} ([[HotPOST]{}]{})*.1em plus 0.5em minus 0.4em, Jun. 2015, pp. 1–6. Z. Yin, M. Gupta, T. Weninger, and J. Han, “A [[Unified Framework]{}]{} for [[Link Recommendation Using Random Walks]{}]{},” in *Proc. 2010 [[International Conference]{}]{} on [[Advances]{}]{} in [[Social Networks Analysis]{}]{} and [[Mining]{}]{} ([[ASONAM]{}]{})*.1em plus 0.5em minus 0.4em, Aug. 2010, pp. 152–159. M. Mohajireen, C. Ellepola, M. Perera, I. Kahanda, and U. Kanewala, “Relational [[Similarity Model]{}]{} for [[Suggesting Friends]{}]{} in [[Online Social Networks]{}]{},” in *Proc. 2011 6th [[International Conference]{}]{} on [[Industrial]{}]{} and [[Information Systems]{}]{} ([[ICIIS]{}]{})*.1em plus 0.5em minus 0.4em, Aug. 2011, pp. 334–339. M. McPherson, L. Smith-Lovin, and J. M. Cook, “Birds of a [[Feather]{}]{}: [[Homophily]{}]{} in [[Social Networks]{}]{},” *Annual Review of Sociology*, vol. 27, pp. 415–444, 2001. R. Blakely. (2016, Sep.) Data scientists target 20 million new voters for [[Trump]{}]{}. <http://www.thetimes.co.uk/article/trump-calls-in-brexit-experts-to-target-voters-pf0hwcts9>. N. Confessore and D. Hakim. (2017, Mar.) Data [F]{}irm [S]{}ays ‘[S]{}ecret [S]{}auce’ [A]{}ided [T]{}rump; [M]{}any [S]{}coff. <https://www.nytimes.com/2017/03/06/us/politics/cambridge-analytica.html>. J. Golbeck, C. Robles, and K. Turner, “Predicting [[Personality]{}]{} with [[Social Media]{}]{},” in *Proc. [[CHI]{}]{} ’11 [[Extended Abstracts]{}]{} on [[Human Factors]{}]{} in [[Computing Systems]{}]{} ([[CHI EA]{}]{})*.1em plus 0.5em minus 0.4em, May 2011, pp. 253–262. S. Burgess, E. Sanderson, and M. Uma[ñ]{}a-Aponte, *School Ties: [[An]{}]{} Analysis of Homophily in an Adolescent Friendship Network*, ser. CMPO Working Paper Series.1em plus 0.5em minus 0.4em, 2011, no. 11/267. E. C. Tupes and R. E. Christal, “,” **, vol. 60, no. 2, pp. 225–251, Jun. 1992. R. R. McCrae and O. P. John, “An [[Introduction]{}]{} to the [[Five]{}]{}-[[Factor Model]{}]{} and [[Its Applications]{}]{},” *Journal of Personality*, vol. 60, no. 2, pp. 175–215, 1992. M. Selfhout, W. Burk, S. Branje, J. Denissen, M. Van Aken, and W. Meeus, “,” **, vol. 78, no. 2, pp. 509–538, Apr. 2010. D. Rawlings and V. Ciancarelli, “Music [[Preference]{}]{} and the [[Five]{}]{}-[[Factor Model]{}]{} of the [[NEO Personality Inventory]{}]{},” *Psychology of Music*, vol. 25, no. 2, pp. 120–132, Oct. 1997. P. J. Rentfrow and S. D. Gosling, “” **, vol. 84, no. 6, pp. 1236–1256, 2003. R. Farahbakhsh, X. Han, A. Cuevas, and N. Crespi, “Analysis of publicly disclosed information in [[Facebook]{}]{} profiles,” in *Proc. 2013 [[IEEE]{}]{}/[[ACM International Conference]{}]{} on [[Advances]{}]{} in [[Social Networks Analysis]{}]{} and [[Mining]{}]{} ([[ASONAM]{}]{})*.1em plus 0.5em minus 0.4em, Aug. 2013, pp. 699–705. H. Liu, “,” **, vol. 13, no. 1, pp. 252–275, Oct. 2007. M. D. Back, J. M. Stopfer, S. Vazire, S. Gaddis, S. C. Schmukle, B. Egloff, and S. D. Gosling, “Facebook [[Profiles Reflect Actual Personality]{}]{}, [[Not Self]{}]{}-[[Idealization]{}]{},” *Psychological Science*, vol. 21, no. 3, pp. 372–374, Jan. 2010. S. D. Gosling, S. Gaddis, and S. Vazire, “Personality [[Impressions Based]{}]{} on [[Facebook Profiles]{}]{},” in *Proc. [[International AAAI Conference]{}]{} on [[Weblogs]{}]{} and [[Social Media]{}]{} ([[ICWSM]{}]{})*.1em plus 0.5em minus 0.4em, 2007, pp. 1–4. S. Butt and J. G. Phillips, “Personality and self reported mobile phone use,” *Computers in Human Behavior*, vol. 24, no. 2, pp. 346–360, Mar. 2008. J. G. Phillips, S. Butt, and A. Blaszczynski, “Personality and [[Self]{}]{}-[[Reported Use]{}]{} of [[Mobile Phones]{}]{} for [[Games]{}]{},” *CyberPsychology & Behavior*, vol. 9, no. 6, pp. 753–758, Dec. 2006. G. Chittaranjan, J. Blom, and D. Gatica-Perez, “Who’s [[Who]{}]{} with [[Big]{}]{}-[[Five]{}]{}: [[Analyzing]{}]{} and [[Classifying Personality Traits]{}]{} with [[Smartphones]{}]{},” in *Proc. 2011 15th [[Annual International Symposium]{}]{} on [[Wearable Computers]{}]{} ([ISWC]{})*.1em plus 0.5em minus 0.4em, Jun. 2011, pp. 29–36. ——, “Mining large-scale smartphone data for personality studies,” *Personal and Ubiquitous Computing*, vol. 17, no. 3, pp. 433–450, Mar. 2013. W. Lane and C. Manner, “The influence of personality traits on mobile phone application preferences,” *Journal of Economics & Behavioral Studies*, vol. 4, no. 5, pp. 252–260, 2012. D. M. Boyd and N. B. Ellison, “,” **, vol. 13, no. 1, pp. 210–230, Oct. 2007. A. Richter and M. Koch, “Functions of [[Social Networking Services]{}]{},” in *Proc. 8th [[International Conference]{}]{} on the [[Design]{}]{} of [[Cooperative Systems]{}]{} ([[COOP]{}]{})*, May 2008, pp. 87–98. J. Heidemann, M. Klier, and F. Probst, “,” **, vol. 56, no. 18, pp. 3866–3878, Dec. 2012. T. Paul, A. Famulari, and T. Strufe, “A survey on decentralized [[Online Social Networks]{}]{},” *Computer Networks*, vol. 75, Part A, pp. 437–452, Dec. 2014. S. G[ö]{}nd[ö]{}r, F. Beierle, S. Sharhan, H. Hebbo, E. Kucukbayraktar, and A. K[ü]{}pper, “[[SONIC]{}]{}: [[Bridging]{}]{} the [[Gap]{}]{} between [[Different Online Social Network Platforms]{}]{},” in *Proc. 2015 [[IEEE International Conference]{}]{} on [[Smart City]{}]{}/[[SocialCom]{}]{}/[[SustainCom]{}]{} ([[SmartCity]{}]{})*.1em plus 0.5em minus 0.4em, Dec. 2015, pp. 399–406. V. A. Rohani and O. S. Hock, “On [[Social Network Web Sites]{}]{}: [[Definition]{}]{}, [[Features]{}]{}, [[Architectures]{}]{} and [[Analysis Tools]{}]{},” *Journal of Computer Engineering*, vol. 1, pp. 3–11, 2009. C. A. Lampe, N. Ellison, and C. Steinfield, “A [[Familiar Face]{}]{}([[Book]{}]{}): [[Profile Elements As Signals]{}]{} in an [[Online Social Network]{}]{},” in *Proc. [[SIGCHI Conference]{}]{} on [[Human Factors]{}]{} in [[Computing Systems]{}]{} ([[CHI]{}]{})*.1em plus 0.5em minus 0.4em, Apr. 2007, pp. 435–444. J. Bao, Y. Zheng, D. Wilkie, and M. Mokbel, “Recommendations in [[Location]{}]{}-based [[Social Networks]{}]{}: [[A Survey]{}]{},” *Geoinformatica*, vol. 19, no. 3, pp. 525–565, 2015. F. Beierle, K. Grunert, S. G[ö]{}nd[ö]{}r, and A. K[ü]{}pper, “Privacy-aware [[Social Music Playlist Generation]{}]{},” in *Proc. 2016 [[IEEE International Conference]{}]{} on [[Communications]{}]{} ([[ICC]{}]{})*.1em plus 0.5em minus 0.4em, May 2016, pp. 5650–5656. V. Pejovic and M. Musolesi, “,” **, vol. 47, no. 3, pp. 1–29, Apr. 2015. M. Hashemi and J. Herbert, “,” in **.1em plus 0.5em minus 0.4em, 2016, pp. 7–8. ——, “A [[Next Application Prediction Service Using]{}]{} the [[BaranC Framework]{}]{},” in *Proc. 2016 [[IEEE]{}]{} 40th [[Annual Computer Software]{}]{} and [[Applications Conference]{}]{} ([[COMPSAC]{}]{})*.1em plus 0.5em minus 0.4em, Jun. 2016, pp. 519–523. R. Wang, G. Harari, P. Hao, X. Zhou, and A. T. Campbell, “[[SmartGPA]{}]{}: [[How Smartphones Can Assess]{}]{} and [[Predict Academic Performance]{}]{} of [[College Students]{}]{},” in *Proc. 2015 [[ACM International Joint Conference]{}]{} on [[Pervasive]{}]{} and [[Ubiquitous Computing]{}]{} ([UbiComp]{})*.1em plus 0.5em minus 0.4em, Sep. 2015, pp. 295–306. H. Xiong, Y. Huang, L. E. Barnes, and M. S. Gerber, “Sensus: [[A Cross]{}]{}-platform, [[General]{}]{}-purpose [[System]{}]{} for [[Mobile Crowdsensing]{}]{} in [[Human]{}]{}-subject [[Studies]{}]{},” in *Proc. 2016 [[ACM International Joint Conference]{}]{} on [[Pervasive]{}]{} and [[Ubiquitous Computing]{}]{} ([UbiComp]{})*.1em plus 0.5em minus 0.4em, Sep. 2016, pp. 415–426. A. Beach, M. Gartrell, S. Akkala, J. Elston, J. Kelley, K. Nishimoto, B. Ray, S. Razgulin, K. Sundaresan, B. Surendar, M. Terada, and R. Han, “[[WhozThat]{}]{}? [[Evolving]{}]{} an [[Ecosystem]{}]{} for [[Context]{}]{}-[[Aware Mobile Social Networks]{}]{},” *IEEE Network*, vol. 22, no. 4, pp. 50–55, 2008. [^1]: <https://developers.google.com/awareness/> [^2]: <http://www.dynamic-project.de>

--- abstract: 'The Large Area Telescope ([*Fermi*]{}/LAT, hereafter LAT), the primary instrument on the [*Fermi Gamma-ray Space Telescope*]{} ([*Fermi*]{}) mission, is an imaging, wide field-of-view, high-energy [$\gamma$-ray]{} telescope, covering the energy range from below 20 MeV to more than 300 GeV. The LAT was built by an international collaboration with contributions from space agencies, high-energy particle physics institutes, and universities in France, Italy, Japan, Sweden, and the United States. This paper describes the LAT, its pre-flight expected performance, and summarizes the key science objectives that will be addressed. On-orbit performance will be presented in detail in a subsequent paper. The LAT is a pair-conversion telescope with a precision tracker and calorimeter, each consisting of a $4\times4$ array of 16 modules, a segmented anticoincidence detector that covers the tracker array, and a programmable trigger and data acquisition system. Each tracker module has a vertical stack of 18 $x,y$ tracking planes, including two layers ($x$ and $y$) of single-sided silicon strip detectors and high-$Z$ converter material (tungsten) per tray. Every calorimeter module has 96 CsI(Tl) crystals, arranged in an 8 layer hodoscopic configuration with a total depth of 8.6 radiation lengths, giving both longitudinal and transverse information about the energy deposition pattern. The calorimeter’s depth and segmentation enable the high-energy reach of the LAT and contribute significantly to background rejection. The aspect ratio of the tracker (height/width) is 0.4, allowing a large field-of-view (2.4 sr) and ensuring that most pair-conversion showers initiated in the tracker will pass into the calorimeter for energy measurement. Data obtained with the LAT are intended to (i) permit rapid notification of high-energy [$\gamma$-ray]{} bursts (GRBs) and transients and facilitate monitoring of variable sources, (ii) yield an extensive catalog of several thousand high-energy sources obtained from an all-sky survey, (iii) measure spectra from 20 MeV to more than 50 GeV for several hundred sources, (iv) localize point sources to 0.3 – 2 arc minutes, (v) map and obtain spectra of extended sources such as SNRs, molecular clouds, and nearby galaxies, (vi) measure the diffuse isotropic [$\gamma$-ray]{} background up to TeV energies, and (vii) explore the discovery space for dark matter.' author: - 'W. B. Atwood, A. A. Abdo, M. Ackermann, B. Anderson, M. Axelsson, L. Baldini, J. Ballet, D. L. Band, G. Barbiellini, J. Bartelt, D. Bastieri, B. M. Baughman, K. Bechtol, D. Bédérède, F. Bellardi, R. Bellazzini, B. Berenji, G. F. Bignami, D. Bisello, E. Bissaldi, R. D. Blandford, E. D. Bloom, J. R. Bogart, E. Bonamente, J. Bonnell, A. W. Borgland, A. Bouvier, J. Bregeon, A. Brez, M. Brigida, P. Bruel, T. H. Burnett, G. Busetto, G. A. Caliandro, R. A. Cameron, P. A. Caraveo, S. Carius, P. Carlson, J. M. Casandjian, E. Cavazzuti, M. Ceccanti, C. Cecchi, E. Charles, A. Chekhtman, C. C. Cheung, J. Chiang, R. Chipaux, A. N. Cillis, S. Ciprini, R. Claus, J. Cohen-Tanugi, S. Condamoor, J. Conrad, R. Corbet, L. Corucci, L. Costamante, S. Cutini, D. S. Davis, D. Decotigny, M. DeKlotz, C. D. Dermer, A. de Angelis, S. W. Digel, E. do Couto e Silva, P. S. Drell, R. Dubois, D. Dumora, Y. Edmonds, D. Fabiani, C. Farnier, C. Favuzzi, D. L. Flath, P. Fleury, W. B. Focke, S. Funk, P. Fusco, F. Gargano, D. Gasparrini, N. Gehrels, F.-X. Gentit, S. Germani, B. Giebels, N. Giglietto, P. Giommi, F. Giordano, T. Glanzman, G. Godfrey, I. A. Grenier, M.-H. Grondin, J. E. Grove, L. Guillemot, S. Guiriec, G. Haller, A. K. Harding, P. A. Hart, E. Hays, S. E. Healey, M. Hirayama, L. Hjalmarsdotter, R. Horn, G. Jóhannesson, G. Johansson, A. S. Johnson, R. P. Johnson, T. J. Johnson, W. N. Johnson, T. Kamae, H. Katagiri, J. Kataoka, A. Kavelaars, N. Kawai, H. Kelly, M. Kerr, W. Klamra, J. Knödlseder, M. L. Kocian, N. Komin, F. Kuehn, M. Kuss, D. Landriu, L. Latronico, B. Lee, S.-H. Lee, M. Lemoine-Goumard, A. M. Lionetto, F. Longo, F. Loparco, B. Lott, M. N. Lovellette, P. Lubrano, G. M. Madejski, A. Makeev, B. Marangelli, M. M. Massai, M. N. Mazziotta, J. E. McEnery, N. Menon, C. Meurer, P. F. Michelson, M. Minuti, N. Mirizzi, W. Mitthumsiri, T. Mizuno, A. A. Moiseev, C. Monte, M. E. Monzani, E. Moretti, A. Morselli, I. V. Moskalenko, S. Murgia, T. Nakamori, S. Nishino, P. L. Nolan, J. P. Norris, E. Nuss, M. Ohno, T. Ohsugi, N. Omodei, E. Orlando, J. F. Ormes, A. Paccagnella, D. Paneque, J. H. Panetta, D. Parent, M. Pearce, M. Pepe, A. Perazzo, M. Pesce-Rollins, P. Picozza, L. Pieri, M. Pinchera, F. Piron, T. A. Porter, L. Poupard, S. Rainò, R. Rando, E. Rapposelli, M. Razzano, A. Reimer, O. Reimer, T. Reposeur, L. C. Reyes, S. Ritz, L. S. Rochester, A. Y. Rodriguez, R. W. Romani, M. Roth, J. J. Russell, F. Ryde, S. Sabatini, H. F.-W. Sadrozinski, D. Sanchez, A. Sander, L. Sapozhnikov, P. M. Saz Parkinson, J. D. Scargle, T. L. Schalk, G. Scolieri, C. Sgrò, G. H. Share, M. Shaw, T. Shimokawabe, C. Shrader, A. Sierpowska-Bartosik, E. J. Siskind, D. A. Smith, P. D. Smith, G. Spandre, P. Spinelli, J.-L. Starck, T. E. Stephens, M. S. Strickman, A. W. Strong, D. J. Suson, H. Tajima, H. Takahashi, T. Takahashi, T. Tanaka, A. Tenze, S. Tether, J. B. Thayer, J. G. Thayer, D. J. Thompson, L. Tibaldo, O. Tibolla, D. F. Torres, G. Tosti, A. Tramacere, M. Turri, T. L. Usher, N. Vilchez, V. Vitale, P. Wang, K. Watters, B. L. Winer, K. S. Wood, T. Ylinen, M. Ziegler' title: 'The Large Area Telescope on the *Fermi Gamma-ray Space Telescope* Mission' --- Introduction {#s1} ============ A revolution is underway in our understanding of the high-energy sky. The early SAS-2 [@Fichtel1975] and COS-B [@Bignami1975] missions led to the EGRET instrument [@Thompson1993] on the *Compton Gamma-Ray Observatory (CGRO)*. EGRET performed the first all-sky survey above 50 MeV and made breakthrough observations of high-energy [$\gamma$-ray]{} blazars, pulsars, delayed emission from [$\gamma$-ray]{} bursts (GRBs), high-energy solar flares, and diffuse radiation from our Galaxy and beyond that have all changed our view of the high-energy Universe. Many high-energy sources revealed by EGRET have not yet been identified. The Large Area Telescope (LAT) on the [*Fermi Gamma-ray Space Telescope*]{} ([*Fermi*]{}), formerly the *Gamma-ray Large Area Space Telescope (GLAST)*, launched by NASA on 2008, June 11 on a Delta II Heavy launch vehicle, offers enormous opportunities for determining the nature of these sources and advancing knowledge in astronomy, astrophysics, and particle physics. In this paper a comprehensive overview of the LAT instrument design is provided, the pre-flight expected performance based on detailed simulations and ground calibration measurements is given, and the science goals and expectations are summarized. The [*Fermi*]{} observatory had been launched shortly before the submission of this paper so no details of in-flight performance are provided at this time, although the performance to date does not deviate significantly from that estimated before launch. The in-flight calibration of the LAT is being refined during the first year of observations and therefore details of in-flight performance will be the subject of a future paper. [*Fermi*]{} follows the successful launch of *Agile* by the Italian Space Agency in April 2007 [@Tavani2008]. The scientific objectives addressed by the LAT include (i) determining the nature of the unidentified sources and the origins of the diffuse emission revealed by EGRET, (ii) understanding the mechanisms of particle acceleration operating in celestial sources, particularly in active galactic nuclei, pulsars, supernovae remnants, and the Sun, (iii) understanding the high-energy behavior of GRBs and transients, (iv) using [$\gamma$-ray]{} observations as a probe of dark matter, and (v) using high-energy [$\gamma$-ray]{}[s]{} to probe the early universe and the cosmic evolution of high-energy sources to $z\ge 6$. These objectives are discussed in the context of the LAT’s measurement capabilities in §\[s3\]. To make significant progress in understanding the high-energy sky, the LAT, shown in Figure \[f1.1\], has good angular resolution for source localization and multi-wavelength studies, high sensitivity over a broad field-of-view to monitor variability and detect transients, good calorimetry over an extended energy band to study spectral breaks and cut-offs, and good calibration and stability for absolute, long term flux measurement. The LAT measures the tracks of the electron ($e^-$) and positron ($e^+$) that result when an incident [$\gamma$-ray]{} undergoes pair-conversion, preferentially in a thin, high-$Z$ foil, and measures the energy of the subsequent electromagnetic shower that develops in the telescope’s calorimeter. Table \[t1.1\] summarizes the scientific performance capabilities of the LAT. Figure \[f1.2\] illustrates the sensitivity and field-of-view (FoV) achieved with the LAT for exposures on various timescales. To take full advantage of the LAT’s large FoV, the primary observing mode of [*Fermi*]{} is the so-called “scanning” mode in which the normal to the front of the instrument ($z$ axis) on alternate orbits is pointed to $+35^\circ$ from the zenith direction and towards the pole of the orbit and to $-35^\circ$ from the zenith on the subsequent orbit. In this way, after 2 orbits, about 3 hours for [*Fermi*]{}’s orbit at $\sim$565 km and $25.5^\circ$ inclination, the sky exposure is almost uniform. For particularly interesting targets of opportunity, the observatory can be inertially pointed. Details of the LAT design and performance are presented in §\[s2\]. The LAT was developed by an international collaboration with primary hardware and software responsibilities at Stanford University, Stanford Linear Accelerator Center, Agenzia Spaziale Italiana, Commissariat à l’Energie Atomique, Goddard Space Flight Center, Istituto Nazionale di Fisica Nucleare, Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules, Hiroshima University, Naval Research Laboratory, Ohio State University, Royal Institute of Technology – Stockholm, University of California at Santa Cruz, and University of Washington. Other institutions that have made significant contributions to the instrument development include Institute of Space and Astronautical Science, Stockholm University, University of Tokyo, and Tokyo Institute of Science and Technology. All of these institutions as well as the Istituto Nazionale di Astrofisica in Italy are making significant contributions to LAT data analysis during the science operations phase of the [*Fermi*]{} mission. Large Area Telescope {#s2} ==================== Technical development path {#s2.1} -------------------------- The LAT is designed to measure the directions, energies, and arrival times of [$\gamma$-ray]{}[s]{} incident over a wide FoV, while rejecting background from cosmic rays. First, the design approach [@Atwood1994] that resulted in the instrument described in detail in §\[s2.2\] made extensive use of detailed simulations of the detector response to signal (celestial [$\gamma$-ray]{}[s]{}) and backgrounds (cosmic rays, albedo [$\gamma$-ray]{}[s]{}, etc.). Second, detector technologies were chosen that have an extensive history of application in space science and high-energy physics with demonstrated high reliability. Third, relevant test models were built to demonstrate that critical requirements, such as power, efficiency, and detector noise occupancy, could be readily met. Fourth, these detector-system models, including all subsystems, were studied in accelerator test beams to validate both the design and the Monte Carlo programs used in the simulations [@Atwood2000]. The modular design of the LAT allowed the construction, at reasonable incremental cost, of a full-scale, fully functional engineering demonstration telescope module for validation of the design concept. This test engineering model was flown on a high-altitude balloon to demonstrate system level performance in a realistic, harsh background environment [@Thompson2002; @Mizuno2004] and was subjected to an accelerator beam test program [@Couto_e_Silva2001]. Particle beam tests were also done on spare flight tracker and calorimeter modules (see (§\[s2.5.1\]). Technical description {#s2.2} --------------------- High-energy [$\gamma$-ray]{}[s]{} cannot be reflected or refracted; they interact by the conversion of the [$\gamma$-ray]{} into an $e^+e^-$ pair. The LAT is therefore a pair-conversion telescope with a precision converter-tracker (§\[s2.2.1\]) and calorimeter (§\[s2.2.2\]), each consisting of a $4\times4$ array of 16 modules supported by a low-mass aluminum grid structure. A segmented anticoincidence detector (ACD, §\[s2.2.3\]) covers the tracker array, and a programmable trigger and data acquisition system (DAQ, §\[s2.2.4\]) utilizes prompt signals available from the tracker, calorimeter, and anticoincidence detector subsystems to form a trigger. The self-triggering capability of the LAT tracker in particular is an important new feature of the LAT design that is possible because of the choice of silicon-strip detectors, which do not require an external trigger, for the active elements. In addition, all of the LAT instrument subsystems utilize technologies that do not use consumables such as gas. Upon triggering, the DAQ initiates the read out of these 3 subsystems and utilizes on-board event processing to reduce the rate of events transmitted to the ground to a rate compatible with the 1 Mbps average downlink available to the LAT. The on-board processing is optimized for rejecting events triggered by cosmic-ray background particles while maximizing the number of events triggered by [$\gamma$-ray]{}[s]{}, which are transmitted to the ground. Heat produced by the tracker, calorimeter, and DAQ electronics is transferred to radiators through heat pipes in the grid. The overall aspect ratio of the LAT tracker (height/width) is 0.4, allowing a large FoV[^1] and ensuring that nearly all pair-conversion events initiated in the tracker will pass into the calorimeter for energy measurement. ### Precision converter-tracker {#s2.2.1} The converter-tracker has 16 planes of high-$Z$ material in which [$\gamma$-ray]{}[s]{} incident on the LAT can convert to an $e^+e^-$ pair. The converter planes are interleaved with position-sensitive detectors that record the passage of charged particles, thus measuring the tracks of the particles resulting from pair conversion. This information is used to reconstruct the directions of the incident [$\gamma$-ray]{}[s]{}. Each tracker module has 18 $x,y$ tracking planes, consisting of 2 layers ($x$ and $y$) of single-sided silicon strip detectors. The 16 planes at the top of the tracker are interleaved with high-$Z$ converter material (tungsten). Figure \[f2.1\] shows the completed 16 module tracker array before integration with the ACD. Table \[t2.1\] is a summary of key parameters of the LAT tracker. See @Atwood2007 for a more complete discussion of the tracker design and performance. We summarize here the features most relevant to the instrument science performance. The single-sided SSDs are AC-coupled, with 384 56-$\mu$m wide aluminum readout strips spaced at 228 $\mu$m pitch[^2]. They were produced on $n$-intrinsic 15-cm wafers by Hamamatsu Photonics, and each has an area of 8.95$\times$8.95 cm$^2$, with an inactive area 1 mm wide around the edges, and a thickness of 400 $\mu$m. Sets of 4 SSDs were bonded edge to edge with epoxy and then wire bonded strip to strip to form “ladders," such that each amplifier channel sees signals from a 35 cm long strip. Each detector layer in a tracker module consists of 4 such ladders spaced apart by 0.2 mm gaps. The delivered SSD quality was very high, with a bad channel rate less than 0.01% and an average total leakage current of 110 nA. The wafer dicing was accurate to better than 20 $\mu$m, to allow all of the assembly to be done rapidly with mechanical jigs rather than with optical references. The support structure for the detectors and converter foil planes is a stack of 19 composite panels, or “trays,” supported by carbon-composite sidewalls that also serve to conduct heat to the base of the tracker array. The tray structure is a low-mass, carbon-composite assembly made of a carbon-carbon closeout, carbon-composite face sheets, and a vented aluminum honeycomb core. Carbon was chosen for its long radiation length, high modulus (stiffness) to density ratio, good thermal conductivity, and thermal stability. The tray-panel structure is about 3 cm thick and is instrumented with converter foils, detectors, and front-end electronics. All trays are of similar construction, but the top and bottom trays have detectors on only a single face. The bottom trays include the mechanical and thermal interfaces to the grid, while the top trays support the readout-cable terminations, mechanical lifting attachments, and optical survey retro-reflectors. Trays supporting thick converter foils have stronger face sheets and heavier core material than those supporting thin foils or no foils. Figure \[f2.2\]a shows a flight tracker tray and Figure \[f2.2\]b shows a completed tracker module with one sidewall removed. The strips on the top and bottom of a given tray are parallel, while alternate trays are rotated $90^\circ$ with respect to each other. An $x,y$ measurement plane consists of a layer of detectors on the bottom of one tray together with an orthogonal detector layer on the top of the tray just below, with only a 2 mm separation. The tungsten converter foils in the first 16 planes lie immediately above the upper detector layer in each plane. The lowest two $x,y$ planes have no tungsten converter material. The tracker mechanical design emphasizes minimization of dead area within its aperture. To that end, the readout electronics are mounted on the sides of the trays and interfaced to the detectors around the $90^\circ$ corner. One fourth of the readout electronics boards in a single tracker module can be seen in Figure \[f2.2\]b. The interface to the data acquisition and power supplies is made entirely through flat cables constructed as long 4-layer flexible circuits, two of which are visible in Figure \[f2.2\]b. As a result, the dead space between the active area of one tracker module and that of its neighbor is only 18 mm. Incident photons preferentially convert in one of the tungsten foils, and the resulting $e^-$ and $e^+$ particles are tracked by the SSDs through successive planes. The pair conversion signature is also used to help reject the much larger background of charged cosmic rays. The high intrinsic efficiency and reliability of this technology enables straightforward event reconstruction and determination of the direction of the incident photon. The probability distribution for the reconstructed direction of incident [$\gamma$-ray]{}[s]{} from a point source is referred to as the Point Spread Function (PSF). Multiple scattering of the $e^+$ and $e^-$ and bremsstrahlung production limit the obtainable resolution. To get optimal results requires that the $e^-$ and $e^+$ directions be measured immediately following the conversion. At 100 MeV the penalty for missing one of the first hits[^3] is about a factor of two in resolution, resulting in large tails in the PSF. Figure \[f2.3\] summarizes these and other considerations in the tracker design that impact the PSF. In particular, it is important that the silicon-strip detector layers have high efficiency and are held close to the converter foils, that the inactive regions are localized and minimized, and that the passive material is minimized. To minimize missing hits in the first layer following a conversion, the tungsten foils in each plane cover only the active areas of the silicon-strip detectors. One of the most complex LAT design trades was the balance between the need for thin converters, to achieve a good PSF at low energy, where the PSF is determined primarily by the $\sim$$1/E$ dependence of multiple scattering, versus the need for converter material to maximize the effective area, important at high energy. The resolution was to divide the tracker into 2 regions, “front” and “back.” The front region (first 12 $x,y$ tracking planes) has thin converters, each 0.03 radiation lengths thick, to optimize the PSF at low energy, while the converters in the back (4 $x,y$ planes after the front tracker section) are $\sim$6 times thicker, to maximize the effective area at the expense of less than a factor of two in angular resolution (at 1 GeV) for photons converting in that region. Instrument simulations show that the sensitivity of the LAT to point-sources is approximately balanced between the front and back tracker sections, although this depends on the source spectral characteristics. The tracker detector performance was achieved with readout electronics designed specifically to meet the LAT requirements and implemented with standard commercial technology [@Baldini2006]. The system is based on two Application Specific Integrated Circuits (ASICs). The first ASIC is a 64-channel mixed-mode amplifier-discriminator chip and the second ASIC is a digital readout controller. Each amplifier-discriminator chip is programmed with a single threshold level, and only a 0 or 1 (i.e., a “hit”) is stored for each channel when a trigger is generated. Each channel can buffer up to 4 events, and the system is able to trigger even during readout of the digital data from previous events. Thus the system achieves high throughput and very low deadtime, and the output data stream is compact and contains just the information needed for effective tracking, with $<$$10^{-6}$ noise occupancy, and with very little calibration required. The system also measures and records the time-over-threshold (TOT) of each layer’s trigger output signal, which provides charge-deposition information that is useful for background rejection. In particular, isolated tracks that start from showers in the calorimeter sometimes range out in the tracker, mimicking a [$\gamma$-ray]{} conversion. The TOT information is effective for detecting and rejecting such background events because at the termination of such tracks the charge deposition is very large, often resulting in a large TOT in the last SSD traversed. The tracker provides the principal trigger for the LAT. Each detector layer in each module outputs a logical OR of all of its 1536 channels, and a first-level trigger is derived from coincidence of successive layers (typically 3 $x,y$ planes). There is no detectable coherent noise in the system, such that the coincidence rate from electronics noise is immeasurably small, while the trigger efficiency for charged particles approaches 100% when all layers are considered. High reliability was a core requirement in the tracker design. The 16 modules operate independently, providing much redundancy. Similarly, the multi-layer design of each module provides redundancy. The readout system is also designed to minimize or eliminate the impact of single-point failures. Each tracker layer has two separate readout and control paths, and the 24 amplifier-discriminator chips in each layer can be partitioned between the two paths by remote command. Therefore, failure of a single chip or readout cable would result in the loss of at most only 64 channels. ### Calorimeter {#s2.2.2} The primary purposes of the calorimeter are twofold: (i) to measure the energy deposition due to the electromagnetic particle shower that results from the $e^+e^-$ pair produced by the incident photon; and (ii) image the shower development profile, thereby providing an important background discriminator and an estimator of the shower energy leakage fluctuations. Each calorimeter module has 96 CsI(Tl) crystals, with each crystal of size $2.7\ {\rm cm} \times 2.0\ {\rm cm} \times 32.6\ {\rm cm}$. The crystals are optically isolated from each other and are arranged horizontally in 8 layers of 12 crystals each. The total vertical depth of the calorimeter is 8.6 radiation lengths (for a total instrument depth of 10.1 radiation lengths). Each calorimeter module layer is aligned $90^\circ$ with respect to its neighbors, forming an $x,y$ (hodoscopic) array [@carson1996]. Figure \[f2.4\] shows schematically the configuration of a calorimeter module and Table \[t2.2\] is a summary of key parameters of the calorimeter. The size of the CsI crystals is a compromise between electronic channel count and desired segmentation within the calorimeter. The lateral dimensions of the crystals are comparable to the CsI radiation length (1.86 cm) and Molière radius (3.8 cm) for electromagnetic showers. Each CsI crystal provides 3 spatial coordinates for the energy deposited within: two discrete coordinates from the physical location of the crystal in the array and the third, more precise, coordinate determined by measuring the light yield asymmetry at the ends of the crystal along its long dimension. This level of segmentation is sufficient to allow spatial imaging of the shower and accurate reconstruction of its direction. The calorimeter’s shower imaging capability and depth enable the high-energy reach of the LAT and contribute significantly to background rejection. In particular, the energy resolution at high energies is achieved through the application of shower leakage corrections. Each crystal element is read out by PIN photodiodes, mounted on both ends of the crystal, which measure the scintillation light that is transmitted to each end. The difference in light levels provides a determination of the position of the energy deposition along the CsI crystal. There are two photodiodes at each end of the crystal, a large photodiode with area 147 mm$^2$ and a small photodiode with area 25 mm$^2$, providing two readout channels to cover the large dynamic range of energy deposition in the crystal. The large photodiodes cover the range 2 MeV – 1.6 GeV, while the small photodiodes cover the range 100 MeV – 70 GeV. Each crystal end has its own front-end electronics and pre-amplifier electronics assembly. Both low and high energy signals go through a pre-amplifier and shaper and then a pair of Track and Hold circuits with gains differing nominally by a factor of eight. An energy domain selection circuit routes the best energy measurement through an analog multiplexer to an Analog to Digital Converter. A calibration charge injection signal can be fed directly to the front end of the pre-amplifiers. The position resolution achieved by the ratio of light seen at each end of a crystal scales with the deposited energy and ranges from a few millimeters for low energy depositions ($\sim$10 MeV) to a fraction of a millimeter for large energy depositions ($>$1 GeV). Simple analytic forms are used to convert the light asymmetry into a position (see Figure \[f2.5\]). Although the calorimeter is only 8.6 radiation lengths deep, the longitudinal segmentation enables energy measurements up to a TeV. From the longitudinal shower profile, an unbiased estimate of the initial electron energy is derived by fitting the measurements to an analytical description of the energy-dependent mean longitudinal profile. Except at the low end of the energy range, the resulting energy resolution is limited by fluctuations in the shower leakage. The effectiveness of this procedure was evaluated in beam tests with the flight-like calibration unit at CERN. Figure \[f2.6\] shows the measured energy loss and the leakage-corrected energy loss in the calorimeter for electron beams of various energies. Further details of the calorimeter are in @Grove2008, @Johnson2001, and @Ferreira2004. Details of the energy reconstruction are discussed in §\[s2.4.2\]. ### Anticoincidence detector {#s2.2.3} The purpose of the ACD is to provide charged-particle background rejection; therefore its main requirement is to have high detection efficiency for charged particles. The ACD is required to provide at least 0.9997 efficiency (averaged over the ACD area) for detection of singly charged particles entering the field-of-view of the LAT. The LAT is designed to measure [$\gamma$-ray]{}[s]{} with energies up to at least 300 GeV. The requirement to measure photon energies at this limit leads to the presence of a heavy calorimeter ($\sim$1800 kg) to absorb enough of the photon-induced shower energy to make this measurement. The calorimeter mass itself, however, creates a problem we call the backsplash effect: isotropically distributed secondary particles (mostly 100–1000 keV photons) from the electromagnetic shower created by the incident high-energy photon can Compton scatter in the ACD and thereby create false veto signals from the recoil electrons. This effect was present in EGRET, where the instrument detection efficiency above 10 GeV was a factor of at least two or more lower than at 1 GeV due to false vetoes caused by backsplash. A design requirement was established that vetoes created by backsplash (self-veto) would reject not more than 20% of otherwise accepted photons at 300 GeV. To suppress the backsplash effect, the ACD is segmented so that only the ACD segment nearby the incident candidate photon may be considered, thereby dramatically reducing the area of ACD that can contribute to backsplash [@Moiseev2004]. In addition, the onboard use of the ACD veto signals is disengaged when the energy deposition in the calorimeter is larger than an adjustable preset energy (10 to 20 GeV). Such events are subsequently analyzed using more complex software than can be implemented on board. Numerous trade studies and tests were performed in order to optimize the ACD, resulting in the design shown schematically in Figure \[f2.7\]. Plastic scintillator tiles were chosen as the most reliable, efficient, well-understood, and inexpensive technology, with much previous use in space applications. Scintillation light from each tile is collected by wavelength shifting fibers (WLS) that are embedded in the scintillator and are coupled to two photomultiplier tubes (PMTs) for redundancy. This arrangement provides uniformity of light collection that is typically better than 95% over each detector tile, only dropping to $>$75% within 1–2 cm of the tile edges. Overall detection efficiency for incident charged particles is maintained by overlapping scintillator tiles in one dimension. In the other dimension, gaps between tiles are covered by flexible scintillating fiber ribbons with $>$90% detection efficiency. To minimize the chance of light leaks due to penetrations of the light-tight wrapping by micrometeoroids and space debris, the ACD is completely surrounded by a low-mass micrometeoroid shield (0.39 g cm$^{-2}$). All ACD electronics and PMTs are positioned around the bottom perimeter of the ACD, and light is delivered from the tiles and WLS fibers by a combination of wavelength-shifting and clear fibers. The electronics are divided into 12 groups of 18 channels, with each group on a single circuit board. Each of the 12 circuit boards is independent of the other 11, and has a separate interface to the LAT central electronics. The PMTs associated with a single board are powered by a High Voltage Bias Supply (HVBS), with redundant HVBS for each board. The tile readout has two thresholds: an onboard threshold of about 0.45 MIP for the initial rejection of charged particles, and a ground analysis threshold of about 0.30 MIP for the final analysis. Further details of the ACD design, fabrication, testing, and performance are given by @Moiseev2007. Table \[t2.3\] is a summary of key parameters of the LAT ACD. ### Data acquisition system (DAQ) and trigger {#s2.2.4} The Data Acquisition System (DAQ) collects the data from the other subsystems, implements the multi-level event trigger, provides on-board event processing to run filter algorithms to reduce the number of downlinked events, and provides an on-board science analysis platform to rapidly search for transients. The DAQ architecture is hierarchical as shown in Figure \[f2.8\]. At the lowest level shown, each of 16 Tower Electronics Modules (TEMs) provides the interface to the tracker and calorimeter pair in one of the towers. Each TEM generates instrument trigger primitives from combinations of tower subsystem (tracker and calorimeter) triggers, provides event buffering to support event readout, and communicates with the instrument-level Event Builder Module that is part of the Global-trigger/ACD-module/Signal distribution Unit (GASU). The GASU consists of (i) the Command Response Unit (CRU) that sends and receives commands and distributes the DAQ clock signal, (ii) the Global-Trigger Electronics Module (GEM) that generates LAT-wide readout decision signals based on trigger primitives from the TEMs and the ACD, (iii) the ACD Electronics Module (AEM) that performs tasks, much like a TEM, for the ACD, and (iv) the Event Builder Module (EBM) that builds complete LAT events out of the information provided by the TEMs and the AEM, and sends them to dynamically selected target Event Processor Units (EPUs). There are two operating EPUs to support on-board processing of events with filter algorithms designed to reduce the event rate from 2–4 kHz to $\sim$400 Hz that is then downlinked for processing on the ground. The on-board filters are optimized to remove charged particle background events and maximize the rate of [$\gamma$-ray]{} triggered events within the total rate that can be downlinked. Finally, the Spacecraft Interface Unit (SIU) controls the LAT and contains the command interface to the spacecraft. Each EPU and SIU utilizes a RAD750 Compact PCI Processor which, when operating at 115.5 MHz, provides 80 to 90 MIPS. The instrument flight software runs only on the EPUs and the SIU. The TEMs and the GASU hardware have software-controlled trigger configuration and mode registers. Not shown in Figure \[f2.8\] is the redundancy of the DAQ system or the LAT’s Power Distribution Unit (PDU). There are two primary EPUs and one redundant EPU, one primary SIU and one redundant SIU, and one primary GASU and one redundant GASU. The PDU, which is also redundant, controls spacecraft power to the TEMs, the GASU, and the EPUs. The feeds from the spacecraft to the PDU are fully cross-strapped. In turn, the TEMs control power to the tracker and the calorimeter modules and the GASU controls power to the ACD. Power to the SIUs is directly provided by the spacecraft. An instrument-level Trigger Accept Message (TAM) signal is issued by the GEM only if the GEM logic is satisfied by the input trigger primitives within the (adjustable) trigger window width. The TAM signal is sent to each TEM and to the AEM with no delays. Upon receipt of the TAM signal, a Trigger Acknowledge (TACK) signal with an adjustable delay is sent by the TEM to the tracker front-ends and a command, also with an adjustable delay, is sent to the calorimeter front-ends. The AEM sends a signal to the ACD front-ends. The TACK causes the entire instrument to be read out (e.g., addresses of hit strips in the tracker and TOT for each layer in each tracker module, and pulse heights for all 3,072 calorimeter channels and 216 ACD channels). Any of the TEMs or the AEM can issue a trigger request to the GEM. The time between a particle interaction in the LAT that causes an event trigger and the latching of the tracker discriminators is 2.3 to 2.4 $\mu$s, much of this delay due to the analog rise times in the tracker front-end electronics. Similarly, the latching of the analog sample-and-holds for the calorimeter and the ACD are delayed (programmable delay of $\sim$2.5 $\mu$s) until the shaped analog signals peak. The minimum instrumental dead time per event readout is 26.50 $\mu$s and is the time required to latch the trigger information in the GEM and send it from the GEM to the EBM. The calorimeter readout can contribute to the dead time if the full four-range CAL readout is requested. During readout of any of the instrument, any TEM and the AEM send a “busy” signal to the GEM. From these signals, the GEM then generates the overall dead time and the system records this information and adds it to the data stream transmitted to the ground. Any of the TEMs can generate a trigger request in several ways: (i) If any tracker channel in the tracker module is over threshold, a trigger request is sent to the module’s TEM which then checks if a trigger condition is satisfied, typically requiring triggers from 3 $x,y$ planes in a row. If this condition is satisfied, the TEM sends a trigger request to the GEM. (ii) If a predetermined low-energy (CAL-LO) or high-energy (CAL-HI) threshold is exceeded for any crystal in the calorimeter module, a trigger request is sent to the GEM. The prompt ACD signals sent to the GEM are of two types: (i) a discriminated signal (nominal 0.4 MIPs threshold) from each of the 97 scintillators (89 tiles and 8 ribbons) of the ACD, used to (potentially) veto tracker triggers originating in any one of the sixteen towers, and (ii) a high-level discriminated signal (nominal 20 MIPs threshold) generated by highly ionizing heavy nuclei cosmic-rays (carbon-nitrogen-oxygen or CNO). The high-level CNO signal is used as a trigger, mostly for energy calibration purposes. During ground testing the CNO signal is only tested through charge injection. In addition, the GEM can logically group tiles and ribbons to form Regions Of Interest (ROIs) for trigger/veto purposes. An ROI can be defined as any combination of the ACD tiles and ribbons. Up to 16 ROIs can be defined through a series of configuration registers. The ROI signal is simply whether any one of the tiles that define the ROI is asserted. Finally, non-detector based trigger inputs to the GEM are used for calibration and diagnostic purposes. The GEM can utilize (i) a periodic signal derived from either the instrument system clock (nominally running at 20 MHz) or the 1 pulse-per-second GPS spacecraft clock (accurate to $\pm$1.5 $\mu$s), and (ii) a solicited trigger signal input that allows the instrument to be triggered through operator intervention. The spacecraft clock is also used to strobe the internal time base of the GEM, thus allowing an accurate measurement of the time of an event relative to the spacecraft clock. Instrument modeling {#s2.3} ------------------- The development and validation of a detailed Monte Carlo simulation of the LAT’s response to signals ([$\gamma$-ray]{}[s]{}) and backgrounds (cosmic-rays, albedo [$\gamma$-ray]{}[s]{}, etc.) has been central to the design and optimization of the LAT. This approach was particularly important for showing that the LAT design could achieve the necessary rejection of backgrounds expected in the observatory’s orbit. The instrument simulation was also incorporated into an end-to-end simulation of data flow, starting with an astrophysical model of the [$\gamma$-ray]{} sky, used to support the pre-launch development of software tools to support scientific data analysis. Figure \[fX\] summarizes the various components of the instrument simulation, calibration, and data analysis. The instrument simulation consists of 3 parts: (i) particle generation and tracking uses standard particle physics simulators of particle interactions in matter to model the physical interactions of [$\gamma$-ray]{}[s]{} and background particle fluxes incident on the LAT. In particular, the simulation of events in the LAT is based on the Geant4 (G4) Monte Carlo toolkit [@Agostinelli2003; @Allison2006], an object-oriented simulator of the passage of particles through matter. G4 provides a complete set of tools for detector modeling. In the LAT application, the simulation is managed by Gleam, our implementation of the Gaudi software framework [@Barrand2001], and so we use only a subset of the G4 tools. (ii) For a given simulated event the instrument response (digitization) is calculated parametrically based on the energy deposition and location in active detector volumes in the anticoincidence detector, tracker, and calorimeter. (iii) From the digitized instrument responses, a set of trigger primitives are computed and a facsimile of the Trigger and On-board Flight Software Filter (see §\[s2.2.4\]) is applied to the simulated data stream. Events that emerge from the instrument simulation (or real data) then undergo event reconstruction and classification (§\[s2.4\]), followed by background rejection analysis (§\[s2.4.3\]). As discussed in §\[s2.4.3\], the background rejection can be tuned depending on the analysis objectives. Information about the detector geometry and materials is stored in a set of structured XML files. These files are used by Gleam to build a G4 representation of the detector (and also to provide information about the detector to our event reconstruction packages). The geometry is quite detailed, particularly for the active elements, namely, the tracker silicon strip detectors, CsI crystals and diodes of the calorimeter, and anticoincidence detector scintillator tiles and ribbons. The current implementation has about 54,000 volume elements, of which about 34,000 are active. G4 contains a full suite of particle interactions with matter, including multiple scattering and delta-ray production for charged particles, pair production and Compton scattering for photons, and bremsstrahlung for $e^-$ and $e^+$, and low-energy interaction with atoms, as well as several models of hadronic interactions. The set of processes implemented is controlled by a “physics list,” which allows for considerable flexibility. In fact, a special version of the model of multiple scattering is used to provide better agreement with our measured data. Detector calibration data (thresholds, gains, non-uniformities, etc.) are used to convert the energy deposited in the active elements to instrument signals. For the tracker, dead channels are removed from the data at this stage, as well as any signals which would have overflowed the electronic buffers. (These same effects are taken into account again during event reconstruction, to aid the pattern recognition.) Event reconstruction and classification {#s2.4} --------------------------------------- The event reconstruction processes the raw data from the various subsystems, correlating and unifying them under a unique event hypothesis. The development of the reconstruction relies heavily on the Monte Carlo simulation of the events. In the following subsections, the basic blocks of the reconstruction are described. We start with track reconstruction, as it is key to developing the subsequent analysis of the other systems: the found tracks serve as guides as to what should be expected in both the calorimeter as well as the ACD for various event types. The analogous reconstruction processing for EGRET, a spark-chamber pair conversion telescope, which did not benefit from a detailed Monte Carlo model of the instrument, is described in @Thompson1993. ### Track reconstruction {#s2.4.1} Spatially adjacent hit tracker strips are grouped together, forming clusters, and the coordinates of these clusters are used in the track finding and fitting. Each cluster determines a precise location in $z$ as well as either $x$ or $y$. Because the planes of silicon detectors are arranged in closely spaced orthogonal pairs, both the $x$ and $y$ determinations can be made, albeit the choice of tracker technology (single-sided silicon strip detectors) imposes the ambiguities associated with projective coordinate readout on the initial pairing of the $x$ and $y$ coordinates when 2 or more particles pass through a detector plane. This ambiguity is resolved for tracks associated with particles that pass through more than one tracker module. For events with tracks confined to one module, the coordinate-pairing ambiguity is resolved for $\sim$90% of these events using calorimeter information. Strictly, resolution of the coordinate-pairing ambiguity is only of secondary importance, having primarily to do with background rejection. At the heart of track-finding algorithms is a mechanism to generate a track hypothesis. A track hypothesis is a trajectory (location and direction) that can be rejected or accepted based on its consistency with the sensor readouts. The generation algorithm is combinatoric, with a significant constraint imposed on the number of trial trajectories considered because of the available computing power. Two algorithms, described below, are used. *Calorimeter-Seeded Pattern Recognition* (CSPR): For most of the LAT science analysis, some energy deposition in the calorimeter is required. When present, both the centroid and shower axis of the calorimeter energy deposition can be computed using a moments analysis (see §\[s2.4.2\]) in most cases. The first and most-often selected algorithm is based on the assumption that the energy centroid lies on the trajectory. The first hit on the hypothesized track, composed of an $x,y$ pair from the layer in the tracker furthest from the calorimeter, is selected at random from the possible $x,y$ pairs. If a subsequent hit is found to be close to the line between the first hit and the location of the energy centroid in the calorimeter, a track hypothesis is generated. The candidate track is then populated with hits in the intervening layers using an adaptation of Kalman fitting [e.g., @Fruhwirth2000]. The process starts from the first hit. A linear projection is made into the next layer. The covariance matrix is also propagated to the layer and provides an estimate of the error ellipse that is searched for a hit to add to the track. The propagation of the covariance matrix includes the complete details of the material crossed, thereby providing an accurate estimate of the error caused by multiple scattering. If a candidate hit exists in the layer, it is incorporated into the trajectory weighted by the covariance matrices. The procedure is then iterated for subsequent layers, allowing for missing hits in un-instrumented regions. Adding more hits to the track is terminated when more than a specified number of gaps (planes without hits associated with the track) have accumulated (nominally 2). The whole process is repeated, starting with each possible $x,y$ pair in the furthest plane from the calorimeter and then continued using pairs from closer layers. After a track of sufficient quality is found and at least two layers have been looped over, the process is terminated. A byproduct of this process is the first Kalman fit to the track, providing the $\chi^2$, the number of hits, the number of gaps, etc. From these quantities a track quality parameter is derived and used to order the candidate tracks from “best” to “worst”. At high energies ($>$1 GeV) the first-hit search is limited to a cone around the direction provided by the calorimeter moments analysis in order to minimize confusion with hits caused by secondary particles generated by backsplash. The cone angle is narrowed as the energy increases, reflecting the improved directional information provided by the calorimeter. Following the completion of the CSPR, only the “best” track found is retained. The biasing caused by the track quality parameters makes this “the longest, straightest track” and hence, for $\gamma$ conversions, preferentially the higher-energy track of the $e^+e^-$ pair. The other tracks are discarded. The hits belonging to the best track are flagged as “used” and a second combinatoric algorithm is then invoked. *Blind Search Pattern Recognition* (BSPR): In this algorithm, calorimeter information is not used for track finding. Events having essentially no energy deposition in the calorimeter are analyzed using this algorithm as well as for subsequent track finding following the stage detailed above. The same procedure described for the CSPR is used, but here the selection of the second hit used to make the initial trajectory is now done at random from the next closest layer to the calorimeter. The trajectory formed by these two hits is projected into the following layer and if a hit in that layer lies sufficiently close to the projection a trial track is generated. The mechanism of populating the track candidate with hits follows that used in the CSPR, but without any estimation of the energy of the track, the multiple scattering errors are set by assuming a minimum energy (default: 30 MeV). Hits are allowed to be shared between tracks if the hit is the first hit on the best track (two tracks forming a vertex) or if the cluster size (number of strips) is larger than expected for the track already assigned to that hit. The total number of tracks allowed to be found is limited (default: 10). The final track fits must await an improved energy estimate to be made using the best track to aid in estimating the fraction of energy deposited in the calorimeter (see §\[s2.4.2\]). Once this is done, the energy is apportioned between the first two tracks according to the amount of multiple scattering observed on each. A subsequent Kalman fit is done but without re-populating the tracks with hits. The final stage of track reconstruction combines tracks into vertices. The process begins with the best track. The second track is selected by simply looping over the other tracks in the event. The distance of closest approach between the best track and the candidate second track is computed and if within a specified distance (default: 6 mm) a vertex solution is generated by covariantly combining the parameters of the two tracks. The $z$-axis location (coordinate along the instrument axis) of the vertex candidate is selected using the detailed topology of the first hits and is assigned either to be in the center of the preceding tungsten foil radiator, in the silicon detector itself, or within the core material of the tracker tray directly above the first hit. A quality parameter is created taking into account the $\chi^2$ for the combination of tracks, the distance of closest approach, etc. The first track is paired with the second track having the best quality parameter. These tracks are marked as “used” and the next unused track is selected and the process repeated. If a track fails to make a satisfactory vertex it is assigned to a vertex by itself. Thus all tracks are represented by a vertex. In addition to the “standard” vertexing discussed above, an additional improvement is possible if calorimeter information is included. In events where either during the conversion process or immediately thereafter much of the energy is in [$\gamma$-ray]{}[s]{} (due to Bremsstrahlung or radiative corrections), the charged tracks can point well away from the incident [$\gamma$-ray]{} direction. However the location of the conversion point is usually well determined and, when combined with the energy centroid location in the calorimeter, can give a fair estimate of the direction. The “best” track as well as the first vertex are combined covariantly with this direction using weights to apportion the total energy between these directions. These “neutral energy” solutions result in significantly reducing the non-gaussian tails of the PSF. ### Energy reconstruction {#s2.4.2} Energy reconstruction begins by first applying the appropriate pedestals and gains to the raw digitized signals. Then, for each calorimeter crystal, the signals from the two ends are combined to provide the total energy in the crystal (independent of location) and the position along the crystal where the energy was deposited. The result is an array of energies and locations. The three-dimensional calorimeter energy centroid is computed along with energy moments (similar to the moment of inertia, but with energy in place of mass). The shower direction is given by the eigenvector with the smallest eigenvalue. Initially, the overall energy is taken to be the sum of the crystal energies (“CALEnergyRaw” in Figure \[f2.6\]). Further improvements must await the completion of the fitted tracks. The trajectory provided by the best track (or best track vertex when available) is used as input to estimate the energy correction necessary to account for leakage out the sides and back of the calorimeter and through the internal gaps between calorimeter modules. Three different algorithms are applied to each event: a parametric correction (PC) based on the barycenter of the shower, a fit to the shower profile (SP) taking into account the longitudinal and transverse development of the shower, and a maximum likelihood (LK) fit based on the correlations of the overall total energy deposited with the number of hits in the tracker and with the energy seen in the last layer. Because the SP method starts to work beyond 1 GeV and the LK method works below 300 GeV, only the PC method covers the entire phase space of the LAT. Figure \[f2.6\] shows the results of the LK method applied to data obtained with electron beams at CERN entering the LAT calibration unit at an angle of 45$^\circ$ to the detector vertical axis. The energy resolutions obtained vary between 4% at 5 GeV and 2% at 196 GeV. At low energy ($\sim$100 MeV), a significant fraction ($\sim$50%) of the energy in a $\gamma$ conversion event can be deposited in the tracker and hence the determination of this contribution to the total energy becomes important. For this purpose the tracker is considered to be a sampling calorimeter where the number of hit silicon strips in a tracker layer provides the estimate of the energy deposition at that depth. The total number of hits in the thin radiator section, the thick radiator section and the non-radiator last layers is computed within a cone with an opening angle which decreases as $E^{-1/2}$, where $E$ is the apparent energy in the calorimeter. The “tracker” energy is added to the corrected calorimeter energy. Because the PC method gives an energy estimate for all events, it is used to iterate the Kalman track fits as mentioned in §\[s2.4.1\]. ### Background rejection {#s2.4.3} The vast majority of instrument triggers and subsequently downlinked data are background events caused by charged particles as well as earth albedo [$\gamma$-ray]{}[s]{}. The task of the hardware trigger is to minimize their effects on the instrumental deadtime associated with reading out the LAT. Subsequently the task of the onboard filter is to eliminate a sufficient number of background events without sacrificing celestial [$\gamma$-ray]{} events such that the resulting data can be transmitted to the ground within the available bandwidth. The final task is for the analysis on the ground to distinguish between background events and [$\gamma$-ray]{} events and minimize the impact of backgrounds on [$\gamma$-ray]{} science. The combination of these 3 elements reduces the background by a factor of almost $10^6$ while preserving efficiency for [$\gamma$-ray]{}[s]{} exceeding 75%. For reference, the average cosmic [$\gamma$-ray]{} event rate in the LAT is $\sim$2 Hz. [ ]{} \[s2.4.3.1\] In order to facilitate the development of the on-board triggering and filtering and subsequent event reconstruction and classification algorithms, a model of the background the LAT encounters in space has been developed. As shown in Table \[t2.4\], the background model includes cosmic rays and earth albedo [$\gamma$-ray]{}[s]{} within the energy range 10 MeV to $10^6$ MeV. Any particles that might either make non-astrophysical [$\gamma$-ray]{}[s]{} and/or need to be rejected as background are included. The model does not include X-rays or soft [$\gamma$-ray]{}[s]{} that might cause individual detectors within the LAT to be activated. The model is meant to be valid outside the radiation belts and the South Atlantic Anomaly (SAA); no particle fluxes from inside the radiation belts are included. The boundaries of the belts are defined to be where the flux of trapped particles is 1 proton cm$^{-2}$ s$^{-1}$ ($E > 10$ MeV). LAT does not take data inside the SAA. The fraction of time spent in the SAA is 14.6%. The AMS [@Aguilar2002] and BESS [@Haino2004] experiments provided important and accurate new measurements of the spectra of the protons and alpha particles, the most abundant of the various galactic cosmic-ray (GCR) components. AMS made detailed latitude-dependent measurements of the splash and reentrant albedo particles ($e^+, e^-$ and protons) in the energy range from $\sim$150–200 MeV up to the cutoff energies where the earth albedo components become lost in the much greater GCR fluxes. These fluxes will be updated with results from the Pamela satellite [@Picozza2007]. For albedo fluxes of particles with energies below $\sim$150 MeV, inaccessible to the AMS and other large instruments, measurements made by NINA and NINA-2 and a series of Russian satellite experiments with an instrument known as Mariya are used. The albedo [$\gamma$-ray]{} fluxes are taken from a reanalysis of the data collected by EGRET when the CGRO satellite was pointed at the Earth. The model is based on empirical fits to the referenced data. No time variability is included. The GCR fluxes are taken to be the same as those observed near solar minimum (maximum GCR intensities). The albedo fluxes may vary with time and be correlated with the GCR fluxes. The fluxes as observed by the NINA and Mariya experiments are used without correcting them for solar cycle variations. While an East-West cutoff variation was included that affects galactic cosmic ray components, all fluxes except albedo protons are assumed to be isotropic. The measurements are not complete enough for us to be able to account for variation in parameters such as the zenith angle of the particles or their pitch angles with respect to the local field. We have attempted to model some the zenith angle dependence for albedo protons, based not on measurements, but on modeling of the albedo [@Zuccon2003]. Further verification and improvement to the model are being done on orbit. The orbit averaged background fluxes in the model are shown in Figure \[f2.9\]. For charged particles, these fluxes are integrated over solid angle. It is straightforward to obtain fluxes per unit solid angle. For galactic cosmic ray components, divide by 8.7 sr, the solid angle of the visible sky that is not blocked by the Earth at [*Fermi*]{}[’s]{} orbital altitude. For the albedo components we have taken the reentrant and albedo fluxes to be the same. \[s2.4.3.2\] After track reconstruction, vertexing, and energy reconstruction, the events are analyzed to determine the accuracy of the energy determinations, the directional accuracy, and whether they are [$\gamma$-ray]{}[s]{}. All of the estimates are based on classification tree (CT) generated probabilities. This statistical tool was found to give the highest efficiency with the greatest purity, exceeding that which we obtained with either a more traditional cut-based analysis or with neural nets. Our usage of classification trees involves training a modest number of trees (a few to $\sim$10) and averaging over the results. The trees are “grown” by minimizing “entropy” as defined in statistics [@Breiman1984]. The final energy estimate for each event is made by first dividing the sample up into subsets according to which energy methods were reporting results (PC+LF+SP, PC+LF, PC+SP, and PC). When more than one energy method is available, the method selected is determined using a CT. The probability that the selected energy is better than the 1$\sigma$ resolution limit is estimated using a second CT. The subsets are then merged, now with a single “best” energy and a probability “knob” that can be used to lessen the presence of tails (both high and low) in the distribution of reconstructed energies at the expense of effective area. The analysis sorts the events according to where they occurred in the LAT tracker. (Events in the thick radiator portion have about a factor of 2 worse angular resolution due to increased multiple scattering.) When there is sufficient energy in the calorimeter (default: $>$10 MeV), the neutral energy solutions are used. If a 2-track vertex is present, a CT determines whether the vertex derived direction or the best track direction is used. As such there are four basic subsets: thin and thick radiator events and vertexed and 1-track events. For each of these subsets the probability that the reconstructed direction is more accurate than the theoretical 68% containment PSF is determined using a CT. The events are re-merged now with a “best” direction solution and associated CT-based probability. This image “knob” can be used to limit the long tails often associated with the PSFs of [$\gamma$-ray]{} instruments. The background rejection is by far the most challenging of all the reconstruction analysis tasks. This is due to the large phase space covered by the LAT and the very low signal-to-noise ratio in the incoming data ($\sim$1:300 for down-linked data). The first task is to eliminate the vast majority of the charged particle flux that enters within the FoV using the ACD in conjunction with the found tracks. One cannot simply demand that there are no triggers from the ACD because high-energy [$\gamma$-ray]{}[s]{} generate a considerable amount of back splash, from the shower that develops in the calorimeter, in the form of hard X-rays that can trigger several ACD tiles. Consequently only the tiles pointed at by the reconstructed tracks are used to establish a veto by the presence of a signal in excess of $\sim$$1/4$ of a minimum ionization event. Because the accuracy of the pointing is energy dependent due to multiple scattering, at low energy, only tiles within the vicinity of the track intersection with the ACD are used, while at high energy the region is restricted to essentially the one tile being pointed to. In addition there are several areas in the ACD where it is not possible to completely cover the acceptance region (e.g. the four vertical edge corners, the screw holes used to mount the tiles, etc.). Since these are known locations, tracks pointing at them must also be eliminated. However, these holes are small and account for a few percent of the surface area and reduce the events sample by $<$2%. The considerations for rejecting backgrounds involve the detailed topology of the events within the tracker and the overall match of the shower profile in 3D in both the tracker and the calorimeter. The tracker provides a clear picture of the initial event topology. For example the identification of a 2-track vertex immediately reduces the background contamination by about an order of magnitude. However a majority of events above 1 GeV don’t contain such a recognizable vertex due to the small opening angle of the $e^+ e^-$ pair along the incoming [$\gamma$-ray]{} direction. The observation of a significant number of extra hits in close proximity to the track(s) indicates they are electrons and hence from the conversion of a [$\gamma$-ray]{} while the presence of unassociated hits or tracks are a strong indicator of background. These as well as other considerations are used for training background rejection CTs. The final discriminator of background is the identification of an electromagnetic shower. Considerations such as how well the tracker solution points to the calorimeter centroid, how well the directional information from the calorimeter matches that of the track found in the tracker, as well as the width and longitudinal shower profile in the various layers of the calorimeter, are important in discrimination of backgrounds. Again the information from the reconstruction is used to train CTs and the resulting probability is used to eliminate backgrounds. The broad range of LAT observations and analysis, from GRBs to extended diffuse radiation, leads to different optimizations of the event selections and different rates of residual backgrounds. For example, in analysis of a GRB, the relatively small region of the sky as well as the very short time window allow the background rejection cuts to be relaxed relative to an analysis of a diffuse source covering a large portion of the sky. Furthermore a key science attribute for GRB observations is the time evolution and the sensitivity of a measurement to rapid time variation scales as the square root of the number of detected burst photons. The background rejection analysis has been constructed to allow analysis classes to be optimized for specific science topics. Table \[t2.5\] lists 3 analysis classes that have been defined based on the backgrounds expected in orbit, current knowledge of the [$\gamma$-ray]{} sky, and the performance of the LAT. Our estimates of LAT performance are given in terms of these analysis classes. Common to all of these analysis classes is the rejection of the charged-particle backgrounds entering within the field of view. The classes are differentiated by an increasingly tighter requirement that the candidate photon events in both the tracker and the calorimeter behave as expected for [$\gamma$-ray]{} induced electromagnetic showers. The loosest cuts apply to the Transient class, for which the background rejection was set to allow a background rate of $<$2 Hz, estimated using the background model described in §\[s2.4.3.1\], which would result in no more than one background event every 5 sec inside a $10^\circ$ radius about a source. The Source class was designed so that the residual background contamination was similar to that expected from the extragalactic [$\gamma$-ray]{} background flux over the entire field of view. Finally, the Diffuse class has the best background rejection and was designed such that harsher cuts would not significantly improve the signal to noise. The various analysis cuts and event selections will be optimized for the conditions found on-orbit during the 1st year all-sky survey phase. Note that these 3 analysis classes are hierarchical; that is all events in the Diffuse class are contained in the Source class and all events in the Source class are in the Transient class. The residuals of background events for the 3 analysis classes are shown in Figure \[f2.10\]. For the Diffuse class, the resulting rejection factor is $\sim$1:$10^6$ at some energies (e.g., $\sim$10 GeV) while retaining $>$80% efficiency for retaining [$\gamma$-ray]{} events. The residual background is worse at low energy particularly for events originating in the thick radiator portion of the tracker. It is here that “splash” backgrounds, entering the backside of the calorimeter can undergo interactions that result in low energy particles which range out in the thick radiators, thus mimicking an event originating in the thick tracker section. In a sense the thick section shields the thin section from this flux and hence the thin section is somewhat cleaner. The leaked background events generally fall into two categories: irreducible events and reducible events. The irreducible events are events in which a background particle interacts in the passive material outside the ACD or within the first $\sim$1 mm of the ACD scintillator and the resulting secondaries contain [$\gamma$-ray]{}[s]{} which enter inside the FoV. This can happen in the case of entering $e^+$ which annihilate to two photons, entering $e^-$ or $e^+$ which bremsstrahlung essentially all their energy to a single photon, and proton interactions that make a $\pi^0$ which decays to 2 photons with the rest of the secondaries either neutral or aimed away from the LAT. In these cases the ACD has no signals and a [$\gamma$-ray]{} is seen in the LAT. There is no way in principle to distinguish and eliminate these events from the celestial [$\gamma$-ray]{} signal and this component is the result of the reality of contemporary instrumentation and the precautionary measures that must be taken to survive in low earth orbit. This irreducible component constitutes $\sim$60% of the residual background events with measured energies above 100 MeV. The reducible background component comprises events that in principle should be identifiable. These events leak through the various filters because they are in the far tails of their parent distributions, overlapping the [$\gamma$-ray]{} (signal) distribution. The filter parameters are chosen to optimize efficiency for [$\gamma$-ray]{}[s]{} versus background rejection. Additional contributions to the reducible background component come from the fact that any real detector will have inefficiencies caused by real world design choices such as gaps in the silicon detector planes of the tracker and in the ACD. This reducible component however is easily monitored by comparing the apparent fluxes of events with and without vertices. The difference is essentially the reducible component because the vertexed event sample has 10 times fewer such reducible background events. Performance of the LAT {#s2.5} ---------------------- The performance of the LAT is basically determined by the design of the LAT hardware, the event reconstruction algorithms (i.e., the accuracy and efficiency with which the low-level event information is used to determine energy and direction), and event selection algorithms (i.e., the efficiency for identifying well reconstructed [$\gamma$-ray]{} events). Figures \[f2.11\] – \[f2.15\] summarize the performance of the LAT. The performance parameters are subject to change as event selection algorithms are further optimized, particularly during the early part of on-orbit operations of [*Fermi*]{}. For the most up-to-date performance parameters go to http://www-glast.slac.stanford.edu/software/IS/glast\_lat\_performance.htm. Figure \[f2.11\] shows the on-axis effective area versus energy for each of the analysis classes defined in Table \[t2.5\]. Contributions from conversions in both the thin and thick sections of the tracker are included, with each contributing about 50% of the effective area. Note that the peak effective area, near 3 GeV, is nearly the same for all 3 analysis classes, while at energies below 300 MeV the effective area for the transient class is a factor of $\sim$1.5 larger than the for the diffuse class. Figure \[f2.12\] shows the effective area for the source class on-axis and at $60^\circ$ off-axis. Figure \[f2.13\] shows the telescope’s acceptance, the average effective area times the field-of-view. Again, the differences between the analysis classes are largest at low energies. Figure \[f2.14\] shows the energy dependence of the 68% containment radius (space angle) for [$\gamma$-ray]{} conversions in the thin section of the tracker that are incident either on-axis or at $60^\circ$ off-axis for the source class. The PSF for [$\gamma$-ray]{}[s]{} converting in the thick section of the tracker is about twice as wide. Figure \[f2.15\] shows the energy resolution of the LAT versus energy for the source class. With a diffuse [$\gamma$-ray]{} background model based on EGRET observations and the instrument performance summarized above, the source sensitivity of the LAT can be estimated. The source sensitivity of course depends not only on the flux of the source but it also depends on the spectrum of the source. Figure \[f2.16\] shows the integral source flux above energy $E$ versus energy corresponding to a $5\sigma$ detection after one year of scanning mode observations. Figure \[f2.17\] shows the differential source flux (in $1/4$ decade bins) corresponding to a $5\sigma$ detection. ### LAT performance tests {#s2.5.1} The design of the LAT was optimized using Monte Carlo simulations. Verification of the design and simulations was done with a series of beam tests at the SLAC, CERN and GSI heavy ion accelerator laboratories. In addition hardware prototypes as well as the flight instrument have been tested using cosmic rays. The early prototype tests at SLAC have already been mentioned. The most extensive beam test was at CERN in 2006. The CERN beams were selected because they cover almost the entire energy range of the LAT for on-orbit operations as well as provide large fluxes of hadrons to verify the modeling of background interactions within the LAT. Because schedule prevented doing beam tests with the entire LAT, a Calibration Unit (CU) consisting of two complete tracker and 3 calorimeter modules was assembled. The CU readout electronics is a copy of the flight instrument data acquisition system. The CU is also instrumented with several ACD scintillator tiles to measure the backsplash response from the calorimeter at high energies. The overall agreement between the Monte Carlo simulations of the CU and the beam test data are excellent, including the overall tracker performance and the PSF, the backsplash into the ACD, and the modeling of hadronic interactions. The largest discrepancies involve the energy calibration in the calorimeter which was found to be low by $\sim$7%. A much more complete discussion of the preliminary beam test results, comparing the CU to the Monte Carlo simulations can be found in @Baldini2007. In addition to the accelerator beam tests, several times during the assembly of the LAT, cosmic ray triggers were recorded to verify the proper functioning of the LAT modules as they were added to the instrument array. Collection of cosmic-ray data, recorded at a trigger rate of $\sim$400 Hz, continued through the environmental testing and pre-launch preparations of the [*Fermi*]{} telescope. While terrestrial cosmic rays are quite messy (e.g., multiple particle types, range of arrival directions) compared to a particle beam from an accelerator, the LAT has sufficient power as a particle detector to provide clean samples of sea-level muons, resulting in relatively large samples of muon events that allow precision testing. With these events, calibrations, efficiencies, and alignment issues were successfully addressed. The first 60 days after launch were a commissioning period for the [*Fermi*]{} spacecraft and the LAT. During part of this period the LAT was subjected to a relatively high rate albedo photon data by pointing at the earth’s limb, and directly observed the “splash” albedo background component with nadir pointed runs, as well as run with modified triggers to allow high-energy cosmic rays to be efficiently collected to verify alignment and efficiencies that may have been affected during the launch. The early operations tests included checks of internal timing and absolute timing, subsystem calibrations, characterization of the perimeter of the South Atlantic Anomaly, tuning the onboard event filters, and commissioning the on-board detection of GRBs. Instrument operations {#s2.6} --------------------- ### Onboard science processing {#s2.6.1} A primary objective of onboard science processing is to provide rapid detection and localization of GRBs. The output of this processing can trigger an autonomous re-pointing of the [*Fermi*]{} to keep the GRB within the LAT FoV for observation of high-energy afterglows and is made available to support follow-up observations of afterglows by other observatories. The [*Fermi*]{} Gamma-ray Burst Monitor also produces onboard detections and localizations, however for burst that trigger the LAT, the LAT’s better point spread function results in significantly improved localization. The onboard estimates of the celestial coordinates of the GRB and the error region are distributed via the Gamma-ray burst Coordinate Network (GCN). The onboard science processing consists of algorithms to (1) select [$\gamma$-ray]{} candidate events, (2) reconstruct directions of [$\gamma$-ray]{} candidate events and (3) search for and localize high energy transients. The information available to the onboard GRB search algorithm differs substantially from that eventually available on the ground. The event selection is based on parameters previously calculated for the onboard filter (described in §\[s2.2.4\]). The onboard software uses fairly simple, computationally efficient algorithms to calculate the directions of candidate [$\gamma$-ray]{} events. The efficiency for successfully reconstructing an event direction is within 25% of what can be achieved with subsequent ground processing however, the reconstructed directions are about a factor of 2 to 5 worse. The onboard GRB detection algorithm utilizes both the temporal and spatial characteristics of GRBs. It works by associating a probability for a cluster of tracks to be located on a small part of the sky during a short interval of time. [$\gamma$-ray]{} candidate events with reconstructed directions are fed to the algorithm in time order. The algorithm searches a list of the $n$ most recent events for the cluster of events that has the smallest probability of occurring in time and space. If the probabilities pass a pre-selected threshold, the time and location of the cluster is passed to a second stage of processing which considers events over a longer time interval. A GRB is declared when cluster probabilities in the second stage exceed a pre-defined threshold. The algorithm will then calculate refined localizations on a configurable sequence of time intervals. The initial burst location and each updated location is sent promptly to the ground via the Tracking and Data Relay Satellite System (TDRSS) and then to the GCN (http://gcn.nasa.gov/). The GCN provides locations of GRBs (the Notices) detected onboard by the LAT or the GBM and reports on follow-up observations (the Circulars and the Reports) made by ground-based and space-based optical, radio, x-ray, TeV, and other observers. Triggering on bursts depends on settable parameter choices. We used a phenomenological burst simulator and background model to guide the initial choice of filter parameter values. These parameters will be optimized once a large enough sample of bursts has been identified via ground reconstruction. ### Pipeline and data products {#s2.6.2} LAT science data arrive at the LAT Instrument Science Operations Center (ISOC) from the [*Fermi*]{} Mission Operations Center at about 3 hour (2 orbits) intervals, 24 hours per day, in approximately 1.5 GB downlinked data sets. Automated processing of the data implements several analysis functions. The primary function is to interpret the event data, via pattern recognition and reconstruction, to indicate the nature of the event as either celestial [$\gamma$-ray]{} photons or background, and determine the direction, arrival time, and energy and provide estimates of the associated errors. During the process of correlating data from all the subsystems, detailed information is available on the operation of the LAT and is collected and trended for monitoring purposes. Once the photons have been isolated, the level-1 data are immediately used to carry out several higher level science analysis tasks that include searching for and refining GRB properties; searching for flaring sources; and tracking the light curves of a pre-selected list of sources. The processing pipeline is designed to allow parallel processing of events, with dependencies enabled so that processes can wait for the parallel processing to finish before aggregating the results. It can process an arbitrary graph of tasks. The pipeline is run in a java application server and interacts with farms of batch processors. About 300 Hz of downlinked on-orbit data can be processed by 100 computing cores within 1-2 hours, allowing processing to finish before the next downlink arrives. Reconstruction inflates the raw science data volume by approximately a factor of 20. Keeping all events processed requires about 150 TB of disk per year. The total LAT pipeline processing compute facility is sized to accommodate prompt processing, reprocessing and simulations. 300 computing cores for reprocessing of data, allow one year of data to be reprocessed in about one month. This is an upper limit on the reprocessing time, since use will be made of the much larger user batch processor pool in the SLAC compute farm. The reconstructed [$\gamma$-ray]{} photon events are then made available, along with instrument response functions and high-level analysis tools, etc., to the [*Fermi*]{} Science Support Center (FSSC) for distribution to the community at the conclusion of the first year on-orbit verification and sky-survey phase and during subsequent mission phases. After completion of the verification and sky survey phase (year 1) of the mission, these data should arrive at the FSSC within about 3-4 hours after arrival of unprocessed data at the ISOC. Automated science processing operates on 3 time scales: per downlink, per week and per month. During year 1 as well as beyond, the ISOC will deliver high-level science data products, resulting from Automated Science Processing (ASP), to the FSSC. These include light curves and GCN notices and circulars for GRBs and AGN flares as well as fluxes, source locations and associated errors for transient or flaring sources. ### Automated science processing {#s2.6.3} Time critical analysis tasks related to detection and characterization of transient sources, referred to as Automated Science Processing (ASP), are performed on the reconstructed and classified events from the level-1 pipeline to facilitate timely follow-up observations by other observatories. The ASP tasks relevant to GRBs (and other impulsive phenomena, such as solar flares) are the refinement of information for GRBs that were detected with onboard processing, the search for untriggered GRBs, and the rapid search for and characterization of [$\gamma$-ray]{} afterglow emission. In this context, untriggered means not triggered by the LAT; however, information about GRBs detected by the GBM and GCN notices from other observatories will be used in conjunction with this search. The baseline for ASP processing uses an unbinned likelihood analysis to determine the position and uncertainty of a GRB and evaluates the spectral index and fluence by fitting a power-law spectrum to the LAT events, also via an unbinned likelihood analysis. The refinement task uses a Bayesian Blocks temporal analysis [@Scargle1998; @Jackson2005] to characterize the prompt burst light curve, and from that analysis, it determines the burst start time and duration. The search for untriggered GRBs uses an algorithm similar to the one developed for onboard detection; but since ASP analysis uses ground processed events, it benefits from a substantially lower residual background rate as well as from more accurate energy and directional reconstructions. Any independently-available information, such as directions and times of GRBs seen by other instruments, is used to increase the sensitivity of the search. The afterglow search uses an unbinned likelihood analysis to fit for a point source at the best-fit GRB position using a likelihood analysis on all data available for $\sim$5 hours after the time of the GRB. The principal products of ASP processing for GRBs are Notices and Circulars released via GCN; for GRB refinements the latency for release of these products will eventually be no more than 15 minutes from the availability of the necessary level-1 data. For GRB and afterglow searches the latency will be less than 1 hour. Overall catalogs of LAT GRBs will be produced by the LAT collaboration. The ASP tasks relevant to blazars and other long-term transient sources relate to monitoring for episodes of flaring. For optimum sensitivity this involves both routinely evaluating the fluxes for a set of sources as well as searching for new transients not already on the list of monitored sources. The flux monitoring task uses an unbinned likelihood analysis to evaluate the fluxes and upper limits of a specified list of sources on daily and weekly bases. The ASP-monitored source list is not static – bright transients will be added as they are found, for example. The general search for flaring sources, to find transients that are not on the list of sources being monitored, is run on daily and downlink ($\sim$3–4 hr) time scales. The baseline algorithm for this search monitors for changes in exposure-corrected maps of counts. Newly-detected transients meeting the detection criteria are released via GCN notices or Astronomers Telegrams (ATELs). The latency for updating daily light curves of monitored sources will eventually be less than 6 hours after the availability of the needed level-1 data. The general search for flaring sources is expected to take less than 1 hour per downlink. The algorithms and event classification cuts used for the ASP analyses are continuing to be refined during flight. The ASP processing tasks are built as part of the general pipeline system in the LAT ISOC, and are extensible as needed. A parallel set of tasks uses the LAT science data for the bright pulsars to validate the instrument response functions and to monitor the high-level performance of the LAT. Key science objectives {#s3} ====================== The LAT is designed to address a number of scientific objectives that include (i) resolving the [$\gamma$-ray]{} sky and determining the origins of diffuse emission and the nature of unidentified sources (§\[s3.1\]), (ii) understanding the mechanisms of particle acceleration in celestial sources (§\[s3.2\]), (iii) studying the high-energy behavior of GRBs and transients (§\[s3.3\]), (iv) probing the nature of dark matter (§\[s3.4\]), and (v) using high-energy [$\gamma$-ray]{}[s]{} to probe the early universe (§\[s3.5\]). The key objectives are largely motivated by the discoveries of EGRET ($\sim$30 MeV – 10 GeV) and of ground-based atmospheric Cherenkov telescopes (ACT) above $\sim$100 GeV. Progress in several areas requires coordinated multi-wavelength observations with both ground and space-based telescopes. The following sections describe how the LAT enables these scientific studies. Resolve the [$\gamma$-ray]{} sky: the origins of diffuse emission and the nature of unidentified sources {#s3.1} -------------------------------------------------------------------------------------------------------- High-energy [$\gamma$-ray]{} sources are seen against a diffuse background of Galactic and extragalactic radiation. Particularly at low Galactic latitudes, the diffuse radiation is bright and highly structured. About 80% of the high-energy luminosity of the Milky Way comes from processes in the interstellar medium (ISM). Because these background emissions are themselves not completely understood, analysis is an iterative process. As sources are discovered and distinguished from the background, the diffuse background model can be improved, thus allowing better analysis of the sources [e.g., @Hunter1997; @Sreekumar1998; @Hartman1999; @Strong2004a]. ### Unidentified EGRET sources {#s3.1.1} Although time signatures allowed identification of many EGRET sources as pulsars or blazars, in the third EGRET catalog [@Hartman1999] 170 of the 271 sources had no firm identifications. Progress towards identifications has been limited primarily by the relatively large EGRET error boxes that often contain many potential counterparts. A wide variety of astrophysical objects have been suggested as possible counterparts for some of these sources. Some examples are: newly-found radio or X-ray pulsars [e.g., @Kramer2003; @Halpern2001], isolated neutron stars [e.g., RX J1836.2+5925 @Mirabal2001; @Reimer2001; @Halpern2002], star forming regions or association of hot and massive stars [e.g., @Kaaret1996; @Romero1999], supernova remnants [e.g., @Sturner1995; @Esposito1996], pulsar wind nebulae [e.g.. @Roberts2001], and microquasars, such as LSI 61$^\circ$303 [@Tavani1998; @Paredes2000]. Figure-of-Merit approaches have increased the number of [$\gamma$-ray]{} sources at high Galactic latitudes identified, with moderate confidence, with blazars [@Mattox2001; @Sowards-Emmerd2003; @Sowards-Emmerd2004; @Sowards-Emmerd2005]. Population studies of the unidentified EGRET sources have also provided clues about their natures. For example, spatial-statistical considerations and variability studies provide evidence for a population of Galactic and variable GeV [$\gamma$-ray]{} emitters among the unidentified EGRET sources [@Nolan2003]. Many sources may be related to star-forming sites in the solar neighborhood or a few kiloparsecs away along the Galactic plane [@Gehrels2000]. These sites harbor compact stellar remnants, SNRs and massive stars, i.e., many likely candidate [$\gamma$-ray]{} emitters. Evidence exists for a correlation with SNRs [@Sturner1995] as well as OB associations [@Romero1999], reviving the SNOB concept of @Montmerle1979 or making the pulsar option attractive. Pulsar populations may also explain a large fraction of unidentified sources close to the Galactic plane [@Yadigaroglu1997] and possibly in the nearby starburst Gould Belt [@Grenier2000]. Other candidate objects among the unidentified sources include radio-quiet neutron star binary systems [@Caraveo1996] and systems with advection-dominated accretion flows onto a black hole such as Cygnus X-1, recently detected as a flaring source by MAGIC [@Albert2007]. With regard to extragalactic sources, understanding the nature of the unidentified sources is important because new [$\gamma$-ray]{} emitting source classes (e.g., normal galaxies, clusters, etc.) are likely to be found in addition to the well-established blazars. A census of these sources is important for establishing their contribution to the extragalactic [$\gamma$-ray]{} background (EGRB; see §\[s3.1.3\]). High-confidence detections and identifications of the first representatives of other extragalactic [$\gamma$-ray]{} sources, such as galaxy clusters [@Dar1995; @Colafrancesco1998; @Totani2000; @Loeb2000; @Gabici2003], will enable comparisons and normalization of theoretical predictions of their contributions to the EGRB. The LAT addresses these challenges with good source localization, energy spectral measurement over a broader range, and nearly continuous monitoring of sources for temporal variability. These capabilities greatly facilitate the source identification process in the following ways: \(1) *Provide good source localization for the majority of [$\gamma$-ray]{} sources, including all of the EGRET detected sources.* For $5\sigma$ one-year LAT survey sources and for EGRET sources (Figure \[f3.1\]), the typical error box sizes (68% confidence radius) are $2.5\arcmin$ and $<$$0.4\arcmin$ respectively, for an $E^{-2}$ source and $12\arcmin$ and $2\arcmin$ respectively, for a source with a spectral cut-off at $\sim$3 GeV, as anticipated for pulsars. More precise source locations and smaller positional uncertainties are a prerequisite for more efficient and conclusive source identifications, with the exception of [$\gamma$-ray]{} variability that is tightly correlated with variability in another band. Small error boxes significantly reduce the number of potential counterparts at other wavebands. Better source localization will also improve spatial-statistical correlation studies by reducing the number of chance coincidences. Finally, a number of unidentified EGRET sources that are likely unresolved composite sources [e.g., @Sowards-Emmerd2003], will be resolved into individual sources. \(2) *Measure source spectra over a broad energy range.* Determining [$\gamma$-ray]{} spectra with the LAT’s resolution will allow investigation of features intrinsic to the sources such as absorption signatures, spectral breaks, transitions, and cutoffs (e.g., attenuation of blazar spectra at high-energy due to $\gamma+ \gamma \to e^+ + e^-$ in the extragalactic background light). The LAT’s wide energy coverage will connect the GeV sky to ground-based very high energy [$\gamma$-ray]{} observations. For example, LAT spans the energy range where the pulsed emission component in pulsars appears to fade out (a few GeV), to be dominated at higher energies by energetic synchrotron nebulae powered by the pulsar. \(3) *Measure [$\gamma$-ray]{} light curves over a broad range of timescales.* The large effective area, wide field of view, stability, and low readout deadtime of the LAT enable measurement of source flux variability over a wide range of timescales. Figure \[f3.2\] illustrates this capability. Coupled with the scanning mode of operation, this capability enables continual monitoring of source fluxes that will greatly increase the chances of detecting correlated flux variability with other wavelengths. It allows periodicity and modulation searches, for example, for orbital modulation in close binaries. LAT sources can be investigated for potential periodicities on time scales of milliseconds to years, encompassing millisecond pulsars, pulsars and binary systems hosting a neutron star. Extrapolating from EGRET analyses of Geminga [e.g., @Mattox1996; @Chandler2001], the LAT sensitivity allows searches in sources as faint as $\sim$$5\times10^{-8}$ photons cm$^{-2}$ s$^{-1}$ ($E > 100$ MeV) without prior knowledge of the period and period derivative from radio, optical, or X-ray observations [@Atwood2006; @Ziegler2008]. Such a capability is crucial for revealing radio-quiet, Geminga-like sources [@Bignami1996] which are expected to contribute significantly to the galactic unidentified source population [@Gonthier2007; @Harding2007]. In general, variability can be a discriminator for different source populations, i.e., expected steadiness in the [$\gamma$-ray]{} emission in the case of molecular-cloud-related CR interactions, [$\gamma$-ray]{}[s]{} from SNRs, starburst galaxies, or in galaxy clusters versus modulated or stochastic variable emission from Active Galactic Nuclei, Galactic relativistic jet sources, black hole or neutron stars in binary systems with massive stars, and pulsar wind nebulae. Population studies for a prospective source class help to select the most promising individual candidate sources for carrying out deep multi-frequency identification campaigns based on their broadband non-thermal properties and also help with investigating common characteristics of the candidate population. For example, galaxy clusters, as a candidate population, can be characterized by mass as deduced from optical richness, by temperature and mass functions, by applying virial mass-over-distance constraints, and by observational characteristics such as the presence or absence of merger activity, the presence or absence of diffuse radio halos or indications of nonthermal spectral components in the hard-X-rays. LAT observations should allow at least several members among each new candidate source populations to be individually discovered and characterized. In view of the large number of expected detections, most probably representing different source classes, confirmation of a given population as [$\gamma$-ray]{} emitters will require a common criteria for statistical assessment [e.g., @Torres2005], as well as dedicated multiwavelength observing campaigns [e.g., @Caraveo2007]. Given the advance for point-source detection provided by the LAT, anticipating new observational features presently unknown in GeV astrophysics is also important. Although speculative at present, GeV [$\gamma$-ray]{} phenomena might be found that initially, or ultimately, have no detectable correspondence in other wavebands (e.g., GeV forming galaxy clusters: @Totani2000; dark matter clumps: @Lake1990 [@Calcaneo-Roldan2000]). ### Interstellar emission from the Milky Way, nearby galaxies, and galaxy clusters {#s3.1.2} The diffuse emission of the Milky Way is an intense celestial signal that dominates the [$\gamma$-ray]{} sky. The diffuse emission traces energetic particle interactions in the ISM, primarily protons and electrons, thus providing information about cosmic-ray spectra and intensities in distant locations [e.g., @Hunter1997]. This information is important for studies of cosmic-ray acceleration and propagation in the Galaxy [@Moskalenko2005]. [$\gamma$-ray]{}[s]{} can be used to trace the interstellar gas independently of other astronomical methods, e.g., the relation of molecular H$_2$ gas to CO molecule [@Strong2004c] and hydrogen overlooked by other methods [@Grenier2000a]. The diffuse emission may also contain signatures of new physics, such as dark matter, or may be used to put restrictions on the parameter space of supersymmetrical particle models and on cosmological models (see §\[s3.4\]). The Galactic diffuse emission must also be modeled in detail in order to determine the Galactic and extragalactic [$\gamma$-ray]{} backgrounds and hence to build a reliable source catalog. Accounting for the diffuse emission requires first a calculation of the cosmic-ray (CR) spectra throughout the Galaxy [@Strong2000]. A realistic calculation that solves the transport equations for CR species must include gas and source distributions, interstellar radiation field (ISRF), nuclear and particle cross sections and nuclear reaction network, [$\gamma$-ray]{} production processes, and energy losses. Finally, the spectrum and spatial distribution of the diffuse [$\gamma$-ray]{}[s]{} are the products of CR particle interactions with matter and the ISRF. One of the critical issues for diffuse emission remaining from the EGRET era is the so-called “GeV excess”. This puzzling excess emission above 1 GeV relative to that expected [@Hunter1997; @Strong2000] has shown up in all models that are tuned to be consistent with directly measured cosmic-ray nucleon and electron spectra [@Strong2004a]. The excess has shown up in all directions, not only in the Galactic plane. The origin of the excess is intensively debated in the literature since its discovery by @Hunter1997. The excess can be the result of an error in the determination of the EGRET effective area or energy response or could be the result of yet unknown physics [for a discussion of various hypotheses see @Moskalenko2005]. Recent studies of the EGRET data have concluded that the EGRET sensitivity above 1 GeV has been overestimated [@Stecker2008] or underestimated [@Baughman2007] or imply different cosmic-ray energy spectra in other parts of the Galaxy compared to the local values [@Strong2004a; @Porter2008]. If these possibilities are eliminated with high confidence then it may be possible to attribute it to exotic processes, e.g., dark matter annihilation products [@de_Boer2005]. See, however, a discussion on limitations in the determination of the diffuse Galactic [$\gamma$-ray]{} emission using EGRET data and a word of caution in @Moskalenko2007. With its combination of good spatial and energy resolution over a broad energy range, the LAT can test different hypotheses. LAT measurements of the Galactic $\gamma$-radiation offer good uniformity and high sensitivity as well. As noted above, understanding the Galactic diffuse emission is critical to analysis of LAT sources and important for cosmic ray and dark matter studies. Optimizing this model over the entire sky will have a high priority in the early phases of the mission. The same basic considerations needed for the development of the model of Galactic diffuse [$\gamma$-ray]{} emission also apply to other galaxies that are candidates for study with the LAT. For example, the LAT will resolve the Large Magellanic Cloud in detail and, in particular, map the massive star-forming region of 30 Doradus. By detecting further members among the normal galaxies in our Local Group, and galaxies with enhanced star formation rates (e.g., Ultra Luminous Infrared Galaxies and starburst galaxies), LAT observations can establish independent measures of cosmic ray production and propagation. Both M31 and the Small Magellanic Cloud are predicted to be detectable with LAT [@Pavlidou2001], and the nearest starburst galaxies as well [@Torres2004]. Galaxy clusters emitting high-energy [$\gamma$-ray]{}[s]{} are, although well hypothesized, observationally not yet established emitters in the GeV sky [@Reimer2003]. Predictions for galaxy clusters as a candidate source class for detectable high-energy emission relate to observations of diffuse radio signatures [@Giovannini1999; @Feretti2004 and references therein], revealing the existence of relativistic electrons in a number of galaxy clusters. Further hints of the presence of nonthermal particles in galaxy clusters arise from observations of hard emission components in case of a few nearby but X-ray bright clusters [@Rephaeli2008 and references therein]. Similarly, large scale cosmological structure formation scenarios predict high-energy [$\gamma$-ray]{} emission from galaxy clusters at a level detectable for [*Fermi*]{}/LAT [@Keshet2003]. Both particle acceleration in merger processes as well as injection of relativistic particles through feedback from AGN as cluster members can provide the mechanism to produce non-thermal particles energized well into the energetic regime of LAT and perhaps beyond. Since galaxy clusters can store cosmic rays [@Berezinsky1997] injected either by AGNs or accelerated by primordial shocks, [$\gamma$-ray]{}[s]{} can be produced in $pp$ interactions via production and decay of neutral pions and from annihilating DM or supersymmetrical particles. However, weak constraints from measurements of the intercluster magnetic field ranging from 0.1 $\mu$G to 1 $\mu$G leave assessments of the total energy content, as well as the relative fraction in relativistic electrons and protons still open to speculation. The first clear detection of high-energy [$\gamma$-ray]{} emission from a galaxy cluster will undoubtedly constrain the baryonic particle content as well as the uncertainly in the estimates of the magnetic field, and consequently enable vastly improved modeling of galaxy clusters over the entire electromagnetic spectrum. ### Extragalactic diffuse emission {#s3.1.3} An isotropic, apparently extragalactic component of the high-energy [$\gamma$-ray]{} sky was studied by EGRET [@Sreekumar1998]. This extragalactic [$\gamma$-ray]{} background (EGRB) is a superposition of all unresolved sources of high-energy [$\gamma$-ray]{}[s]{} in the universe plus any truly diffuse component. A list of the contributors to the EGRB includes “guaranteed” sources such as blazars and normal galaxies [@Bignami1979; @Pavlidou2002], and potential sources such as galaxy clusters [@Ensslin1997], shock waves associated with large scale cosmological structure formation [@Loeb2000; @Miniati2002], distant [$\gamma$-ray]{} burst events [@Casanova2007], pair cascades from TeV [$\gamma$-ray]{} sources and UHE cosmic rays at high redshifts (so-called Greisen-Zatsepin-Kuzmin cut-off). A consensus exists that a population of unresolved AGN certainly contribute to the EGRB inferred from EGRET observations; however predictions range from 25% up to 100% of the EGRB [@Stecker1996; @Mukherjee1999; @Chiang1998; @Mucke2000]. A number of exotic sources that may contribute to the EGRB have also been proposed: baryon-antibaryon annihilation phase after the Big Bang [@Stecker1971; @Gao1990; @Dolgov1993], evaporation of primordial black holes [@Hawking1974; @Page1976; @Maki1996], annihilation of so-called weakly interacting massive particles (WIMPs) [@Silk1984; @Rudaz1991; @Jungman1996; @Bergstrom2001; @Ullio2002; @Elsasser2005], and strings [@Berezinsky2001]. The EGRB is difficult to disentangle from the intense Galactic diffuse foreground (see previous section) because it is relatively weak and has a continuum spectrum with no strongly distinguishing features. Indeed, determination of the EGRB spectrum depends on the adopted model for the Galactic diffuse emission spectrum, which itself is not yet firmly established. Even at the Galactic poles, the EGRB does not dominate over the Galactic component, with its flux comparable to the Galactic contribution from inverse Compton scattering of the interstellar radiation from stars and dust near the Galactic plane and the cosmic microwave background [@Strong2000; @Moskalenko2000]. The determination of the EGRB is thus model dependent and influenced by the adopted size of the Galactic halo, the electron spectrum there, and the spectrum of low-energy background photons which must be determined independently. Recent studies suggest that there are two more diffuse emission components originating nearby in the solar system: [$\gamma$-ray]{} emission due to inverse Compton scattering of solar photons on cosmic-ray electrons [@Moskalenko2006; @Orlando2007; @Orlando2008] and a [$\gamma$-ray]{} glow around the ecliptic due to the albedo of small solar system bodies (produced by cosmic-ray interactions) in the Main Asteroid Belt between the orbits of Mars and Jupiter and Kuiper Belt beyond Neptune’s orbit [@Moskalenko2008], for more details see §\[s3.2.3\]. Extensive work has been done [@Sreekumar1998] to derive the spectrum of the EGRB from EGRET data. Sreekumar et al. (1998) used the relation of modeled Galactic diffuse emission to total measured diffuse emission to determine the EGRB, as the extrapolation to zero Galactic contribution of the total diffuse emission. The derived spectral index $-2.10\pm0.03$ appears to be close to that of [$\gamma$-ray]{} blazars. Using a different approach, @Dixon1998 concluded that the derived EGRB is affected by a significant contribution from a Galactic halo component. A new detailed model of the Galactic diffuse emission [@Strong2004a] includes an anisotropic Inverse Compton cross section, which brightens the high-latitude IC intensity. This re-analysis [@Strong2004b] gives a new estimate of the EGRB that is lower in flux and steeper than found by @Sreekumar1998 and is not consistent with a power-law. The sensitivity and resolution of the LAT allow it to resolve many more individual sources, such as AGNs, not resolved by EGRET and that contribute to current estimates of the EGRB. Other components of the remaining EGRB will therefore become more important. Accurate calculations of the “guaranteed background” from conventional sources will make the limits and constraints imposed on exotic processes more reliable. Estimating point-source contributions to the EGRB requires statistical information about the particular population under consideration, e.g., luminosity function, evolutionary properties, etc. This analysis has been done for the [$\gamma$-ray]{} blazar population using a luminosity function derived from EGRET observations to estimate the contribution of unresolved point sources to the EGRB [e.g., @Chiang1998] as $>$25%. The improved sensitivity of the LAT will reduce the uncertainty of the LAT blazar luminosity function significantly, and at the same time probe the blazar evolution to the redshifts of their expected birth. This approach will enable LAT observations to place interesting constraints on the cosmological blazar formation rate. So far GeV-photon absorption in the cosmic background radiation field has not been taken into account in any diffuse source background model. With LAT’s sensitivity in a much broader energy range as compared to previous pair conversion telescopes, the expected absorption imprints on the diffuse spectrum may provide information on both the source population as well as the background radiation field. With the large number of extragalactic sources resolved by the LAT, the extragalactic component of the diffuse flux will be reduced accordingly; predictions of the reduction due to radio-loud AGN are in the range $\sim$15%–40%. Fluctuation analysis, where signatures of excess variance are searched for in the surface brightness of the EGRB, is a very general approach to estimating the contribution of any isotropically distributed source population to the diffuse flux. Application of this method to the EGRET data set revealed a point source contribution to the EGRB of 5%–100% [@Willis1996] from analysis on an angular scale of $3.5^\circ \times 3.5^\circ$, the scale of the @Hunter1997 Galactic diffuse emission model. With LAT’s sensitivity, point spread function and more uniform exposure, smaller spatial scales can be probed, thereby improving the detectability of a signal from contributing point sources to the EGRB. Understand the mechanisms of particle acceleration in celestial sources {#s3.2} ----------------------------------------------------------------------- [$\gamma$-ray]{} observations are a direct probe of particle acceleration mechanisms operating in astrophysical systems. Advances with LAT observations in our understanding of these non-thermal processes can be anticipated by reference to discoveries made with EGRET in several important source categories: blazars, pulsars, supernovae remnants, and the Sun. ### Blazar AGN jets {#s3.2.1} With high-confidence detections of more than 60 AGN, almost all of them identified with BL Lacs or Flat Spectrum Radio Quasars (FSRQs) [@Hartman1999], EGRET established blazars as a class of powerful but highly variable [$\gamma$-ray]{} emitters, in accord with the unified model of AGN as supermassive black holes with accretion disks and jets. Although blazars comprise only several per cent of the overall AGN population, they largely dominate the high-energy extragalactic sky. This is because most of the non-thermal power, which arises from relativistic jets that are narrowly beamed and boosted in the forward direction, is emitted in the [$\gamma$-ray]{} band (Figure \[f3.3\]), whereas the presumably nearly-isotropic emission from the accretion disk is most luminous at optical, UV, and X-ray energies.  Most extragalactic sources detected by the LAT are therefore expected to be blazar AGNs, in contrast with the situation at X-ray frequencies, where most of the detected extragalactic sources are radio-quiet AGN. The estimated number of blazars that [*Fermi*]{}/LAT will detect ranges from a thousand [@Dermer2007] to several thousand (@Stecker1996 [@Chiang1998; @Mucke2000]: see Figure \[f3.4\]). Such a large and homogeneous sample will greatly improve our understanding of blazars and will be used to perform detailed population studies and to carry out spectral and temporal analyses on a large number of bright objects. In particular, the very good statistics will allow us to a) extend the $\log N-\log S$ curve to fluxes about 25 times fainter than EGRET, b) estimate the luminosity function and its cosmological evolution, and c) calculate the contribution of blazars and radio galaxies to the extragalactic [$\gamma$-ray]{} background (see previous section). These observations will chart the evolution and growth of supermassive black holes from high-redshifts to the present epoch, probe a possibly evolutionary connection between BL Lacs and FSRQs, verify the unified model for radio galaxies and blazars [@Urry1995], and test the “blazar sequence” [@Fossati1998]. The redshift dependence of spectral parameters of blazars in the LAT energy band, together with the measurements or limits from ground-based TeV instruments, will be used to measure the evolution of the Extragalactic Background Light (see §\[s3.5\]). Finally, LAT blazar detections will be essential in determining if a truly diffuse component of extragalactic [$\gamma$-ray]{} emission is required, or if such background can be accounted for by a superposition of various classes of discrete objects. The LAT’s wide field of view will allow AGN variability to be monitored on a wide range of time scales. Rapid flares as bright as those observed by EGRET from 3C 279 [@Kniffen1993] and by [*Swift*]{} and *Agile* from 3C454.3 [@Giommi2006; @Vercellone2008] will be measurable with [*Fermi*]{} at [$\gamma$-ray]{} energies on time scales of hours (e.g., see Figure \[f3.2\]). In addition, the duty cycle of flaring of a large number of blazars will be determined with good accuracy. Measurements of the short variability time scale for luminous [$\gamma$-ray]{} emission will place lower limits on the Doppler factor of the jet plasma. The values of the Doppler factor can be correlated with [$\gamma$-ray]{} intensity states for a specific blazar and correlated with membership in different subclasses for many blazars. The Doppler factors can also be compared with values obtained from superluminal motion radio observations in order to infer the location of the [$\gamma$-ray]{} emission site, with the goal to study the evolution of jet Lorentz factor with distance from the black hole. Most viable current models of formation and structure of relativistic jets involve conversion of the gravitational energy of matter flowing onto a central supermassive black hole. [$\gamma$-ray]{} flares are most likely related to the dissipation of magnetic accretion energy or extraction of energy from rotating black holes [e.g., @Blandford1977]. However, the conversion process itself is not well understood, and many questions remain about the jets, such as: How are they collimated and confined? What is the composition of the jet, both in the initial and in the radiative phase? Where does the conversion between the kinetic power of the jet into radiation take place, and how? What role is played by relativistic hadrons. There are also questions about the role of the magnetic field, such as whether the total kinetic energy of the jet is, at least initially, dominated by Poynting flux. The first step in answering these questions is to determine the emission mechanisms in order to infer the content of the luminous portions of jets. This understanding should, in turn, shed light on the jet formation process and its connection to the accreting black hole. Determining the emission mechanisms, whether dominated by synchrotron self-Compton, external Compton, or hadronic processes, requires sensitive, simultaneous multiwavelength observations. Such observations can uncover the causal relationships between the variable emissions in different spectral bands and provide detailed modeling of the time-resolved, broadband spectra. The sensitivity and wide bandpass of the LAT, coupled with well-coordinated multiwavelength campaigns, are essential. Figure \[f3.4\] shows representative spectral energy distributions of [$\gamma$-ray]{} blazars and the detection pass-band and sensitivity of the LAT. Broadband campaigns have been organized to measure the total jet power as compared with accretion power, and the spectra from these observations should reveal whether a single zone structure is sufficient or whether multiple zones are required. Furthermore, the content of the inner part of the jet will be tightly constrained by broadband X-ray spectra and by temporal correlations between the X-ray and [$\gamma$-ray]{} variability; this is because the radiative energy density in the vicinity of black holes in AGN can be reliably estimated from contemporaneous broadband data, and this circumnuclear radiation must Compton-scatter with all “cold” charged particles contained in the jet [e.g., @Sikora2000; @Moderski2004]. Finally, the detection of anomalous [$\gamma$-ray]{} spectral features will indicate the importance of hadronic processes, with significant implications for the origin of ultra-high-energy cosmic rays. ### Pulsars, pulsar wind nebulae and supernova remnants {#s3.2.2} Pulsars, with their unique temporal signature, were the only definitively identified EGRET population of Galactic point sources. There were five young radio pulsars detected with high significance, along with the radio-quiet pulsar Geminga and one likely millisecond pulsar [for a summary, see @Thompson2001]. A number of other pulsars had lower significance pulse detections and many of the bright, unidentified [$\gamma$-ray]{} sources are coincident with known radio pulsars. Surrounding young pulsars are bright non-thermal pulsar wind nebulae (PWNe). In the case of the Crab pulsar, EGRET detected a clear signature of PWN emission on off-pulse phases; several other EGRET sources near young pulsars/PWNe show strong variability, possibly connected with variations in the wind shock termination. Even more encouraging has been the success in detecting PWN Compton emission in the TeV band [@Aharonian2005b] from a number of PWNe. Finally, it has long been noticed [@Montmerle1979; @Kaaret1996; @Yadigaroglu1997] that [$\gamma$-ray]{} sources are spatially correlated with massive star sites, including supernova remnants (SNRs). While EGRET was not able to make definitive associations with SNRs, the LAT has the spatial and spectral resolution to do so. [ ]{} \[s3.2.2.1\] Rotation-induced electric fields in charge-depleted regions of pulsar magnetospheres (“gaps”) accelerate charges to ten’s of TeV and produce non-thermal emission across the electromagnetic spectrum. The coherent radio emission, through which most pulsars are discovered, is however a side-show, representing a tiny fraction of the spin-down power. In contrast $\sim$GeV peak in the pulsed power can represent as much as 20-30% of the total spin-down. This emission, with its complex pulse profile and phase-varying spectrum, thus gives the key to understanding these important astrophysical accelerators. And, despite 40 years of pulsar studies, many central questions remain unanswered. A basic issue is whether the high energy emission arises near the surface, close to the classical radio emission [“the polar cap” model, @Daugherty1996] or at a significant fraction of the light cylinder distance [“outer gap” models, @Cheng1986; @Romani1996]. In addition to geometrical (beam-shape) differences, the two scenarios predict that different physics dominates the pair production. Near the surface $\gamma + B \to e^+ + e^-$ is important, while in the outer magnetosphere $\gamma + \gamma \to e^+ + e^-$ dominates; these result in substantially different predictions for the high energy pulsar spectrum (see Figure \[f3.5\]). There are a number of pulsar models estimating detailed pulse profiles and spectral variation with pulse phase . Some also predict emission between the polar cap and outer magnetosphere extremes [@Muslimov2003; @Dyks2003]. The improved statistics, energy resolution and high energy sensitivity provided by the LAT enable serious tests of these models for individual bright pulsars. Also, with predicted numbers ranging from dozens to hundreds, the LAT survey of the Galactic pulsar population will provide additional key tests of massive star populations and pulsar evolution. An extensive campaign of pulsar timing using radio telescopes at Parkes, Jodrell Bank, Nancay, Green Bank, and Arecibo, plus X-ray timing with the Rossi X-ray Timing Explorer has been started in order to provide contemporaneous ephemerides with the [$\gamma$-ray]{} observations [@Smith2008]. As discussed in §\[s3.1.1\], LAT’s high sensitivity also allows searches for pulsations in many sources independent of external timing information. Finding a larger population of radio-quiet pulsars is another test of pulsar models [e.g., @Gonthier2007] as well as a new window on the neutron star population of the Galaxy. Indeed, shortly after in-orbit activation, [*Fermi*]{}/LAT detected a radio-quiet pulsar in the supernova remnant CTA 1 [@Abdo2008]. \[s3.2.2.2\] For the Crab pulsar, EGRET detected unpulsed, possibly variable, emission below $\sim$150 MeV (likely synchrotron) and Compton-scattered PWN emission at higher energies [@de_Jager2006]. In this and other pulsars the connection with the IC flux observed in the TeV band is particularly valuable in constraining the PWN B field and the injected particle spectrum. Recent successes with detecting PWN at TeV energies show that the Galactic plane contains an abundance of such sources. To illustrate the capability of the LAT for advancing PSR/PWN physics, we have simulated one particularly interesting source, the “Kookaburra/Rabbit” complex [@Ng2005]. EGRET data suggested that the source was composite and now X-ray [@Ng2005] and TeV [@Aharonian2006a] studies show that the source contains two PWNe. One contains the young energetic radio pulsar PSR J1420-6048, for the other radio pulsations are not known and the source may be Geminga-like. We have simulated, see Figure \[f3.6\], a plausible PWN spectrum for the two sources (following the HESS morphology) along with a Vela-like pulsed emission for PSR J1420-6048, in the K3 region, and Geminga-like emission for a pulsar in the Rabbit. At high energies the simulation indicates that the two PWNe can be resolved. \[s3.2.2.3\] Cosmic rays with energy $\le10^{15}$ eV have long been thought to be shock-accelerated in supernova remnants. For some time, non-thermal X-ray emission has implied a significant population of electrons accelerated to TeV energies [@Allen1997]. Moreover, recently the HESS experiment has had great success in detecting TeV emission from Galactic SNR [@Aharonian2005b]. However the origin of this emission – inverse Compton scattering from a leptonic component or $\pi^0$ decay from a hadronic component – is still uncertain. The [*Fermi*]{}/LAT has the spatial and spectral sensitivity to resolve this question and thus constrain the origin of cosmic rays. Particularly interesting sources are G0.9+0.1 [@Aharonian2005a] and RX J1713.7-3946 [@Aharonian2006b]. In the case of G0.9+0.1, LAT observations will probe the inverse Compton emission mechanism and the interstellar radiation field at the Galactic center [@Porter2006]. In case of RX J1713.7-3946, extended TeV emission matches well spatially the lower energy X-ray emission. This match might implicate inverse Compton emission from $e^+e^-$ populations [@Porter2006] or can be easily accommodated by a $\pi^0$ model. In the GeV band, well covered by the LAT, the spectra differ (see Figure \[f3.7\]), and can be distinguished. In the particular case of RX J1713.7-3946 and for perhaps a dozen additional objects, careful analysis of LAT observations should be able to resolve the emission at $E > 10$ GeV – such spatial-spectral studies can further constrain the particle acceleration physics and may isolate shell SNR emission from core PWN emission in composite sources. ### [$\gamma$-ray]{} emission from the Sun and solar system bodies {#s3.2.3} The 2005 January 20 solar flare produced one of the most intense, fastest rising, and hardest solar energetic particle events ever observed in space or on the ground. [$\gamma$-ray]{} measurements of the flare [@Share2006; @Grechnev2008] revealed what appear to be two separate components of particle acceleration at the Sun: i) an impulsive release lasting $\sim$10 min with a power-law index of $\sim$3 observed in a compact region on the Sun and, ii) an associated release of much higher energy particles having an spectral index $\le$2.3 interacting at the Sun for about two hours. Pion-decay [$\gamma$-ray]{}[s]{} appear to dominate the latter component. Such long-duration high-energy events have been observed before, most notably on 1991 June 11 when the EGRET instrument on CGRO observed $>$50 MeV emission for over 8 hours [@Kanbach1993]. It is possible that these high-energy components are directly related to the particle events observed in space and at Earth. Solar activity is expected to rise in 2008 with a peak occurring as early as 2011. During normal operations [*Fermi*]{} will be able to observe the Sun about 20% of the time with the possibility of increasing that to about 60% during heightened solar activity. With LAT’s large effective area and field-of-view, and its low deadtime it is expected to observe tens of these high-energy events from the Sun. For intense events LAT may be able to localize the source to about $30\arcsec$, sufficient to determine if it originates from the flare’s X-ray footpoints or from a different location that might be expected if the high-energy particles were accelerated in a shock associated with a coronal mass ejection. The quiet Sun is also a source of [$\gamma$-ray]{}[s]{} which will be detectable by LAT. Estimates of the cosmic-ray proton interactions with the solar atmosphere (solar albedo) were made by @Seckel1991, it is expected that LAT will observe a flux of $\sim$$10^{-7}$ cm$^{-2}$ s$^{-1}$ above 100 MeV from pion decays that is at the limit of EGRET sensitivity [@Thompson1997b]. In addition, a diffuse emission component with maximum in the direction of the Sun due to the inverse Compton scattering of solar photons on cosmic-ray electrons was predicted to be detected by LAT [@Moskalenko2006; @Orlando2007; @Orlando2008]. A detailed analysis of the EGRET data [@Orlando2008] yielded the flux of these two solar components at 4$\sigma$, consistent with the predicted level. Observations of the inverse Compton scattering of solar photons will allow for continuous monitoring of the cosmic-ray electron spectrum from the close proximity of the solar surface to Saturn’s orbit at 10 AU, important for heliospheric cosmic-ray modulation studies. The fluxes of these components will vary over the solar cycle as solar modulation increases, thus we can expect the highest fluxes to be observed early in the [*Fermi*]{} mission. Recent studies suggest that LAT will be able to see another diffuse emission component originating nearby in the solar system: a [$\gamma$-ray]{} glow around the ecliptic due to the albedo of small solar system bodies (produced by cosmic-ray interactions) in the Main Asteroid Belt between the orbits of Mars and Jupiter and Kuiper Belt beyond Neptune’s orbit [@Moskalenko2008]. Observations of the albedo of small bodies can be used to derive their size distribution. Additionally [$\gamma$-ray]{} albedo of Kuiper Belt objects could be used to probe the cosmic-ray spectrum in the far outer solar system close to the heliospheric boundary. Since the ecliptic is projected across the Galactic center, and passes through high Galactic latitudes, both diffuse emission components (inverse Compton scattering of solar photons and the albedo of small solar system bodies) are important to take into account when studying the sources in the direction of the Galactic center and extragalactic diffuse emission (see also §§\[s3.1.2\], \[s3.4\]). The Moon is also a source of [$\gamma$-ray]{}[s]{} due to CR interactions with its surface and has been detected by EGRET [@Thompson1997b]. However, contrary to the CR interaction with the gaseous atmospheres of the Earth and the Sun, the Moon surface is solid, consisting of rock, making its albedo spectrum unique. The spectrum of [$\gamma$-ray]{}[s]{} from the Moon is very steep with an effective cutoff around 3–4 GeV (600 MeV for the inner part of the Moon disk) and exhibits a narrow pion-decay line at 67.5 MeV, perhaps unique in astrophysics [@Moskalenko2007a]. Apart from other astrophysical sources, the albedo spectrum of the Moon is well understood, including its absolute normalization; this makes it a useful “standard candle” for [$\gamma$-ray]{} telescopes. The steep albedo spectrum also provides a unique opportunity for energy calibration of [$\gamma$-ray]{} telescopes such as LAT. Finally, the brightest [$\gamma$-ray]{} source on the sky is the Earth’s atmosphere due to its proximity to the spacecraft. The Earth’s albedo due to the cosmic-ray interactions with the atmosphere has been observed by EGRET [@Petry2005]. Its observations can provide important information about interactions of cosmic rays and solar wind particles with Earth’s magnetic field and the atmosphere. Study the high energy behavior of GRBs and transients {#s3.3} ----------------------------------------------------- Over the last decade the study of X-ray, optical, and radio afterglows of [$\gamma$-ray]{} bursts (GRBs) has revealed their distance scale, helping to transform the subject from phenomenological speculation to quantitative astrophysical interpretation. We now know that long-duration GRBs ($\tau > 2$ s) and at least some short-duration GRBs lie at cosmological distances and that both classes involve extremely powerful, relativistic explosions. Long GRBs are associated with low metallicity hosts with high star formation rates, and have nuclear offsets of $\sim$10 kpc [@Bloom2002]. Long-duration bursts are typically found in star-forming regions of galaxies and are sometimes associated with supernovae, indicating that the burst mechanism is associated with the collapse of very massive stars [@Zhang2004a]. Short-duration bursts are often located in much lower star-formation rate regions of the host galaxy, suggesting that in some cases these bursts arise from the coalescence of compact objects [@Bloom2006; @Nakar2007]. For the $\sim$30% of long-duration bursts seen by [*Swift*]{} that have measured redshifts, the redshift distribution peaks near $z \sim 2.8$ [@Jakobsson2006], comparable to Type 2 AGN. The sparse distribution for short bursts with spectroscopic redshifts spans a much lower range, $z \sim 0.1 - 1.1$. However, a photometric study of the host galaxies of short bursts without spectroscopically determined redshifts indicates that the fainter hosts tend to lie at redshifts $z > 1$ [@Berger2007]. The standard picture that has emerged of GRB physics is that an initial fireball powers a collimated, super-relativistic blast wave with initial Lorentz factor $\sim 10^2 - 10^3$. Prompt [$\gamma$-ray]{} and X-ray emission from this “central engine” may continue for few $\times 10^3$ s. Then external shocks arising from interaction of the ejecta with the circumstellar environment at lower Lorentz factors give rise to afterglows in the X-ray and lower-energy bands that are detected for hours to months. The physical details – primal energy source and energy transport, degree of blast wave collimation, and emission mechanisms – remain for debate [@Zhang2004]. The LAT will help constrain many uncertainties in these areas. EGRET detected two components of high-energy [$\gamma$-ray]{} emission from GRBs: $>$100 MeV emission contemporaneous with the prompt pulsed emission detected in the 10–1000 keV band, and a delayed component extending to GeV energies that lasted more than an hour in the case of GRB 940217 [@Hurley1994]. Analogous components were detected in the short burst GRB 930131 [@Sommer1994]. Most importantly, EGRET detected one burst (GRB 941017) in which a third power-law component was evident above the usual Band function spectrum [@Band1993], with an inferred peak in $\nu F(\nu)$ above 300 MeV during most of the prompt emission phase [@Gonzalez2003]. This indicates that some bursts occur for which the bulk of the energy release falls in the LAT energy band. The prompt pulsed component in these bursts was poorly measured by EGRET since the severe spark chamber deadtime ($\sim$100 ms/event) was comparable to or longer than pulse timescales. The LAT is designed with low deadtime ($\sim$26 $\mu$s/event) so that even very intense portions of bursts will be detected with very little ($<$ few %) deadtime. The delayed-emission component will also be much better measured because of LAT’s increased effective area, larger FoV, and low self-veto at supra-GeV energies. These observations will test models of delayed GeV emission, for example, those involving production of [$\gamma$-ray]{}[s]{} from ultra-high-energy cosmic rays [@Bottcher1998], impact of a relativistic wind from the GRB on external matter [@Meszaros1994], and synchrotron self-Compton radiation [@Dermer2000]. Internal and external shock models [@Zhang2004] are currently constrained primarily by spectral and temporal behavior at sub-MeV energies [@Fenimore1999], where the most detailed observations have been made. But these observations span only a relatively narrow energy range. The LAT’s sensitivity will force comparison of models with observations over a dynamic range in energy of $\sim 10^3-10^4$, and a factor of $\sim$$10^6$ including joint GBM observations. The LAT can provide time-dependent spectral diagnostics of bright bursts and will be able to measure high-energy exponential spectral cutoffs expected for moderately high redshift GRBs caused by $\gamma\gamma$ absorption in the cosmic UV-optical background (complementing AGN probes). The LAT will distinguish such attenuation from $\gamma\gamma$ absorption internal to the sources. Internal absorption is expected to produce time-variable breaks in power-law energy spectra. Signatures of internal absorption will constrain the bulk Lorentz factor and adiabatic/radiative behavior of the GRB blast wave as a function of time for sufficiently bright bursts [@Baring1997; @Lithwick2001; @Baring2006]. To estimate the LAT sensitivity to GRB, a phenomenological GRB model is adopted that assumes the spectrum of the GRB is described by the Band function, and the high-energy power law extends up to LAT energies. In order to compare the LAT sensitivity to GRB with the BATSE catalog of GRB, we compute the fluence of GRBs in the 50–300 keV energy band. Figure \[f3.8\] shows the minimum detectable fluence as a function of the localization accuracy, for different viewing angles and for different high-energy spectral indexes keeping the peak energy and the low energy spectral index of the Band model fixed (to 500 keV and –1, respectively). The plot showns the expected relation between the fluence and the localization accuracy, which scales as the inverse of the square root of the burst fluence. Detailed simulations, based on extrapolations from the BATSE-detected GRBs, and adopting the distribution of Band parameters of the catalog of bright BATSE bursts [@Kaneko2006], suggest that the LAT may detect one burst per month, depending on the GRB model for high energy emission. These estimations are in good agreement with the observed number of GRBs. In the first few months of operations LAT has already detected high-energy emission from four GRBs: GRB 080825C [@Bouvier2008 GCN: 8183], the bright GRB 080916C , GRB 081024B [@Omodei2008 GCN 8407] and GRB 081215A, [@McEnery2008 GCN 8684]. For more than one-third of LAT-detected bursts, LAT localizations should be sufficiently accurate for direct X-ray and optical counterpart searches. For instance, $\sim$50% of the LAT bursts are projected to have localization errors commensurate with the field of view of [*Swift*]{}’s XRT ($23\arcmin$), which very efficiently detects afterglows with few arc-second error radii. Burst positions are also calculated rapidly onboard, albeit with less initial accuracy, by the LAT flight software, as well as on the ground by the science analysis software pipeline, and distributed via the GCN network. Searches are conducted during ground analysis for fainter bursts not detected by the on-board trigger of the LAT. Simulations show that LAT observations may constrain quantum gravity scenarios that give rise to an energy-dependent speed of light and consequent energy-dependent shifts of GRB photon arrival times [@Amelino-Camelia1998; @Alfaro2002]. Short-duration GRBs, which exhibit negligible pulse spectral evolution above $\sim$10 keV may represent the ideal tool for this purpose [@Scargle2008]. The LAT properties important for such measurements are its broad energy range, sensitivity at high energies, and $<$10 $\mu$s event timing. The LAT’s low deadtime and simple event reconstruction, even for multi-photon events, enable searches for evaporation of primordial black holes with masses of $\sim$$10^{17}$ gm [@Fichtel1994]. Probe the nature of dark matter {#s3.4} ------------------------------- Compelling evidence for large amounts of nonbaryonic matter in the Universe is provided by the rotation curves of galaxies, structure-formation arguments, the dynamics and weak lensing of clusters of galaxies, and, most recently, WMAP measurements of the CMB (@Spergel2007, for review see e.g., @Bergstrom2000). One of the most attractive candidates for Dark Matter is the Weakly Interacting Massive Particle (WIMP). Several theoretical candidates for WIMPs are provided in extensions of the Standard Model of Particle Physics such as Super-Symmetry. Searches for predicted particle states of these theories are one of the prime goals of accelerator-based particle physics, in particular the experiments at the Large Hadron Collider (LHC), which is planned to be operational in 2008. Annihilations of WIMPs can lead to signals in radio waves, neutrinos, antiprotons and positrons and [$\gamma$-ray]{}[s]{}. [$\gamma$-ray]{} observations have the advantage over charged particles that the direction of the [$\gamma$-ray]{}[s]{} points back to the source, and they are not subject to additional flux uncertainties such as unknown trapping times [@Bergstrom2001; @Ullio2002]. However, predicted rates are subject to significant astrophysical uncertainties. Substructure in Dark Matter Halos is especially uncertain, with the predicted flux, for a given annihilation cross section, varying by several orders of magnitude. Observations of the [$\gamma$-ray]{} signal of WIMPs may not only constrain the particle nature of these particles but also, in the case that the LHC experiments discover a WIMP candidate, establish the connection between those particles and the Dark Matter. If the Dark Matter is identified, the LAT may be able to image the distribution of Dark Matter in the Galaxy which would constrain scenarios for structure formation. Two types of WIMP annihilation signals into [$\gamma$-ray]{}[s]{} are possible: a spectrally *continuous* flux below $m_\chi$ the mass of the annihilating particle, resulting mainly from the decay of $\pi^0$ mesons produced in the fragmentation of annihilation final states, and *monoenergetic* [$\gamma$-ray]{} lines resulting from WIMP annihilations into two-body final states containing two photons or a Z boson and a photon. Generally, the continuous signal has a much larger rate, but with a signature that is difficult to separate from the other Galactic diffuse foreground contributions, while the monoenergetic line is a much smaller signal, but, if detected, is more easily distinguished. The basic quantities that LAT observations can constrain are the total velocity-averaged annihilation cross section, the branching fraction in different final states, and the mass of the WIMPs. Different astrophysical sources can be used to search for a signal from WIMP annihilations, each with advantages and challenges. Table \[t3.1\] summarizes the different search strategies that we have studied. Detailed calculations of LAT sensitivities to Dark Matter are described in a separate paper [@Baltz2008]. Generally, sensitivities are in the cosmologically interesting region of $\langle\sigma v\rangle \sim10^{-26}-10^{-25}$ cm$^3$ s$^{-1}$, in the mass range between 40 and 200 GeV. Figure \[f3.10\] shows the expected number of halo clumps vs. detection significance for a generic WIMP of mass 100 GeV and $\langle\sigma v\rangle = 2.3\times10^{-26}$ cm$^3$ s$^{-1}$ assuming the distribution of halo clumps given by @Taylor2005a ([-@Taylor2005a; -@Taylor2005b]) in which about 30% of the halo mass is concentrated in halo clumps. The diffuse background was assumed to consist of an isotropic extragalactic component [@Sreekumar1998] and a Galactic component [@Strong2000]. The intensity needed to detect a [$\gamma$-ray]{} line with $5\sigma$ significance is in the vicinity of $10^{-9}$ ph cm$^{-2}$ s$^{-2}$ sr$^{-1}$ for an annulus around the Galactic Center (masking the galactic plane to $\pm15^\circ$). In @Baltz2006, information obtainable with [*Fermi*]{} is compared with what may be learned at upcoming accelerator-based experiments, for a range of particle Dark Matter models. Over sizable ranges of particle model parameter space, [*Fermi*]{} has significant sensitivity and will provide key pieces of the puzzle. The challenge will be to untangle the annihilation signals from the astrophysical backgrounds due to other processes. Use high-energy [$\gamma$-ray]{}[s]{} to probe the early universe {#s3.5} ----------------------------------------------------------------- Photons above 10 GeV can probe the era of galaxy formation through absorption by near UV, optical, and near IR extragalactic background light (EBL). The EBL at IR to UV wavelengths is accumulated radiation from structure and star formation and its subsequent evolution in the universe with the main contributors being the starlight in the optical to UV band, and IR radiation from dust reprocessed starlight [see e.g., @Madau1996; @MacMinn1996; @Primack2001; @Hauser2001]. Since direct measurements of EBL suffer from large systematic uncertainties due to contamination by the bright foreground (e.g., interplanetary dust, stars and gas in the Milky Way, etc.), the indirect probe provided by absorption of high-energy [$\gamma$-ray]{}[s]{} via pair production ($\gamma + \gamma \to e^+ + e^-$), emitted from blazars, during their propagation in the EBL fields, can be a powerful tool for probing the EBL density. For example, observations of relatively nearby TeV blazars by the HESS atmospheric Cherenkov telescope [@Aharonian2006c] have placed significant limits on the EBL at IR energies in the local universe. The photon-photon pair production cross section has a pronounced maximum at $E_\gamma \approx 0.8$ TeV (1 eV/EEBL) (interaction angle averaged), close to the pair production threshold. Hence the LAT energy range extending to greater than 300 GeV is ideal for probing the EBL in the largely unexplored optical-UV band. According to current EBL models [e.g., @Primack1999; @Stecker2006; @Kneiske2004], absorption breaks in the LAT energy range are expected for sources located at $z \ge 0.5$. This offers for the first time the opportunity to constrain the *evolution* of the EBL. For this purpose, at least two methods have been developed: probing the horizon of extragalactic [$\gamma$-ray]{}[s]{} through measurements of either the ratio of absorbed to unabsorbed flux versus redshift [@Chen2004], or detection of the e-folding cutoff energy $E(\tau_{\gamma\gamma} = 1)$ as a function of redshift [@Fazio1970; @Kneiske2004] in a large number of suitable sources. With the expected LAT flux sensitivity the number of detected [$\gamma$-ray]{} loud blazars will increase to potentially several thousand sources (see §\[s3.2.1\]) with redshifts up to $z \sim 5-6$. Such a large number of sources will be required for a statistically meaningful search for evolutionary behavior of spectral absorption features in bright and hard-spectrum AGNs. Any of the analysis methods employed requires disentangling source intrinsic opacity effects, particularly if they are evolutionary with redshift, from the absorption due to EBL. Absorption in the local environment of AGN but external to the jet radiation fields has been shown to mimic an absorption pattern similar to what is expected from EBL attenuation of [$\gamma$-ray]{}[s]{} [@Reimer2007], i.e. higher [$\gamma$-ray]{} opacities from higher redshift sources. Careful source selection and a statistical assessment of the radiation field density at the [$\gamma$-ray]{} source site will be an integral part of the analysis. Monitoring of external photon fields in AGN (e.g., broad-line region lines) and correlating with the observed [$\gamma$-ray]{} cutoff energy may offer verification, and possibly quantification, of this effect. Summary {#s4} ======= The Large Area Telescope, the primary instrument on the [*Fermi Gamma-ray Space Telescope*]{}, is a state-of-the-art, high-energy [$\gamma$-ray]{} telescope. The LAT’s combination of wide field-of-view, large effective area, excellent single photon angular resolution (particularly at high energies), good energy resolution, excellent time resolution and low instrumental dead time, will push back several frontiers in high-energy astrophysics. Data from the LAT and software analysis tools will be available to the entire scientific community. The [*Fermi*]{}/LAT Collaboration acknowledges the generous ongoing support of a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l’Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Instituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council, and the Swedish National Space Board in Sweden. Additional support from the following agencies is also gratefully acknowledged: the Istituto Nazionale di Astrofisica in Italy and the K. A. Wallenberg Foundation for providing a grant in support of a Royal Swedish Academy of Sciences Research fellowship for JC. Abdo, A., et al. 2008, Science, 322, 1218 Agostinelli, S., et al. 2003, Nucl. Instr. Meth. A, 506, 250 Aguilar, M., et al. 2002, , 366, 331 Aharonian, F. A., et al. 2005a, , 432, L25 Aharonian, F. A., et al. 2005b, , 437, L7 Aharonian, F. A., et al. 2006a, , 456, 245 Aharonian, F. A., et al. 2006b, , 449, 223 Aharonian, F. A., et al. 2006c, , 440, 1018 Albert, J., et al. 2007, , 665, L51 Alfaro, J., Morales-Tecotl, H. A., & Urrutia, L. F. 2002, , 65, 103509 Allen, G. E., et al. 1997, , 487, L97 Allison, J., et al. 2006, [IEEE Trans. Nuc. Sci.]{}, 53, 270 Amelino-Camelia, G., Ellis, J., Mavromatos, N.E., Nanopoulos, D.V., & Sarkar, S. 1998, , 395, 525 Atwood, W. B., et al. 1994, Nucl. Instr. Meth. A, 342, 302 Atwood, W. B., et al. 2000, Nucl. Instr. Meth. A, 446, 444 Atwood, W. B., Ziegler, M., Johnson, R. P., & Baughman, B. M. 2006, , 652, L49 Atwood, W. B., et al. 2007, [APh]{}, 28, 422 Baldini, L., et al. 2006, [IEEE Trans. Nuc. Sci.]{}, 53, 466 Baldini, L., et al. 2007, in Proc. of $1^{\rm st}$ GLAST Symposium, eds.S. Ritz, P. Michelson, & C. Meegan, (NY: AIP), AIP Conf. Proc. 921, 190 Baltz, E. A., Battaglia, M., Peskin, M. E., & Wizansky, T. 2006, , 74, 103251 Baltz, E. A., et al. 2008, [JCAP]{}, 07, 013 Band, D., et al. 1993, , 413, 281 Baring, M. G., & Harding, A. K. 1997, , 491, 663 Baring, M. G. 2006, , 650, 1004 Barrand, G., et al. 2001, Computer Phys. Comm., 140, 45 Baughman, B. M., Atwood, W. B., Johnson, R. P., Porter, T. A., & Ziegler M., 2007, in Proc. 30th Int. Cosmic Ray Conf., in press (arXiv:0706.0503) Berezinsky, V. S., Blasi, P., & Ptuskin, V. S. 1997, , 487, 529 Berezinsky, V., Hnatyk, B., & Vilenkin, V. 2001, , 64, 043004 Berger, E., et al. 2007, , 660, 496 Bergström, L. 2000, Rept. Prog. Phys., 63, 793 Bergström, L., Edsjö, J., & Ullio, P. 2001, , 87, 251301 Bidoli, V., et al. 2002, Annales Geophysicae, 20, 1693 Bignami, G. F. et al. 1975, Space Sci. Instr., 1, 245 Bignami, G. F., Fichtel, C. E., Hartman, R. C., & Thompson D. J. 1979, , 323, 649 Bignami, G. F., & Caraveo, P. A. 1996, , 34,331 Blandford, R. D., & Znajek, R. L. 1977, , 179, 433 Bloom, J. S., Kulkarni, S. R., & Djorgovski, S. G. 2002, , 123, 1111 Bloom, J. S., et al. 2006, , 638, 354 Böttcher, M., & Dermer, C. D. 1998, , 499, L131 Bouvier, A., et al. 2008, GCN Circular 8183 Breiman, L., Friedman, J., Stone, C. J., & Olshen, R. A. 1984, Clasification and Regression Trees, Wadsworth International Group, Belmont, CA Calcáneo-Roldán, C., & Moore, B. 2000, , 62, 123005 Caraveo, P. A., Bignami, G. F., & Trümper, J., 1996, , 7, 209 Caraveo, P. A., & Reimer, O. 2007, in Proc. of 1st GLAST Symposium, eds. S. Ritz, P. Michelson & C. Meegan, (NY:AIP), AIP Conf. Proc. 921, 289 Carlson, P., et al. 1996, Nucl. Instr. Meth. A, 376, 271 Casanova, S., Dingus, B. L., & Zhang, B. 2007, , 656, 306 Chandler, A. M., et al. 2001 , 556, 59 Chen, A., Reyes, L.C., & Ritz, S. 2004, 608, 686 Cheng, K. S., Ho, C., & Ruderman, M. 1986, , 300, 500 Chiang, J., & Mukherjee, R. 1998, , 496, 752 Colafrancesco, S., 2002, , 396, 31 Colafrancesco, S., & Blasi, P. 1998, [APh]{}, 9, 227 Couto é Silva, E. do, et al. 2001, Nucl. Instr. Meth. A, 474, 19 Dar, A., & Shaviv, N. 1995, , 75, 3052 Daugherty, J. K., & Harding, A. K. 1996, , 458, 278 de Boer, W., et al. 2005, , 95, 209001 de Jager, O. C., et al. 2006, , 120, 441 Dermer, C. D., Chiang, J., & Mitman, K. E. 2000, , 537, 785 Dermer, C. D. 2007, , 659, 958 Dixon, D. D., Hartmann, D. H., Kolaczyk, E. D., Samimi, J., Diehl, R., Kanbach, G., Mayer-Hasselwander, H., & Strong, A. W. 1998, [New Astron.]{}, 3, 539 Dolgov, A., & Silk, J. 1993, , 47, 4244 Dyks, J., & Rudak, B. 2003, , 598, 1201 Elsässer, D., & Mannheim, K. 2005, , 94, 171302 Engelmann, J. J., et al. 1990, , 233, 96 Ensslin, T. A., Biermann, P. L., Kronberg, P. P., & Wu, X.-P., 1997, , 477, 560 Esposito, J. A., et al. 1996, , 461, 820 Fazio, G. G., & Stecker, F. W. 1970, , 226, 135 Fenimore, E. E., Ramirez-Ruiz, E., & Wu, B. 1999, , 518, 73 Feretti, L., Burigana, C., & Ensslin, T. 2004, [New Astron.]{}, 48, 1134 Ferreira, O., et al. 2004, [Nucl. Instr. Meth. Phys. Res. A]{}, 530, 323 Fichtel, C. E., et al. 1975, , 198, 163 Fichtel, C. E., et al. 1994, , 434, 557 Fossati G., et al. 1998, , 299, 433 Frühwirth, R., Regler, M., Bock, R. K., Grote, H. & Notz, D. 2000, Data Analysis Techniques for High-Energy Physics, 2nd edition, Cambridge University Press Fryer, C. L., & Heger, A. 2005, , 623, 302 Funk, S., et al. 2008, , 679, 1299 Gabici, S., & Blasi, P. 2003, [APh]{}, 19, 679 Gao, Y.-T., Stecker, F. W., Gleiser, M., & Cline, D. B. 1990, , 361, 37 Gehrels, N., et al. 2000, , 404, 363 Giommi, P., et al. 2006, , 456, 911 Giovannini, G., Tordi, M., & Feretti, L. 1999, [New Astron.]{}, 4, 141 Gonthier, P. L., et al. 2007, , 309, 245 Gonzalez, M. M., et al. 2003, , 424, 74 Grechnev, V. V. 2008, , 252, 149 Grenier, I. A. 2000, , 364, L93 Grenier, I. A., Casandjian, J.-M., & Terrier, R. 2005, [[Science]{}]{}, 307, 1292 Grove, E., et al. 2008, in preparation Haino, S., et al. 2004, [Phys. Lett. B]{}, 594, 35 Halpern, J. P., et al. 2001, , 552, L125 Halpern, J. P., et al. 2002, , 573, L41 Harding, A. K., Grenier, I. A., & Gonthier, P. L. 2007, , 309, 221 Hartman, R. C., et al. 1999, , 123, 79 Hauser, M. G., & Dwek, E. 2001, , 39, 249 Hawking, S. W., 1974, , 248, 30 Hunter, S. D., et al. 1997, , 481, 205 Hurley, K. 1994, , 372, 652 Jackson, B., et al. 2005, IEEE Signal Processing Letters, 12, 105 Jakobsson, P., et al. 2006, , 447, 897 Johnson, W. N., et al. 2001, [IEEE Trans. Nuc. Sci.]{}, 48, 1182 Jungman, G., Kamionkowski, M., & Griest, K. 1996, , 267, 195 Kaaret, P., & Cottam, J. 1996, , 462, L35 Kaneko, Y., et al. 2006, , 166, 298 Kanbach, G., et al. 1993, , 97, 349 Keshet, U., et al. 2003, , 585, 128 Kneiske, T. et al. 2004, , 413, 807 Kniffen, D. A., et al. 1993, , 411, 133 Kramer, M., et al. 2003, , 342, 1299 Lake, G. 1990, , 346, 39 Lithwick, Y., & Sari, R. 2001, , 555, 540 Loeb, A., & Waxman, E. 2000, , 405, 156 MacMinn, D., & Primack, J. R. 1996, , 75, 413 Madau, P., & Phinney, E. S. 1996, , 456, 124 Maki, K., Mitsui, T., & Orito, S. 1996, , 76, 3474 Mattox, J., et al. 1996, , 120, 95 Mattox, J., Hartman, R. C., & Reimer, O. 2001, , 135, 155 Mészáros, P., Rees, M. J., & Papathanassion, H. 2004, , 432, 181 McEnery, J., et al. 2008, GCN Circular 8407 Mikhailov, V., et al. 2002, J. Mod. Phys. A, 17, 1695 Miniati, F. 2002, , 337, 199 Mirabal, N., & Halpern, J. P. 2001, , 547, L137 Mizuno, T., et al. 2004, , 614, 1113 Moderski, R., et al. 2004, , 611, 77 Moiseev, A. A., et al. 2004, [APh]{}, 22, 275 Moiseev, A. A., et al. 2007, [APh]{}, 27, 339 Montmerle, T. 1979, , 231, 95 Moskalenko, I. V., & Strong, A. W. 2000, , 528, 357 Moskalenko, I. V., & Strong, A. W. 2005, in Proc. Int. Conf. on Astrophysical Sources of High Energy Particles and Radiation, eds. T. Bulik, et al. (NY: AIP), AIP Conf. Proc. 801, 57 Moskalenko, I. V., Porter, T. A., & Digel, S. W. 2006, , 652, L65 Moskalenko, I. V., & Porter, T. A. 2007, , 670, 1467 Moskalenko, I. V., Digel, S. W., Porter, T. A., Reimer, O., & Strong A. W. 2007, Nucl. Phys. B (Proc. Suppl.), 173, 44 Moskalenko, I. V., Porter, T. A., Digel, S. W., Michelson, P. F., & Ormes, J. F. 2007, , 681, 1708 Mücke, A., & Pohl, M. 2000, , 312, 177 Mukherjee, R., & Chiang, J. 1999, [APh]{}, 11, 213 Muslimov, A., & Harding, A. K. 2003, , 588, 430 Nakar, E. 2007, , 442, 166 Narumoto T., & Totani, T. 2006, , 643, 81 Ng, C.-Y., Roberts, M. S. E., & Romani, R. W. 2005, , 627, 904 Nolan, P. L., et al. 2003, , 597, 615 Norris, J. P., et al. 1996, , 459, 393 Omodei, N., et al. 2008, GCN Circular 8407 Orlando, E., & Strong, A. W. 2007, , 309, 359 Orlando, E., & Strong, A. W. 2008, , 480, 847 Page, D. N., & Hawking, S. W. 1976, , 206, 1 Paredes, J. M., Martí, J., Ribó, M., & Massi, M. 2000, [[Science]{}]{}, 288, 2340 Pavlidou, V., & Fields, B. D. 2001, , 558, 63 Pavlidou, V., & Fields, B. D. 2002, , 575, L5 Picozza, P., et al. 2007, [APh]{}, 27, 296 Petry, D. 2005, in High Energy Gamma Ray Astronomy: 2nd International Symposium, Heidelberg, July, 2004, eds. F. A. Aharonian, H. J. Volk, & D. Horns, (NY:AIP), AIP Conf. Proc. 745, 709 Porciani, C., & Madau, P. 1999, , 526, L522 Porter, T. A., Moskalenko, I. V., & Strong A. W. 2006, , 648, L29 Porter, T. A., Moskalenko, I. V., Strong A. W., Orlando, E., & Bouchet, L. 2008, , 682, 400 Preece, R. D., et al. 2000, , 126, 19 Primack, J. R., Bullock, J. S., Sommerville, R. S. & MacMinn, D. 1999, [APh]{}, 11, 93 Primack, J. R., Somerville, R. S., Bullock, J. S. & Devriendt, J. E. G. 2001, AIP, 558, 463 Primack, J. R., Bullock, J. S., & Sommerville, R. S. 2005, in Proc. of the Gamma 2004 Symposium on High Energy Gamma Ray Astronomy (NY:AIP), AIP Conf. Proc. 745, 23 Reimer, A. 2007, , 665, 1023 Reimer, O., et al. 2001, , 324, 772 Reimer, O., Pohl, M., Sreekumar, P., & Mattox, J. R. 2003, , 558, 155 Rephaeli, Y., Nevalainen, J., Ohashi, T., & Bykov, A. M. 2008, , 134, 71 Roberts, M. S. E., Romani, R. W., & Kawai, N. 2001, , 133, 451 Romani, R. W. 1996, , 470, 469 Romero, G. E., et al. 1999, , 348, 868R Rudaz, S., & Stecker, F. W. 1991, 368, 406 Scargle, J. D. 1998, , 504, 405 Scargle, J. D., Norris, J. P., & Bonnell, J. T. 2008, , 673, 972 Seckel, D., Stanev, T., & Gaisser, T. K. 1991, , 382, 652 Selesnik, R. S., Looper, M. D., & Mewaldt, R. A. 2007, Space Weather, 5, S04003 Share, G.H., et al. 2006, , 38, 255 Sikora, M., & Madejski, G. 2000, , 534, 109 Silk, J., & Srednicki, M., 1984, , 53, 624 Smith, D. A., et al. 2008, , to be published Sommer, M., et al. 1994, , 422, L63 Sowards-Emmerd, D., Romani, R. W., & Michelson, P. F. 2003, , 590, 109 Sowards-Emmerd, D., et al. 2004, , 609, 564 Sowards-Emmerd, D., et al. 2005, , 626, 95 Spergel, D. N., et al. 2007, , 170, 377 Sreekumar, P., et al. 1998, , 494, 523 Stecker, F. W., Vette, J. I., & Trombka, J. I. 1971, , 231, 122 Stecker, F., & Salamon, M. 1996, , 464, 600 Stecker, F. W., Malkan, A., & Scully, S. T. 2006, , 648, 774; and erratum: , 658, 1392 (2007) Stecker, F., Hunter, S. D., & Kniffen, D. A. 2008, [APh]{}, 29, 25 Strong, A. W., Moskalenko, I. V., & Reimer, O. 2000, , 537, 763 Strong, A. W., Moskalenko, I. V., & Reimer, O. 2004a, , 613, 962 Strong, A. W., Moskalenko, I. V., & Reimer, O. 2004b, , 613, 956 Strong, A. W., Moskalenko, I. V., Reimer O., Digel S., & Diehl R. 2004c, , 422, L47 Sturner, S. J., & Dermer, C. D. 1995, , 291, L17 Tajima, H., et al. 2008, GCN Circular 8246 Takata, J., et al. 2006, , 366, 1310 Tavani, M., et al. 1998, , 497, L89 Tavani, M., et al. 2008, [Nucl. Instr. Meth. Phys. Res. A]{}, 588, 52 Taylor, J. E., & Babul, A. 2005a, , 364, 515 Taylor, J. E., & Babul, A. 2005b, , 364, 535 Thompson, D. J., et al. 1993, , 86, 629 Thompson, D. J., et al. 1997a, Proc. 4th Compton Symp., ed. C.D. Dermer, et al. (New York, AIP), pp.39-51 Thompson, D. J., Bertsch, D. L., Morris, D. J., & Mukherjee, R. 1997b, , 102, 14735 Thompson, D. J., et al. 2002, [IEEE Trans. Nuc. Sci.]{}, 49, 1898 Thompson, D. J. 2001, Proc. of High-Energy Gamma-ray Astronomy Symposium, eds. F.A. Aharonian & H.J. Völk (New York, AIP), AIP Conf. Proc., 558, 103 Thompson, D. J. 2004, in Cosmic Gamma-Ray Sources, eds. K. S. Cheng, & G. E. Romero (Dordrecht Boston London: Kluwer), 149 Torres, D. F., et al. 2004, , 607, L99 Torres, D. F., & Reimer, O. 2005, , 629, L141 Totani, T., & Kitayama, T. 2000, , 545, 572 Ullio, P., et al. 2002, , 66, 123502 Urry, M., & Padovani, P. 1995, , 107, 803 Vercellone, S., et al. 2008, , 676, L13 Voronov, S. A., et al. 1991, Cosmic Research (English Translation) 29(4), 567 Willis, T. D. 1996, PhD thesis, Stanford University Yadigaroglu, I. A., & Romani, R. W. 1997, , 476, 347 Zhang, B., & Mészáros, P. 2004, Intl. J. Mod. Phys. A, 19, 238 Zhang, W., Woosley, S. E., & Heger, A. 2004, , 608, 365 Ziegler, M., Baughman, B. M., Johnson, R. P., & Atwood, W. B. 2008, , 680, 620 Zuccon, P., et al. 2003, [APh]{}, 20, 221 [lc]{} Energy range & 20 MeV – 300 GeV\ Effective area at normal incidence & 9,500 cm$^2$\ Energy resolution (equivalent Gaussian $1\sigma$):\ $\quad$ 100 MeV – 1 GeV (on axis) & 9%–15%\ $\quad$ 1 GeV – 10 GeV (on axis) & 8%–9%\ $\quad$ 10 GeV – 300 GeV (on-axis) & 8.5%–18%\ $\quad$ $>$10 GeV ($>$$60^\circ$ incidence) & $\le$6%\ Single photon angular resolution (space angle)\ on-axis, 68% containment radius:\ $\quad$ $>$10 GeV & $\le$$0.15^\circ$\ $\quad$ 1 GeV & $0.6^\circ$\ $\quad$ 100 MeV & $3.5^\circ$\ $\quad$ on-axis, 95% containment radius & $<3\times\theta_{68\%}$\ $\quad$ off-axis containment radius at $55^\circ$ & $<1.7\times$ on-axis value\ Field of View (FoV) & 2.4 sr\ Timing accuracy & $<10$ $\mu$sec\ Event readout time (dead time) & $26.5$ $\mu$sec\ GRB location accuracy on-board & $<10\arcmin$\ GRB notification time to spacecraft & $<$5 sec\ Point source location determination & $<0.5\arcmin$\ Point source sensitivity ($>$100 MeV) & $3\times10^{-9}$ ph cm$^{-2}$ s$^{-1}$ [lcl]{} Noise occupancy (fraction of channels with noise hits per trigger) & $10^{-6}$ & Trigger rate, data volume, track reconstruction. The requirement, driven by the trigger rate, is $<10^{-4}$ \ Single channel efficiency for minimum ionizing particle (MIP), within fiducial volume & $>$99% & PSF, especially at low energy. It is important to measure the tracks in the first 2 planes following the conversion point. \ Ratio of strip pitch to vertical spacing between tracker planes & 0.0071 & High-energy ($>$1 GeV) PSF \ Silicon-strip detector pitch (center-to-center distance between strips) & 228 $\mu$m & Small value needed to maintain a small pitch-to-plane-spacing ratio without destroying the FoV. \ Aspect ratio (height/width) & 0.4 & Large FoV for photons with energy determination \ Front converter foil thickness in radiation lengths (100% W) & $12\times0.03$\ (0.010 cm/foil) & Minimize thickness per plane for low-energy PSF, but not so much that support material dominates. Maximize total thickness to maximize effective area. \ Back converter foil thickness in radiation lengths (93% W) & $4\times0.18$\ (0.072 cm/foil) & Effective area and FoV at high energies \ Support material and detector material per $x-y$ plane (radiation lengths) & 0.014 & Stable mechanical support is needed, but much of this material is in a non-optimal location for the PSF. Minimize to limit PSF tails from conversions occurring in support material. [lcl]{} [ ]{} Depth, including tracker (radiation lengths) & 10.1 & Calorimeter depth is a compromise in shower containment against maximum permitted mass. Use segmentation and shower profile analysis to improve energy measurement at high energies \ Sampling (angle dependent) & $>$90% active & Energy loss in passive material causes low-energy tails on measured energy and affects energy resolution. \ Longitudinal segmentation & 8 segments & Shower profile analysis permits estimation of and correction for energy leakage \ Lateral segmentation & $\sim$1 Molière radius & Correlation of energy deposition in calorimeter with extrapolated tracks in tracker is critical part of background rejection. [lcl]{} [ ]{} Segmentation into tiles & $<$1000 cm$^2$ each & Minimize self-veto, especially at high energy. This value is for the top. Side tiles are smaller, to achieve a similar solid angle, as seen from the calorimeter. \ Efficiency of a tile for detecting a MIP & $>$0.9997 & Cosmic ray rejection, to meet a requirement of 0.99999 when combined with the other subsystems. \ Number of layers & 1 & Minimize material, mass, and power. Dual readout on each tile for redundancy. \ Micrometeoroid / thermal blanket thickness & 0.39 g cm$^{-2}$ & Small value needed to minimize [$\gamma$-ray]{} production in this passive material from cosmic-ray interactions. \ Total thickness (radiation lengths) & 10.0 mm (0.06) & Minimize absorption of incoming gamma radiation [lccc]{} [ ]{} Galactic Cosmic Rays\ $\quad$ protons + antiprotons & AMS\ $\quad$ electrons & AMS\ $\quad$ positrons & AMS\ $\quad$ He & AMS\ $\quad$ $Z > 2$ nuclei & HEAO–3\ Splash Albedo\ $\quad$ protons & & AMS & Nina\ $\quad$ electrons & & AMS & Mariya\ $\quad$ positrons & & AMS & Mariya\ Re-entrant Albedo\ $\quad$ protons & & Nina\ $\quad$ electrons & & Mariya\ $\quad$ positrons & & Mariya\ Earth albedo [$\gamma$-ray]{}[s]{} &\ Neutrons & [lcl]{} Transient & 2 & Maximize effective area, particularly at low energy, at the expense of higher residual background rate; suitable for study of localized, transient sources \ Source & 0.4 & Residual background rate comparable to extragalactic diffuse rate estimated from EGRET; suitable for study of localized sources \ Diffuse & 0.1 & Residual background rate comparable to irreducible limit and tails of PSF at high-energy minimized; suitable for study of the weakest diffuse sources expected [lll]{} Galactic center & Large number of photons & Disturbance by many point sources, uncertainty in diffuse background prediction. \ Satellites, sub-halos & Low celestial diffuse background, good identification of source & Low number of photons. \ Milky way halo & Large number of photons & Uncertainty in Galactic diffuse background prediction \ Extragalactic & Large number of photons & Astrophysical uncertainties, uncertainty in Galactic diffuse contribution. \ Spectral lines & No astrophysical uncertainties, smoking gun signal & Very low number of photons. ![Schematic diagram of the Large Area Telescope. The telescope’s dimensions are $1.8\ {\rm m} \times 1.8\ {\rm m} \times 0.72$ m. The power required and the mass are 650 W and 2,789 kg, respectively. []{data-label="f1.1"}](f1.ps){width="3.5in"} ![LAT source sensitivity for exposures on various timescales. Each map is an Aitoff projection in galactic coordinates. In standard sky-survey mode, nearly uniform exposure is achieved every 2 orbits, with every region viewed for $\sim$30 min every 3 hours. []{data-label="f1.2"}](f2.ps){width="5in"} ![Completed tracker array before integration with the ACD. []{data-label="f2.1"}](f3.ps){width="3.5in"} ![(a) A flight tracker tray and (b) a completed tracker module with one sidewall removed. []{data-label="f2.2"}](f4a.ps "fig:"){width="3.5in"}![(a) A flight tracker tray and (b) a completed tracker module with one sidewall removed. []{data-label="f2.2"}](f4b.ps "fig:"){width="3.5in"} ![Illustration of tracker design principles. The first two points dominate the measurement of the photon direction, especially at low energy. (Note that in this projection only the $x$ hits can be displayed.) (a) Ideal conversion in W: Si detectors are located as close as possible to the W foils, to minimize the lever arm for multiple scattering. Therefore, scattering in the 2nd W layer has very little impact on the measurement. (b) Fine detector segmentation can separately detect the two particles in many cases, enhancing both the PSF and the background rejection. (c) Converter foils cover only the active area of the Si, to minimize conversions for which a close-by measurement is not possible. (d) A missed hit in the 1st or 2nd layer can degrade the PSF by up to a factor of two, so it is important to have such inefficiencies well localized and identifiable, rather than spread across the active area. (e) A conversion in the structural material or Si can give long lever arms for multiple scattering, so such material is minimized. Good 2-hit resolution can help identify such conversions. []{data-label="f2.3"}](f5.eps){width="3.5in"} ![LAT calorimeter module. The 96 CsI(Tl) scintillator crystal detector elements are arranged in 8 layers, with the orientation of the crystals in adjacent layers rotated by $90^\circ$. The total calorimeter depth (at normal incidence) is 8.6 radiation lengths. []{data-label="f2.4"}](f6.ps){width="3.5in"} ![Light asymmetry measured in a typical calorimeter crystal using sea level muons. The light asymmetry is defined as the logarithm of the ratio of the outputs of the diodes at opposite ends of the crystal. The width of the distribution at each position is attributable to the light collection statistics at each end of the crystal for the $\sim$11 MeV energy depositions of vertically incident muons used in the analysis. This width scales with energy deposition as $E^{-1/2}$. []{data-label="f2.5"}](f7.eps){width="3.5in"} ![Energy resolution as a function of electron energy as measured with the LAT calibration unit in CERN beam tests. Each panel displays a histogram of the total measured energy (hatched peak) and the reconstructed energy (solid peak), using the LK method, at beam energies of 5, 10, 20, 50, 99.7 and 196 GeV, respectively. The beams entered the calibration unit at an angle of $45^\circ$ to the detector vertical axis. As long as shower maximum is within the calorimeter, the energy measurement and resolution are considerably improved by the energy reconstruction algorithms. The measured energy resolutions ($\Delta E/E$) are indicated in the figure. []{data-label="f2.6"}](f8.ps){width="6.0in"} ![LAT Anticoincidence Detector (ACD) design. The ACD has a total of 89 plastic scintillator tiles with a $5\times5$ array on the top and 16 tiles on each of the 4 sides. Each tile is readout by 2 photomultipliers coupled to wavelength shifting fibers embedded in the scintillator. The tiles overlap in one dimension to minimize gaps between tiles. In addition, 2 sets of 4, scintillating fiber ribbons are used to cover the remaining gaps. The ribbons, which are under the tiles, run up the side, across the top, and down the other side. Each ribbon is readout with photomultipliers at both ends. []{data-label="f2.7"}](f9.ps){width="3.5in"} ![LAT Data Acquisition System (DAQ) architecture. The Global-trigger/ACD-module/Signal distribution Unit (GASU) consists of the ACD Electronics Module, the Global Trigger Module (GTM), the Event Builder Module (EBM), and the Command Response Unit (CRU). The trigger and data readout from each of the 16 pairs of tracker and calorimeter modules is supported by a Tower Electronics Module (TEM). There are two primary Event Processing Units (EPU) and one primary Spacecraft Interface Unit (SIU). Not shown on the diagram are the redundant units (e.g. 1 SIU, 1 EPU, 1 GASU). []{data-label="f2.8"}](f10.ps){width="3.5in"} ![Components of the instrument simulation, calibration, and data analysis. []{data-label="fX"}](f11.ps){width="3.5in"} ![Orbit averaged background fluxes of the various components incident on the LAT used in the background model. The fluxes are shown as a function of total kinetic energy of the particles: protons (green filled triangles up), He (purple filled triangles up), electrons (filled red squares), positrons (light blue squares), Earth albedo neutrons (black squares), and Earth albedo [$\gamma$-ray]{}[s]{} (dark blue filled triangles down). The effect of geomagnetic cutoff is seen at 3 GeV for protons and electrons, and at higher energy for helium nuclei. At low energies the curves show the sum of re-entrant and splash albedo for electrons and positrons. []{data-label="f2.9"}](f12.ps){width="3.5in"} ![Ratio of the residual background to the extragalactic diffuse background inferred from EGRET observations [@Sreekumar1998] for each of the three analysis classes. The integral EGRET diffuse flux is $1.45\times10^{-7}$ ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$ above 100 MeV. []{data-label="f2.10"}](f13.eps){width="3.5in"} ![ Effective area versus energy at normal incidence for Diffuse (dashed curve), Source (solid curve), and Transient (dotted curve) analysis classes. []{data-label="f2.11"}](f14.eps){width="3.5in"} ![ Effective area versus energy at normal incidence (solid curve) and at $60^\circ$ off-axis (dashed curve) for Source analysis class. []{data-label="f2.12"}](f15.eps){width="3.5in"} ![ Acceptance versus energy for Diffuse (dashed curve), Source (solid curve), and Transient (dotted curve) analysis classes. []{data-label="f2.13"}](f16.eps){width="3.5in"} ![ 68% containment radius versus energy at normal incidence (solid curve) and at $60^\circ$ off-axis (dashed curve) for conversions in the thin section of the tracker. []{data-label="f2.14"}](f17.eps){width="3.5in"} ![Energy resolution versus energy for normal incidence (solid curve) and at $60^\circ$ off-axis (dashed curve). []{data-label="f2.15"}](f18.eps){width="3.5in"} ![Integral source sensitivity for $5\sigma$ detection for 1 year sky survey exposure. The source is assumed to have a power law differential photon number spectrum with index –2.0 and the background is assumed to be uniform with integral flux (above 100 MeV) of $1.5 \times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$ (dotted curve) and spectral index –2.1, typical of the diffuse background at high galactic latitudes. The background is 10 times higher and 100 times higher for the dashed and solid curves, respectively, representative of the diffuse background near or on the galactic plane. []{data-label="f2.16"}](f19.eps){width="3.5in"} ![Differential source sensitivity in 1/4 decade bins for $5\sigma$ detection for 1 year sky survey exposure. The source is assumed to have a power law differential photon number spectrum with index –2.0 and the background is assumed to be uniform with integral flux (above 100 MeV) of $1.5 \times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$ (dotted curve) and spectral index –2.1, typical of the diffuse emission at high galactic latitudes. The background is 10 times higher and 100 times higher for the dashed and solid curves, respectively, representative of the diffuse background near or on the galactic plane. []{data-label="f2.17"}](f20.eps){width="3.5in"} ![LAT 68% confidence radii localizations for a source with integral flux (above 100 MeV) of $10^{-7}$ ph cm$^{-2}$ s$^{-1}$ versus source spectral index for a source detected in the one-year sky survey. The variation of angular resolution with energy and viewing angle from the instrument axis is taken into account. In effect, the source viewing angle is averaged over in sky-survey mode. The source is assumed to be located in a region with uniform background with integral diffuse flux (above 100 MeV) of $1.5 \times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$ and spectral index –2.1. The source localization radius scales as (flux)$^{-1/2}$. []{data-label="f3.1"}](f21.eps){width="3.5in"} ![Minimum time necessary to detect a source at high latitude with $5\sigma$ significance (thick solid curve), to measure its flux with an accuracy of 20% (thin solid curve) and its spectral index with an uncertainty of 0.1 (dashed curve), as a function of source flux. A photon spectral index of 2.0 is assumed. The steps at short times are due to the discontinuous source coverage due to the observatory survey mode. []{data-label="f3.2"}](f22.eps){width="3.5in"} ![(a) Cumulative number distribution of EGRET detected blazars measured over two-week intervals (FSRQs: blue curves, BL Lac objects: red curves) and various model predictions (@Stecker1996: long-dashed line; @Mucke2000: dashed-dotted lines; @Dermer2007: dashed lines). The predicted number of radio-loud AGN ranges from $\sim$$10^3$ up to $\sim$$10^4$ sources. (b) The predicted power distribution of radio-loud AGN for the respective models, with the solid black line representing the model of @Narumoto2006, the black dotted line corresponds to the predicted power distribution of @Stecker1996. The colored histograms correspond to the predicted power distributions for BL Lacs (blue) and FSRQs (red) of @Mucke2000, and the colored curves correspond to the power distributions of BL Lacs (blue) and FSRQs (red) of @Dermer2007. The main contribution to the extragalactic diffuse [$\gamma$-ray]{} background is predicted to come from sources at the peak of the respective model distribution. []{data-label="f3.3"}](f23a.eps "fig:"){width="3.5in"} ![(a) Cumulative number distribution of EGRET detected blazars measured over two-week intervals (FSRQs: blue curves, BL Lac objects: red curves) and various model predictions (@Stecker1996: long-dashed line; @Mucke2000: dashed-dotted lines; @Dermer2007: dashed lines). The predicted number of radio-loud AGN ranges from $\sim$$10^3$ up to $\sim$$10^4$ sources. (b) The predicted power distribution of radio-loud AGN for the respective models, with the solid black line representing the model of @Narumoto2006, the black dotted line corresponds to the predicted power distribution of @Stecker1996. The colored histograms correspond to the predicted power distributions for BL Lacs (blue) and FSRQs (red) of @Mucke2000, and the colored curves correspond to the power distributions of BL Lacs (blue) and FSRQs (red) of @Dermer2007. The main contribution to the extragalactic diffuse [$\gamma$-ray]{} background is predicted to come from sources at the peak of the respective model distribution. []{data-label="f3.3"}](f23b.eps "fig:"){width="3.5in"} ![Spectral energy distributions (SEDs) of four [$\gamma$-ray]{} blazars: 3C 279 (a typical FSRQ, $z = 0.5362$, top); W Com (a low energy peaked BL Lac object, LBL, $z = 0.102$) and PKS 2155-304 (a high energy peaked BL Lac object, HBL, $z = 0.116$) middle; M 87 (a FR-I radio galaxy, $z = 0.00436$, bottom). Included in the SEDs are multiwavelength data points collected in different epochs (different brightness states) for each source (errors bars not represented for clarity). A qualitative representation of the average expected LAT pass band and sensitivity for 1 year of observations is shown. The LAT integral sensitivity shows the minimum needed for a 20% determination of the flux after a one-day (yellow/upper bowties), one-month (orange/middle bowties), and one-year (red/bottom bowties) exposure of in all-sky survey mode for a blazar with a $E^{-2}$ [$\gamma$-ray]{} spectrum. The resulting significance at each of these levels is about $8\sigma$, the spectral index is determined to about 6%, and the bowtie shape indicates the energy range that contributes the most to the sensitivity. To make a measurement at that level or better, a flat spectral energy density curve must lie above the axis of the bowtie. []{data-label="f3.4"}](f24.eps){width="3.0in"} ![The observed EGRET Vela pulsar spectrum, along with realizations of the expected spectrum after one year of GLAST LAT sky survey observations for two pulsar models. The expected sensitivity allows discrimination between the two models and allows tests of the emission zone structure through phase resolved spectra. []{data-label="f3.5"}](f25.eps){width="3.5in"} ![Simulation of the K3/Rabbit PWN complex. (a): simulated smoothed LAT count maps from 5 years of sky survey-mode observation -– the K3 region is at upper left and the Rabbit nebula is at lower right. Shown are the full $E > 100$ MeV emission and the $E > 3$ GeV non-pulsed emission (obtainable by gating off of the PSR J1420-6048 pulse). The green contours show the HESS TeV emission. At high energies the two PWNe are clearly resolved. (b): the HESS spectrum of Compton emission from the PWNe along with simulated LAT spectra from five years of sky-survey type observations (red points). The blue points show the simulated off-pulse spectrum measured with the LAT, indicating a clear detection of the synchrotron component of the PWNe. Also shown are two pulsar-like spectra. The brighter pulsar model is for Vela-like emission from PSR J1420-60438; the fainter (dashed line) is for an unknown Geminga-like pulsar in the Rabbit. []{data-label="f3.6"}](f26a.eps "fig:"){width="3.5in"}![Simulation of the K3/Rabbit PWN complex. (a): simulated smoothed LAT count maps from 5 years of sky survey-mode observation -– the K3 region is at upper left and the Rabbit nebula is at lower right. Shown are the full $E > 100$ MeV emission and the $E > 3$ GeV non-pulsed emission (obtainable by gating off of the PSR J1420-6048 pulse). The green contours show the HESS TeV emission. At high energies the two PWNe are clearly resolved. (b): the HESS spectrum of Compton emission from the PWNe along with simulated LAT spectra from five years of sky-survey type observations (red points). The blue points show the simulated off-pulse spectrum measured with the LAT, indicating a clear detection of the synchrotron component of the PWNe. Also shown are two pulsar-like spectra. The brighter pulsar model is for Vela-like emission from PSR J1420-60438; the fainter (dashed line) is for an unknown Geminga-like pulsar in the Rabbit. []{data-label="f3.6"}](f26b.ps "fig:"){width="1.5in"} ![The HESS spectrum of the shell SNR RX J1713.7-3946, with plausible leptonic and hadronic models. Note that EGRET was not able to distinguish the SNR from the relatively bright nearby point source 3EG J1714-3857 and upper limits to the SNR flux from this source’s spectrum did not allow EGRET to distinguish these possibilities. However, 5y of LAT observations at typical sky survey duty cycle can. (a) Simulated LAT spectra for the two cases [@Funk2008]. A differential spectral index of $\gamma=2$ has been assumed for both the parent proton and electron energy distributions. The observed [$\gamma$-ray]{} spectrum is sensitive to that assumption, which limits the ability to differentiate between parent species. (b) Result of a Lucy-Richardson deconvolution of the simulated LAT counts map, after cleaning of the point source. The SNR is clearly resolved, although the bright background of the Galactic plane limits the S/N of the detection. []{data-label="f3.7"}](f27a.eps "fig:"){width="3.0in"}![The HESS spectrum of the shell SNR RX J1713.7-3946, with plausible leptonic and hadronic models. Note that EGRET was not able to distinguish the SNR from the relatively bright nearby point source 3EG J1714-3857 and upper limits to the SNR flux from this source’s spectrum did not allow EGRET to distinguish these possibilities. However, 5y of LAT observations at typical sky survey duty cycle can. (a) Simulated LAT spectra for the two cases [@Funk2008]. A differential spectral index of $\gamma=2$ has been assumed for both the parent proton and electron energy distributions. The observed [$\gamma$-ray]{} spectrum is sensitive to that assumption, which limits the ability to differentiate between parent species. (b) Result of a Lucy-Richardson deconvolution of the simulated LAT counts map, after cleaning of the point source. The SNR is clearly resolved, although the bright background of the Galactic plane limits the S/N of the detection. []{data-label="f3.7"}](f27b.ps "fig:"){width="3.0in"} ![[$\gamma$-ray]{} burst localization with the LAT. The lines correspond to the scaling law between the location accuracy (at $1\sigma$) and the intensity of the burst, expressed as fluence in the 50–300 keV band. Solid lines correspond to GRB at normal incidence, and dashed lines to $60^\circ$ off-axis. Different sets of lines are for different high-energy spectral indexes (assuming the Band function describes the GRB spectral energy distribution). The starting points of the lines, (filled circles for on-axis, and empty for off-axis) correspond to the minimum fluence required to detect a burst (at least 10 counts in the LAT detector). []{data-label="f3.8"}](f28.eps){width="3.5in"} ![Number of clumps observed by [*Fermi*]{}/LAT vs. number of $\sigma$ significance in 5 years of LAT observations in all-sky scanning mode (solid line); 1 year of observation (dashed line). A generic WIMP of mass 100 GeV and $\langle\sigma v\rangle = 2.3\times10^{-26}$ cm$^3$ s$^{-1}$, a halo clump distribution from @Taylor2005a ([-@Taylor2005a; -@Taylor2005b]) and diffuse [$\gamma$-ray]{} backgrounds according to @Sreekumar1998 and @Strong2000 have been assumed. []{data-label="f3.10"}](f29.eps){width="3.5in"} [^1]: FoV $=\int A_{\rm eff}(\theta,\phi) d\Omega/A_{\rm eff}(0,0)=2.4$ sr at 1 GeV, where $A_{\rm eff}$ is the effective area of the LAT after all analysis cuts for background rejections have been made. [^2]: pitch = distance between centers of adjacent strips. [^3]: The term “hit” refers to the detection of the passage of a charged particle through a silicon strip and the recording of the strip address.

--- address: | CSSM and Department of Physics and Mathematical Physics, University of Adelaide, Australia 5005\ E-mail: [awilliam@physics.adelaide.edu.au]{} author: - ', FREDERIC D.R. BONNET, PATRICK O. BOWMAN, DEREK B. LEINWEBER, JON IVAR SKULLERUD, AND JAMES M. ZANOTTI' title: 'GLUONS, QUARKS, AND THE TRANSITION FROM NONPERTURBATIVE TO PERTURBATIVE QCD ' --- =cmr8 1.5pt \#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{} Introduction {#sec:intro} ============ Lattice gauge theory is currently the only known “first principles” approach to studying nonperturbative QCD. It is therefore important for lattice QCD to provide constraints and guidance for the construction of quark-based models[@DSE_review] and to provide an indication of the momentum regime at which we can expect perturbative QCD to become applicable. The quark and gluon propagators are two of the most fundamental quantities in QCD. There has been considerable interest in the infrared behavior of the gluon propagator as a probe into the mechanism of confinement and by studying the scalar part of the quark propagator, the mass function, we can gain insight into the mechanisms of chiral symmetry breaking. Both are used as input for other quark-model calculations. Gluon Propagator ================ We use an ${\cal O}(a^2)$ tree-level, tadpole-improved action[@Weisz83] and for the tadpole (mean-field) improvement parameter we use the plaquette measure[@tadpole]. A full description and discussion of the gluon propagator results summarized here can be found elsewhere.[@LandauGaugeDE; @long_glu; @big_vol_glu] Dimensions $\beta$ $a$ (fm) Volume $\text{(fm}^4\text{)}$ Configurations ---- ----------------- --------- ---------- ------------------------------- ---------------- 1w $16^3\times 32$ 5.70 0.179 $2.87^3 \times 5.73$ 100 1i $16^3\times 32$ 4.38 0.166 $2.64^3 \times 5.28$ 100 2 $10^3\times 20$ 3.92 0.353 $3.53^3 \times 7.06$ 100 3 $8^3 \times 16$ 3.75 0.413 $3.30^3 \times 6.60$ 100 4 $16^3\times 32$ 3.92 0.353 $5.65^3 \times 11.30$ 100 5 $12^3\times 24$ 4.10 0.270 $3.24^3 \times 6.48$ 100 6 $32^3\times 64$ 6.00 0.099 $3.18^3 \times 6.34$ 75 : Details of the lattices used to calculate the gluon propagator. Lattices 1w and 1i have the same dimensions and approximately the same lattice spacing, but were generated with the Wilson and improved actions respectively. Lattice 6 was generated with the Wilson action.[]{data-label="table:latlist"} Gauge fixing on the lattice is achieved by maximizing a functional, the extremum of which implies the gauge fixing condition. The usual Landau gauge fixing functional implies that $\sum_\mu {\partial}_\mu A_\mu = 0$ up to [${\cal O}(a^{2})$]{}. To ensure that gauge dependent quantities are also [${\cal O}(a^{2})$]{} improved, we implement the analogous [${\cal O}(a^{2})$]{} improved gauge fixing.[@LandauGaugeDE] The dimensionless lattice gluon field $A_{\mu}(x)$ is calculated from the link variables in the usual way, which agrees with the continuum to ${\cal O}(a^2)$. We then calculate the scalar part of the propagator $$D(x-y) = \sum_\mu \langle A_\mu(y) A_\mu(x) \rangle \, .$$ To isolate the nonperturbative behavior of the gluon propagator, we can divide the propagator by its lattice tree level form (i.e., that of lattice perturbation theory).[@long_glu] For the momentum space gluon propagator $D(q^2)$, we see that in the continuum $q^2D(q^2)$ will approach a constant up to logarithmic corrections as $q^2\to\infty$ because of asymptotic freedom. The continuum tree-level propagator is $1/q^2$. We also expect asymptotic freedom on the lattice despite finite lattice spacing artefacts. We [*define*]{} the lattice $q_\mu$ such that the lattice $D^{\rm tree}(q)\equiv 1/q^2$, and use this momentum throughout. This is referred to as tree-level correction and we have seen that it significantly reduces discretization arrors at large momenta. For the two actions considered here, this means that we work with the momentum variables defined as $$q_{\mu}^W \equiv \frac{2}{a} \sin\frac{{\hat{q}}_{\mu} a}{2}, \hskip1cm q_\mu^I \equiv \frac{2}{a}\sqrt{ \sin^2 \Bigl( \frac{{\hat{q}}_\mu a}{2} \Bigr) + \frac{1}{3}\sin^4 \Bigl( \frac{{\hat{q}}_\mu a}{2} \Bigr) } \, , \label{eq:latt_momenta}$$ for the Wilson and improved actions respectively. All figures (quark and gluon propagators) have a cylinder cut imposed upon them, i.e. all momenta must lie close to the lattice diagonal. In Table \[table:latlist\] we show the various lattices that we have studied for the gluon propagator. In Fig. \[fig:Comp1i\_6\] we plot $q^2 D(q^2)$ for a fine unimproved Wilson action and for our finest improved action. Despite having very different lattice spacings the agreement is excellent for the entire intermediate and high-momentum regime. The small discrepancy in the deep infrared due to finite volume effects is not apparent in this way of plotting that data. We plot $D(q^2)$ for five different lattice in Fig. \[fig:AllProps\] and see pleasing agreement for the results. Note that we are plotting bare quantities only and there is thus an overall wavefunction renormalization for the gluon propagator (i.e., $Z_3(\mu,a)$ for the renomalization point $\mu$). The vertical scale is thus unimportant and only the variation with momentum is relevant. This way of presenting the data shows that there is a small residual finite volume dependence, where the infrared gluon propagator is [*decreasing*]{} with increasing lattice volume $V$. We have performed a fit as a function of $1/V$ and have seen that the large volume $\beta=3.92$ lattice gives results which are very close to the infinite volume limit. In Fig. \[fig:pert\_vs\_latt\] we plot $D(q^2)$ in the intermediate and ultraviolet regime and compare with the three-loop perturbative QCD form. We see that the above $q\sim 2$ GeV the agreement is excellent, but that below this momentum scale nonperturbative effects are becoming apparent. Quark Propagator ================ The Landau gauge quark propagator results summarized here have been presented and discussed in more detail elsewhere.[@quark_prop] All ${\cal O}(a)$ errors in the fermion action can be removed by adding appropriate terms to the Lagrangian[@Luscher:1996sc; @Dawson:1997gp]. It is then usual to perform appropriate field transformations to improve the quark operators as well.[@Heatlie:1991kg] (14,7) (0,0) (7,7)(-0.9,-0.4)[[ ]{}]{} (7,0) (7,7)(-0.9,-0.4)[[ ]{}]{} In the continuum, the quark propagator has the following general form, $$S(p) = \frac{1}{i{\not\!p}A^c(p) + B^c(p)} \equiv \frac{Z^c(p)}{i{\not\!p}+M^c(p)}.$$ We expect the lattice quark propagator to have a similar form, but with ${\not\!k}$ replacing ${\not\!p}$: $$S(p) = \frac{Z(p)}{i{\not\!k}+ M(p)}$$ where $k$ is a new ‘lattice momentum’, $k_\mu = \frac{1}{a}\sin(\hat p_\mu a)$. We do not have sufficient space here to describe the hybrid tree-level correction that was used for the quark propagator results presented here, but a detailed description has recently been given.[@quark_prop] We again use the cylinder cut to further reduce hypercubic discretization artefacts. As for the gluon propagator the results for $Z(p)$ are for the bare quantity only and contain an overall renormalization constant $Z_2(\mu,a)$. In Fig. \[fig:z\_np\_compare\] the vertical scales have been adjusted so that the two sets of results are renormalized and hence coincide at 2.1 GeV. In this figure we see the charactreistic dip in the infrared for $Z(p)$, which occurs also in model Dyson-Schwinger equation studies[@DSE_review] of dynamical chiral symmetry breaking. This dip has essentially disappeared by around 2 GeV. The improved action correspondning to $S_R$ is the one we prefer and it gives the more expected ultraviolet behavior of the renormalized $Z(p)$, i.e., it tends towards a constant. In Fig. \[fig:m\_np\_phys\] we see the characteristic behavior of the quark mass function familiar from quark model studies[@DSE_review] and that the transition to the perturbative regime is occuring at approximately 2 GeV. Note that there is no renormalization of the quark mass function. At large momenta the mass function should become the running quark mass of perturbative QCD. Finally in Fig. \[fig:Mall-chiral\] we present a simple quadratic extrapolation to the chiral limit for the available $S_I$ data. The slight dip is not statistically significant and is almost certainly a residual lattice artefact. The infrared mass (i.e., at $p=0$) in the chiral limit is approximately $330\pm30$ MeV, which is characteristic of the constituent quark mass scale. Summary and Conclusions ======================= The gluon propagator has been calculated on fine unimproved lattices and on a variety of improved lattices with an [${\cal O}(a^{2})$]{} improved action in [${\cal O}(a^{2})$]{} improved Landau gauge. The infrared behavior of this propagator strongly suggests the the Landau gauge gluon propagator is infrared finite. We have ruled out the $1/q^{4}$ behavior popular in some Dyson-Schwinger quark model studies[@DSE_review] and indeed any infrared singularity appears to be very unlikely. The possible effects of lattice Gribov copies remains a very interesting question and we are currently carrying out similar studies across a variety of lattices in Laplacian gauge, which is a Landau-like smooth gauge fixing, but is free of Gribov copies. We have used two different definitions of the ${\cal O}(a)$ improved quark propagator, corresponding to the quark propagators denoted $S_I$ and $S_R$. We make use of asymptotic freedom to factor out the tree level behaviour, replacing it with the ‘continuum’ tree level behaviour $Z(p)=1, M(p)=m$. This tree-level correction dramatically improves the data. We find that $M(0)$ approaches a value of $300\pm30$ MeV in the chiral limit, which is very much in keeping with the concept of a “constituent quark mass” and agrees with the infrared values of the quark mass commonly used in model studies.[@DSE_review] We also find a significant dip in the value for $Z(p)$ at low momenta. This is again entirely consistent with what is found in model studies of dynamical chiral symmetry breaking [@DSE_review]. An examination of Figs. \[fig:Comp1i\_6\], \[fig:pert\_vs\_latt\], \[fig:z\_np\_compare\], \[fig:m\_np\_phys\], and \[fig:Mall-chiral\] provide a clear indication that perturbative QCD behavior is not becoming dominant in the gluon and quark propagators until we reach momenta of order $Q^2\simeq 4$ GeV$^2$. References {#references .unnumbered} ========== [99]{} C.D. Roberts and A.G. Williams, Prog. Part. Nucl. Phys. [**33**]{}, 477 (1994). P. Weisz, Nucl. Phys. B [**212**]{}, 1 (1983). G.P. Lepage & P.B. Mackenzie, Phys. Rev. D [**48**]{}, 2250 (1993). F.D.R. Bonnet, P.O. Bowman, D.B. Leinweber, A.G. Williams & D.G. Richards, Aust. J. Phys. [**52**]{}, 939 (1999); Nucl. Phys. B (Proc. Suppl.) [**83–84**]{}, 905–907 (2000). D.B. Leinweber, J.I. Skullerud, A.G. Williams, and C. Parrinello, Phys. Rev. D [**58**]{}, 031501 (1998); [*ibid.*]{} [**60**]{}, 094507 (1999); Erratum – [*ibid.*]{} [**61**]{}, 079901 (2000); D.B. Leinweber, C. Parrinello, J.I. Skullerud, and A.G. Williams Nucl. Phys. B (Proc. Suppl.) [**73**]{}, 629-631 (1999); [*ibid.*]{}, 626-628 (1999). F.D.R. Bonnet, P.O. Bowman, D.B. Leinweber, and A.G. Williams, Phys. Rev. D [**62**]{}, 051501 (2000); [*ibid.*]{} [**64**]{}, 034501 (2001). M. Lüscher [*et al.*]{}, Nucl. Phys. B [**478**]{}, 365 (1996). C. Dawson [*et al.*]{}, Nucl. Phys. B (Proc. Suppl.) [**63**]{}, 877 (1998). G. Heatlie [*et al.*]{}, Nucl. Phys. B [**352**]{}, 266 (1991). J.I. Skullerud and A.G. Williams, Nucl. Phys. B (Proc. Suppl.) [**83–84**]{}, 209–211 (2000); Phys. Rev. D [**63**]{}, 054508 (2001); J.I. Skullerud, D.B. Leinweber, and A.G. Williams, to appear in Phys. Rev. D, hep-lat/0102013.

--- author: - | P. Cea\ Dipartimento Interateneo di Fisica, Università di Bari and INFN - Sezione di Bari,\ I-70126 Bari, Italy\ E-mail: - | M. Consoli\ INFN - Sezione di Catania, I-95123 Catania, Italy\ E-mail: - | L. Cosmai\ INFN - Sezione di Bari, I-70126 Bari, Italy\ E-mail: title: 'Large logarithmic rescaling of the scalar condensate: a subtlety with substantial phenomenological implications' --- Introduction {#Introduction} ============ Recent lattice data [@Cea:2004ka], collected near the critical line of a 4D Ising model, support the large logarithmic rescaling of the scalar condensate predicted in an alternative description of symmetry breaking in $\Phi^4$ theories, see Refs.[@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni]. This result, while confirming previous numerical indications obtained in Refs. [@Cea:1998hy; @Cea:1999kn; @Cea:1999zu], would have a substantial phenomenological implication: one cannot use ‘triviality’ to place upper bounds on the Higgs boson mass. This point of view has been challenged in a recent paper [@Balog:2004zd] by Balog et al.. These authors, referring just to Ref.[@Cea:2004ka], while otherwise ignoring the previous numerical indications of Refs. [@Cea:1998hy; @Cea:1999kn; @Cea:1999zu], draw the opposite conclusion: the standard interpretation, as they say the ‘Conventional Wisdom’ (CW), is completely consistent with all lattice data. The aim of this paper is to respond to their criticism, recapitulate in a unified framework the results of Ref.[@Cea:2004ka] and Refs. [@Cea:1998hy; @Cea:1999kn; @Cea:1999zu], and reiterate our conclusion: ‘triviality’, by itself, cannot be used to place upper bounds on the Higgs boson mass. The rescaling of the scalar condensate {#The rescaling} ====================================== Before entering the details of the controversy, we shall first remind once again in this section why in a spontaneously broken phase there are [*two*]{} basically different definitions of the field rescaling. In fact, the widespread skepticism concerning the interpretation of ‘triviality’ proposed in Refs.[@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni] originates from a non sufficient appreciation of this crucial point. To this end, let us introduce the bare ‘lattice’ field $\Phi_B(x)=\Phi_{\text{latt}}(x)$ (i.e. as defined at a locality scale fixed by the ultraviolet cutoff $\Lambda\sim \pi/a$, $a$ being the lattice spacing) and its expectation value (the ‘scalar condensate’) $$\label{vB} v_B=\langle\Phi_B\rangle$$ Connecting to the stability analysis, the values $\pm v_B$ represent the absolute minima of the effective potential $V_{\text{eff}}(\varphi_B)$ of the theory. We shall also introduce the bare shifted fluctuation field $$\label{hB} h_B(x)=\Phi_B(x)- \langle\Phi_B\rangle,$$ whose expectation value vanishes by definition. Now, a first natural definition of the field rescaling, say $Z=Z_{\text{prop}}$, is obtained from the residue of the shifted-field propagator $$\label{pole} G_{\text{pole}}(s)\sim \frac{Z_{\text{prop}}}{ s - m^2_H}$$ near the physical mass-shell $s=m^2_H$. This can be used to define a renormalized fluctuation field $h_R(x)$ $$\label{hr} h_B(x)=\sqrt{Z_{\text{prop}}} h_R(x)$$ whose propagator has the canonical form. This first definition of the field rescaling is constrained by the Kállen-Lehmann representation to lie in the range $0 < Z_{\text{prop}}\leq 1$, the free-field limit corresponding to the case $Z_{\text{prop}} = 1$. This can be rigorously established for the local, Lorentz-covariant theory. If this is viewed as the continuum limit of the cutoff theory, one expects [@Zimmermann:1970aa] $$\label{zimmermann} Z^{-1}_{\text{prop}}=1+\int^{\Lambda^2}_{s_o}ds\rho_\Lambda(s)$$ where $s_o$ denotes the continuum threshold ($s_o=(2m_H)^2$ in perturbation theory) and $\rho_\Lambda(s)$ the spectral function. For this reason, in the continuum limit $\Lambda \to \infty$, where according to ‘triviality’ the spectral function should tend to $\delta(s-m^2_H)$, $Z_{\text{prop}}$ should tend to unity. Another definition of the field rescaling, say $Z\equiv Z_\varphi$, is peculiar of a broken-symmetry phase. It indicates the rescaling that is needed to relate the [*physical*]{} vacuum field $v_R$ to the bare $v_B$, i.e. $$\label{vR} v_R= \frac{v_B}{\sqrt{Z_\varphi}} \,.$$ By [*physical*]{}, we mean that the second derivative of the effective potential $V''_{\text{eff}}(\varphi_R)$ evaluated at $\varphi_R=\pm v_R$, is precisely given by the physical Higgs boson mass squared, i.e. $$V''_{\text{eff}}(v_R)= m^2_H$$ This is very simple to understand. $V''_{\text{eff}}(\varphi_R)$ represents the renormalized 2-point function evaluated at an external 4-momentum $p_{\mu}=0$. For $\varphi_R=v_R$, this should match the inverse of the renormalized connected propagator at zero momentum. In a ‘trivial’ theory, in the continuum limit, this has the simple free-field form $G_R(0)= 1/m^2_H$. Therefore, this other definition is equivalent to the relation $$\label{z1phi} Z_\varphi= \frac{ V''_{\text{eff}}(v_R)} {V''_{\text{eff}}(v_B)}=m^2_H \chi_2 (0)$$ where $\chi_2(0)=1/V''_{\text{eff}}(v_B)$ is the bare zero-momentum susceptibility. Let us now explain why $Z_{\text{prop}}$ and $Z_\varphi$ are completely different physical quantities. To this end, we shall consider the class of ‘triviality compatible’, gaussian-like approximations to the effective potential, say $V_{\text{eff}}(\varphi_B)\equiv V_{\text{triv}}(\varphi_B)$, where the shifted fluctuating field is governed by an effective quadratic hamiltonian. In this class, either $Z_{\text{prop}} = 1$ identically or it tends to unity in the continuum limit $\Lambda \to \infty$. In spite of this, the rescaling of the vacuum field $$\label{z2phi} Z_\varphi= m^2_H \chi_2 (0) \sim \ln \Lambda$$ diverges logarithmically. This result is easily recovered noticing that the class $V_{\text{triv}}$ includes the one-loop potential, the gaussian approximation and the infinite set of post-gaussian calculations where the effective potential reduces to the sum of a classical background energy and of the zero-point energy of the [*free*]{} massive shifted field with a $\varphi_B-$dependent mass, $\Omega=\Omega(\varphi^2_B)$. For instance, at one loop one finds $\Omega^2(\varphi^2_B)=m^2_B+\frac{\lambda_B\varphi^2_B}{2}$, $m^2_B$ and $\lambda_B$ denoting respectively the bare mass squared and the bare fourth-order self-coupling. In gaussian and post-gaussian calculations, the functional dependence $\Omega=\Omega(\varphi^2_B)$ is determined upon a minimization procedure of the energy functional in a suitable class of quantum trial states. In all cases, the basic mass parameter of the broken-symmetry phase is obtained through the relation $m_H=\Omega(v^2_B)$. Therefore, the various approximations belonging to $V_{\text{triv}}$ share the same simple general structure, up to non-leading terms that vanish faster when $\Lambda \to \infty$ (see Ref.[@Consoli:1999ni] for the general argument, Ref.[@Branchina:1993rj] for the gaussian approximation and Ref.[@Ritschel:1994vr] for the case of a post-gaussian approximation). For the particularly simple case of the classically scale-invariant theory (the ‘Coleman-Weinberg regime’) this is given by $$\label{vtriv} V_{\text{triv}}(\varphi_B)= \frac{ \lambda_{\text{eff}} \varphi^4_B}{4!} + \frac{\Omega^4(\varphi^2_B) }{64\pi^2} (\ln\frac{\Omega^2(\varphi^2_B)}{\Lambda^2} -\frac{1}{2})$$ with $$\label{omega} \Omega^2(\varphi_B)= \frac{\lambda_{\text{eff}} \varphi^2_B}{2}$$ all differences among the various approximations being isolated in the relation between the effective coupling $\lambda_{\text{eff}}=\lambda_{\text{eff}}(\Omega)$ and the bare coupling. For instance, at one loop $\lambda_{\text{eff}}=\lambda_B$ while in the gaussian approximation (see Ref.[@Branchina:1993rj]) $\lambda_{\text{eff}}$ corresponds to resum all one-loop bubbles with mass $\Omega=\Omega(\varphi^2_B)$ so that $$\label{leff} \lambda_{\text{eff}}(\Omega)=\frac{\lambda_B} {1+ \frac{\lambda_B}{16\pi^2} \ln\frac{\Lambda}{\Omega} }$$ Now, since all approximations display the same general structure in Eqs.(\[vtriv\]) and (\[omega\]), upon minimization and after setting $\varphi_B=\pm v_B$, one finds the same leading logarithmic trend of the effective coupling $$\label{ltriv} \lambda_{\text{eff}}(m_H) \sim \frac{ 16\pi^2}{ 3 \ln \frac{\Lambda}{m_H}}$$ Formally, this is precisely the same relation obtained in perturbation theory from the leading-order (LO) renormalized coupling $$\label{lambdaR} \lambda_{\text{LO}}(m_H) =\frac{\lambda_B}{1+ \frac{3\lambda_B}{ 16\pi^2}\ln \frac{\Lambda}{m_H} }$$ after sending $\lambda_B \to \infty$ at the Landau pole $$\label{lambdaR2} \lambda_{\text{LO}}(m_H) \sim \frac{ 16\pi^2}{ 3 \ln \frac{\Lambda}{m_H}}$$ Therefore, one might be tempted to conclude that, at least at the leading logarithmic level, the alternative picture of Refs.[@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni], is equivalent to the CW. However, there is one notable difference: the quadratic shape of the ‘trivial’ effective potential Eq.(\[vtriv\]) at its absolute minima, namely the inverse susceptibility $V''_{\text{triv}}(v_B)$, is [*not*]{} given by $m^2_H=\Omega^2(v^2_B)$. Rather, one finds $$\label{shape} V''_{\text{triv}}(v_B)\sim \lambda_{\text{eff}} m^2_H \sim \frac{m^2_H}{ \ln \Lambda }$$ thus leading to Eq.(\[z2phi\]). Eq. (\[z2phi\]) has substantial phenomenological implications. In fact, using Eq. (\[vR\]) with a logarithmically divergent $Z_\varphi$ as in Eq. (\[z2phi\]), one finds $$\label{vrvb} v^2_R \sim \frac{ v^2_B }{ \ln \Lambda }$$ instead of the conventional relation obtained for a trivial unit rescaling $$\label{vrvbpt} \left[ v^2_R \right]_{\text{CW}} \sim v^2_B$$ Now, whatever the $v_R-v_B$ relation, the Higgs boson mass squared goes like $$\label{stable} m^2_H \sim \frac{ v^2_B }{ \ln \Lambda }$$ This becomes $$\label{usual} m^2_H \sim \frac{ \left[ v^2_R \right]_{\text{CW}} }{ \ln \Lambda }$$ by using Eq.(\[vrvbpt\]) but becomes $$\label{golden} m^2_H \sim v^2_R$$ if one defines $v_R$ through Eq.(\[vrvb\]). In the latter case, $m_H$ and $v_R$ scale uniformly in the continuum limit. Therefore, assuming to relate the $v_R$ of Eq.(\[vrvb\]) (and [*not*]{} the conventional $\left[ v_R \right]_{\text{CW}} \sim v_B$ of Eq.(\[vrvbpt\])) to some physical scale (say 246 GeV), a measurement of $m_H$ would not provide any information on the magnitude of $\Lambda$ since, according to Eq.(\[golden\]), the ratio $C=m_H/v_R$ is now a cutoff-independent quantity. Moreover, in this approach, the quantity $C$ does not represent the measure of any [*observable*]{} interaction (see the Conclusions of Ref.[@Agodi:1995qv]). We emphasize that the difference between $Z_\varphi$ and $Z_{\text{prop}}$ has a precise physical meaning being a distinctive feature of the Bose condensation phenomenon [@Consoli:1999ni]. In the class of ‘triviality-compatible’ approximations to the effective potential, one finds $C=m_H/v_R=2 \pi \sqrt{2 \zeta}$, with $0< \zeta \leq 2$ [@Consoli:1999ni], $\zeta$ being a cutoff-independent number determined by the quadratic shape of the effective potential $V_{\text {eff}}(\varphi_R)$ at $\varphi_R=0$. For instance, $\zeta=1$ corresponds to the classically scale-invariant case or ‘Coleman-Weinberg regime’. As for the standard interpretation of ‘triviality’, the gaussian effective potential approach can also be extended to any number N of scalar field components [@Stevenson:1987nb]. In particular, when studying the continuum limit in the large-N limit of the theory, one has to take into account the non-uniformity of the two limits, cutoff $\Lambda \to \infty$ and $N \to \infty$ [@Ritschel:1992ss; @Ritschel:1994vr]. This is crucial to understand the difference with respect to the standard large-N analysis. Lattice tests of the alternative interpretation of ‘triviality’. Part 1. {#latticetest1} ======================================================================== To test the alternative picture of ‘triviality’ of Refs. [@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni] against the CW one can run numerical simulations of the theory and check the scaling properties of the squared Higgs lattice mass against those of the inverse zero-momentum lattice susceptibility. The numerical simulations of Ref.[@Cea:2004ka] (and of Refs.[@Cea:1998hy; @Cea:1999kn; @Cea:1999zu]) were performed in the Ising limit that traditionally has been chosen as a convenient laboratory for the numerical analysis of the theory. In this limit, a one-component $\Phi^4_4$ theory becomes governed by the lattice action $$\label{ising} S_{\text{Ising}} = -\kappa \sum_x\sum_{\mu} \left[ \phi(x+\hat e_{\mu})\phi(x) + \phi(x-\hat e_{\mu})\phi(x) \right]$$ where $\phi(x)$ takes only the values $\pm 1$. Using the Swendsen-Wang [@Swendsen:1987ce], and Wolff [@Wolff:1989uh] cluster algorithms we computed the bare magnetization: $$\label{baremagn} v_B=\langle |\phi| \rangle \quad , \quad \phi \equiv \sum_x \phi(x)/L^4$$ (where $\phi$ is the average field for each lattice configuration) and the bare zero-momentum susceptibility: $$\label{chi} \chi_{\text{latt}}=L^4 \left[ \left\langle |\phi|^2 \right\rangle - \left\langle |\phi| \right\rangle^2 \right] .$$ We report in Table 1 our determinations of $v_B$ and $\chi_{\text{latt}}$. Other values, obtained over the years by different authors, are reported in Table 1 of Ref. [@Balog:2004zd]. Checks of the logarithmic trend predicted in Eq.(\[z2phi\]) can be performed using different methods. In a first indirect approach, discussed in this Section, one can use the fact that, both in perturbation theory and according to Refs. [@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni], the bare field expectation value $v_B$ is predicted to diverge logarithmically in units of the physical Higgs boson mass $m_H$, i.e. $$\label{v2Blog} \frac{v^2_B}{m^2_H} \sim |\ln(\kappa-\kappa_c)| \,.$$ Therefore, following CW, where the zero-momentum susceptibility is predicted to scale uniformly with the inverse squared Higgs mass $$\label{mrchitau} \left[ m^2_H \chi_2(0) \right]_{\text{CW}}\sim 1$$ one expects $$\label{v2Bchi2pt} \left[ v^2_B \chi_2(0) \right]_{\text{CW}}\sim |\ln(\kappa-\kappa_c)|$$ On the other hand, in the approach of Refs. [@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni] one predicts $Z_\varphi=\chi_2(0)m^2_H \sim \ln(\Lambda)$ so that, in this case, one would rather expect (CS=Consoli-Stevenson) $$\label{v2Bchi2cs} \left[ v^2_B \chi_2(0) \right]_{\text{CS}}\sim |\ln(\kappa-\kappa_c)|^2$$ The two leading-order predictions in Eq. (\[v2Bchi2pt\]) and in Eq. (\[v2Bchi2cs\]) were directly compared with the lattice data for the product $v^2_B \chi_2(0)$ reported in Table 3 of Ref.[@Cea:2004ka]. These data were fitted to a 3-parameter form $$\label{Ffit} \alpha |\ln(\kappa-\kappa_c)|^\gamma$$ where $\alpha$ is a normalization constant and one can set the exponent $\gamma=1$, according to Eq. (\[v2Bchi2pt\]), or $\gamma=2$ according to Eq. (\[v2Bchi2cs\]). Using this type of functional form the results of the fit to the lattice data single out unambiguously the alternative picture of Refs. [@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni], i.e. $\gamma=2$, over the value $\gamma=1$ (see Fig.2 of Ref.[@Cea:2004ka]). However, it has been pointed out by the authors of Ref.[@Balog:2004zd] that the lattice data can also be reproduced for $\gamma=1$, once the scale within the log is left as a free parameter, thus effectively simulating the presence of next-to-leading corrections. Indeed, in this case one gets a good fit in both cases with precise determinations of the critical point, namely $\kappa_c=0.074833(17)$ for $\gamma=1$ and $\kappa_c=0.074819(15)$ for $\gamma=2$, in good agreement with the value $\kappa_c=0.074834(15)$ obtained by Gaunt et al. [@Gaunt:1979aa] from the symmetric phase. In this sense, we can agree with Balog et al.: a definitive test to decide between the two asymptotic behaviours $\gamma=1$ and $\gamma=2$ has to be postponed to simulations performed closer to the critical point where the non-leading terms associated with the scale of the logs should become unessential. However, although one can certainly get a good fit with $\gamma=1$, the agreement between lattice data and the prediction based on 2-loop renormalized perturbation theory is [*not*]{} good. This can be checked considering the expression reported by Balog et al. ($l=|\ln (\kappa -\kappa_c)|$) $$\label{2loop} \left[ v^2_B \chi_2(0) \right]_{\text{2-loop}}= a_1( {l} - \frac{25}{27} \ln { l}) +a_2$$ together with the theoretical relations [@Balog:2004zd] $$\label{c2} a_1=\frac{9(C'_2)^2}{32\pi^2}$$ and $$\label{cc2} \frac{a_2}{a_1}=\ln(C'_3)+2\ln(C'_1)-1.6317$$ Using the input values reported by Lüscher and Weisz (LW) in Table 1 of Ref .[@Luscher:1988ek] and the relations $C'_1=e^{1/6}C_1$, $C'_2=C_2$ and $C'_3=C_3$ (see Eqs.(4.37) and (4.38) of Ref.[@Luscher:1988ek]), the predictions for the Ising model ($\bar{\lambda}=1$) are: $C'_2=6.49(7)$, $2\ln(C'_1)=1/3+3.0(4)$ and $\ln(C'_3)=-3.0(1)$ or $a_1=1.20(3)$, $a_2=-1.6(5)$. However, the data for $v^2_B \chi_2(0)$ require $a_1=1.267(14)$ and $a_2=-2.89(8)$, see Ref.[@Balog:2004zd]. Therefore, the quality of the 2-loop fit is poor (see Fig.1). Nevertheless, one can adopt a pragmatic point of view, ignoring possible problems related to the matching conditions with the symmetric phase, and try to extract from the data for $v^2_B \chi_2(0)$ a new set of constants $C'_i$. In particular the precise determination from $a_1=1.267(14)$ implies $$\label{ccc2} C'_2=6.67(4)$$ that will be used in the following. Thus we can summarize the results of this section as follows:            1) leaving out the scale within the logs as a free parameter (and thus allowing effectively for the presence of next-to-leading corrections) the lattice data for $v^2_B\chi_2(0)$ are unable to distinguish between the powers $\gamma=1$ and $\gamma=2$. In this sense, a definitive test has to be postponed to simulations performed closer to the critical point where the non-leading terms associated with the scale of the logs should become unessential.            2) a 2-loop fit as in Eq.(\[2loop\]), with $a_1$ and $a_2$ deduced consistently from the LW tables, does [*not*]{} provide a good description of the data (see Fig.1). However, within the effective 2-loop formula Eq.(\[2loop\]), when $a_1$ and $a_2$ are left as free parameters, one can obtain precise determinations of the critical point $\kappa_c=0.074833(17)$ and of the integration constant $C'_2=6.67(4)$. These will be used in the next section to check whether the lattice observables are consistent with the critical behaviour predicted by Renormalization Group (RG) analysis. Lattice tests of the alternative interpretation of ‘triviality’. Part 2. {#latticetest2} ======================================================================== Additional numerical evidences concerning the relative scaling of $m_H$ and $\chi_2(0)$ can be obtained by comparing again with the predictions of perturbation theory. To this end, we performed in Ref.[@Cea:2004ka] a test of the logarithmic trend predicted in Eq.(\[z2phi\]) assuming as input entries the full 3-loop values $m_{\text{input}}\equiv m_R$ reported in the first column of Table 3 of Ref. [@Luscher:1988ek] at the various values of $\kappa$. These input mass values, to leading order, follow the scaling law $$\label{mrtau} m_H = A \sqrt{\kappa -\kappa_c}\cdot |\ln(\kappa-\kappa_c)|^{-1/6}$$ so that one can check whether the quantity $$\label{zphi} Z_\varphi\equiv 2\kappa m^2_{\text{input}} \chi_{\text{latt}}$$ tends to unity or grows logarithmically when approaching the continuum limit. Now, using in Eq.(\[zphi\]) the central values of $m_{\text{input}}\equiv m_R$ reported in the LW Table and the values of $\chi_{\text{latt}}$ reported in Table 1, the conclusion is unambiguous: the $Z_\varphi$ in Eq.(\[zphi\]) becomes larger and larger approaching the continuum limit along the RG curve $m_{\text{input}}=m_{\text{input}}(\kappa)$ and the observed increase is completely consistent with the logarithmic trend predicted in Eq.(\[z2phi\]) (see Fig.2). The discrepancy with the perturbative predictions can also be checked noticing that for $m_{\rm input}=0.08$ Ref. [@Luscher:1988ek] predicts $\kappa=0.07481(8)$, i.e. [*smaller*]{} than $0.0749$. Therefore, by inspection of Table 1, the relevant lattice susceptibility will be definitely [*larger*]{} than its value for $\kappa=0.0749$, $\chi_{\rm latt}\sim 1100$, so that using Eq. (\[zphi\]), one gets the [*lower bound*]{} $Z_\varphi > 1.05$ which cannot be reconciled with the perturbative predictions. Balog et al. object to our conclusions that “..the crucial question is whether the estimates $m_{\text{input}}$ of $m_R$ are reliable”. According to these authors, “..the measured values of $m_R$ are considerably lower than the corresponding estimates $m_{\rm input}$” (i.e. the $m_R$’s that we took from the LW table). As a matter of fact, by replacing the LW $m_R$’s with the results of their simulations, one gets a remarkably constant value $2\kappa m^2_R\chi_{\text{latt}}\sim 0.88$. Thus their objection does not concern our strategy but rather the validity of the LW entries themselves as reliable estimates of the Higgs mass parameter $m_H$. Actually, as we shall illustrate in the following, their conclusion does not apply: to check the true behaviour of $Z_\varphi=2\kappa m^2_H\chi_{\text{latt}}$ one should first identify correctly the operative definition of $m_H$ on the lattice. The mass values reported by Balog et al. are not [*reliable*]{} determinations of the physical Higgs mass if one requires the theoretical consistency of the adopted definition of ‘mass’. To fully appreciate what is going on, we need to go back to Refs. [@Cea:1998hy; @Cea:1999kn; @Cea:1999zu] (otherwise ignored by the authors of Ref.[@Balog:2004zd]). In those calculations, one was fitting the lattice data for the connected propagator to the (lattice version of the) two-parameter form $$\label{gprop} G_{\text{fit}}(p)= \frac{Z_{\text{prop}}}{ p^2 + m^2_{\text{latt}} } \,.$$ This is a clean and simple strategy: there is no reason to restrict the analysis of the propagator to the limit $p\to 0$. In fact, in a free-field theory (the lattice version of) Eq.(\[gprop\]) is valid in the full range $0\leq p^2 \leq \Lambda^2$ (with $\Lambda\sim \pi/a$). In a ‘trivial’ theory one expects a two-parameter fit to the propagator data to have small residual corrections. These, however, should become smaller and smaller by approaching the continuum limit. In this way, after computing the lattice zero-momentum susceptibility $\chi_{\text{latt}}$, it becomes possible to compare the measured value of $Z_\varphi \equiv 2\kappa m^2_{\text{latt}} \chi_{\text{latt}}$ with the fitted $Z_{\text{prop}}$, both in the symmetric and broken phases. While no difference was found in the symmetric phase (see Fig. 3), $Z_\varphi$ and $Z_{\text{prop}}$ were found to be sizeably different in the broken phase. The discrepancy was found to become larger and larger by approaching the continuum limit and thus cannot be explained in terms of residual perturbative corrections to Eq.(\[gprop\]). In particular, $Z_{\text{prop}}$ was very slowly varying and steadily approaching unity from below in the continuum limit consistently with Kállen-Lehmann representation and ‘triviality’. $Z_\varphi$, on the other hand, was found to rapidly increase [*above*]{} unity in the same limit. The observed trend was consistent with the logarithmically increasing trend predicted in Eq.(\[z2phi\]). This conclusion was based on the values of $m_{\text{latt}}$ extracted by skipping the lowest 3-4 $p^2-$values in the fit to the propagator data. In fact, differently from the symmetric-phase simulations, where Eq.(\[gprop\]) reproduces the data in the full range $0\leq p^2 \leq \Lambda^2$, in the broken-symmetry phase the two-parameter form Eq. (\[gprop\]) does not reproduce the propagator data down to $p=0$, see Fig.4 (and Figs.3,5,6 of Ref.[@Cea:1999kn]). Thus one has to choose. Either a) to restrict to the lowest 3-4 $p^2-$ data, and obtain from the fit a pair of values $(Z_{\text{low}},m_{\text{low}})$ or b) skip these first few data and fit the much larger sample of higher momentum data thus obtaining the other pair $(Z_{\text{high}},m_{\text{high}})$. The two sets of mass values differ sizeably. For instance for $\kappa=0.076,0.07512,0.07504$ the results of Refs. [@Cea:1999kn] were respectively $m_{\text{high}}=0.4286(46),0.2062(41),0.1723(34)$. On the other hand, the alternative values from the lowest momentum data are respectively $m_{\text{low}}= 0.392(4),0.1737(24),0.1419(17)$ for the same $\kappa$’s. Numerically, the typical $m_{\text{low}}$’s are extremely close to the other mass definition $m_{\text{TS}}({\bf{k=0}})$, the mass extracted at zero 3-momentum from the exponential decay (TS=‘Time Slice’) of the connected two-point correlator $$\label{corr} C_1(t,0; {\bf k})\equiv \langle S_c(t;{\bf k})S_c(0;{\bf k})+ S_s(t;{\bf k})S_s(0;{\bf k}) \rangle _{\rm conn} ,$$ where $$\label{cos} S_c(t; {\bf k})\equiv \frac{1}{L^3} \sum _{ { \bf x} } \phi({\bf x}, t) \cos ({\bf k} \cdot {\bf x}) ,$$ $$\label{sin} S_s(t;{ \bf k})\equiv \frac{1}{L^3} \sum _ {{\bf x}} \phi({\bf x}, t) \sin ({\bf k} \cdot {\bf x}) .$$ Here, $t$ is the Euclidean time; ${\bf x}$ is the spatial part of the site 4-vector $x^{\mu}$; ${\bf k}$ is the lattice momentum ${\bf k}=(2\pi/L) (n_x,n_y,n_z$), with $(n_x,n_y,n_z)$ non-negative integers; and $\langle ...\rangle_{\rm conn}$ denotes the connected expectation value with respect to the lattice action, Eq. (\[ising\]). In this way, parameterizing the correlator $C_1$ in terms of the energy $E_k$ as ($L_t$ being the lattice size in time direction) $$\label{fitcor} C_1(t,0;{\bf k})= A \, [ \, \exp(-E_k t)+\exp(-E_k(L_t-t)) \, ] \,,$$ the mass can be determined through the lattice dispersion relation $$\label{disp} m^2_{\rm TS}({\bf k}) = ~2 (\cosh E_k -1)~~ -~~2 \sum ^{3} _{\mu=1}~ (1-\cos k_\mu) \,.$$ In a free-field theory $m_{\rm TS}$ is independent of [**k**]{} and coincides with $m_{\text{latt}}$ from Eq. (\[gprop\]). Now, using the corresponding susceptibility values $\chi_{\text{latt}}=$37.85(6), 193.1(1.7), 293.4(2.9) reported in Ref. [@Cea:1999kn] and the above values of $m_H=m_{\text{high}}$ one obtains a set of logarithmically increasing values (see Fig.5): $Z_\varphi \equiv (2\kappa) m^2_H \chi_{\text{latt}}= 1.05(2),1.23(5),1.31(5)$ as predicted in Refs.[@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni]. This is in contrast with the values of the other quantity $Z_{\text{low}}\sim \hat{Z}_R\equiv (2\kappa) m^2_{\text{low}} \chi_{\text{latt}}= 0.884(18),0.875(24),0.887(21)$ which remain remarkably stable (see the corresponding entries for $\hat{Z}_R$ shown in Table 4 of Ref.[@Balog:2004zd]). Therefore, the crucial question raised by the propagator data is the following. Which is the ‘true’ lattice definition of $m_H$ ? Is the $p\to 0$ choice (adopted by Balog et al.) and represented by $m_{\text{low}}\sim m_{\text{TS}}({\mathbf{k=0}})$, or that obtained from $m_{\text{high}}$, as proposed in Refs. [@Cea:1999kn]? As discussed in Ref.[@Cea:1999kn], there are several arguments that suggest the correct definition of $m_H$ to be obtained from $m_{\text{high}}$, thus regarding the other values extracted from the very low-momentum region as a symptom of the distinct dynamics of the scalar condensate. Here we list a few:             i) in the continuum theory, the shifted fluctuation field is defined as the $p_\mu\neq 0 $ projection of the full quantum field. However in a lattice simulation with periodic boundary conditions, where the momenta $p_i$ are proportional to an integer number $n_i$ times the inverse lattice size $1/L$, the notion $p_\mu\neq 0$ is ambiguous. In fact, [*any*]{} finite set of integers will evolve onto the $p_\mu=0$ state in the limit $L \to \infty$. This means that, for a given lattice mass, by increasing the lattice size, to separate unambiguously the genuine finite-momentum fluctuation field from the ‘condensate’ itself, one should increase correspondingly the set of integers $n_i$. This gives a clean physical meaning to the fit obtained by skipping the lowest momentum propagator data and to the pair of parameters $(Z_{\text{high}},m_{\text{high}})$.             ii) the values obtained in Ref. [@Cea:1999kn] (respectively for $\kappa=$0.076, 0.07512,0.07504) $m_{\text{high}}=$0.4286(46), 0.2062(41), 0.1723(34) give a physical mass $m_H$ that scales as expected. This can easily be checked from the remarkable agreement between these values and the predicted trend Eq.(\[mrtau\]) which is valid at the leading-log level both in perturbation theory and in the alternative picture of Refs.[@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni], see Eqs.(\[ltriv\]) and (\[lambdaR2\]). In this way one finds $A=17.282(364)$, $\kappa_c=0.074836(15)$ and predicts $m_H\sim$0.499, 0.408, 0.293, 0.199 for $\kappa=$0.0764, 0.0759, 0.0754, 0.0751 in good agreement with the values $0.5, 0.4, 0.3, 0.2$ reported in the LW Table.           iii) the identification of $m_{\text{low}}\sim m_{\text{TS}}({\mathbf{k=0}})$ as the physical $m_H$ contradicts the observed dependence of $m_{\text{TS}}({\mathbf{k}})$ on $|{\mathbf{k}}|$ in the limit ${\mathbf{k}}\to 0$. In fact, in the symmetric phase (see Fig. 6) the energy spectrum has the form $\sqrt{ {\mathbf{k}}^2+ m^2 }$ up to momenta ${\mathbf{k}}^2 \sim 25 m^2$(!). However in the broken phase the energy is not reproduced by the (lattice version of the) form $\sqrt{ {\mathbf{k}}^2+ {\text{const}} }$ (see Fig.7) and thus the very notion of ‘mass’ becomes problematic. At larger ${\mathbf{k}}$, where $m_{\text{TS}}({\mathbf{k}})$ becomes insensitive to ${\mathbf{k}}$, it agrees well with $m_{\text{high}}$. The very different behaviour between symmetric and broken phase shown in Figs. 6 and 7 has no counterpart in the conventional picture.            iv) the values of $Z_{\text{prop}}$ obtained from the fit to the higher-momentum data in Ref. [@Cea:1999kn] for $\kappa=0.076,0.07512,0.07504$, $Z_{\text{high}}=$0.9321(44), 0.9551(21), 0.9566(13), exhibit a monotonical increase toward unity (from below) as expected on the base of the Kállen-Lehmann representation in a ‘trivial’ theory. This confirms that fitting the propagator skipping the lowest momentum data gives consistent results. On the contrary, the values for $Z_{\text{low}}\sim \hat{Z}_R$ obtained from the fit to the lowest momentum points alone, remain constant to $\sim 0.88$ in the limit $\kappa \to \kappa_c$ (see also the values in Table 4 of Ref.[@Balog:2004zd]). Assuming this definition as the correct one to be used in the Källen-Lehmann representation, one would find an inner contradiction with the non-interacting nature of the shifted fluctuation field in the continuum limit. No trace of this discussion is found in Ref.[@Balog:2004zd]. These authors, ignoring the evident difference between Fig.3 and Fig.4, as well as between Fig.6 and Fig.7, do not address the physical consistency of the mass definition extracted from the very low-momentum region of the lattice data. They just limit themselves to the remark that, for the $\lambda\Phi^4$ case, the lattice propagator is consistently reproduced by Eq.(\[gprop\]) for $p \to 0$. However, this agreement is not relevant here since the Ising limit is known to anticipate much better the true aspects of the theory in finite lattices. In the Ising case, in fact, the fit is not good and Balog et al. are forced to extract the lattice mass by fitting just to the three lowest momentum points, precisely as with the parameter $m_{\text{low}}$ of Ref. [@Cea:1999kn]. It is also surprising their claim (see the Abstract of Ref.[@Balog:2004zd]) that the lattice data are consistent with the perturbative predictions. We have seen in the previous section that, if one requires consistency between 2-loop predictions and the data for $v^2_B\chi_2(0)$, one has to replace the integration constant $C'_2$ given in the LW’s Table 1 with the value $C'_2=6.67(4)$ given in Eq.(\[ccc2\]). Therefore, using the perturbative relation reported in Eq.(2.14) of Ref.[@Balog:2004zd], namely ($\alpha_R=\frac{g_R}{16\pi^2}$) $$\label{ZRhat} \hat{Z}_R=(2\kappa)C'_2 ( 1- \frac{7}{36} \alpha_R + {\cal O}(\alpha^2_R))$$ one can compare this $\hat{Z}_R$ with the other estimate extracted from the product $m^2_R\chi_2(0)$ and reported in Table 4 of Ref.[@Balog:2004zd]. In this case, for $\kappa=0.076$ (where the relevant value is $g_R= 30.37(28)$) Eq.(\[ZRhat\]) predicts $\hat{Z}_R= 0.976(6)$ while the value reported in Table 4 by Balog et al. is $\hat{Z}_R= 0.896(4)$. Analogously, for $\kappa=0.0751$ (where the relevant values is $g_R= 20.51(74)$) Eq.(\[ZRhat\]) predicts again $\hat{Z}_R= 0.976(6)$ while the value reported in Table 4 by Balog et al. is $\hat{Z}_R= 0.883(17)$. These discrepancies, that are at the level of $\sim 10\sigma$ and $\sim 5\sigma$, can hardly be considered indicative of theoretical consistency. Quite independently, the authors of Ref.[@Balog:2004zd] have not shown that the $m_R$’s reported in their Table 4 lie on well defined RG trajectories $m_R=m_R(\kappa)$ using the [*same*]{} $\kappa_c=0.074833(17)$ obtained from the 2-loop fit to the data for $v^2_B\chi_2(0)$. In particular, this means that their value $m_R=0.395(1)$ for $\kappa=0.076$ has to come out consistent with the other value $m_R=0.1688(15)$ for $\kappa=0.0751$. To clarify this issue, they should quote the new values of the mass that replace the LW entries for all values of $\kappa$. Summary and conclusions {#summary} ======================= Let us now try to summarize the various theoretical and numerical points addressed in this paper. We shall start by observing that the field rescaling is usually viewed as an ‘operatorial statement’ between bare and renormalized fields operators of the type $$\label{operator} "~\Phi_B(x)= \sqrt{Z} \Phi_R(x)~"$$ As pointed out in Ref.[@Agodi:1995qv], this relation is a consistent short-hand notation in a theory where the field operator admits an asymptotic Fock representation, as in QED. In the presence of spontaneous symmetry breaking it has no rigorous basis since the Fock representation exists only for the [*shifted*]{} fluctuating field $h_B(x)= \Phi_B(x) -\langle \Phi_B \rangle$, the one with a vanishing expectation value. For this reason, following Refs.[@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni] (see the discussion given in Section 2), one can consider theoretical frameworks that are fully consistent with ‘triviality’ but where such an operatorial relation is [*not*]{} valid. In fact, the physical conditions used to determine the $h_B-h_R$ relation, through a $Z=Z_{\text {prop}}\sim 1$ in Eq.(\[hr\]), can be basically different from those used in the $v_B-v_R$ case, through a $Z=Z_\varphi\sim \ln \Lambda$ in Eq.(\[vR\]). In this case, if one wants to introduce a renormalized field operator $\Phi_R(x)$ (whose vacuum expectation value is what one defines by $v_R$ and whose fluctuating part is what one defines by $h_R$), this cannot be related to $\Phi_B(x)$ by means of Eq.(\[operator\]). As reviewed at the end of Section 2, this ‘subtlety’ has a substantial phenomenological implication: ‘triviality’, by itself, cannot be used to place upper bounds on the Higgs boson mass. Thus, the importance of the issue requires dedicated lattice simulations to check the validity of the logarithmic trend predicted in Eq.(\[z2phi\]) by measuring the relative scaling of the physical Higgs mass squared $m^2_H$ vs. the zero-momentum susceptibility. Clearly, the answer to this question depends on the given procedure adopted on the lattice to extract $m_H=m_{\text{latt}}$. Our point is that a correct choice requires to fulfill several consistency tests, taking into account the free-field nature of the fluctuation field in the continuum limit. Thus one should first find a 4-momentum region where the connected propagator is described by the (lattice version of the) two-parameter form $$G_{\text{fit}}(p)= \frac{Z_{\text{prop}}}{ p^2 + m^2_{\text{latt}} } \,.$$ One should also check that the fitted value $m_H=m_{\text{latt}}$ controls the energy eigenvalues $E=E({\mathbf{k}})$ governing the exponential time decay of the connected correlator in a given region of 3-momentum ${\mathbf{k}}$. In fact, where the energy is not reproduced by the (lattice version of the) form $\sqrt{ {\mathbf{k}}^2 + { \text{const.}} }$, the very notion of mass becomes problematic. Finally, one should also check that the fitted values of $Z_{\text{prop}}$ approach unity from below in the continuum limit as required by the Kállen-Lehmann representation in a ‘trivial’ theory. Now, the usual assumption is to extract $m_H=m_{\text{latt}}$ from the $p \to 0$ and/or ${\mathbf{k}} \to 0$ limits. This is certainly valid in a simulation performed in the symmetric phase (see Figs.3 and 6) where the whole 3- and 4-momentum regions give the same indications. However, in the broken-symmetry phase, where the vacuum is some sort of ‘condensate’, there might be reasons to avoid the strict zero-momentum limit to extract the physical particle mass. For instance, as mentioned in Sect.4, in the continuum theory, the shifted fluctuation field is defined as the $p_\mu\neq 0 $ projection of the full quantum field. However in a lattice simulation with periodic boundary conditions, where the momenta $p_i$ are proportional to an integer number $n_i$ times the inverse lattice size $1/L$, the notion $p_\mu\neq 0$ is ambiguous. In fact, [*any*]{} finite set of integers will evolve onto the $p_\mu=0$ state in the limit $L \to \infty$. Therefore, for a given lattice mass, by increasing the lattice size, to separate unambiguously the genuine finite-momentum fluctuation field from the ‘condensate’ itself, one should increase correspondingly the set of integers $n_i$. In addition, there are precise physical motivations suggested by the non-relativistic limit of a broken-symmetry $\lambda\Phi^4$ theory: the low-temperature phase of a hard-sphere Bose gas [@Huang:1957]. In this case, the low-lying excitations for ${\mathbf{k}}\to 0$ are phonons, i.e. collective oscillations of the hard-sphere system whose energy grows linearly $E_{\text{ph}}({\mathbf{k}})\sim c_s |{\mathbf{k}}|$, $c_s$ being the speed of sound. Only at [*larger*]{} $|{\mathbf{k}}|$ does the energy spectrum grow quadratically. Therefore, a determination of the effective hard-sphere mass through the non-relativistic 1-particle relation $E_{\text{1-part}}({\mathbf{k}})\sim \frac { {\mathbf{k}}^2 }{2 m_{\text{eff}}} $ cannot be obtained from the ${\mathbf{k}} \to 0$ limit of the energy spectrum which is dominated by the phonon branch. This type of problems were preliminarily considered in Ref. [@Cea:1999kn]. The result of that investigation was that the physical mass $m_H=m_{\text{latt}}$ is obtained from the propagator data after skipping the lowest 3-4 momentum points. The mass value $m_{\text{low}}$, obtained from the lowest momentum points, which is numerically close to the other mass extracted from the exponential decay of the connected correlator at zero 3-momentum, does not fulfill the same consistency checks. Now, for $\kappa=0.076,0.07512,0.07504$ the results of Ref. [@Cea:1999kn] were respectively $m_H=m_{\text{latt}}=0.4286(46),0.2062(41),0.1723(34)$. In this way, using the susceptibility values $\chi_{\text{latt}}=$37.85(6), 193.1(1.7), 293.4(2.9) reported in Ref. [@Cea:1999kn] one obtains a set of logarithmically increasing values: $Z_\varphi \equiv (2\kappa) m^2_H \chi_{\text{latt}}= 1.05(2),1.23(5),1.31(5)$ as predicted in Refs.[@Consoli:1994jr; @Consoli:1997ra; @Consoli:1999ni] (see Fig.5). This is in contrast with the values obtained using $m_{\text{low}}$ for which the quantity $\hat{Z}_R\equiv (2\kappa) m^2_{\text{low}} \chi_{\text{latt}}\sim 0.88$ remains remarkably stable. To provide [*further*]{} evidence, we replaced in Ref.[@Cea:2004ka] the direct evaluation of $m_H=m_{\text{latt}}$ with the theoretical input values predicted in the LW Tables. This is not in contradiction with the previous strategy, in fact the values obtained in Ref. [@Cea:1999kn] $m_{\text{latt}}=$0.4286(46), 0.2062(41), 0.1723(34) give a physical mass $m_H$ that scales as in Eq.(\[mrtau\]) and that is in good agreement with the values $m_H=m_{\text{input}}(\kappa) $ reported in the LW Table. Using the values of $\chi_{\text{latt}}$ reported in our Table 1 and the LW entries for the mass, this additional test confirms that the quantity $Z_\varphi\equiv 2\kappa m^2_{\text{input}} \chi_{\text{latt}}$ increases logarithmically when approaching the continuum limit (see Fig.2). Now, Balog et al., being aware that the above numerical evidences have “..serious non standard implications for the Higgs sector of the Standard Model” (see the Conclusions of Ref.[@Balog:2004zd]), have performed a new analysis. They claim that the LW entries for $m_R$ should be replaced by new values (whose consistency with RG trajectories $m_R=m_R(\kappa)$, however, has not been shown). As far as we can see, they have essentially re-discovered the result of Ref. [@Cea:1999kn] that the quantity $\hat{Z}_R\equiv (2\kappa) m^2_{\text{low}} \chi_{\text{latt}}$ is a constant $\sim 0.88$. However, ignoring Ref. [@Cea:1999kn], they fail to appreciate why their values $m_R=m_{\text{low}}$ do not represent a consistent lattice definition of $m_H$. A point where we accept their criticism concerns the lattice data for $v^2_B \chi_2(0)$. Restricting to this observable, a definitive test of the leading-logarithmic trend has to be postponed to data taken closer to the critical point where the non-leading terms associated with the scale of the logs should become unessential. However, as discussed at the end of Sect.3, within the perturbative framework, the data for $v^2_B \chi_2(0)$ give precise information on the integration constants $C'_i$ (i=1,2,3) that should be used for the matching with the symmetric phase. Using the precise outcome of the fit $C'_2=6.67(4)$ Eq. (\[ZRhat\]) predicts $\hat{Z}_R\sim 0.976(6)$ with 5-10$\sigma$ discrepancies with respect to the values reported by Balog et al in their Table 4. This is precisely the same discrepancy pointed out by Jansen et al. Ref.[@Jansen:1989cw] whose origin cannot be understood ignoring the main point of Ref. [@Cea:1999kn] and of this paper. In the broken phase, a naive zero 4-momentum limit of the connected propagator and/or a naive zero 3-momentum limit of the energy eigenvalue controlling the exponential decay of the connected correlator, do not provide consistent estimates of the physical mass parameter $m_H$. The phenomenological implications are substantial. Once the quadratic shape of the effective potential (the inverse susceptibility), that is a pure zero-momentum quantity, does not scale uniformly with $m^2_H$, although the finite-momentum fluctuation field becomes free in the continuum limit, one cannot use ‘triviality’ to place upper bounds on $m_H$. [10]{} P. Cea, M. Consoli, and L. Cosmai, [*Large logarithmic rescaling of the scalar condensate: New lattice evidences*]{}, [[hep-lat/0407024]{}](http://xxx.lanl.gov/abs/hep-lat/0407024). M. Consoli and P. M. Stevenson, [*The non trivial effective potential of the ’trivial’ [$\lambda \phi^4$]{} theory: a lattice test*]{}, [*Z. Phys.*]{} [ **C63**]{} (1994) 427–436, \[[[ http://arXiv.org/abs/hep-ph/9310338]{}](http://xxx.lanl.gov/abs/http://arXiv.org/abs/hep-ph/9310338)\]. M. Consoli and P. M. Stevenson, [*Mode-dependent field renormalization and triviality in [$\lambda \phi^4$]{} theory*]{}, [*Phys. Lett.*]{} [**B391**]{} (1997) 144–149. M. Consoli and P. M. Stevenson, [*Physical mechanisms generating spontaneous symmetry breaking and a hierarchy of scales*]{}, [*Int. J. Mod. Phys.*]{} [ **A15**]{} (2000) 133, \[[[ http://arXiv.org/abs/hep-ph/9905427]{}](http://xxx.lanl.gov/abs/http://arXiv.org/abs/hep-ph/9905427)\]. P. Cea, M. Consoli, and L. Cosmai, [*First lattice evidence for a non-trivial renormalization of the [Higgs]{} condensate*]{}, [*Mod. Phys. Lett.*]{} [**A13**]{} (1998) 2361–2368, \[[[ http://arXiv.org/abs/hep-lat/9805005]{}](http://xxx.lanl.gov/abs/http://arXiv.org/abs/hep-lat/9805005)\]. P. Cea, M. Consoli, L. Cosmai, and P. M. Stevenson, [*Further lattice evidence for a large re-scaling of the [Higgs]{} condensate*]{}, [*Mod. Phys. Lett.*]{} [**A14**]{} (1999) 1673–1688, \[[[ http://arXiv.org/abs/hep-lat/9902020]{}](http://xxx.lanl.gov/abs/http://arXiv.org/abs/hep-lat/9902020)\]. P. Cea, M. Consoli, and L. Cosmai, [*Large rescaling of the [Higgs]{} condensate: Theoretical motivations and lattice results*]{}, [*Nucl. Phys. Proc. Suppl.*]{} [**83**]{} (2000) 658–660, \[[[hep-lat/9909055]{}](http://xxx.lanl.gov/abs/hep-lat/9909055)\]. J. Balog, A. Duncan, R. Willey, F. Niedermayer, and P. Weisz, [*The 4d one component lattice [$phi^4$]{} model in the broken phase revisited*]{}, [[hep-lat/0412015]{}](http://xxx.lanl.gov/abs/hep-lat/0412015). W. Zimmermann, [*[Local Operator Products and Renormalization in Quantum Field Theory]{}*]{}, vol. 1 of [*Brandeis University Summer Institute in Theoretical Physics 1970*]{}, p. 395. MIT Press, 1970. V. Branchina, M. Consoli, and N. M. Stivala, [*Renormalization of massless lambda [$phi^4$]{} theories: Asymptotic freedom and spontaneous symmetry breaking*]{}, [*Z. Phys.*]{} [**C57**]{} (1993) 251–266. U. Ritschel, [*[NonGaussian]{} corrections to [Higgs]{} mass in autonomous [$\lambda \varphi^4$]{} in (3+1)-dimensions*]{}, [*Z. Phys.*]{} [**C63**]{} (1994) 345–350, \[[[ hep-ph/9210206]{}](http://xxx.lanl.gov/abs/hep-ph/9210206)\]. A. Agodi, G. Andronico, and M. Consoli, [*Lattice [$\Phi^4_4$]{} effective potential giving spontaneous symmetry breaking and the role of the [Higgs]{} mass*]{}, [*Z. Phys.*]{} [**C66**]{} (1995) 439–452, \[[[hep-lat/9410001]{}](http://xxx.lanl.gov/abs/hep-lat/9410001)\]. P. M. Stevenson, B. Alles, and R. Tarrach, [*[O(N)]{} symmetric [$\lambda \varphi^4$]{} theory: the [Gaussian]{} effective potential approach*]{}, [*Phys. Rev.*]{} [**D35**]{} (1987) 2407. U. Ritschel, I. Stancu, and P. M. Stevenson, [*Unconventional large [N]{} limit of the [Gaussian]{} effective potential and the phase transition in [$\lambda \varphi^4$]{} theory*]{}, [*Z. Phys.*]{} [**C54**]{} (1992) 627–634. R. H. Swendsen and J.-S. Wang, [*Nonuniversal critical dynamics in [Monte Carlo]{} simulations*]{}, [*Phys. Rev. Lett.*]{} [**58**]{} (1987) 86–88. U. Wolff, [*Collective [Monte Carlo]{} updating for spin systems*]{}, [*Phys. Rev. Lett.*]{} [**62**]{} (1989) 361. D. S. Gaunt, M. F. Sykes, and S. McKenzie, [*[Susceptibility and fourth-field derivative of the spin $\frac{1}{2}$ Ising model for $T>T_c$ and $d=4$]{}*]{}, [*J. Phys.*]{} [**A12**]{} (1979) 871. M. [Lüscher]{} and P. Weisz, [*Scaling laws and triviality bounds in the lattice [$\phi^4$]{} theory. 2. [One]{} component model in the phase with spontaneous symmetry breaking*]{}, [*Nucl. Phys.*]{} [**B295**]{} (1988) 65. I. Montvay and P. Weisz, [*Numerical study of finite volume effects in the four- dimensional ising model*]{}, [*Nucl. Phys.*]{} [**B290**]{} (1987) 327. T. D. Lee, K. Huang, and C. N. Yang, [*[Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties]{}*]{}, [*Phys. Rev.*]{} [**106**]{} (1957) 1135–1145. K. Jansen, T. Trappenberg, I. Montvay, G. Munster, and U. Wolff, [*Broken phase of the four-dimensional [Ising]{} model in a finite volume*]{}, [*Nucl. Phys.*]{} [**B322**]{} (1989) 698.

--- abstract: 'We present an extended version of the recently proposed “LLOB” model for the dynamics of latent liquidity in financial markets. By allowing for finite cancellation and deposition rates within a continuous reaction-diffusion setup, we account for finite memory effects on the dynamics of the latent order book. We compute in particular the finite memory corrections to the square root impact law, as well as the impact decay and the permanent impact of a meta-order. [ The latter is found to be linear in the traded volume and independent of the trading rate, as dictated by no-arbitrage arguments]{}. In addition, we consider the case of a spectrum of cancellation and deposition rates, which allows us to obtain a square root impact law for moderate participation rates, as observed empirically. Our multi-scale framework also provides an alternative solution to the so-called price diffusivity puzzle in the presence of a long-range correlated order flow.' author: - 'M. BENZAQUEN$^{\ast}$$\dag$${\ddag}$[^1] and J.-P. BOUCHAUD${\ddag}$\' bibliography: - 'bibs.bib' title: 'Market impact with multi-timescale liquidity' --- Market microstructure; price formation; limit order book; market impact Introduction {#intro .unnumbered} ============ Understanding the price formation mechanisms is undoubtably among the most exciting challenges of modern finance. *Market impact* refers to the way market participants’ actions mechanically affect prices. Significant progress has been made in this direction during the past decades [@Hasbrouk2007; @bouchaud2008markets; @weber2005order; @Bouchaud_impact_2010]. A notable breakthrough was the empirical discovery that the aggregate price impact of a meta-order[^2] is a concave function (approximately square-root) of its size $Q$ [@Grinold; @Almgren2005; @Toth2011; @Donier2015]. In the recent past, so called “latent” order book models [@Toth2011; @mastromatteo2014agent; @MPRL; @DonierLLOB] have proven to be a fruitful framework to theoretically address the question of market impact, among others.\ As a precise mathematical incarnation of the latent order book idea, the zero-intelligence LLOB model of Donier *et al.* [@DonierLLOB] was successful at providing a theoretical underpinning to the square root impact law. The LLOB model is based on a continuous mean field setting, that leads to a set of reaction-diffusion equations for the dynamics of the latent bid and ask volume densities. In the infinite memory limit (where the agents intentions, unless executed, stay in the latent book forever and there are no arrivals of new intentions), the latent order book becomes exactly linear and impact exactly square-root. Furthermore, this assumption leads to zero permanent impact of uninformed trades, and an inverse square root decay of impact as a function of time. While the LLOB model is fully consistent mathematically, it suffers from at least two major difficulties when confronted with micro-data. First, a strict square-root law is only recovered in the limit where the execution rate $m_0$ of the meta-order is larger than the normal execution rate $J$ of the market itself – whereas most meta-order impact data is in the opposite limit $m_0 \lesssim 0.1 J$. Second, the theoretical inverse square-root impact decay is too fast and leads to significant short time mean-reversion effects, not observed in real prices.\ The aim of the present paper is to show that introducing different timescales for the renewal of liquidity allows one to cure both the above deficiencies. In view of the way financial markets operate, this step is very natural: agents are indeed expected to display a broad spectrum of timescales, from low frequency institutional investors to High Frequency Traders (HFT). We show that provided the execution rate $m_0$ is large compared to the low-frequency flow, but small compared to $J$, the impact of a meta-order crosses over from a linear behaviour at very small $Q$ to a square-root law in a regime of $Q$s that can be made compatible with empirical data. We show that in the presence of a continuous, power-law distribution of memory times, the temporal decay of impact can be tuned to reconcile persistent order flow with diffusive price dynamics (often referred to as the *diffusivity puzzle*) [@bouchaud2008markets; @bouchaud2004fluctuations; @Lillo2004]. We argue that the permanent impact of uninformed trades is fixed by the slowest liquidity memory time, beyond which mean-reversion effects disappear. Interestingly, the permanent impact is found to be linear [ in the executed volume $Q$ and independent of the trading rate]{}, as dictated by no-arbitrage arguments.\ Our paper is organized as follows. We first recall the LLOB model of [@DonierLLOB] in Section \[llobrecall\]. We then explore in Section \[fincandep\] the implications of finite cancellation and deposition rates (finite memory) in the reaction-diffusion equations, notably regarding permanent impact (Section \[permimp\]). We generalize the reaction-diffusion model to account for several deposition and cancellation rates. In particular, we analyse in Section \[multifsec\] the simplified case of a market with two sorts of agents: long memory agents with vanishing deposition and cancellation rates, and short memory high frequency agents (somehow playing the role of market makers). Finally, we consider in Section \[densnusec\] the more realistic case of a continuous distribution of cancellation and deposition rates and show that such a framework provides an alternative way to solve the diffusivity puzzle (see [@BenzaquenFLOB]) by adjusting the distribution of cancellation and deposition rates. Many details of the calculations are provided in the Appendices. Locally linear order book model {#llobrecall} =============================== We here briefly recall the main ingredients of the locally linear order book (LLOB) model as presented by Donier *et al.* [@DonierLLOB]. In the continuous “hydrodynamic” limit we define the latent volume densities of limit orders in the order book: $\varphi_{\mathrm{b}}(x,t)$ (bid side) and $\varphi_{\mathrm{a}}(x,t)$ (ask side) at price $x$ and time $t$. The latter obey the following set of partial differential equations: \_t \_ &=& D\_[xx]{}\_ -\_ + (x\_t-x) - R\_(x)\ \_t \_ &=& D\_[xx]{}\_ -\_ + (x-x\_t) - R\_(x)  ,  where the different contributions on the right hand side respectively signify (from left to right): heterogeneous reassessments of agents intentions with diffusivity $D$ (diffusion terms), cancellations with rate $\nu$ (death terms), arrivals of new intentions with intensity $\lambda$ (deposition terms), and matching of buy/sell intentions (reaction terms). The price $x_t$ is conventionally defined through the equation $ \varphi_{\mathrm{b}}(x_t,t)= \varphi_{\mathrm{a}}(x_t,t)$. The non-linearity arising from the reaction term in Eqs. and can be abstracted away by defining $ \phi(x,t) = \varphi_{\textrm b}(x, t) - \varphi_{\textrm a}(x, t)$, which solves: $$\begin{aligned} \partial_t \phi &=& D \partial_{xx} \phi -\nu\phi + s(x,t) \ ,\label{firsteqsrc}\end{aligned}$$ where the source term reads $s(x,t) = \lambda \,{\textrm{sign}}(x_t-x)$ and the price $x_t$ is defined as the solution of $$\begin{aligned} \phi(x_t,t) &=& 0 \ . \label{priceeq}\end{aligned}$$ Setting $\xi=x-x_t$, the stationary order book can easily be obtained as: $\phi^\mathrm{st}(\xi)=-({\lambda}/{\nu}) \, {\textrm{sign}}(\xi) [1-\exp(-|\xi|/\xi_{\mathrm c})]$ where $\xi_{\mathrm c}=\sqrt{D\nu^{-1}}$ denotes the typical length scale below which the order book can be considered to be linear: $\phi^\mathrm{st}(\xi) \approx -\mathcal L \xi$ (see Fig. \[Obstat\]). The slope $\mathcal L := \lambda/\sqrt{\nu D}$ defines the [*liquidity*]{} of the market, from which the total execution rate $J$ can be computed since: $$\begin{aligned} J := \left. \partial_\xi \phi^\mathrm{st}(\xi) \right|_{\xi=0} = D \mathcal{L}.\end{aligned}$$ Donier *et al.* [@DonierLLOB] focussed on the *infinite memory* limit, namely $\nu, \lambda \rightarrow 0$ while keeping $\mathcal L \sim \lambda {\nu}^{-1/2}$ constant, such that the latent order book becomes exactly linear since in that limit $\xi_{\mathrm c} \to \infty$. This limit considerably simplifies the mathematical analysis, in particular concerning the impact of a meta-order. An important remark must however be introduced at this point: although the limit $\nu \to 0$ is taken in [@DonierLLOB], it is assumed that the latent order book is still able to reach its stationary state $\phi^\mathrm{st}(\xi)$ before a meta-order is introduced. In other words, the limit $\nu \to 0$ is understood in a way such that the starting time of the meta-order is large compared to $\nu^{-1}$. Price trajectories with finite cancellation and deposition rates {#fincandep} ================================================================ As mentioned in the introduction we here wish to explore the effects of non-vanishing cancellation and deposition rates, or said differently the behaviour of market impact for executiong times larger than $\nu^{-1}$. The general solution of Eq.  is given by: $$\begin{aligned} \phi(x,t) &=& \left( \mathcal G_\nu \star \phi_0\right)(x,t) + \int \text d y\int_0^\infty \text d \tau\, \mathcal G_\nu(x-y,t-\tau) s(y,\tau) \ , \label{convol}\end{aligned}$$ where $\phi_0(x) =\phi(x,0)$ denotes the initial condition, and where $\mathcal G_\nu (x,t) = e^{-\nu t}\mathcal G (x,t)$ with $\mathcal G$ the diffusion kernel: $$\begin{aligned} \mathcal G(x,t) &=& \Theta(t) \frac{e^{-\frac{x^2}{4Dt}}}{\sqrt{4\pi Dt}} \ .\end{aligned}$$ Following Donier *et al.* [@DonierLLOB], we introduce a buy (sell) meta-order as an extra point-like source of buy (sell) particles with intensity rate $m_t$ such that the source term in Eq.  becomes: $s(x,t) = m_t \delta(x-x_t)\cdot \mathds{1}_{[0,T]} +\lambda \,{\textrm{sign}}(x_t-x)$, where $T$ denotes the time horizon of the execution. In all the following we shall focus on buy meta-orders – without loss of generality since within the present framework everything is perfectly symmetric. Performing the integral over space in Eq.  and setting $\phi_0(x)=\phi^{\mathrm{st}}(x)$ yields: $$\begin{aligned} \phi(x,t) &=& \phi^\mathrm{st}(x)e^{-\nu t} + \int_0^{\min (t,T)} \text d \tau\, m_\tau \mathcal G_\nu(x-x_\tau,t-\tau) -\lambda\int_0^{t } \text d \tau \, {\textrm{erf}}\left[ \frac{x-x_\tau}{\sqrt{4D(t-\tau)}} \right] e^{-\nu(t-\tau)} \ .\label{mastereq}\end{aligned}$$ The equation for price, , is not analytically tractable in the general case, but different interesting limit cases can be investigated. In particular, focussing on the case of constant participation rates $m_t = m_0$, one may consider: - (*i*) Small participation rate $m_0\ll J$ *vs* large participation rate $m_0\gg J$. - (*ii*) Fast execution $\nu T\ll 1$ (the particules in the book are barely renewed during the meta-order execution) *vs* slow execution $\nu T\gg 1$ (the particles in the book are completely renewed, and the memory of the initial state has been lost). - (*iii*) Small meta-order volumes $Q:=m_0 T\ll Q_\mathrm{lin.}$ (for which the linear approximation of the stationary book is appropriate, see Fig. \[Obstat\]) *vs* large volumes $Q \gg Q_\mathrm{lin.}$ (for which the linear approximation is no longer valid). So in principle, one has to consider $2^3 = 8$ possible limit regimes. However, some regimes are mutually exclusives so that only 6 of them remain. A convenient way to summarize the results obtained for each of the limit cases mentioned above is to expand the price trajectory $x_t$ up to first order in $\sqrt{\nu}$ as:[^3] $$\begin{aligned} x_t &=& \alpha \left[ z_t^0+\sqrt{\nu} z_t^1+O(\nu)\right] \ ,\label{alphaz0z1}\end{aligned}$$ where $z_t^0$ and $z_t^1$ denote respectively the 0th order and 1st order contributions. Table 1 gathers the results for fast execution ($\nu T\ll 1$) and small meta-order volumes ($Q \ll Q_\mathrm{lin.}$). Note that the leading correction term $z_t^1$ is negative, i.e. the extra incoming flux of limit orders acts to lower the impact of the meta-order, see Fig. \[pricetraj\]. The price trajectory for slow execution and/or large meta-order volumes, on the other hand, simply reads: $$\begin{aligned} x_t &=& \frac{m_0 \nu}{\lambda} t \ . \end{aligned}$$ The corresponding calculations and explanations are given in Appendix A. Permanent impact as a finite memory effect {#permimp} ========================================== As mentioned in the introduction, the impact relaxation following the execution is an equally important question. We here compute the impact decay after a meta-order execution. In the limit of small cancellation rates, we look for a scaling solution of the form $z^1_t= T F(\nu t)$ (see Eq. ) where $F$ is a dimensionless function. We consider the case where $\nu T \ll 1$ and $Q \ll Q_\mathrm{lin.}$. Long after the end of the execution of the meta-order, i.e. when $t\gg T$, Eq.  together with Eqs.  and becomes (to leading order): $$\begin{aligned} 0&=& -\frac{\lambda\alpha T}{\sqrt{D}}F(\nu t)e^{-\nu t} - 2\lambda\alpha\int_0^{t } \text d \tau \, \frac{z_t^0-z_\tau^0}{\sqrt{4\pi D(t-\tau)}} e^{-\nu(t-\tau)} \nonumber \\ && \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad -2\lambda \alpha T\sqrt{\nu} \int_0^t \text d \tau \, \frac{F(\nu t)-F(\nu \tau )}{\sqrt{4\pi D(t-\tau)}} e^{-\nu(t-\tau)} \ .\label{}\end{aligned}$$ Letting $u = \nu t$ and $z_t^0 = \beta/\sqrt{u}$ (see Table \[tableimpact\]) yields: $$\begin{aligned} 0&=&\sqrt{\pi}e^{-u} F(u) + \beta \int_0^{u } \text d v \, \frac{\sqrt{v} -\sqrt{u}}{\sqrt{uv(u-v)}} e^{v-u} + \int_0^{u } \text d v \, \frac{F(u)-F(v) }{\sqrt{u-v}} e^{v-u} \ .\label{intequ}\end{aligned}$$ Finally seeking $F$ asymptotically of the form $F(u) = F_\infty +Bu^{-\gamma}+Cu^{-\delta}e^{-u}$ one can show that: $$\begin{aligned} F(u)&=&F_\infty -\frac{\beta}{\sqrt{u}}\left[1-e^{-u}\right] \qquad (u \gg 1)\ ,\label{Fu}\end{aligned}$$ with the permanent component given by $F_\infty = \beta\sqrt{\pi}$, where $\beta$ depends on the fast/slow nature of the execution (see Table \[tableimpact\]).\ Injecting the solution for $F(u)$ in Eq. , and taking the limit of large times, one finds that the $t^{-1/2}$ decay of the 0th order term is exactly compensated by the $\beta u^{-1/2}$ term coming from $F(u)$, showing that the asymptotic value of the impact, given by $I_\infty= \alpha \sqrt{\nu} { T} F_\infty$, is reached exponentially fast as $\nu t \to \infty$ (see Fig. \[pricetraj\]). This result can be interpreted as follows. At the end of execution (when the peak impact is reached), the impact starts decaying towards zero in a slow power law fashion (see [@DonierLLOB]) until approximately $t \sim \nu^{-1}$, beyond which all memory is lost (since the book has been globally renewed). Impact cannot decay anymore, since the previous reference price has been forgotten. Note that in the limit of large meta-order volumes and/or slow executions, all memory is already lost at the end of the execution and the permanent impact trivially matches the peak impact (see Fig. \[pricetraj\]).\ An important remark is in order here. Using Table \[tableimpact\], one finds that $I_\infty { = \frac12} \xi_c (Q/Q_\mathrm{lin.})$ in both the small and large participation regime. In other words, we find that the permanent impact is [*linear*]{} in the executed volume $Q$, as dictated by no-arbitrage arguments [ [@huberman2004price; @gatheral2010no]]{} and compatible with the classical Kyle framework [ [@kyle1985continuous]]{}. Impact with fast and slow traders {#multifsec} ================================= Set up of the problem --------------------- As stated in the introduction, one major issue in the impact results of the LLOB model as presented by Donier *et al.* [@DonierLLOB] is the following. Empirically, the impact of meta-orders is only weakly dependent on the participation rate $m_0/J$ (see e.g. [@Toth2011]). The corresponding *square root law* is commonly written as: $$\begin{aligned} I_Q &:=& \langle x_T \rangle = Y \sigma \sqrt{\frac Q V} \ , \label{empiricalimp}\end{aligned}$$ where $\sigma$ is the daily volatility, $V$ is the daily traded volume, and $Y$ is a numerical constant of order unity. Note that $I_Q$ only depends on the total volume of the meta-order $Q=m_0 T$, and not on $m_0$ (or equivalently on the time $T$).\ As one can check from Table \[tableimpact\], the independence of impact on $m_0$ only holds in the large participation rate limit ($m_0\gg J$). However, most investors choose to operate in the opposite limit of small participation rates $m_0 \ll J$, and all the available data is indeed restricted to $m_0/J \lesssim 0.1$. Here we offer a possible way out of this conundrum. The intuition is that the total market turnover $J$ is dominated by high frequency traders/market makers, whereas resistance to slow meta-orders can only be provided by slow participants on the other side of the book. More precisely, consider that only two sorts of agents co-exist in the market (see Section \[densnusec\] for a continuous range of frequencies): 1. Slow agents with vanishing cancellation and deposition rates: $\nu_{\text{s}} T \rightarrow 0$, while keeping the corresponding liquidity $\mathcal L_{\text{s}}:= \lambda_{\text{s}}/\sqrt{\nu_{\text{s}} D}$ finite; and 2. Fast agents with large cancellation and deposition rates, $\nu_{\text{f}} T \gg 1$, such that $\mathcal L_{\text{f}}:= \lambda_{\text{f}}/\sqrt{\nu_{\text{f}} D} \gg \mathcal L_{\text{s}}$. The system of partial differential equations to solve now reads: \_t \_ &=& D \_[xx]{} \_ -\_\_ +s\_(x,t)\ \_t \_ &=& D \_[xx]{} \_ -\_\_ +s\_(x,t)  , where $s_k(x,t) = \lambda_k \,{\textrm{sign}}(x_{kt}-x) + m_{kt}\delta(x-x_{kt}) $, together with the conditions: $$\begin{aligned} m_{\text{s}t}+ m_{\text{f}t} &=& m_0 \label{ratesequal} \\ x_{\text{s}t}=x_{\text{f}t} &=& x_t \label{priceequal}\ .\end{aligned}$$ Equation  means that the meta-order is executed against slow and fast agents, respectively contributing to the rates $m_{\text{s}t}$ and $m_{\text{f}t}$. Equation  simply means that there is a unique transaction price, the same for slow and for fast agents. The total order book volume density is then given by $\phi =\phi_{\text{s}}+\phi_{\text{f}}$. In particular, in the limit of slow/fast agents discussed above the stationary order book is given by the sum of $\phi_{\text{s}}^\mathrm{st}(x) \approx -\mathcal L_{\text{s}} x$ and $\phi_{\text{f}}^\mathrm{st}(x) \approx - (\lambda_{\text{f}}/\nu_{\text{f}}){\textrm{sign}}(x)$ (see Fig. \[Obstat\_multi\]). The total transaction rate now reads $$\begin{aligned} J = \ D \left|\partial_x\left[ \phi_{\text{s}}^\mathrm{st}+\phi_{\text{f}}^\mathrm{st}\right]\right|_{x=0}=J_{\text{s}}+J_{\text{f}},\end{aligned}$$ where $J_{\text{f}} \gg J_{\text{s}}$ (which notably implies that $J \approx J_{\text{f}}$). From linear to square-root impact --------------------------------- We now focus on the regime where the meta-order intensity is large compared to the average transaction rate of slow traders, but small compared to the total transaction rate of the market, to wit: $J_{\text{s}} \ll m_0 \ll J$. In this limit Eqs.  and , together with the corresponding price setting equations $\phi_k(x_{kt},t) \equiv 0$ yield (see Appendix B): x\_[t]{} &=&(2[L\_]{} \_0\^t d m\_ )\^[1/2]{}\ x\_[t]{} &=& \_0\^t d m\_  . \[x1tx2t\] Differentiating Eq.  with respect to time together with Eqs.  and using Eq.  yields: $$\begin{aligned} m_{\text{f}t} &=&\frac{m_0}{\sqrt{1+\frac{t}{t^\star}}}, \quad \text{with} \quad t^\star:=\frac{1}{2\nu_{\text{f}}} \frac{J_{\text{f}}^2}{J_{\text{s}}m_0}, \label{m2}\end{aligned}$$ and $m_{\text{s}t}=m_0-m_{\text{f}t}$. Equation indicates that most of the incoming meta-order is executed against the rapid agents for $t < t^\star$ but the slow agents then take over for $t>t^\star$ (see Fig. \[pricetraj\_multi\]). The resulting price trajectory reads: $$\begin{aligned} x_{t} &=&\frac{\lambda_{\text{f}}}{\mathcal L_{\text{s}} \nu_{\text{f}}}\left(\sqrt{1+\frac{t}{t^\star}}-1\right) \, , \label{ptrajmultif}\end{aligned}$$ which crosses over from a linear regime when $t \ll t^\star$ to a square root regime for $t \gg t^\star$ (see Fig. \[pricetraj\_multi\]). For a meta-order of volume $Q$ executed during a time interval $T$, the corresponding impact is linear in $Q$ when $T < t^\star$ and square-root (with $I_Q$ independent of $m_0$) when $T > t^\star$. This last regime takes place when $Q > m_0 t^\star$, which can be rewritten as: $$\begin{aligned} \frac{Q}{V_{\text{d}}} > \frac{1}{\nu_{\text{f}} T_{\text{d}}} \frac{J}{J_{\text{s}}},\end{aligned}$$ where $V_{\text{d}}$ is the total daily volume and $T_{\text{d}}$ is one trading day. Numerically, with a HFT cancellation rate of – say – $\nu_{\text{f}} = 1$ sec$^{-1}$ and $J_{\text{s}} = 0.1 J$, one finds that the square-root law holds when the participation rate of the meta-order exceeds $3 \, 10^{-4}$, which is not unreasonable when compared with impact data. Interestingly the cross-over between a linear impact for small $Q$ and a square-root for larger $Q$ is consistent with the data presented by Zarinelli *et al.* [@Zarinelli] (but note that the authors fit a logarithmic impact curve instead).\ Impact decay ------------ Regarding the decay impact for $t > T$, the problem to solve is that of Eqs. , and only where Eq.  becomes: $$\begin{aligned} m_{\text{s}t}+m_{\text{f}t} &=& 0 \ .\label{ratessumzero}\end{aligned}$$ The solution behaves asymptotically ($t\gg T$) to zero as $x_t \sim t^{-1/2}$ (see Appendix B). Given the results of Section \[permimp\] in the presence of finite memory agents, the absence of permanent impact may seem counter-intuitive. In order to understand this feature of the double-frequency order book model in the limit $\nu_{\text{s}}\, T \rightarrow 0$, $\nu_{\text{f}}\, T\gg 1$, one can look at the stationary order book. As one moves away from the price the ratio of slow over fast volume fractions ($\phi_{\text{s}}/\phi_{\text{f}}$) grows linearly to infinity. Hence, the shape of the latent order book for $|x| \gg x^\star$ matches that of the infinite memory single-agent model originally presented by Donier *et al.* [@DonierLLOB] (see Fig. \[Obstat\_multi\]). This explains the mechanical return of the price to its initial value before execution, encoded in the slow latent order book. Note that in the limit of very small but finite $\nu_{\text{s}}$, the permanent impact is of order $\sqrt{\nu_{\text{s}}}$, as obtained in Section \[permimp\].\ The linear regime ----------------- The regime of very small participation rates for which $m_0 \ll J_{\text{s}},J_{\text{f}}$ is also of conceptual interest. In such a case Eq.  must be replaced with: $$\begin{aligned} x_{\text{s}t} &=&\frac{1}{\mathcal L_{\text{s}}} \int_0^t \textrm d \tau \, \frac{m_{\text{s}\tau}}{\sqrt{4\pi D (t-\tau)}} \label{x1tspecbis} \ , $$ which together with Eqs. , and yields, in Laplace space (see Appendix B): $$\begin{aligned} \widehat m_{1p} &=&\frac1p \frac{m_0}{1+\sqrt{pt^\dagger}} \ , \label{m1p}\end{aligned}$$ where $t^\dagger = (m_0/\pi J_{\text{s}}) t^\star$, with $t^\star$ defined in Eq. . For small times ($t \ll t^\dagger$) one obtains $m_{\text{s}t}= 2m_0 \sqrt{t/t^\dagger}$ while for larger times ($t^\dagger \ll t < T$), $m_{\text{s}t}=m_0[1-\sqrt{t^\dagger/(\pi t)}]$. Finally using again Eqs. , and yields $x_t = (\nu_{\text{f}}/\lambda_{\text{f}})m_0 t$ for $t \ll t^\dagger$ and $x_t = (\nu_{\text{f}}/\lambda_{\text{f}})m_0\sqrt{t t^\dagger/\pi}$ for $t^\dagger \ll t < T$, identical in terms of scaling to the price dynamics observed in the case $J_{\text{s}} \ll m_0 \ll J_{\text{f}}$ discussed above. The asymptotic impact decay is identical to the one obtained in that case as well.\ Multi-frequency order book {#densnusec} ========================== The double-frequency framework [presented in Sec. \[multifsec\]]{} can be extended to the more realistic case of a continuous range of cancellation and deposition rates. Formally, one has to solve an infinite set of equations, labeled by the cancellation rate $\nu$: $$\begin{aligned} \partial_t \phi_\nu = D \partial_{xx} \phi_\nu -\nu\phi_\nu +s_\nu(x,t)\ , \label{phinuc}\end{aligned}$$ where $\phi_\nu(x,t)$ denotes the contribution of agents with typical frequency $\nu$ to the latent order book, and $s_\nu(x,t) = \lambda_\nu \,{\textrm{sign}}(x_{ \nu t}-x) + m_{\nu t}\delta(x-x_{ \nu t})$, with $\lambda_\nu =\mathcal L_{\nu}\sqrt{\nu D}$. Equation must then be completed with: \_0\^d() m\_[t]{} &=&m\_t\ x\_[t]{} &=&x\_[t]{} , \[densnutauxboth\] where $\rho(\nu)$ denotes the distribution of cancellation rates $\nu$, and where we have allowed for an arbitrary order flow $m_t$. Solving exactly the above system of equations analytically is too ambitious a task. In the following, we present a simplified analysis that allows us to obtain an approximate scaling solution of the problem for a power law distribution of frequencies $\nu$. The propagator regime {#diffusivitypuz} --------------------- We first assume, for simplicity, that the order flow $J_\nu$ is independent of frequency (see later for a more general case), and consider the case when $m_t \ll J$, $\forall t$. Although not trivially true, we assume (and check later on the solution) that this implies $m_{\nu t}\ll J$ $\forall \nu$, such that we can assume linear response for all $\nu$. Schematically, there are two regimes, depending on whether $t \gg \nu^{-1}$ – in which case the corresponding density $\phi_\nu(x,t)$ has lost all its memory, or $t \ll \nu^{-1}$. In the former case the price trajectory follows Eq. , while in the latter case it is rather Eq.  that rules the dynamics. One thus has: t1 x\_[ t]{} &=& \_0\^t d\ t1 x\_[ t]{} &=& \_0\^t d m\_  . \[densnux\] Inverting Eqs.  and defining $\Psi(t) := 2/\sqrt{\pi t}$ yields (see Appendix B and in particular Eq. ): t1 m\_[t]{} &=& [ L]{} \_0\^t d (t-)[x\_[ ]{}]{}\ t1 m\_[t]{} &=& L x\_[ t]{}  . \[nupetitgrand\] Our approximation is to assume that $m_{\nu t}$ in Eq.  is effectively given by Eq.  as soon as $\nu<1/t$ and by Eq.  when $\nu>1/t$ such that Eq.  becomes: $$\begin{aligned} \int_0^{1/t} \textrm d\nu \rho(\nu)\bigg[\int_0^t \textrm d\tau \Psi(t-\tau){\dot x_{ \tau }}\bigg]+ \int_{1/t}^\infty \textrm d\nu \rho(\nu) \bigg[\nu^{-1/2} \dot x_{ t} \bigg] &=&\frac {m_t}{\mathcal L \sqrt{D}} \ . \label{densnucentral}\end{aligned}$$ Equation  may be conveniently re-written as[^4] $\int_0^t \textrm d\tau \big[G(t) \Psi(t-\tau) + H(t)t_{\textrm c}^{1/2} \delta(t-\tau) \big] \dot x_\tau= {m_t}/({\mathcal L \sqrt{D}})$. Formally inverting the kernel $M(t,\tau):=\big[ G(t) \Psi(t-\tau) + H(t)t_{\textrm c}^{1/2} \delta(t-\tau) \big]$ then yields the price dynamics $\dot x_t$ as a linear convolution of the past order flow $m_{\tau \leq t}$. Note that when $m_t \to 0$, $\dot x_t$ is also small and hence, using Eqs. , all $m_{\nu t}$ are all small as well, justify our use of Eqs.  for all frequencies. Resolution of the “diffusivity puzzle” -------------------------------------- Let us now compute the functions $G$ and $H$ for a specific power-law distribution $\rho(\nu)$ defined as: $$\begin{aligned} \rho(\nu)&=& Z \nu^{\alpha-1} e^{-\nu t_{\textrm c}} \ , \label{rhonudens}\end{aligned}$$ where $\alpha>0$, $t_{\textrm c}$ is a high-frequency cutoff, and $Z=t_{\textrm c}^\alpha/\Gamma(\alpha)$.[^5] For such a distribution, one obtains $G(t) = 1- \Gamma(\alpha,t_{\textrm c}/t)/\Gamma(\alpha)$ and $H(t) = \Gamma(\alpha-1/2,t_{\textrm c}/t)/\Gamma(\alpha)$. In the limit $t\ll t_{\textrm c}, \ G(t)\approx 1$ and $H(t)\approx 0$. In the limit $t\gg t_{\textrm c}, \ G(t)\approx (t/t_{\textrm c})^{-\alpha}/[\alpha\Gamma(\alpha)]$, and the dominant term in the first order expansion of $H(t)$ depends on whether $\alpha \lessgtr 1/2$. One has $H(t|_{\alpha<1/2})\approx 2 (t/t_{\textrm c})^{1/2-\alpha}/[\Gamma(\alpha)(1-2\alpha)]$ and $H(t|_{\alpha>1/2})\approx \Gamma(\alpha-1/2)/\Gamma(\alpha)$. Focussing on the interesting case $\alpha < 1/2$, one finds (see Fig. \[fig:kernel\]) that inversion of the kernel $M(t,\tau)$ is dominated, at large times, by the first term $G(t) \Psi(t-\tau)$. Hence, one finds in that regime:[^6] $$\begin{aligned} x_{t}&\approx&\frac{\alpha \Gamma(\alpha)}{\mathcal L t_{\textrm c}^{\alpha} \sqrt{D} } \int_0^t \textrm d \tau \, \frac{m_\tau \tau^{\alpha}}{\sqrt{4\pi (t-\tau)} } \ . \label{densnuprop}\end{aligned}$$ Let us now show that this equation can lead to a diffusive price even in the presence of a long-range correlated order flow. Assuming that $\langle m_t m_{t'}\rangle \sim |t -t'|^{-\gamma}$ with $0 < \gamma < 1$ (defining a long memory process, as found empirically [@bouchaud2004fluctuations; @bouchaud2008markets]), one finds from Eq. (\[densnuprop\]) that the mean square price is given by: $$\begin{aligned} \langle x_t^2 \rangle \propto \iint_0^t \textrm d \tau \textrm d \tau' \frac{ \langle m_\tau m_{\tau'}\rangle {(\tau \tau')}^{\alpha}}{\sqrt{(t-\tau)(t-\tau')} } \ . \end{aligned}$$ Changing variables through $\tau \to tu$ and $\tau' \to tv$ easily yields $\langle x_t^2 \rangle \propto t^{1+2\alpha-\gamma}$. Note that the LLOB limit corresponds to a unique low-frequency for the latent liquidity. This limit can be formally recovered when $\alpha \to 0$. In this case, we recover the “disease” of the LLOB model, namely a mean-reverting, subdiffusive price $\langle x_t^2 \rangle \propto t^{1-\gamma}$ for all values of $\gamma > 0$. Intuitively, the latent liquidity in the LLOB case is too persistent and prevents the price from diffusing. Imposing price diffusion, i.e. $\langle x_t^2 \rangle \propto t$ finally gives a consistency condition similar in spirit to the one obtained in [@bouchaud2004fluctuations]: $$\begin{aligned} \alpha&=& \frac{\gamma}{2} < \frac12 \ . \label{alphasgamma} \end{aligned}$$ Equation  states that for persistent order flow to be compatible with diffusive price dynamics, the long-memory of order flow must be somehow buffered by a long-memory of the liquidity, which makes sense. The present resolution of the diffusivity puzzle – based on the memory of a multi-frequency self-renewing latent order book – is similar to, but different from that developed in [@BenzaquenFLOB]. In the latter study we assumed the reassessment time of the latent orders to be fat-tailed, leading to a “fractional” diffusion equation for $\phi(x,t)$. Metaorder impact ---------------- We now relax the constraint that $\lambda_\nu \propto \sqrt{\nu}$ and define $J_{\nu} := J_{\text{hf}} (\nu t_c)^{\zeta}$ with $\zeta>0$, meaning that HFT is the dominant contribution to trading, since in this case $$\begin{aligned} J&=& \int_{0}^\infty \textrm d\nu \rho(\nu) J_{\nu} = J_{\text{hf}} \frac{\Gamma(\zeta+\alpha)}{\Gamma(\alpha)}. \end{aligned}$$ (The case $\zeta<0$ could be considered as well, but is probably less realistic).\ We consider a meta-order with constant execution rate $m_0 \ll J_{\text{hf}}$. Since $J_\nu$ decreases as the frequency decreases, there must exist a frequency $\nu^\star$ such that $m_0 = J_{\nu^\star}$, leading to $\nu^\star t_c = (m_0/J_{\text{hf}})^{1/\zeta}$. When $\nu \ll \nu^\star$, we end up in the non-linear, square-root regime where $m_0 \gg J_\nu$ and Eq.   holds. Proceeding as in the previous section, we obtain the following approximation for the price trajectory: $$G_\zeta(t) \bigg[\int_0^{t }\textrm d\tau \Psi(t-\tau)\dot x_{\tau}\mathds{1}_{\{ t\leq \nu^{\star -1}\}} + \frac{x_t \dot x_t}{2\sqrt D} \mathds{1}_{\{ t>\nu^{\star -1}\}} \bigg]+t_c^{1/2} H_\zeta(t) \dot x_{t} =\frac{m_0 \sqrt{D}}{J_{\text{hf}}} \ . \label{densnucentral_s}$$ where, in the limit $t\gg t_c$ and $\alpha + \zeta < 1/2$: G\_(t)&:=& \_0\^[1/t]{} d() (t\_c)\^ ()\^[+]{} 1[()(+s)]{}\ H\_(t)&:=& \_[1/t]{}\^d() (t\_c)\^[-1/2]{} (t)\^[+-1/2]{} 1[()(1/2 - -s)]{} . At short times $t \ll \nu^{\star -1}$, Eq.  boils down to Eq.  with $\alpha \rightarrow \alpha+\zeta$ and one correspondingly finds: $$x_t \propto x_c \frac{m_0}{J_{\textrm{hf}}} \left(\frac{t}{t_c}\right)^{\frac12+\alpha+\zeta} \ ,$$ where $x_c := \sqrt{Dt_c}$. For $t \gg \nu^{\star -1}$, the second term in Eq.  dominates over both the first and the third terms, leading to a generalized square-root law of the form: $$x_t \propto x_c \sqrt{\frac{m_0}{J_{\text{hf}}}} \, \left(\frac{t}{t_c}\right)^{\frac{1+\alpha+\zeta}2} \ ,$$ Compatibility with price diffusion imposes now that $\alpha + \zeta = \gamma/2$, which finally leads to (see Fig. \[pricetraj\_multidens\]): x\_t && x\_c () \^, t t\_c ()\^[1/]{}\ x\_t && x\_c ()\^, t t\_c ()\^[1/]{}  . In the latter case, setting $\gamma = 1/2$ and $Q = m_0 T$, one finds an impact $I_Q:=x_T$ behaving as[^7] $Q^{5/8}$ as soon as $Q > \upsilon (J_{\text{hf}}/m_0)^{(1-\zeta)/\zeta}$, where we have introduced an elementary volume $\upsilon := J_{\text{hf}} t_c$, which is the volume traded by HFT during their typical cancellation time. Conclusion {#concl} ========== In this work, we have extended the LLOB latent liquidity model [@DonierLLOB] to account for the presence of agents with different memory timescales. This has allowed us to overcome several conceptual and empirical difficulties faced by the LLOB model. We have first shown that whenever the longest memory time is finite (rather than divergent in the LLOB model), a permanent component of impact appears, even in the absence of any “informed” trades. This permanent impact is [*linear*]{} in the traded quantity [ and independent of the trading rate]{}, as imposed by no-arbitrage arguments. We have then shown that the square-root impact law holds provided the meta-order participation rate is large compared to the trading rate of “slow” actors, which can be small compared to the total trading rate of the market – itself dominated by high-frequency traders. In the original LLOB model where all actors are slow, a square-root impact law independent of the participation rate only holds when the participation rate is large compared to the total market rate, which is not consistent with empirical data. Finally, the multi-scale latent liquidity model offers a new resolution of the diffusivity paradox, i.e. how an order flow with long-range memory can give rise to a purely diffusive price. We show that when the liquidity memory times are themselves fat-tailed, mean-reversion effects induced by a persistent order book can exactly offset trending effects induced by a persistent order flow.\ We therefore believe that the multi-timescale latent order book view of markets, encapsulated by Eqs.  and , is rich enough to capture a large part of the subtleties of the dynamics of markets. It suggests an alternative framework to build agent based models of markets that generate realistic price series, that complement and maybe simplify previous attempts [@Toth2011; @mastromatteo2014agent]. A remaining outstanding problem, however, is to reconcile the extended LLOB model proposed in this paper with some other well known “stylized facts” of financial price series, namely power-law distributed price jumps and clustered volatility. We hope to report progress in that direction soon. Another, more mathematical endeavour is to give a rigorous meaning to the multi-timescale reaction model underlying Eqs.  and and to the approximate solutions provided in this paper. It would be satisfying to extend the no-arbitrage result of Donier et al. [@DonierLLOB], valid for the LLOB model, to the present multi-timescale setting.\ We thank J. Bonart, A. Darmon, J. de Lataillade, J. Donier, Z. Eisler, A. Fosset, S. Gualdi, I. Mastromatteo, M. Rosenbaum and B. Tóth for extremely fruitful discussions. Appendix A {#sec:Appendix1 .unnumbered} ========== We here provide the calculations that link Eq.  and Table \[tableimpact\] during a meta-order execution ($t\leq T$); the impact decay computations ($t>T$) are given and discussed in Section \[permimp\].\ In the limit of slow execution of the meta-order, one has ${(x_t-x_\tau)^2}\ll {{4D(t-\tau)}}$ such that Eq.  together with Eq.  becomes: $$\begin{aligned} 0 &=& \phi^\mathrm{st}(x_t)e^{-\nu t} + \int_0^{t} \text d \tau\, \frac{m_0}{\sqrt{4\pi D(t-\tau)} } e^{-\nu(t-\tau) } -{2 \lambda}\int_0^{t } \text d \tau \, \frac{x_t-x_\tau}{\sqrt{4\pi D(t-\tau)} } e^{-\nu(t-\tau)} \ .\label{slowshort}\end{aligned}$$ Interestingly, slow and short execution is only compatible with small meta-order volume[^8] (indeed, combining $m_0\ll J$ and $\nu T\ll 1$ implies $m_0 T \ll J \nu^{-1}$). Thus for slow and short execution, using the linear approximation $\phi^\mathrm{st}(x_t)=-\mathcal L x_t$ and letting Eq.  into Eq.  yields: 0&=& -L z\^0\_t +m\_0\ 0&=& -L z\^1\_t - 2\_0\^t d  . Equation yields $\alpha = m_0/(\mathcal L\sqrt{\pi D})$ and $z_t^0=\sqrt{t}$, and it follows from Eq.  that $z_t^1 = - kt$ where $k=\sqrt{4/\pi} - \sqrt{\pi/4}$.\ In the limit of fast execution, one has ${(x_t-x_\tau)^2}\gg {{4D(t-\tau)}}$ such that the meta-order term can be approximated through the saddle point method. Letting $x_\tau \approx x_t- (t-\tau)\dot x_t$ into the price equation now yields: $$\begin{aligned} 0 &=& \phi^\mathrm{st}(x_t)e^{-\nu t} + \int_0^{t} \text d \tau\, m_0 \frac{e^{-\frac{\dot x_t^2(t-\tau)}{4D}}}{\sqrt{4\pi D(t-\tau)} } e^{-\nu(t-\tau) } -{ \lambda}\int_0^{t } \text d \tau \, e^{-\nu(t-\tau)} \ .\label{fastshort}\end{aligned}$$ Letting $u=t-\tau$ and given ${4D}/{\dot x_t^2}\ll t$ such that $\int_0^t \mathrm du \approx \int_0^\infty \mathrm du$, Eq.  becomes: $$\begin{aligned} 0 &=& \phi^\mathrm{st}(x_t)e^{-\nu t} + \frac{m_0}{\sqrt{\dot x_t^2+4D\nu}} +\frac{ \lambda}\nu\left( e^{-\nu t}-1\right) \, .\label{fastshortbis} $$ For short execution with small meta-order volume (we use $\phi^\mathrm{st}(x_t)=-\mathcal L x_t$), letting Eq.  into Eq.  yields: 0&=& -L z\^0\_t +\ 0&=& -L z\^1\_t - -t  . Equation yields $\alpha = \sqrt{{2m_0}/{\mathcal L}}$ and $z_t^0=\sqrt{t}$, and thus Eq.  becomes $\dot z_t^1 + {z_t^1}/({2t}) = - \frac12\sqrt{J/({2m_0})}$. It follows that $z_t^1 = - \frac t3\sqrt{J/(2m_0)} $. For a fast, short and large meta-order, $x_t$ is expected to go well beyond the linear region of the order book such that in a hand-waving static approach (consistent with fast and short execution) one can match $m_0 t$ and the area of a rectangle of sides $x_t$ and $\lambda\nu^{-1}$ (see Fig. \[Obstat\]). Letting $x_t=b t$ yields $b = m_0\nu /\lambda$. Note that this result can be recovered by letting $x_t=b t$ and $\phi^\mathrm{st}(x_t)=-\lambda \nu^{-1}$ into Eq. . Indeed, at leading order one obtains: $$\begin{aligned} 0&=& -\frac{\lambda}\nu + \frac{m_0}{ |\dot x_t|} \ ,\label{}\end{aligned}$$ from which the result trivially follows.\ For long execution ($\nu T\gg1$) the memory of the initial book is rapidly lost and one expects Markovian behaviour. Letting again $x_t=b t$ into the price equation and changing variables through $\tau =t(1-u)$ yields: $$\begin{aligned} 0 &=& m_0 \sqrt{t} \int_0^1 \text d u\,\frac{ e^{-\frac{b ^2tu}{4D}}}{\sqrt{4\pi Du}} e^{-\nu t u} -\lambda\int_0^1 \text d u \textstyle\,e^{-\nu t u} \, {\textrm{erf}}\sqrt{\frac{b^2tu}{4D}} \nonumber \\ &=& \left(m_0 - \frac{\lambda b }{\nu}\right)\frac{1}{\sqrt{b ^2+4D\nu }}\, {\textrm{erf}}\,\textstyle \sqrt{\left( \frac{b ^2}{4D}+\nu\right) t }\ . \label{longall}\end{aligned}$$ Interestingly, Eq.  yields $b = m_0\nu /\lambda$ (regardless of execution rate and meta-order size), which is exactly the result obtained above in the case of fast and short execution of a large meta-order but for different reasons. Appendix B {#sec:Appendix2 .unnumbered} =========== We here provide the calculations underlying the double-frequency order book model presented in Section \[multifsec\]. In particular [for the case $J_{\text{s}}\ll m_0\ll J_{\text{f}}$]{}, Eqs.  are obtained as follows. In the limit of large trading intensities the saddle point methods (as detailed in Appendix A) can also be applied to the case of nonconstant execution rates (one lets $m_\tau \approx m_t$ about which the integrand is evaluated, see [@DonierLLOB]), in particular one obtains (equivalent to Eq. ): $$\begin{aligned} \mathcal L_{\text{s}} x_{\text{s}t}|\dot x_{\text{s}t}|&=& {m_{\text{s}t}}\ , \label{} \end{aligned}$$ which yields Eq. . For the rapid agents ($\nu_{\text{f}}T\gg 1$) we must consider the case of long execution. In particular, an equation tantamount to Eq.  can also be derived in the case of nonconstant execution rates. Proceeding in the same manner, one easilly obtains: $$\begin{aligned} 0 &=& \left( m_{\text{f}t} - \frac{\lambda_{\text{f}} \dot x_{\text{f}t} }{\nu_{\text{f}}} \right) \frac{1}{\sqrt{\dot x_{\text{f}t} ^2+4D\nu_{\text{f}} }}\, {\textrm{erf}}\,\textstyle \sqrt{\left( \frac{\dot x_{\text{f}t}^2}{4D}+\nu_{\text{f}}\right) t }\ , \quad\quad \label{} $$ which yields ${ \dot x_{\text{f}t} } = m_{\text{f}t}\nu_{\text{f}} /\lambda_{\text{f}}$ and thus Eq. . Then, as mentioned in Section \[multifsec\], the asymptotic impact decay is obtained from Eqs. , and only where for $t>T$ we replace Eq.  with Eq. . Using Eq.  together with Eq.  in the limit $\nu_{\text{s}}T\rightarrow 0$, and $\nu_{\text{f}}T\gg 1$ together with yields ($t>T$): L\_ x\_t &=& \_0\^T + \_T\^t d\ 0&=& \_0\^T + \_T\^t d . \[decaymultifreq\] Asymptotically ($t\gg T$) the system of Eqs.  becomes: L\_ x\_t &=& \_0\^T +\_T\^t\ 0&=&\_0\^t d. \[decaymultifreqsimp\] We expect the asymptotic impact decay to be of the form $x_t = x_\infty + B/\sqrt{t}$. In addition Eq.  indicates that $m_{\text{f}t} \sim \dot x_t$. We thus let $m_{\text{s}t}=-m_{\text{f}t}=C/t^{3/2}$. Injecting into Eq.  yields $x_\infty = 0$ (no permanent impact) and: $$\begin{aligned} \frac{\mathcal L_{\text{s}} B}{\sqrt{t}} &=&\frac1{\sqrt{t}} \left[ \frac{m_0f_{T}}{\sqrt{4\pi D}} +\frac{C}{\sqrt{\pi DT}} \right] \ , \label{l1Bsqrtt}\end{aligned}$$ where $ f_T=T$ if $t^\star\ll T$ and $f_T = T^2/(3t^\star)$ if $ t^\star \gg T$. On the other hand, letting $u=t-\tau$ in Eq.  and using $x_t-x_s \approx (t-s)\dot x_t$ yields at leading order: $$\begin{aligned} 0&=&\int_0^\infty \textrm du \,\frac{e^{-\nu_{\text{f}}u}}{\sqrt{u}}\left[ -\frac C{t^{3/2}} + \frac{\lambda_{\text{f}}B u}{t^{3/2}} \right] = \sqrt{\frac{\pi}{\nu_{\text{f}}t^{3}}}\left[ -C + \frac{\lambda_{\text{f}} B}{\nu_{\text{f}}}\right]\label{} \ ,\end{aligned}$$ which combined with Eq.  easily leads to the values of $B$ and $C$.\ For the case $m_0\ll J_{\text{s}},J_{\text{f}}$, the calculations are slightly more subtle. Inverting Eq.  in Laplace space yields: $$\begin{aligned} m_{\text{s}t} &=&2{\mathcal L_{\text{s}}} \sqrt{D} \int_0^t \textrm d \tau \, \frac{ \dot x_{\text{s}\tau}}{\sqrt{\pi (t-\tau)} } \label{m1tx1point} \ .\end{aligned}$$ One can easily check this result by re-injecting Eq.  into Eq. . In turn, inverting Eq.  is straightforward and yields $ m_{\text{f}t} =({\lambda_{\text{f}}}/{\nu_{\text{f}}})\dot x_{\text{f}t}$. Injecting $\dot x_{\text{s}t}=\dot x_{\text{f}t}$ into Eq.  and using Eq.  yields: $$\begin{aligned} m_{\text{s}t} &=&\frac 1{\sqrt{t^\dagger}} \int_0^t \textrm d \tau \, \frac{ m_0- m_{\text{s}\tau}}{\sqrt{\pi(t-\tau)} } \label{} \ ,\end{aligned}$$ [which can be written as: $$\begin{aligned} \int_0^t \textrm d \tau \, { m_{\text{s}\tau}}\Phi(t-\tau) = 2m_0 \sqrt{t}\ , \quad \text{with} \ \Phi(t) := {\delta(t)}{\sqrt{\pi t^\dagger}} +\frac{\theta(t)}{\sqrt{t}} \ . \label{realtolapm1} \end{aligned}$$ Taking the Laplace transform of Eq.  one obtains $\widehat \Phi(p) \widehat m_{sp}=m_0\sqrt{\pi}/p^{3/2}$ with $\widehat \Phi(p)=\sqrt{\pi t^\dagger}+\sqrt{\pi/p}$, which in turn yields Eq. .]{} [^1]: $^\ast$Corresponding author. Email: michael.benzaquen@polytechnique.edu [^2]: A “meta-order” (or parent order) is a bundle of orders corresponding to a single trading decision. A meta-order is typically traded incrementally through a sequence of child orders. [^3]: Note that working at constant $\mathcal L$ implies $\lambda=O\big(\sqrt{\nu}\big)$. [^4]: We have implicitly defined the dimensionless functions $G(t) = \int_0^{1/t} \textrm d\nu \rho(\nu)$ and $H(t) =t_{\textrm c}^{-1/2} \int_{1/t}^\infty \textrm d\nu \rho(\nu) \nu^{-1/2}$. [^5]: Note that rigorously one should also introduce a low frequency cutoff $\nu_{\textrm{LF}}$ to ensure the existence of a stationary state of the order book in the absence of meta-order. Otherwise, $\langle\nu^{-1}\rangle=\infty$ when $\alpha \leq 1$ and the system does not reach a stationary state (see the end of Section \[llobrecall\] and [@BenzaquenFLOB] for a further discussion of this point). [^6]: Taking into account the $H(t)$ contribution turns out not to change the following scaling argument. [^7]: [Note that $5/8\approx 0.6$ is very close close to the empirical impact results reported by Almgren *et al.* and Brockmann *et al.* [@Almgren2005; @Brockmann2015] in the case of equities, for which $\gamma$ is usually close to 1/2.]{} [^8]: Equivalently, rapid and long execution is only consistent with large meta-order volume (combining $m_0\gg J$ and $\nu T\gg 1$ implies $m_0 T \gg J \nu^{-1}$).

--- abstract: 'X-ray observations with [*Chandra*]{} and [*XMM-Newton*]{} have shown that there are relatively narrow cores to the iron K$\alpha$ emission lines in active galactic nuclei (AGN). Plausible origins for this core emission include the outer regions of an accretion disk, a pc-scale molecular torus, and the optical broad-line region (BLR). Using data from the literature it is shown that no correlation exists between the Fe K$\alpha$ core width and the BLR (specifically H$\beta$) line width. This shows that in general the iron K$\alpha$ core emission does not arise from the BLR. There is a similar lack of correlation between the width of the Fe K$\alpha$ core and black hole mass. The average K$\alpha$ width is about a factor of two lower than the H$\beta$ width. It therefore seems likely that in many cases the narrow core arises in the torus. There is a very wide range of observed Fe K$\alpha$ core widths, however, and this argues for multiple origins. The simplest explanation for the observed line profiles in AGN is that they are due to a mixing of very narrow emission from the inner edge of the torus, and broadened emission from the accretion disk, in varying proportions from object-to-object.' author: - | K. Nandra$^{1}$\ $^1$Astrophysics Group, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AW, UK title: 'On the origin of the iron K$\alpha$ line cores in Active Galactic Nuclei' --- \[firstpage\] galaxies: active – galaxies: nuclei – galaxies: Seyfert – quasars: emission lines – X-rays: galaxies INTRODUCTION {#Sec:Introduction} ============ Iron K$\alpha$ emission lines are extremely common in the X-ray spectra of AGN and provide a potentially unique probe of the circumnuclear environment. The very earliest detections of the emission line were found in heavily obscured objects (e.g. Mushotzky et al. 1978). The emission was attributed to optical broad-line region clouds which, if in the line-of-sight, could simultaneously account for the heavy X-ray absorption (e.g. Holt et al. 1980). Later, evidence was found for Fe K$\alpha$ lines in unobscured AGN (e.g. Pounds et al. 1990; Nandra & Pounds 1994). These have been attributed to optically thick material out of the line of sight, and are accompanied by a Compton “reflection" continuum component (Guilbert & Rees 1988; Lightman & White 1988; George & Fabian 1991; Matt et al. 1991). This matter may be identified with the accretion disk (Fabian et al. 1989), or the molecular torus envisaged in orientation-dependent unification schemes for Seyfert galaxies (Krolik & Kallman 1987; Awaki et al. 1991; Ghisellini, Haardt & Matt 1994; Krolik, Madau & Zycki 1994). Iron K$\alpha$ line production in AGN could therefore plausibly originate in three separate sites: the accretion disk, the torus, and the BLR. It should be noted that in the present work we refer to the BLR simply as the region in which the optical broad lines originate, without necessarily identifying it with a physical structure. While the BLR has traditionally been envisaged as a system of “clouds" , it is possible that it can be identified with a disk wind (e.g. Murray & Chiang 1995; Elvis 2000), or even the outer regions of the accretion disk (Collin-Souffrin 1987). The clearest way to distinguish between the possible origins for the iron K$\alpha$ line is by measurements of its profile, as the various models posit emission from widely differing distances from the central black hole. X-ray spectra from the ASCA satellite (Tanaka, Inoue & Holt 1994) were the first to have sufficient spectral resolution to address the issue. These spectra supported an origin for the lines in the accretion disk, as heavily broadened and redshifted emission was observed in a number of Seyfert galaxies (Tanaka et al. 1995; Nandra et al. 1997). Data with higher resolution and/or signal-to-noise ratio are now available thanks to the [*Chandra*]{} and [*XMM-Newton*]{} satellites. The existence of relativistic accretion disk lines has been confirmed in some cases (e.g. Fabian et al. 2002), and questioned in others (e.g. Gondoin et al. 2001; Reeves et al. 2004), but in general they are difficult to confirm unambiguously, leaving the situation uncertain (Bianchi et al. 2004). One thing that is clear based on the new data is that relatively narrow “core” emission is observed in a large number of objects (Yaqoob et al. 2001; Kaspi et al. 2001). This raises the interesting question of the origin of the core emission, regardless of how often broader lines are seen. ------------- ------------ ------------- ------- ------------------------ ---------------- --------------------------- ------------ Object RA DEC $z$ $M_{\rm BH}$ FWHM($H\beta$) FWHM(Fe K$\alpha$) References (J2000) (J2000) $10^{6} M_{\rm \odot}$ km s$^{-1}$ km s$^{-1}$ (1) (2) (3) (4) (5) (6) (7) (8) Fairall 9 01 23 45.8 $-58$ 48 20 0.047 $255 \pm 56$ $6270\pm290$ $17040^{+55960}_{-14270}$ 1, 2, 3 3C120 04 33 11.1 $+05$ 21 16 0.033 $55.5 \pm 31.4$ $2360\pm 170$ $2000^{+2950}_{-2000}$ 1, 2, 3 NGC 3516 11 06 47.5 $+72$ 34 07 0.009 $42.7\pm 14.6$ $3353 \pm 310$ $1930^{+1380}_{-1380}$ 1, 1, 3 NGC 3783 11 39 0.17 $-37$ 44 19 0.010 $29.8 \pm 5.4$ $3570\pm190$ $1720 \pm 360$ 1, 2, 4 NGC 4051 12 03 09.6 $+44$ 31 53 0.002 $1.91 \pm 0.78$ $1072 \pm 112$ $6330^{+7740}_{-3310}$ 1, 1, 3 NGC 4593 12 39 39.4 $-05$ 20 39 0.009 $5.36 \pm 9.37$ $5320\pm610$ $2140^{+8370}_{-1230}$ 1, 2, 3 MCG-6-30-15 13 35 53.8 $-34$ 17 44 0.008 $1.5 \pm 0.30$ $1700 \pm 170$ $3250^{+5230}_{-3250}$ 5, 6, 2 IC4329A 13 49 19.2 $-30$ 18 34 0.016 $9.9 \pm 17.9$ $5620\pm 200$ $15090^{+12430}_{-9950}$ 1, 2, 3 Mrk 279 13 53 0.34 $+69$ 18 30 0.030 $34.9 \pm 9.2$ $5430\pm 180$ $4200^{+3350}_{-2950}$ 1, 2, 7 NGC 5548 14 17 59.5 $+25$ 08 12 0.017 $67.1 \pm 2.6$ $5830\pm 230$ $1700\pm 1500$ 1, 2, 8 H1821+643 18 21 57.3 $+64$ 20 36 0.297 ... $6620\pm 720$ $11700^{+6400}_{-4100}$ ..., 2, 9 Mrk 509 20 44 09.7 $-10$ 43 25 0.034 $143 \pm 12$ $3430\pm 240$ $2820^{+268-}_{-2800}$ 1, 2, 3 MR 2251-178 22 54 05.8 $-17$ 34 55 0.064 $102 \pm 20$ $6810\pm 460$ $390^{+260}_{-390}$ 10, 2, 11 NGC 7469 23 03 15.6 $+08$ 52 26 0.016 $12.2 \pm 1.4$ $2650\pm 220$ $6310^{+1580}_{-1580}$ 1, 2, 12 ------------- ------------ ------------- ------- ------------------------ ---------------- --------------------------- ------------ The most comprehensive and systematic study of the iron K$\alpha$ line core emission has been presented by Yaqoob & Padmanhaban (2004 hereafter YP04) using data from the [*Chandra*]{} High-Energy Transmission Grating (HETG) spectrometer. The core emission is invariably observed close to the rest energy for neutral iron, i.e. 6.4 keV, implying a very low ionization state for the originating material. The equivalent widths range from a few tens to $\sim 200$ eV, indicating both a large covering fraction, and reasonably high optical depth (say $\tau >0.1$; Awaki et al. 1991; Leahy & Creighton 1993). The lines are often resolved, but YP04 also presented clear evidence for differences in the velocity width of the cores from source to source, from $1,000-15,000$ km s$^{-1}$. A possible explanation for this is that there are varying contributions from the accretion disk and molecular torus (e.g. Reeves et al. 2001; YP04; Zhou & Wang 2005). If the former dominates, the core is expected to be relatively broad. In the latter case, due to the large distance, it would likely be unresolved even at the HETG resolution. Another alternative is clearly suggested by the range of line widths, as these fall exactly in the observed range for the optical BLR (e.g., Yaqoob et al. 2001; Bianchi et al. 2003). One might therefore hypothesize that the cores of the iron K$\alpha$ lines originate in that region. The detailed properties of the Fe K$\alpha$ lines might then shed light on the BLR’s physical structure. If iron line cores do indeed arise from the BLR there is a clear prediction that the velocity width of the iron lines should be correlated with, and roughly equal to, the width of the optical BLR emission lines. It is the purpose of this [*Letter*]{} to present a test of this prediction. Data and Results ================ To test the prediction of the BLR model for the iron K$\alpha$ line we clearly require profile data in both the X-ray and the optical. The former is more restrictive, as the only instrumental currently capable of resolving lines to sufficient accuracy is the [*Chandra*]{} HETG. We must also restrict ourselves to objects where it is possible to measure the BLR line width. We have therefore chosen a sample of type I Seyferts for which Fe K$\alpha$ line widths measured with HETG are available in the literature. The resulting sample consists of 14 AGN, listed in Table 1. Our primary resource for the iron line measurements is YP04 and in general we use those data in preference to others in the literature due to the uniformity of the analysis. Exceptions are NGC 3783 and NGC 5548, where Kaspi et al. (2002) and Steenbrugge et al. (2005) have presented line width measurement based on much longer observations than those presented in YP04. We neglect objects were the line width cannot be constrained by the data due to low signal-to-noise ratio. The errors on the Fe K$\alpha$ width are in general two sided. Following YP04 we adopt the larger of the two-sided errors. In cases where there are multiple observations we take the weighted mean of the measurements and combine the errors. All 14 Seyferts have published H$\beta$ width measurements. We adopt Marziani et al. (2003 hereafter M03) as the primary resource for these again for the sake of uniformity, but also because those authors have taken care to deconvolve the broad emission from any narrow components. Uncertainties on the FWHM values were determined from Table 4 of M03, by combining in quadrature the appropriate error on the blue and red wavelengths at half-maximum. Again, the larger of the two-sided errors was then adopted. In two cases, NGC 3516 and NGC 4051, we take H$\beta$ widths and uncertainties from Peterson et al. (2004 hereafter P04), noting that these authors measured the line widths from the RMS, rather than the integrated, spectrum. In one case, MCG-6-30-15, no uncertainty is given for FWHM H$\beta$ in the primary reference so we adopt a nominal 10 per cent uncertainty, typical of the upper end of the M03 values. We note, however, that the Fe K$\alpha$ measurement errors are generally much larger, so this assumption, and the adoption of the RMS widths in two cases, should not significantly affect our results. An additional note of caution is that the measurements are non-simultaneous. Any variability in one width and not the other would introduce scatter into the correlation. The plot of iron K$\alpha$ versus H$\beta$ width is shown in Fig. \[fig: hbeta\]. It is immediately clear from this figure that there is no correlation between the X-ray and optical line widths. Spearman (rank) and Pearson (linear) correlation tests both give statistics with high probabilites ($>50$ per cent) of being observed by chance. While the sample is small, and the measurement errors are relatively large, we can be confident that statistical effects have not contributed to destroying a true correlation between the two quantities. Several things are worth pointing out in this regard. Firstly, the object with the lowest FWHM H$\beta$, NGC 4051 (which is a so-called Narrow-Line Seyfert 1; NLS1 Osterbrock & Pogge 1985) has a relatively broad iron K$\alpha$ line core. The broadest H$\beta$ line object is seen in MR 2251-178, but this has the narrowest Fe K$\alpha$ core in the sample, and its value is relatively well determined. Furthermore, removing either (or indeed both) of these extreme objects from the sample fails to reveal a correlation. Finally, it is clear that objects with similarly broad optical lines have Fe core widths which are very different. For example, the widths in MR 2251-178 and H1821+643, the two objects with the broadest optical lines (FWHM H$\beta > 6500$ km s$^{-1}$), have K$\alpha$ widths spanning from the very narrowest to the very broadest observed values. Fig \[fig:mass\] shows the relationship between the iron K$\alpha$ line width and the black hole mass. The latter were taken largely from the compilation of O’Neill et al. (2005), which mostly originate in P04. Primary references are given in Table 1. For masses not taken from P04 we assign a nominal 20 per cent error. It is clear both that the FWHM measurements are not consistent with a constant, are not correlated, and nor are they consistent with $M \propto v^{2}$. The first is expected if the line arises from the same radius with respect to the gravitational radius (i.e. $r/r_{\rm g}$ is constant) and the latter two if the line arises from the same physical radius (i.e. $r$ is constant). DISCUSSION ========== The Fe K$\alpha$ core and BLR line widths in AGN differ substantially from object-to-object and cover a wide range. We have shown in the above analysis, however, that there is no relationship between the two. The most immediate conclusion is therefore that the Fe K$\alpha$ core does not generally arise from the optical BLR. In the above analysis, we have concentrated exclusively on the H$\beta$ width as a measure of the BLR velocity. It has become well established from reverberation mapping experiments (e.g. Clavel et al. 1991; Peterson et al. 1991) that there is not a single “BLR". Indeed this work has shown that the broad optical lines in AGN arise from a range of radii in rough proportion to the ionization potential of the species. One can image the BLR either as a series of clouds, or a continuous structure such as a wind, with a range of ionization states as a function of radius. Because iron is likely to survive at some level in all of the material responsible for the various broad lines, Fe K$\alpha$ emission in the BLR should occur at all radii, with the overall width representing an optical-depth weighted average over the whole region. As H$\beta$ is emitted at relatively large radii, one would expect the Fe K$\alpha$ cores to be, on average, broader than H$\beta$. There is no evidence for this, and if anything the latter is the case. The weighted mean H$\beta$ width for the sample is $3200 \pm 60$ km$s^{-1}$, whereas the Fe K$\alpha$ line is less than half that width, with weighted mean FWHM $1350 \pm 250$. The latter value is in agreement with that presented by YP04. The implication is that, typically, the Fe K$\alpha$ core comes from [*outside*]{} the BLR and hence is plausibly identified with the torus responsible for obscuring the BLR in the unification schemes (Antonucci & Miller 1985). A torus origin for the neutral Fe K$\alpha$ line is thought to be most likely in Seyfert 2 galaxies. The line cores are unresolved at the HETG resolution (e.g. Sambruna et al. 2001; Ogle et al. 2003) but there is a “Compton shoulder” which shows unambiguously that they originate in optically thick material (Iwasawa et al. 1997; Bianchi et al. 2002; Matt et al. 2004). A tentative detection of the Compton shoulder has been made in the Seyfert 1 NGC 3783 (Kaspi et al. 2002; Yaqoob et al. 2005), supporting a torus origin for the line core in that object also. If the line originates universally from the inner edge of the torus one [*still*]{} might expect the line widths to be correlated, as the H$\beta$ emission radius and the inner edge of the torus (which may represent the dust sublimation radius) should both be controlled by the power of the central source. Alternatively, the inner edge of the torus might occur at a fixed number of gravitational radii. In the latter case, however, one would predict a constant Fe K$\alpha$ core width. The details of this are, however, strongly dependent on the precise geometry and physical parameters, because the iron K$\alpha$ emission is expected to arise from a range of radii in the torus. It seems clear, however, that in objects where the iron line core width exceeds the H$\beta$ width, one can rule out the torus as a possible origin. While the inner BLR remains a possible site for line emission, in these objects the Fe K$\alpha$ emission core seems most likely to be associated with the outer regions of the accretion disk, given the compelling evidence for the origin of the broadest Fe K$\alpha$ emission in the disk (e.g. Vaughan & Fabian 2004; Ponti et al. 2004). Three notes of caution should be sounded about the analysis described above. Firstly, the sample size is currently very small. It will clearly be important to see if the conclusions remain robust when more objects are included. This requires further high signal-to-noise ratio spectra of Seyferts to be obtained with [*Chandra*]{}. Secondly, it is well known in spectroscopy that the continuum parameterization can affect the line parameters significantly. These effects are exacerbated when the signal-to-noise ratio is low, as is the case with some of the spectra of Seyferts used in this [*Letter*]{}. When considering the narrow core of the line and where high resolution data are available, however, the uncertainties due to continuum modeling should be minimised. Finally, as noted earlier, the Fe K$\alpha$ and H$\beta$ measurements are not simultaneous, hence variability of the widths could introduce scatter into their relationship. A particularly well-studied case is that of NGC 5548. P04 have presented a series of measurements of the H$\beta$ width, which show excess RMS variability of $\sim 15$ per cent. This is larger than the typical measurement error in FWHM H$\beta$, but the uncertainties in the X-ray line widths still dominate. Variability should not therefore strongly affect our conclusions, but obtaining strictly simultaneous, or at least contemporaneous, X-ray and optical spectroscopy would clearly be highly desirable. Overall it appears that, based on the present data, the structures responsible for the production of the optical broad lines (e.g. clouds, disk wind) are a negligible source of iron K$\alpha$ emission in Seyfert galaxies. This may not be at all surprising given the physical parameters expected for the BLR. Assuming “typical” values for BLR clouds of $N_{\rm H} \sim 10^{23}$ cm$^{-2}$ and a covering fraction of 10 per cent one would predict an equivalent width of order only 10 eV for the Fe K$\alpha$ line (Leighy & Creighton 1993). Similarly, the column densities in the outflows responsible for the UV and X-ray absorption lines in Seyferts (e.g. Mathur, Wilkes & Elvis 1998; Crenshaw, Kraemer & George 2004) are typically $\sim 10^{22}$ cm$^{-2}$. The optical depth of this outflowing gas, which may be identified with the disk wind, are therefore too low to produce significant iron K$\alpha$ emission even if the covering fraction is high. In individual cases it will clearly be very difficult to rule out a BLR contribution to Fe K$\alpha$. Our conclusion, however, is that the simplest approach towards the modeling the iron K$\alpha$ line complex in Seyferts is to assume a mix of contributions from both the torus and accretion disk, in differing proportions depending on the source. The key questions that now remain relate to the reasons why those proportions differ and, particularly, why in some cases one or other of the components of the iron K$\alpha$ line appear to be absent. Acknowledgements {#acknowledgements .unnumbered} ================ The author gratefully acknowledges the support of the Leverhulme Trust in the form of a research fellowship. I am indebted to Ian George and Brad Peterson for helpful discussions. I thank the referee, Kazushi Iwasawa, for constructive comments. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration and of the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center. Awaki, H., Koyama, K., Inoue, H., Halpern, J.P., 1991, PASJ, 43, 195 Bianchi S., Matt G., Fiore F., Fabian A.C., Iwasawa K., Nicastro F., 2002, A&A, 396, 793 Bianchi S., Matt G., Balestra I., Perola G. C., 2003, A&A, 407, L21 Bianchi S., Matt G., Balestra I., Guainazzi M., Perola G. C., 2004, A&A, 422, 65 Clavel, J., et al., 1991, ApJ, 366, 64 Collin-Souffrin, S., 1987, A&A, 179, 60 Crenshaw D. M., Kraemer S. B., George I. M., 2003, ARA&A, 41, 117 Elvis M., 2000, ApJ, 545, 63 Fabian A. C., Rees M. J., Stella L., White N. E., 1989, MNRAS, 238, 729 Fabian A. C., et al., 2002, MNRAS, 335, L1 George I. M., Fabian A. C., 1991, MNRAS, 249, 352 Ghisellini G., Haardt F., Matt G., 1994, MNRAS, 267, 743 Gondoin, P., Barr, P., Lumb, D., Oosterboek, T., Orr, A., Parmar, A.N., 2001, A&A, 378, 806 Guilbert P. W., Rees M. J., 1988, MNRAS, 233, 475 Holt, S.S., Mushotzky, R.F., Boldt, E.A., Serlemitsos, P.J., Becker, R.H., Szymkowiak, A.E., White, N.E., 1980, ApJ, 241, L13 Iwasawa K., Fabian A.C., Matt G., 1997, MNRAS, 289, 443 Kaspi, S., et al., 2001, ApJ, 554, 216 Kaspi, S., et al., 2002, ApJ, 574, 643 Krolik J. H., Madau P., Zycki P. T., 1994, ApJ, 420, L57 Leahy D. A., Creighton J., 1993, MNRAS, 263, 314 Lightman, A.P., White, T.R., 1988, ApJ, 355, 57 Mathur, S., Wilkes, B.J., Elvis, M., 1998, ApJ, 503, L23 Marziani, P., Sulentic, J.W., Zamanov, R., Calvani, M., Dultzin-Hacyan, D., Bachev, R., Zwitter, T., 2003, ApJS, 145, 199 Matt, G., Perola, G.C., Piro, L., 1991, A&A, 247, 25 Matt G., Bianchi S., Guainazzi M., Molendi S., 2004, A&A, 414, 155 Murray, N., Chiang, J., 1995, ApJ, 454, 105 Nandra K., Pounds K. A., 1994, MNRAS, 268, 405 Nandra K., George I. M., Mushotzky R. F., Turner T. J., Yaqoob T., 1997, ApJ, 477, 602 Peterson, B.M., et al., 1991, ApJ, 368, 119 Peterson, B.M., et al., 2004, ApJ, 613, 682 (P04) Ponti G., Cappi M., Dadina M., Malaguti G., 2004, A&A, 417, 451 Pounds, K.A., Nandra, K., Stewart, G.C., George, I.M., Fabian, A.C., 1990, Nature, 344, 132 Reeves J. N., Turner M. J. L., Pounds K. A., O’Brien P. T., Boller T., Ferrando P., Kendziorra E., Vercellone S., 2001, A&A, 365, L134 Reeves J. N., Nandra K., George I. M., Pounds K. A., Turner T. J., Yaqoob T., 2004, ApJ, 602, 648 Tanaka Y., et al., 1995, Nature, 375, 659 Tanaka, Y., Inoue, H., Holt, S.S., 1994, PASJ, 46 Vaughan S., Fabian A. C., 2004, MNRAS, 348, 1415 Yaqoob T., George I. M., Nandra K., Turner T. J., Serlemitsos P. J., Mushotzky R. F., 2001, ApJ, 546, 759 Yaqoob T., Padmanabhan U., 2004, ApJ, 604, 63 (YP04) Yaqoob T., Reeves J.N., Markowitz A., Serlemitsos P.J., Padmanabhan U., 2005, ApJ, 627, 156 Zhou, X.-L., Wang, J.-M., 2005, ApJ, 618, L83 \[lastpage\]

--- abstract: 'The ability of a robot to detect and respond to changes in its environment is potentially very useful, as it draws attention to new and potentially important features. We describe an algorithm for learning to filter out previously experienced stimuli to allow further concentration on novel features. The algorithm uses a model of habituation, a biological process which causes a decrement in response with repeated presentation. Experiments with a mobile robot are presented in which the robot detects the most novel stimulus and turns towards it (‘neotaxis’).' author: - | Stephen Marsland, Ulrich Nehmzow and Jonathan Shapiro\ Department of Computer Science\ University of Manchester\ Oxford Road\ Manchester M13 9PL\ `{smarsland, ulrich, jls}@cs.man.ac.uk` bibliography: - 'thebib.bib' - 'Manbib.bib' title: 'Novelty Detection for Robot Neotaxis [^1]' --- Introduction ============ Many animals have the ability to detect novelty, that is to recognise new features or changes within their environment. This paper describes an algorithm which learns to ignore stimuli which are presented repeatedly, so that novel stimuli stand out. A simple demonstration of the algorithm on an autonomous mobile robot is given. We term the robot’s behaviour of following the most novel stimulus [*neotaxis*]{}, meaning ‘turn towards new things’, taken from the Greek ([*neo*]{} = new, [*taxis*]{} = follow). A number of different versions of the novelty filter are described and compared to find the best for the particular data used. Attending to more novel stimuli is a useful ability for a mobile robot as it can limit the amount of data which the robot has to process in order to deal with its environment. It can be used to recognise when perceptions are new and must therefore be learned. In addition, it means that the robot can be used as an inspection agent, so that after training to learn common features it will highlight any ‘novel’ stimuli, i.e., those which it has not seen previously. Related Work ------------ A number of novelty detection methods have been proposed within the neural network literature, but they are mostly trained off-line. Particularly noteworthy is the Kohonen Novelty Filter [@Kohonen76; @Kohonen93], which is an auto-encoder neural network trained by back-propagation of error. After training, any presentation to the network produces one of the trained outputs, and the bitwise difference between the input and output shows the novel parts of the input. This work has been extended by a number of authors. For example, Aeyels [@Aeyels90] adds a ‘forgetting’ term into the equations. Ho and Rouat [@Ho98] use a biologically inspired model that times how long an oscillatory network takes to converge to a stable output, reasoning that previously seen inputs should converge faster than novel ones. Finally, Levine and Prueitt [@Levine92] use the gated dipole proposed by Grossberg [@Grossberg72; @Grossberg72a] to compare inputs with pre-defined ones, novel features causing greater output values. The Novelty Filter ================== Habituation ----------- Habituation is a reduction in behavioural response that occurs when a stimulus is presented to an organism repeatedly. It is present in many animals, from the sea slug [*Aplysia*]{}  [@Bailey83; @Greenberg87] through toads [@Ewert78; @Wang92] and cats [@Thompson86] to humans [@OKeefe77]. It has been modelled by Groves [@Groves70], Stanley [@Stanley76] and Wang and Hsu [@Wang90]. Habitation differs from other processes which decrement synaptic efficacy, such as fatigue, in that a change in stimulus restores the response to its original levels. This process is called dishabituation. There is also a ‘forgetting’ effect, where a stimulus which has not been presented for a long time recovers its response. Further details can be found in [@Thompson66; @Wang95]. The habituation mechanism used in the system described here is Stanley’s model. The synaptic efficacy, $y(t)$, decreases according to the following equation: $$\tau \frac{dy(t)}{dt} = \alpha \left[ y_0 - y(t) \right] - S(t), \label{HabEqn}$$ where $y_0$ is the original value of $y$, $\tau$ and $\alpha$ are time constants governing the rate of habituation and recovery respectively, and $S$ is the stimulus presented. The effects of the equation are shown in figure \[curves\]. The principal difference between this and the model of Wang and Hsu is that the latter allows for long-term memory, so repeated training causes faster learning. ![ []{data-label="curves"}](habit.eps){width=".45\textwidth"} Figure \[curves\] shows the synaptic efficacy increasing again at time 150, when the stimulus is removed. This is effectively a ‘forgetting’ effect, and is caused by a dishabituation mechanism which increases the strength of synapses that do not fire. In the implementation described here this effect can be removed. The experiments reported in section \[Results\] investigate effects of the filter both with and without forgetting. Using Habituation for a Novelty Filter\[NF\] -------------------------------------------- ![ []{data-label="hsom"}](HSOM.eps){width=".45\textwidth"} The principle behind the novelty filter is that perceptions are classified by some form of clustering network, whose output is modulated by habituable synapses, so that the more frequently a neuron fires, the lower the efficacy of the synapse becomes. This means that only novel features will produce any noticeable output. If the habituable synapses receive zero input (rather than none) during turns when their neuron does not fire, the synapses will ‘forget’ the inhibition over time, providing that this forgetting mechanism (or dishabituation) is turned on. The choice of clustering algorithm is very important and depends on the data being classified. In this paper, we compare the performance of three different networks, described below, on the robot application. The three networks described were chosen because they performed best on sample data that was selected to be similar to that they would see on the robot. In addition to those described below, the Neural Gas [@Martinetz93] network also performed well, but computational constraints means that it was not possible to run it on the robot. Some Possible Clustering Networks \[NNs\] ----------------------------------------- ### Kohonen’s Self-Organising Map (SOM) Kohonen’s Self-Organising Map [@Kohonen93] works in the following way:\ Every element of the input vector is connected to every node of the map by a modifiable connection. The distance $d$ between the input and each of the neurons in the field is calculated using $$d = \sum_{i=0}^{N-1} \left[ \mathbf{v} (t) - \mathbf{w}_i (t) \right] ^2$$ where $\mathbf{v} (t)$ is the input vector at time $t$ and $\mathbf{w}_{i}$ the weight between input $i$ and the neuron. In a Learning Vector Quantiser [@Kohonen93], used here, the neuron with the minimum $d$ is selected and the weight for that neuron and its topological neighbours are updated by: $$\mathbf{w}_{i} (t+1) = \mathbf{w}_{i} (t) + \eta (t) \left[ \mathbf{v} (t) - \mathbf{w}_{i} (t) \right]$$ where $\eta$ is the learning rate, $0 \leq \eta \leq 1$. Usually, a two-dimensional SOM is used, but in the implementation described here a ring-shaped network, effectively a line with the end neurons linked together, was used. The neighbourhood size and learning rate remained constant so that the system was always learning. The neighbourhood comprised only the nearest neighbours of each neuron, and $\eta$ was fixed at 0.25. ### The Temporal Kohonen Map (TKM) This self-organising map, proposed by Chappell and Taylor [@Chappell93], is based on Kohonen’s SOM, but uses “leaky integrator” neurons whose activity decays exponentially over time. The exponential decay is controlled by a time constant ($\gamma$ in equations \[TKMeqn\] and \[TKMupdate\] below). This is similar to a short-term memory, allowing previous inputs to have some effect on the processing of the current input, so that the neurons which have won recently are more likely to win again. In the experiments reported here the value $\gamma = 0.4$ was used, meaning that only the previous 2 or 3 winners had any influence in deciding the current winner. The activity of the neurons is calculated using $$a_i (t) = \gamma \cdot a_i (t-1) + e^{ \left( - \frac{1}{2} \right) \left[ \mathbf{v} (t) - \mathbf{w}_i (t) \right]^2}, \label{TKMeqn}$$ and, in a similar way to the SOM, the neuron with the largest activity $a$ is chosen as winner, and its weights and those of its topological neighbours updated using the following weight update rule ($\eta$ and the neighbourhood remained the same): $$\mathbf{w}_i (t+1) = \mathbf{w}_{i} (t) + \eta \sum_{k=0}^{n} \gamma^k \left[ \mathbf{v} (t-k) - \mathbf{w}_i (t-k) \right]. \label{TKMupdate}$$ ### The $K$–Means Clustering Algorithm One of the simplest ways to cluster data is by using the $K$–means algorithm [@Bishop95b]. A pre-determined number of prototypes, $\mathbf{\mu}$, are chosen to represent the data, so that it is partitioned into $K$ clusters. The positions of the prototypes are chosen to minimise the sum-of-squares clustering function, $$J = \sum_{j=1}^{K} \sum_{n \in S_j} \| \mathbf{x}^n - \mathbf{\mu}_j \|^2$$ for data points $\mathbf{x}^n$. This separates the data into $K$ partitions $S_j$. The algorithm can be carried out as an on-line or batch procedure, with the on-line version, used here, having the update rule $$\Delta \mathbf{\mu}_j = \eta \left( \mathbf{x}^n - \mathbf{\mu}_j \right).$$ Using the Novelty Filter on a Mobile Robot\[impl\] ================================================== The robot implementation was designed to show that the novelty filter described in section \[NF\] can be used to detect new stimuli. The novelty filter was incorporated into a system where a robot detects and turns towards new stimuli. It was implemented on a Fischer Technik mobile robot, which uses a Motorola 68HC11 microcontroller. The robot has a two wheel differential drive system and four light sensors facing in the cardinal directions. {width=".3\textwidth"}  In the experiments described below, the robot received a number of different light stimuli, which varied in the frequency of the flashes. It classified these stimuli autonomously and decided whether or not to respond (turn towards the source) according to how novel they were. Each of the sensors on the robot, in this case four light sensors, had its own novelty filter, as shown in figure \[SysLayout\]. At each cycle, the current reading on each sensor was concatenated with the previous five to form a six element input vector, known as a delay line or lag vector. This vector was classified by the novelty filter and an output produced. In the case of the TKM, which keeps an internal history of previous inputs, only the most recent reading was needed as input. The output of the filter was a function of how many times that neuron had fired before, due to the habituating synapse. Each of the four novelty filters fed their output to a comparator function which propagated the strongest signal, providing that it was above a pre-defined threshold, to the action mechanism. If none of the stimuli were strong enough, the cycle repeated. Owing to memory constraints, the clustering mechanism was limited to just twelve neurons arranged in a ring. All three of the networks described in section \[NNs\] were the same size. A bypass function was associated with each sensor. If a neuron had not fired before (that is, its synapse had not been habituated) the comparator function favoured it, so that the system responded rapidly to new signals. If two new signals were detected simultaneously, the stronger one was used. Experiments and Results \[Results\] =================================== Three separate experiments were carried out. The first, the results of which are shown in figure \[ExpLayout\] and table \[Tab1\], was designed to test the forgetting mechanism as well as the general ability to turn towards novel stimuli. The robot was initially placed in a featureless environment. A light was introduced to project onto one of the light sensors. Once the robot had turned to face this light source, a second, slowly flashing light was added. As this light was more novel, the robot turned towards it. A further, faster flashing light was then introduced, which the robot again faced. Finally, the constant light was switched off and, in the case where a ‘forgetting’ mechanism was used, the robot perceived this lack of stimulus as novel and turned back towards it. Otherwise it did not respond. ![ []{data-label="ExpLayout"}](exp1.eps "fig:"){width=".2\textwidth"}![ []{data-label="ExpLayout"}](exp2.eps "fig:"){width=".2\textwidth"} ![ []{data-label="ExpLayout"}](exp3.eps "fig:"){width=".2\textwidth"}![ []{data-label="ExpLayout"}](exp4.eps "fig:"){width=".2\textwidth"} In the second experiment, steps (a) and (b) of figure \[ExpLayout\] were again followed. However, instead of a faster flash being shown in the third stage, a second flashing light of the same (slow) frequency was shown. If the flashing light was still novel, the robot turned towards this as it was a newer version of the most novel stimulus. However, if the flashing light had ceased to be novel, the robot ignored it. Finally, instead of a second flashing light in part (c), a second constant light was introduced. Whether or not the robot responded to this depended on whether or not the forgetting mechanism was switched on and which sensor it was on – if it was a sensor which had not previously seen it, the robot responded. Table \[Tab1\] shows the reactions of the robot in the three experiments, both with forgetting turned on and off. The constants used for the experiments were: $\tau = 0.1$, $\alpha=0.5$, $\beta=0.1$ and a boredom threshold (i.e., the value below which a stimuli ceased to be novel) of 0.4. The parameters of the networks were kept at the levels found to be optimal in simulations. The overall qualitative results were the same for all three networks, although the SOM took longer to produce consistent output when a new pattern was introduced (owing to the changes in the spatial pattern in the lag vector) while the TKM responded to them quickly. Experiment Forgetting Stage Action ------------ ------------ ------------------ -------------------------------------------------- 1 On Constant On Robot turns towards it Slow Flashing On Robot turns towards it Fast Flashing On Robot turns towards it Constant Off Robot turns towards it Off Constant On Robot turns towards it Slow Flashing On Robot turns towards it Fast Flashing On Robot turns towards it Constant Off Robot does not respond 2 On Constant On Robot turns towards it Slow Flashing On Robot turns towards it Slow Flashing On If on a different sensor, robot turns towards it Off Constant On Robot turns towards it Slow Flashing On Robot turns towards it Slow Flashing On If on a different sensor, robot turns towards it 3 On Constant On Robot turns towards it Slow Flashing On Robot turns towards it Constant On If on a different sensor, robot turns towards it Off Constant On Robot turns towards it Slow Flashing On Robot turns towards it Constant On If on a different sensor, robot turns towards it In table \[Tab1\] it can be seen that particular inputs caused the robot to move even when the stimulus had been seen before. This occurred because the stimulus was on a sensor which had not perceived it previously. This meant that the robot’s attention was changing unnecessarily, so a method to rectify this was devised. When a stimulus is marked as novel the robot rotates through $360^\circ$, pausing every $90^\circ$, so that each of the novelty filters learns to recognise all the stimuli. This means that the robot reacts to stimuli in the same way regardless of which sensor they impinge on. This functionality can be produced in other ways, such as using one novelty filter to monitor all the sensors and adding additional memory of what each sensor was seeing to turn the robot in the appropriate direction. The output of the network took a few iterations to stabilise for each new input, and the SOM in particular occasionally generated spurious readings, caused by misreading the signals so that the input vector varied. This was usually because the sensor polling could not be precisely timed, so that occasionally the time between readings varied and so an unexpected input was received. Further Experiments ------------------- In the experiments described previously, all three clustering networks showed similar qualitative results. For this reason, further tests were designed to try and discriminate between the networks. The additional experiments performed involved using flashing lights which flashed at varying speeds. The neotaxis behaviour of the robot remained fixed. Two additional patterns of flashing lights were used, short–short–long–long and short-long-short-long, which the K[-]{}Means network and Temporal Kohonen Map both recognised more accurately than the SOM. The TKM in particular dealt with all the stimuli very well, but the SOM was occasionally subject to errors and took longer to respond. The number of patterns which it is possible for the robot to learn and recognise is limited by the size of the network. Conclusions and Future Work \[Disc\] ==================================== The mechanism described here is capable of recognising features which vary in time and habituating to those that are seen repeatedly. In this way it successfully acts as a novelty filter, highlighting those stimuli which are new and directing attention towards them. This is a useful ability, since it can reduce the amount of data which the robot needs to process in order to deal with its environment. However, in the application described here, the inputs are fairly clean, the environment being designed to produce differentiable inputs. One of the assumptions that is made in this paper is that the clustering networks used will reliably separate the inputs so that new stimuli cause a new neuron to win, and old stimuli activate the same neuron each time. This is not necessarily true, and the potential problems this highlights need to be investigated. Using a growing network such as the Growing Neural Gas of Fritzke [@Fritzke95] is one solution, as is using a Mixture of Experts [@Jordan94] in place of the clustering network, each expert recognising a different part of the input space. In addition, the sensors used here, photocells, are crude and do not give a great deal of information, and the robot has very limited memory. To produce a system which is capable of interacting with real world environments it will be necessary to use more and better sensors. The next step will be to transfer the system onto the Manchester Nomad 200 robot, [*FortyTwo*]{}, and take advantage of the sensors available, viz. sonar, infra-red and a monochrome CCD camera. Before the novelty filter can deal with this information, sensor inputs will have to be extensively preprocessed, with features extracted from the images. Work using sonar scans taken whilst the robot is exploring an environment have shown success in applying the novelty filter to a real world problem (work to be published). However, once data about the surrounding environment can be interpreted, the novelty filter presented here can be used in an inspection agent which learns a representation of an environment and can then explore and detect new or changed features within both that and similar environments. This is the ultimate aim of this research. Acknowledgements {#acknowledgements .unnumbered} ================ This research is supported by a UK EPSRC Studentship. [^1]: In Proceedings of the 2nd International Symposium on Neural Computation, pages 554 - 559, 2000

--- abstract: 'In this paper we propose a multi-state model for the evaluation of the conversion option contract. The multi-state model is based on age-indexed semi-Markov chains that are able to reproduce many important aspects that influence the valuation of the option such as the duration problem, the time non-homogeneity and the ageing effect. The value of the conversion option is evaluated after the formal description of this contract.' address: - 'Department of Pharmacy, University “G. d’Annunzio” of Chieti-Pescara, Chieti, Italy' - 'Department of Econometrics, Statistics and Economics, University of Barcelona, Barcelona, Spain' - 'MEMOTEF Department, University “La Sapienza”, Rome, Italy' - 'Department of Business, University of Cagliari, Cagliari, Italy' author: - - - - title: 'Multi-state models for evaluating conversion options in life insurance' --- ./style/arxiv-vmsta.cfg Introduction {#sec1} ============ The conversion option is an option that allows the policyholder to convert his original temporary insurance policy (TIP) to permanent insurance policy (PIP) before the initial policy is due. Insurance companies may find convenient this kind of contract because it may be much less expensive to convert the initial policy instead of issuing a new one. On the other side the policyholder may be interested in converting the contract because, at the time of conversion, insurance companies do not require any evidence of insurability and calculate the new premium according to the age at the issue of the original contract. However, at the time of conversion the insured individual has to pay the difference of cash value between the original TIP and converted PIP. The literature on conversion option is not large and the main reference is represented by the article [@su10] where a valuation model was constructed based on mortality tables and then extended to a Lee–Carter model of mortality. A related article is [@no08] where the author considered an exchange option that is available in Norway. In general, insurance companies collect data in form of sequences of events concerning the health status of the policyholders. Therefore they can evaluate survival probabilities taking into account for the health evolution of the insured person. This means that the adoption of a multi-state model can improve the evaluation process of policy-linked contracts like the conversion option when compared with information extracted from simple mortality tables. Indeed, as argued in [@kwjo08], mortality rates are limited to accurately predict the dynamics of mortality. Moreover recent literature includes contributions where multi-state models, based on Markov chains, have been advanced as a valuable alternative to traditional mortality models see, e.g., [@lili07; @lili12; @toma91; @kwjo06]. A general approach based on semi-Markov processes has been applied to problems of disability insurance also in recent years, see [@stmasi07; @daguma09; @daguma13; @ma13]. Their appropriateness is due to the rejection of the geometric (exponential in continuous time model) distribution hypothesis for modeling the waiting times in a health status before making a transition in another state. Indeed, the geometric (exponential) hypothesis results in the lack of memory property that is very convenient from a mathematical point of view but is rarely supported by empirical evidence. In this paper we focus on the evaluation of the conversion options when an age-indexed semi-Markov multi-state model describes the evolution of the health status of the policyholder. To this end we first derive transition probabilities for the model and then we develop the evaluation procedure by analyzing the TIP and PIP contracts and the conversion option. The obtained results represent the generalization of the results of [@su10] in a more general framework. Particularly, we show that the value of the conversion option depends on many parameters that are contemporary managed by our model such as the health status evolution of the policyholder, the age of the policyholder and the chronological time effect due to medical-scientific progress. We start in Section \[sec2\] by describing the age indexed semi-Markov model. In we explain the valuation procedure of the conversion option and we calculate its value. The paper ends with some conclusions and suggestions for further research. Age-indexed semi-Markov model {#sec2} ============================= Following the approach of [@jama97] it is possible to give a tractable extension of discrete time non-homogeneous semi-Markov chains useful to consider different aspects that are relevant for the evaluation of the conversion option like the duration problem, the non-homogeneity and the ageing effect. This approach has been further generalized in [@da11; @dape11; @dape12] where general indexed semi-Markov processes were investigated and applied to different problems. On a complete probability space $ (\varOmega, \mathcal{F}, \mathbb{P})$ we consider two sequences of random variables that evolve jointly:\ $$J_{n}:\varOmega\rightarrow E=\{1,2,\ldots,D\},$$ $$T_{n}:\varOmega\rightarrow\mathbb{N}.$$ $J_{n}$ represents the state at the $n$-th transition which can be identified with one of the mutually exclusive elements of the set $E$. In our framework, the set $E$ contains all possible values of the health-status of the policyholder, included the death state denoted by $D$. The quantity $T_{n}$ denotes the time of the $n$-th transition, i.e. the time when the policyholder enters in the health-status $J_{n}$. We define the age-index process by the relation: $$\label{age-index} A_{n}=A_{n-1}+T_{n}-T_{n-1},\quad n\in\mathbb{N},$$ where $A_{0}$ is known. From now on we will set $A_{0}=a$ and as usually $T_{0}=0$. This implies that by recursive substitution $A_{n}=a+T_{n}$, that is the age at the time of the $n$-th transition is given by the initial age ($A_{0}=a$) plus the time of occurrence of the $n$-th transition ($T_{n}$). The key assumption is to consider the triple $(J_{n}, T_{n}, A_{n})$ like a non-homogeneous Markov Renewal Process with index: $$\begin{aligned} &\mathbb{P}\bigl[J_{n+1}=j, T_{n+1}\leq t \bigm\vert\sigma(J_{h},T_{h},A_{h}, \, h\leq t), J_{n}=i, T_{n}=s, A_{n}=a+s\bigr]\notag\\ & \quad =\mathbb{P}[J_{n+1}=j, T_{n+1}\leq t \mid J_{n}=i, T_{n}=s, A_{n}=a+s]=\, \,^{a}Q_{ij}(s;t),\label{due}\end{aligned}$$ where $\sigma(J_{h},T_{h},A_{h},\,h\leq t)$ is the natural filtration of the three-variate process $(J_{h},\break T_{h},A_{h})_{h\in \mathbb{N}}$. Relation (\[due\]) affirms that the knowledge of the values $J_{n}, T_{n}, A_{n}$ is sufficient to give the conditional distribution of the couple $J_{n+1}, T_{n+1}$ whatever the values of the past variables might be. Let us denote by $^{a}p_{\mathit{ij}}(s)$ transition probabilities of the embedded non-homogeneous age indexed Markov chain: $$^{a}p_{ij}(s):=\mathbb{P}[J_{n+1}=j\mid J_{n}=i, T_{n}=s, A_{n}=a+s]=\lim _{t\rightarrow\infty}\,^{a}Q_{ij}(s;t).$$ Furthermore, it is necessary to introduce the probability that the process will remain in the state $i$ up to the time $t$ given the entrance in $i$ at time $s$: $$^{a}\overline{H}_{i}(s;t)=\mathbb{P}[T_{n+1}>t \mid J_{n}=i, T_{n}=s, A_{n}=a+s]=1-\sum _{j\in E}\,^{a}Q_{ij}(s;t).$$ Now it is possible to define the distribution function of the waiting time in each state $i$, given that the state successively occupied is known $$\begin{aligned} ^{a}G_{ij}(s;t):=&\, \mathbb{P}[T_{n+1}\leq t \mid J_{n+1}=j, J_{n}=i, T_{n}=s, A_{n}=a+s] \\[-1pt] =&\,{\left}\{ \begin{matrix} \frac{^{a}Q_{ij}(s;t)}{^{a}p_{ij}(s)} & \textrm{if} \ ^{a}p_{ij}(s) \neq0 ,\\ 1 & \textrm{if} \ ^{a}p_{ij}(s) = 0 . \end{matrix} {\right}. \end{aligned} $$ The main advantage of semi-Markov models as compared to Markovian models is that in a semi-Markovian environment the probability distribution functions $ ^{a}G_{\mathit{ij}}(s;\cdot)$ can be of any type. On the contrary, in a Markovian model they should be geometrically distributed. Since disability data have shown rejection of the geometricity of the waiting time distributions (see, e.g. [@hapi99; @stmasi07; @daguma09; @dadijama11]), semi-Markovian models are more appropriate to describe the dynamics of evolution in time. Let us denote by $^{a}N(t)=\sup\{n\in\mathbb{N}: T_{n}\leq t \mid A_{0}=a\}$ the process counting the number of transitions up to time $t$ and define consequently the age-indexed semi-Markov chain by $$^{a}Z(t)=J_{\,^{a}N(t)}.$$ In the valuation procedure it will be useful to introduce the backward recurrence time process $B(t)=t-T_{\,^{a}N(t)}$. It denotes the time elapsed from the last transition of the system. The relevance of this process in the disability insurance modeling has been described in [@daguma09]. To characterize the probabilistic evolution of the system we introduce the following transition probability function: The age-indexed semi-Markov transition probability function with initial and final backward is the matrix-valued function $$^{a+s-u}\boldsymbol{\varPhi}\bigl(u,s;u',t\bigr)= \bigl( \,^{a+s-u}\phi _{ij}\bigl(u,s;u',t\bigr) \bigr), \quad i,j\in E,\,\, u,s,u',t\in\mathbb{N},$$ whose generic element $^{a+s-u}\phi_{ij}(u,s;u',t)$ expresses the probability $$\label{prob} \mathbb{P}\bigl[\,^{a}Z(t)\!=\!j, B(t)\!= \!u' \bigm\vert \,^{a}Z(s)\!=\!i, B(s)\!=\!u, A_{\,^{a}N(s)}\!= \!a+T_{\,^{a}N(s)}\bigr].$$ In disability insurance the probability (\[prob\]) can be interpreted as the probability that an insured will be at time $t$ in a disability of degree $j$ and duration $u'$ given that at time $s$ she/he was in a disability of degree $i$ and duration $u$ and of age $a+s$. The age-indexed semi-Markov transition probability function with initial and final backward satisfy the following recursive system of equations $$\begin{aligned} ^{a+s-u}\phi_{ij} \bigl(u,s;u',t\bigr)&=1_{\{i=j\}}1_{\{u'=t-s+u\}} \frac {^{a+s-u}\overline{H}_{i}(s-u;t)}{^{a+s-u}\overline{H}_{i}(s-u;s)}\notag \\[-2pt] & \quad +\sum_{k\in E}\sum_{\theta=s+1}^{t-u'} \frac {^{a+s-u}q_{ik}(s-u;\theta)}{^{a+s-u}\overline{H}_{i}(s-u;s)}\,\cdot \,^{a+\theta}\phi_{kj}\bigl(0, \theta;u',t\bigr)\xch{,}{.}\label{numero}\end{aligned}$$ where $$\begin{aligned} ^{a+s}q_{ij}(s;t)&=\mathbb{P}[J_{n+1}=j, T_{n+1}= t \mid J_{n}=i, T_{n}=s, A_{n}=a+s]\notag\\[-1pt] & ={\left}\{ \begin{matrix} ^{a+s}Q_{ij}(s;t)-\,^{a+s}Q_{ij}(s;t-1) & \textrm{if} \ t > s, \\ 0 & \textrm{if} \ \xch{t = s.}{t = s} \end{matrix} {\right}.\label{kernel}\end{aligned}$$ Let us denote by $\mathbb{P}_{(i,s-u,a+s-u)}(\cdot)$ the probability measure $$\mathbb{P}\bigl(\cdot\bigm\vert\,^{a}Z(s)\!=\!i, T_{\,^{a}N(s)}=s-u, A_{\, ^{a}N(s)}\!=\!a+s-u\bigr),$$ and by $\mathbb{P}_{(i,s-u,a+s-u, >s)}(\cdot)$ the probability measure $$\mathbb{P}\bigl(\cdot\bigm\vert\,^{a}Z(s)\!=\!i, T_{\,^{a}N(s)}=s-u, A_{\, ^{a}N(s)}\!=\!a+s-u, T_{\,^{a}N(s)+1}>s\bigr).$$ Observe that the information set $\{^{a}Z(s)\!=\!i, B(s)\!=\!u, A_{\,^{a}N(s)}\!=\!a+T_{\,^{a}N(s)}\}$ is equivalent to $\{^{a}Z(s)\!=\! i, T_{\,^{a}N(s)}=s-u, T_{\,^{a}N(s)+1}>s, A_{\,^{a}N(s)}\!=\!a+s-u\}$, so that the age-indexed semi-Markov transition probability function can be denoted by $$\begin{aligned} ^{a+s-u}\phi_{ij}\bigl(u,s;u',t\bigr)&=\mathbb{P}_{(i,s-u,a+s-u,>s)}\xch{\bigl[\, ^{a}Z(t)\!=\!j, B(t) = u'\bigr]}{\bigl[\, ^{a}Z(t)\!=\!j, B(t) = u'\bigr].}\notag\\ &= \mathbb{P}_{(i,s-u,a+s-u,>s)}\bigl[\,^{a}Z(t)\!=\!j, T_{\,^{a}N(t)}=t-u', T_{\,^{a}N(s)+1}>t\bigr]\notag\\ &\quad {+}\, \mathbb{P}_{(i,s-u,a+s-u,>s)}\bigl[\,^{a}Z(t)\!=\!j,\! T_{\,^{a}N(t)}\,{=}\,t\,{-}\,u', T_{\,^{a}N(s)+1}\,{\leq}\, \xch{t\bigr].}{t\bigr]} \label{trans}\end{aligned}$$ The first summand of (\[trans\]) can be represented as follows: $$\begin{aligned} & \frac{\mathbb{P}_{(i,s-u,a+s-u,>s)}[\,T_{\,^{a}N(s)+1}>t , ^{a}Z(t)\!=\!j, T_{\,^{a}N(t)}=t-u' ]}{\mathbb{P}_{(i,s-u,a+s-u,>s)}[T_{\,^{a}N(s)+1}>s]}\\ & \quad = \frac{1}{\mathbb{P}_{(i,s-u,a+s-u,>s)}[T_{\,^{a}N(s)+1}>s]}\\ & \qquad \cdot \bigl(\mathbb{P}_{(i,s-u,a+s-u,>s)}\bigl[\,T_{\,^{a}N(s)+1}>t , ^{a}Z(t)\!=\!j, T_{\,^{a}N(t)}=t-u'\bigr]\\ & \qquad \cdot\mathbb{P}_{(i,s-u,a+s-u,>s)}\bigl[\,T_{\,^{a}N(t)}=t-u'\bigr]\\ & \qquad \cdot\mathbb{P}\bigl[T_{\,^{a}N(s)+1}>t \bigm\vert \, ^{a}Z(s)\!=\!i, T_{\, ^{a}N(s)}=s-u, A_{\,^{a}N(s)}\!=\!a+s-u\bigr] \bigr)\\ & \quad = \frac{1}{^{a+s-u}\overline{H}_{i}(s-u;s)}\cdot \bigl(1_{\{i=j\}} \cdot1_{\{u'=t-s+u\}}\cdot^{a+s-u}\overline{H}_{i}(s-u;t) \bigr).\end{aligned}$$ The second summand of (\[trans\]) can be represented as follows: $$\begin{aligned} &\frac{\mathbb{P}_{(i,s-u,a+s-u)}[\,^{a}Z(t)=j, T_{\,^{a}N(t)}=t-u', s<T_{\,^{a}N(s)+1}\leq t]}{\mathbb{P}_{(i,s-u,a+s-u,>s)}[T_{\, ^{a}N(s)+1}>s]}\\ & \quad =\frac{1}{^{a+s-u}\overline{H}_{i}(s-u;s)}\sum_{k\in E}\sum_{\theta=s+1}^{t-u'}\mathbb{P}_{(i,s-u,a+s-u)}\bigl[\,^{a}Z(t)=j, T_{\,^{a}N(t)}=t-u',\\ & \quad \quad \quad J_{\,^{a}N(s)+1}=k, T_{\,^{a}N(s)+1}= \theta\bigr]\\ & \quad =\frac{1}{^{a+s-u}\overline{H}_{i}(s-u;s)}\\ & \quad \quad \!\!\cdot\!\! \sum_{k\in E}\sum_{\theta=s+1}^{t-u'}\!\!\!\mathbb {P}_{(i,s-u,a+s-u)}\bigl[\,^{a}Z(t)\!=\!j,\! T_{\,^{a}N(t)}\,{=}\,t\,{-}\,u' \bigm\vert J_{\,^{a}N(s)+1}\!=\!k, T_{\,^{a}N(s)+1}\!=\! \theta\bigr]\\ & \quad \quad \!\!\cdot\mathbb{P}_{(i,s-u,a+s-u)}[\, J_{\,^{a}N(s)+1}=k, T_{\,^{a}N(s)+1}=\theta]\\ & \quad = \sum_{k\in E}\sum_{\theta=s+1}^{t-u'}\frac{^{a+s-u}q_{ik}(s-u;\theta)}{^{a+s-u}\overline{H}_{i}(s-u;s)}\,\cdot \,^{a+\theta}\phi_{kj}\bigl(0,\theta;u',t\bigr).\end{aligned}$$ The last equality is obtained using the assumption (\[due\]) on the Markovianity of the triple $(J_{n}, T_{n}, A_{n})$ with respect to transition times $T_{n}$ and the definition of the age-indexed semi-Markov kernel given in formula (\[kernel\]). The above-presented transition probabilities generalize the corresponding transition probabilities with initial backward derived in [@dajama11] by including the dependence on the final backward. Moreover they generalize the transition probabilities with initial and final backward given in [@daguma09] by including the dependence on the age-index process. In the sequel of the paper we need to consider survival functions for our age-indexed model. To this end we introduce the hitting time of state $D$ (death of the policyholder) given the occupancy of state $i$ at time $s$ with age $a + s$ and duration in the state equal to $u$: $$^{a+s-u}T_{i,D}(u,s) := \inf\bigl\{t>s:\, ^{a}Z(t)=D \bigm\vert \,^{a}Z(s)=i, B(s)=u\bigr\}.$$ The survival function of the age-indexed semi-Markov chain is the vector valued function $^{a+s-u}\mathbf{S}(u,s;t)= (\,^{a+s-u}S_{i}(u,s;t) )$, $i\in E$, $u,s,t\in\mathbb{N}$ with generic element given by: $$^{a+s-u}S_{i}(u,s;t):=\mathbb{P}\bigl[\,^{a}T_{i,D}(u,s)>t \bigr].$$ It denotes the probability to not enter state $D$ in the time interval $(s,t]$ given the occupancy of state $i$ at time $s$ being aged $a+s$ with entrance in this state with last transition $u$ periods before. This function can be calculated using the following relation: $$^{a+s-u}S_{i}(u,s;t)=\sum_{j\neq D}\sum _{u'=0}^{t-s+u}\,^{a+s-u}\phi _{ij}\bigl(u,s;u',t\bigr).$$ It is simple to note that $$\begin{aligned} \mathbb{P}\bigl[\,^{a+s-u}T_{i,D}(u,s)=t \bigr]&=\,^{a+s-u}S_{i}(u,s;t-1)-\, ^{a+s-u}S_{i}(u,s;t)\notag\\ & =:\varDelta^{a+s-u}S_{i}(u,s;t-1).\end{aligned}$$ The conversion option in life insurance {#sec3} ======================================= Let us consider the general situation where a female insured aged $x$ at the initial time $0$ with a health state $i\in E$ buys an $n$-year term insurance policy (TIP). When the policy is almost due, if she is still alive she decides to extend the policy for the rest of her life. The extension can be done by converting the initial TIP into a PIP or buying a new PIP. In \[conversion\] we report a diagram that summarizes the time schedule of a conversion option contract. It should be remarked that at time $n$, the decision to convert the TIP into a PIP or to purchase a new PIP should be taken considering the new health state of the policyholder ($^{a}Z(n)$), the duration in this state ($B(n)$) and the age ($x+n$). ![A conversion option diagram[]{data-label="conversion"}](78f01) The valuation of the conversion option needs the study of two kinds of contracts involved here: the TIP and the PIP contracts. Temporary insurance policy contract ----------------------------------- Term insurance policies provide coverage for a limited time ($n$ years) and gives to the policyholder a benefit in case of death. In this paper without loss of generality we assume that the benefit is set to 1 Euro. The possession of this coverage is subordinated to the payment, by the policyholder, of an yearly premium until the occurrence of the death event or the expiry of the contract whichever occur before. For the TIP contract, let us introduce the random variable (r.v.) [*conditional Present Value of Death Benefit*]{} denoted by $(\mathit{PVDB})_{i,u,x}$. It takes value $\delta^{s}$ when the death of the policyholder occurs at any time $s\leq n$. Given the initial conditions $\{^{a}Z(0)=i, B(0)=u, A(0)=x\}$, the death event may occur at time $s$ with probability $^{x}S_{i}(u,0;s-1)-\, ^{x}S_{i}(u,0;s)$, then it results in $$\begin{aligned} \mathcal{A}_{i,u}(x,0,n):=&\,\mathbb{E} \bigl[(\mathit{PVDB})_{i,u,x}\bigr]=\sum_{s=1}^{n} \mathbb{P}\bigl[^{x}T_{i,D}(u,0)=s\bigr]\cdot1\cdot \delta^{s} \\ =&\,\sum_{s=1}^{n} \varDelta^{x}S_{i}(u,0;s-1) \delta^{s}. \end{aligned} $$ Let us introduce the r.v. [*conditional Present Value of Unitary Premiums*]{} denoted by $(\mathit{PVUP})_{i,u,x}$. Since premiums are paid in the due case, the r.v. $(\mathit{PVUP})_{i,u,x}$ takes value $\sum_{r=0}^{s-1}\delta^{r}$ when the death of the policyholder occurs at time $s\leq n-1$ and value $\sum_{r=0}^{n}\delta^{r}$ if she will survive time $n$. Let us denote by $p_{i,u}(x,0)$ the annual premium $n$-TIP with $1$ Euro payable at the year of death of an insured of age $x$, in health state $i$ obtained $u$ years before. Then the r.v. [*conditional Present Value of Premiums*]{} denoted by $(\mathit{PVP})_{i,u,x}$ is simply defined by $$\begin{aligned} & (\mathit{PVP})_{i,u,x}:=p_{i,u}(x,0)\cdot(\mathit{PVUP})_{i,u,x}, \quad \textrm{for}\, i\neq \xch{D,}{D}\notag\\ & (\mathit{PVP})_{i,u,x} := 0, \quad \textrm{for}\ \xch{i=D,}{i=D.}\end{aligned}$$ and then it results in $$\begin{aligned} \mathcal{P}_{i,u}(x,0,n)&:=\mathbb{E}\bigl[(\mathit{PVP})_{i,u,x}\bigr]\\ &\,=\sum_{s=1}^{n-1} \Biggl(p_{i,u}(x,0)\sum_{r=0}^{s-1}\delta^{r} \Biggr)\varDelta\,^{x}S_{i}(u,0;s-1)\\ &\,\quad + \Biggl(p_{i,u}(x,0)\sum_{r=1}^{n}\delta ^{r} \Biggr)\,^{x}S_{i}(u,0;n).\end{aligned}$$ Furthermore if we assume that premiums are fixed according to the equivalence principle, i.e. in a way such that the actuarial present value of premiums should be equal to the actuarial present value of benefits (see e.g. [@hapi99]), then we have that: =4.5pt =4.5pt $$\mathcal{A}_{i,u}(x,0,n)= \mathcal{P}_{i,u}\xch{(x,0,n),}{(x,0,n)}$$ from which we recover the fair premium $$p_{i,u}(x,0)=\frac{\sum_{s=1}^{n} \varDelta\,^{x}S_{i}(u,0;s-1) \delta ^{s}}{\sum_{s=1}^{n-1}\sum_{r=1}^{s-1}\delta^{r} \varDelta\, ^{x}S_{i}(u,0;s-1)+\sum_{r=1}^{n}\delta^{r}\,^{x}S_{i}(u,0;n)}.$$ Permanent insurance policy -------------------------- Permanent insurance policies provide coverage for an unlimited time horizon and gives to the policyholder a benefit of 1 Euro in case of death. The possession of this coverage is subordinated to the payment, by the policyholder, of an yearly premium until the occurrence of the death event. Relatively to the PIP contract let us introduce the r.v. [*conditional Present Value of Death Benefits*]{} denoted by $(\widetilde {\mathit{PVDB}})_{i,u,x}$. It takes value $\delta^{s}$ when the death of the policyholder occurs at time $s\in\mathbb{N}$. In analogy with the TIP case it results in $$\begin{aligned} \tilde{\mathcal{A}}_{i,u}(x,0)&:=\mathbb{E} \bigl[(\widetilde {\mathit{PVDB}})_{i,u,x}\bigr]=\sum_{s=1}^{\infty} \mathbb {P}\bigl[^{x}T_{i,D}(u,0)=s\bigr]\cdot1\cdot \delta^{s} \\ & \,=\sum_{s=1}^{\infty} \varDelta^{x}S_{i}(u,0;s-1) \delta^{s}. \end{aligned} $$ Let us introduce the r.v. [*conditional Present Value of Unitary Premiums*]{} denoted by $(\widetilde{\mathit{PVUP}})_{i,u,x}$. Premiums are paid until the occurrence of the death of the policyholder, formally the r.v. $(\widetilde{\mathit{PVUP}})_{i,u,x}$ assumes value $\sum_{r=1}^{s-1}\delta^{r}$ when the death of the policyholder occurs at time $s\in\mathbb{N}$. Let us denote by $\tilde{p}_{i,u}(x,0)$ the annual premium for a PIP with $1$ Euro payable at the year of death of an insured of age $x$, in health state $i$ obtained $u$ years before. Then the r.v. [*conditional Present Value of Premiums*]{} denoted by $(\widetilde{\mathit{PVP}})_{i,u,x}$ is simply defined by $$\begin{aligned} & (\widetilde{\mathit{PVP}})_{i,u,x}:=\tilde{p}_{i,u}(x,0)\cdot\xch{(\widetilde {\mathit{PVUP}})_{i,u,x},}{(\widetilde {\mathit{PVUP}})_{i,u,x}}\quad \textrm{for}\ i\neq D,\notag\\ & (\widetilde{\mathit{PVP}})_{i,u,x} := 0, \quad \textrm{for}\ \xch{i=D,}{i=D.}\end{aligned}$$ and then it results in $$\tilde{\mathcal{P}}_{i,u}(x,0):=\mathbb{E}\bigl[(\widetilde {\mathit{PVP}})_{i,u,x}\bigr]=\sum_{s=1}^{\infty} \tilde{p}_{i,u}(x,0)\sum_{r=1}^{s-1} \delta^{r}\varDelta\,^{x}S_{i}(u,0;s-1).$$ Furthermore if we assume that premiums are fixed according to the equivalence principle we have that: $$\tilde{\mathcal{A}}_{i,u}(x,0)= \tilde{\mathcal{P}}_{i,u}(x,0),$$ from which we recover the fair premium $$\tilde{p}_{i,u}(x,0)=\frac{\sum_{s=1}^{\infty}\delta^{s} \varDelta\, ^{x}S_{i}(u,0;s-1)}{\sum_{s=1}^{\infty}\sum_{r=1}^{s-1}\delta ^{r}\varDelta\,^{x}S_{i}(u,0;s-1)}.$$ Valuation of the conversion option ---------------------------------- In this subsection we develop the valuation procedure for conversion options when survival probability functions are derived from a multi-state model of the policyholder’s health. The valuation makes use of the random variables introduced for describing the TIP and PIP contracts and what we called [*exercise set*]{} of the option. The introduction of the exercise set is a prerogative of our model and was not present in earlier studies on conversion options. We remember that the policyholder possesses a TIP issued at time zero with maturity $n$ and at time $n$ should decide to prolong the insurance coverage either by means of converting the TIP into a PIP or purchasing a new PIP. We define the r.v. [*conditional Conversion Gain*]{} as $$(\mathit{CG})_{i,u,x}= \bigl[(\mathit{PVDB})_{i,u,x} \bigr]- \bigl[(\mathit{PVP})_{i,u,x} \bigm\vert\textrm {conversion} \bigr],$$ where $ [(\mathit{PVDB})_{i,u,x} ]$ is the r.v. denoting the present value of death benefits and$ [(\mathit{PVP})_{i,u,x}\mid\textrm {conversion} ]$ is the r.v. describing the present value of premiums when the policyholder possesses an option to convert the original TIP into a PIP before the expiry of the TIP. They are both conditional on the information set $\{^{a}Z(0)=i, B(0)=u,\break A(0)=x\}$ describing the initial health conditions of the policyholder at the inception time zero. The formal definition of the r.v. $ [(\mathit{PVP})_{i,u,x}\mid\textrm{conversion} ]$ is given in Definition \[def4\] below. Similarly it is possible to define the r.v. [*conditional No Conversion Gain*]{} as $$(\mathit{NCG})_{i,u,x}= \bigl[(\mathit{PVDB})_{i,u,x} \bigr]- \bigl[(\mathit{PVP})_{i,u,x} \bigm\vert \textrm{no conversion} \bigr],$$ where $ [(\mathit{PVP})_{i,u,x}\mid\textrm{no conversion} ]$ is the r.v. denoting the present value of premiums when the policyholder does not possess an option to convert the original TIP into a PIP and then must purchase a new PIP at time $n$ if she wants to extend the insurance protection. The formal definition of the r.v. $ [(\mathit{PVP})_{i,u,x}\mid\textrm{no conversion} ]$ is given in Definition \[def3\] below. The difference between the Conversion Gain and the No Conversion Gain define the r.v. [*conditional Net Gain*]{}, i.e.: $$(G)_{i,u,x}=(\mathit{CG})_{i,u,x}-(\mathit{NCG})_{i,u,x},$$ and its expected value is called conditional [*Value of the Conversion Option*]{}, i.e.: $$(\mathit{VCO})_{i,u,x}=\mathbb{E}\bigl[(G)_{i,u,x}\bigr].$$ It is simple to realize that $$\label{vco} (\mathit{VCO})_{i,u,x}=\mathbb{E} \bigl[(\mathit{PVP})_{i,u,x}\bigm\vert \textrm{no conversion} \bigr]-\mathbb{E} \bigl[(\mathit{PVP})_{i,u,x}\bigm\vert\textrm {conversion} \bigr].$$ Therefore, we need to calculate the expectations on the right hand side of  (\[vco\]). To do this we proceed first to the formal definition of the two random variables involved in the computation. This requires the introduction of some auxiliary concepts. Let us consider a time $n\in\mathbb{N}$, then the triple $(i,u,x)$ is called an [*n-scenario*]{} if $^{a}Z(n)=i$, $B(n)=u$, $A_{N(n)}=x-n+u$. We say that the 0-scenario $(i,u,x)$ is state-unchanged at time $n$ if the $n$-scenario will be $(i,u,x+n)$. Two state-unchanged scenarios share the same health state and duration in this state but are characterized by different ages of the policyholder. The conditional [*cash Value*]{} is defined by $$V_{i,u}(x+n,n):=\bigl[\tilde{p}_{i,u}(x+n,n)-p_{i,u}(x+n,0) \bigr]\cdot \widetilde{\mathit{PVUP}}_{i,u,x}\cdot\delta^{n}.$$ The expectation of the cash value is the quantity the policyholder has to pay at the time of conversion to the insurance company: $$\begin{aligned} \mathcal{V}_{i,u}(x+n,n)&:=\mathbb{E}\bigl[V_{i,u}(x+n,n)\bigr]\\ &\,= \bigl[\tilde{p}_{i,u}(x+n,n)-p_{i,u}(x+n,0)\bigr]\cdot\sum_{h=n+1}^{\infty}\delta^{h}\,\varDelta\,^{x+n}S_{i}(u,n;h-1). \end{aligned} $$ The quantity $\mathcal{V}_{i,u}(x+n,n)$ expresses the gain the policyholder expect to realize buying the conversion option under the hypothesis of an unchanged $n$-scenario. This quantity is greater or equal than zero because $$\tilde{p}_{i,u}(x+n,n)\geq p_{i,u}(x+n,0),$$ that is, the premiums for a PIP are greater than the corresponding premium for a TIP given the same $n$-scenario $(i,u,x+n)$. In analogy with the financial options, we can define a set where it is convenient to exercise the conversion option. This is a prerogative of the adopted multi-state model because in the paper [@su10], if the insured person was still alive at the conversion time it was always convenient to prolong the coverage by exercising the option. However, in our more general framework, this is not the case, because given the initial 0-scenario $(i,u,x)$ it is possible after $n$ years that the insured person improves considerably the health state and the prospective expectation of a prolonged life. This has been observed in the evolution of several diseases like HIV infection, see e.g. [@dadijama11].=1 Given the 0-scenario $(i,u,x)$, we define the [*exercise set*]{} as $$\begin{aligned} C_{i,u}(x,n):=& \bigl\{\bigl(j,u'\bigr)\in E\times\mathbb{N} :\notag\\ & \mathbb{E}\bigl[p_{i,u}(x,0)\cdot\widetilde {\mathit{PVUP}}_{i,u,x}+V_{i,u}(x+n,n)\bigr]\leq\tilde{\mathcal {P}}_{j,u'}(x+n,n) \bigr\}.\end{aligned}$$ The set $C_{i,u}(x,n)$ comprehends all couples of health states and durations where it is convenient for the policyholder to exercise the conversion option. Indeed, if the expected payment to face by converting the option $\mathbb{E}[p_{i,u}(x,0)\cdot\widetilde {\mathit{PVUP}}_{i,u,x}+V_{i,u}(x+n,n)]$ is smaller than the expected present value of premiums to be paid for a new PIP in the new $n$-scenario $(j,u',x+n)$ it is convenient to convert the option because with an inferior cost the policyholder guarantees to herself the same insurance protection. Therefore, if $(j,u')\in C_{i,u}(x,n)$ the policyholder will convert the option; on the contrary, if $(j,u')\in C_{i,u}^{c}(x,n)$ the policyholder will not convert the option.=1 Now we are in the position to define the random variables $$\bigl[(\mathit{PVP})_{i,u,x}\bigm\vert\textrm{no conversion}\bigr], \,\,\,\, \bigl[(\mathit{PVP})_{i,u,x}\bigm\vert\textrm{conversion}\bigr].$$ \[def3\] The r.v. $[(\mathit{PVP})_{i,u,x}\mid\textrm{no conversion}]$ is defined by the following relation: $$\bigl[(\mathit{PVP})_{i,u,x}\bigm\vert\textrm{no conversion}\bigr]:=(\mathit{PVP})_{i,u,x}+( \widetilde {\mathit{PVP}})_{\,^{a}Z(n),B(n),A(n)}\cdot\delta^{n}.$$ Then, the conditional present value of premiums given no conversion is equal to the conditional present value of premiums from the TIP contract plus the conditional present value of premiums of the subsequent PIP calculated under the $n$-scenario $(^{a}Z(n),B(n),A(n))$ and discounted at time zero. It is possible to calculate its expectation that is given here below: $$\begin{aligned} \mathbb{E}[\mathit{PVP}\mid\textrm{no conversion}]&=\mathcal {P}_{i,u}(x,0,n) \\ & \quad + \sum_{j\in E} \sum_{u'\geq0} \,^{x}\phi_{ij}\bigl(u,0;u',n\bigr)\cdot \delta^{n}\cdot\tilde{\mathcal{P}}_{j,u'}(x+n,n). \end{aligned} $$ \[def4\] The r.v. $[(\mathit{PVP})_{i,u,x}\mid\textrm{conversion}]$ is defined by the following relation: $$\begin{aligned} \bigl[(\mathit{PVP})_{i,u,x}\bigm\vert\textrm{conversion}\bigr]&:=(\mathit{PVP})_{i,u,x}\notag\\ &\quad +\delta^{n} (\widetilde{\mathit{PVP}})_{\,^{a}Z(n),B(n),A(n)}\cdot1_{\{(^{a}Z(n),B(n))\in C_{i,u}^{c}(x,n)\}}\notag\\ &\quad + \delta^{n} \bigl[\bigl(p_{i,u}(x,0)\widetilde{\mathit{PVUP}}\bigr) + V_{i,u}(x+n,n)\bigr]\notag\\ &\quad \cdot 1_{\{(^{a}Z(n),B(n))\in C_{i,u}(x,n)\}}.\label{pvpconv}\end{aligned}$$ Then, the conditional present value of premiums given the possibility to convert is equal to the conditional present value of premiums from the TIP contract plus the conditional present value of premiums from the PIP calculated under the $n$-scenario $(^{a}Z(n),B(n),A(n))$ and discounted at time zero if this scenario does not belong to the exercise set plus the expected payment to face by converting the option if the $n$-scenario belongs to the exercise set. It is possible to calculate the expectation of (\[pvpconv\]) that is given here below: $$\begin{aligned} \mathbb{E}[\mathit{PVP}\mid\textrm{conversion}]&=\mathcal{P}_{i,u}(x,0,n)\notag\\ & \quad + \sum_{(j,u')\in C_{i,u}^{c}(x,n)}\,^{x}\phi_{ij}\bigl(u,0;u',n\bigr)\cdot \delta^{n}\cdot\tilde{\mathcal{P}}_{j,u'}(x+n,n)\notag\\ & \quad + \sum_{(j,u')\in C_{i,u}(x,n)}\,^{x}\phi_{ij}\bigl(u,0;u',n\bigr)\cdot \delta^{n}\cdot\Biggl[V_{i,u}(x+n,n)\notag\\ & \quad +\sum_{h=n+1}^{\infty}p_{i,u}(x,0)\sum_{r=n+1}^{h}\delta ^{r}\varDelta\,^{x+n}S_{j}\bigl(u',n;h\bigr) \xch{\Biggr].}{\Biggr]}\label{pvpnoconv}\end{aligned}$$ Now we are in the position of computing the value of the conversion option by substituting (\[pvpconv\]) and (\[pvpnoconv\]) in Formula (\[vco\]). Some algebra gives the following representation: $$\begin{aligned} (\mathit{VCO})_{i,u,x}&=\sum_{(j,u')\in C_{i,u}(x,n)}\!\!\!\!\!\!\!\,^{x}\phi _{ij}\bigl(u,0;u',n\bigr)\delta^{n}\cdot \Biggl[\tilde{\mathcal{P}}_{j,u'}(x+n,n)- V_{i,u}(x+n,n)\\ &\quad -\sum_{h=n+1}^{\infty}p_{i,u}(x,0)\sum_{r=n+1}^{h}\delta^{r} \bigl(\,^{x+n}S_{j}\bigl(u',n;h\bigr){-}\,^{x+n}S_{j}\bigl(u',n;h+1\bigr) \bigr) \!\Biggr],\end{aligned}$$ from which we realize that $\mathit{VCO}\geq0$ because on the exercise set $C_{i,u}(x,n)$ the term within square brackets is nonnegative. We would like to remark that the value of the conversion option is nonnegative unless the exercise set is empty. Moreover the value does depend on the dynamics of the health state of the policyholder and therefore, in our model, it is sensitive to the duration of permanence in the health state, to the chronological time and to the age of the policyholder. Conclusions {#sec4} =========== The valuation of conversion options in life insurance is an important subject in modern actuarial mathematics. This study accomplished several goals. First, we proposed a general multistate model that can reproduce important aspects in the modeling of life insurance contracts and we calculated transition probability function for the model. Second, we defined the main variables necessary to the description of the contract and we calculated the value of the conversion option in a very general framework. As particular cases we obtain formulas for the valuation of temporary insurance policy and permanent insurance policy that are embedded in the conversion option contract. This paper leaves several points opened. First of all the application to real data of the model is by far the most urgent task to be accomplished. This task can be accomplished once a reliable dataset is obtained and adequate computer programmes are built. Then, the possibility to extend the results to more complex models is also relevant, in this light a possible extension to subordinated semi-Markov chains is worth mentioning. [18]{} : . , – (). : . , – () , : . , () , : A semi-markov model with memory for price changes. J. Stat. Mech. Theory Exp. **P12009** (2011) , : . , () , : Weighted-indexed semi-markov models for modeling financial returns. J. Stat. Mech. Theory Exp. **P07015** (2011) , , : . , – (). , , : . , – () , , : . , – (). , , : . , – () , , : . , – () , , : . , – () , , , : . (), – (). , , , : . , – () , : . , (). , : . , () , : . , – () , : . , – () , : . , – (). , : . , – () , : . , – (). , : . , – () , : . , – (). , : . , – () , : . , – (). , : . , – () : . , – (). : . , – () : . , – (). : . , – () , , : . , – (). , , : . , – () : . , – (). : . , – () , : . , – () , : . , – ()

--- abstract: 'We study the effect of a domain wall on the electronic transport in ferromagnetic quantum wires. Due to the transverse confinement, conduction channels arise. In the presence of a domain wall, spin up and spin down electrons in these channels become coupled. For very short domain walls or at high longitudinal kinetic energy, this coupling is weak, leads to very few spin flips, and a perturbative treatment is possible. For very long domain wall structures, the spin follows adiabatically the local magnetization orientation, suppressing the effect of the domain wall on the total transmission, but reversing the spin of the electrons. In the intermediate regime, we numerically investigate the spin-dependent transport behavior for different shapes of the domain wall. We find that the knowledge of the precise shape of the domain wall is not crucial for determining the qualitative behavior. For parameters appropriate for experiments, electrons with low longitudinal energy are transmitted adiabatically while the electrons at high longitudinal energy are essentially unaffected by the domain wall. Taking this co-existence of different regimes into account is important for the understanding of recent experiments.' author: - 'Victor A. Gopar' - Dietmar Weinmann - 'Rodolfo A. Jalabert' - 'Robert L. Stamps' title: | Electronic transport through domain walls in ferromagnetic nanowires:\ Co-existence of adiabatic and non-adiabatic spin dynamics --- Introduction ============ A new kind of electronic devices taking advantage of the electron spin have been developed during the last years. The influence of the spin on electronic transport attracts considerable interest since early experiments in multi-layered magnetic structures have shown that the resistance is considerably increased in the case of an anti-parallel magnetization of the layers, as compared to a parallel configuration [@baibich; @binasch]. This is at the base of the so-called Giant Magneto-Resistance (GMR), which is already used in the read-heads of commercial high performance hard-disks. In magnetic configurations that are obtained when one substitutes the non-magnetic spacer layer between the ferromagnets by domain walls [@gregg], the effect of a magnetic domain wall on the electronic transport properties has become a subject of great interest. In particular, the effect of a [*single domain wall*]{} on the resistance of a ferromagnetic nanowire has been measured for electro-deposited cylindrical Co wires down to 35 nm in diameter [@ebels], and thin polycrystalline Co films having a thickness of 42 nm and a width down to 150 nm [@dumpich]. The results of both experiments indicate that, besides negative contributions from the anisotropic magneto-resistance, the domain wall scattering yields a positive contribution to the resistance. The prospect of interesting technological applications of magneto-electronic devices exploiting the spin degree of freedom of the electrons together with its importance from the fundamental point of view has strongly motivated theoretical studies of spin-dependent electronic transport. Many efforts have been made in order to explain the electronic transport, in particular the enhancement of the magneto-resistance in these spin-dependent electronic devices. For example, a Boltzmann equation has been applied to study the resistance of multi-layered magnetic/non-magnetic structures when the spin-diffusion length is larger than the mean free path [@valet]. While the electronic spin is expected to follow adiabatically a very slowly varying magnetization [@stern], the deviations from this adiabatic behavior, which are due to the finite length of the domain wall, lead to a so-called mistracking of the spin, and result in a GMR-like enhancement of the magneto-resistance in multi-domain wall configurations [@gregg]. Including spin-dependent scattering, the mistracking and the resulting magneto-resistance have been calculated for such a system, within the so-called two-band model [@levy], which consists in a simplification of the complicated band structure of a ferromagnetic metal. An outstanding problem in magnetism is a fully consistent description of transport and thermal properties in terms of electronic states calculated from first principles. Despite remarkable progress in the past twenty years, such a description does not yet exist in a form suitable for predicting features such as domain wall structures in non-equilibrium situations. As such, it is reasonable to search for suitably simplified model descriptions that capture the essence of the important physics involved. In the present case, a two-band model is useful for the study of how the geometrical characteristics of a magnetic domain wall affect electron transport. We make a distinction between spatially extended electronic states that contribute strongly to conduction and more localized states that contribute strongly to the formation of local magnetic moments. For the transition metals, this model assumes that exchange correlations between electrons in primarily $d$-like orbitals are largely responsible for the formation of magnetic moments leading to the microscopic magnetization. The $s$-like orbitals contribute much less strongly to the magnetization and instead interact relatively weakly with the local moments via a contact interaction term. These types of $s$-$d$ interaction models have proven very useful in the past for discussions of indirect exchange interactions in magnetic transition metal multilayers. In our work we therefore assume that the domain wall represents a stable magnetic state with an energy above the ferromagnet ground state. The exact shape and dimensions of the wall are determined by exchange correlation energies and spin orbit interactions primarily affecting electrons associated with the magnetization of the wall. These interactions are small perturbations on the conduction electron states, and the interaction between conduction electrons and the domain wall is represented by a simple contact potential. In a single electron picture, the wall appears as a spatially varying spin dependent potential for the conduction electrons. The magnitude of the splitting between the spin up and spin down potentials is taken as a free parameter that is related to the exchange correlation energy of the electrons involved in forming the wall, and the contact potential describing interaction of conduction electrons with the effective potential associated with the wall structure. In the following we refer to this contact potential between conduction electrons and the magnetization as an ’exchange interaction’ although it is quite distinct from the exchange interaction used to parameterize the interactions leading to magnetic ordering. In thin ballistic quantum wires and narrow constrictions or point contacts, the lateral confinement of the electronic wave-functions leads to the emergence of quantized transport channels. As a consequence, the conductance is quantized and exhibits steps of $e^2/h$ as a function of the Fermi energy [@wees; @wharam]. Nakanishi and Nakamura [@nakanishi] considered the conductance of very narrow quantum wires including the effect of a domain wall. A perturbative approach allowed them to study the effect of a very short domain wall on the conductance steps. Imamura and collaborators [@imamura] calculated the conductance of a point contact connecting two regions of a ferromagnet having parallel or anti-parallel magnetization directions. Within an $s-d$ two-band model, they numerically obtained a non-monotonic dependence of the domain wall contribution to the resistance on the width of the point contact. There have been attempts to compare different approaches to the calculation of the domain wall magneto-resistance (DWMR), which point towards the importance of including more realistic band structures. In the ballistic case, van Hoof and collaborators [@hoof] calculated the DWMR for an adiabatic model where the magnetization direction changes very slowly along the wire, using an extension of the standard band structure calculation to include an infinite spin spiral, as well as a “linear” model, where the magnetization turns in a finite region at a constant rate. These two approaches yield a much larger effect than a two-band model, where corrections with respect to an infinitely long domain wall are calculated. On the other hand, first principle calculations for the case of abrupt magnetization interfaces yield a DWMR which is orders of magnitude larger than for the other models that take into account realistic domain wall lengths. It then seems necessary to develop more accurate treatments within the two-band model in order to understand the crossover from the abrupt domain wall situation to the adiabatic regime. Since we are working with nanowires, obvious transverse quantization effects appear, which are more easily tractable within a two-band model. Moreover, the two-band model allows to easily obtain the transmission coefficients with and without spin flip, making it possible to study the mistracking effect in finite length domain walls and its GMR-like consequences. Finally, given the typical experimental parameters, it would be important to go beyond the ballistic limit. Taking into account disorder within a two-band model seems much more doable than in the framework of a band structure calculation. In this paper, we study the effect of a domain wall on the conductance of a nanowire within the two-band model, comparing different shapes and sizes of the domain wall. Focusing on the contribution of the spin-dependent scattering of the domain wall, we do not consider material-dependent contributions to the resistance like the anisotropic magneto-resistance. After presenting our model in section \[sec:model\], we consider the perturbative regime of weak spin coupling induced by the domain wall in section \[sec:weak\_coupling\]. From a comparison of different domain wall shapes, we shall extract the relevant parameters governing the transmission through the domain wall with and without spin-flip processes. While this applies to short domain walls, in section \[sec:strong\_coupling\] we study the transport in the general case of a strong spin coupling induced by the domain wall, which allows to treat domain walls of arbitrary length. We present the deviations from the full transmission with spin rotation following the local magnetic structure expected for infinitely long domain walls, which is due to finite domain wall length, and treat the most interesting crossover regime. This case is relevant since typical experiments [@ebels; @dumpich] are far from the thin wall regime, but not really in the adiabatic limit in which the length of the domain walls is large. Model {#sec:model} ===== We consider a wire along the $z$ axis with a domain wall, and choose the origin ($z=0$) in the middle of the wall (see Fig. \[fig:wallsketch\]). As explained above, we work within the two-band model, where the $d$ electrons are responsible for the magnetization and the current is carried by the $s$ electrons. Therefore, we write for the latter an effective Hamiltonian $$\label{eq:H} H=-\frac{\hbar^2}{2m}\nabla^2 +\frac{\Delta}{2} \vec{f}(\vec{r})\cdot\vec{\sigma}\, ,$$ where $\Delta$ is the spin splitting of the $s$ electrons due to the exchange coupling with the $d$ electrons and $\vec{\sigma}$ is the vector of the Pauli matrices. The unit vector $\vec{f}$ represents the direction of the local magnetization. Its functional dependence describes the shape of the domain wall. The lateral confinement present in a nanowire may have a considerable influence on this shape, leading to domain walls which are altered as compared to the case of bulk domain walls [@dumpich; @bruno; @prejbeanu1; @prejbeanu2; @hausmanns]. In addition, a spin-polarized current through the domain wall creates a torque which can alter its shape [@waintal]. Working in the linear response regime of low current, we do not need to take into account this back-action of the conduction electrons on the magnetic structure. ![\[fig:wallsketch\]Sketch of a ferromagnetic quantum wire containing a domain wall (grey region), for the example geometry of a square cross-section. The arrows indicate the magnetization directions far from the domain wall, for the case of a Néel wall.](fig1-GWJS.eps){width="\columnwidth"} Assuming that the magnetization only depends on $z$, we will therefore be interested in comparing different functional forms of the kind $\vec{f}=\{f_x(z), 0, f_z(z)\}$. This choice does not imply a loss of generality and corresponds to a so-called “Néel wall”, where the magnetization is parallel to the wire axis in the leads far from the domain wall (arrows in Fig. \[fig:wallsketch\]), and turns inside the $x$–$z$ plane parallel to this wire axis when going through the wall. The assumption $\vec{f}(\vec{r})=\vec{f}(z)$ allows us to separate the transverse and longitudinal parts of the Hamiltonian (\[eq:H\]). The transverse quantization gives rise to transport channels with quantum numbers $n_1$ and $n_2$, and an energy $E_{n_1,n_2}$. The density of channels $\rho=2\pi m A/\hbar^2$ (where $A$ is the cross-section of the transverse area of the wire) is equal to that of a two-dimensional system, and thus independent of the energy and the shape of the cross-section. For the example of a wire with a square cross-section of side $w$, we have $$E_{n_x,n_y}=\frac{\hbar^2}{2m}\left(\left(\frac{\pi n_x}{w}\right)^2+\left(\frac{\pi n_y}{w}\right)^2\right)\, .$$ Far away from the domain wall, the orbital parts of the eigenstates are products of transverse channels and longitudinal plane waves. Since $\lim_{z \to \pm\infty}f_z(z)= \pm 1$, the associated eigenenergies for spin up are $$E_\uparrow = E_{n_1,n_2}+\frac{\hbar^2 k_z^2}{2m}\pm \frac{\Delta}{2}\, ,$$ while for spin down we have $$E_\downarrow = E_{n_1,n_2}+\frac{\hbar^2 k_z^2}{2m}\mp \frac{\Delta}{2}\, .$$ The domain wall leads to the scattering of these states and the conductance (in units of $e^2/h$) is given by the Landauer formula $$\label{eq:landauer} g=\sum_{n_1,n_2} \sum_{\sigma,\sigma'} T_{n_1,n_2}^{\sigma,\sigma'}(E_{\textrm{F}})\, ,$$ where the sum is done over the occupied channels. $T_{n_1,n_2}^{\sigma,\sigma'}$ is the transmission coefficient in the channel $(n_1,n_2)$, for scattering of electrons with spin $\sigma$ into spin $\sigma'$. Such a coefficient only depends on the longitudinal energy $$\epsilon=E_{\textrm{F}}-E_{n_1,n_2}$$ as $$T_{n_1,n_2}^{\sigma,\sigma'}(E_{\textrm{F}})=T^{\sigma,\sigma'}(\epsilon)\, .$$ Therefore, for each channel $(n_1,n_2)$, we are left with an effective one-dimensional problem at energy $\epsilon$. In order to determine the transmission probability $T^{\sigma,\sigma'}(\epsilon)$, we write the spinor wave-function in the up-down basis (with fixed, $z$-independent spin orientations) as $$|\psi(z)\rangle= \phi_\uparrow(z)|z,\uparrow\rangle+\phi_\downarrow(z)|z,\downarrow\rangle \, ,$$ where $|z,\uparrow\rangle$ has to be interpreted as the tensor product of the position eigenvector $|z\rangle$ and the spin up state $|\uparrow\rangle$. The Schrödinger equation with the Hamiltonian (\[eq:H\]) leads to a system of coupled differential equations for the components $\phi_\uparrow(z)$ and $\phi_\downarrow(z)$: \[eq:cde\] $$\label{eq:df1} \frac{d^2}{dz^2}\phi_\uparrow+ \frac{2m}{\hbar^2}\left(\epsilon-\frac{\Delta}{2} f_z \right)\phi_\uparrow =\frac{2m}{\hbar^2}\frac{\Delta}{2} f_x \phi_\downarrow$$ $$\label{eq:df2} \frac{d^2}{dz^2}\phi_\downarrow+ \frac{2m}{\hbar^2}\left(\epsilon+\frac{\Delta}{2} f_z \right)\phi_\downarrow =\frac{2m}{\hbar^2}\frac{\Delta}{2} f_x \phi_\uparrow \, .$$ While the term containing $f_z$ plays the role of a spin-dependent potential, the transverse component $f_x$ of the wall profile is responsible for the coupling between the spinor components $\phi_\uparrow$ and $\phi_\downarrow$. The scattering solutions of Eq. (\[eq:cde\]) are then needed to calculate the transmission coefficients, and therewith the conductance through the domain wall. For the extreme case of an abrupt domain wall, when $f_z$ has a jump from $-1$ to $1$, the right-hand-side of Eqs. (\[eq:cde\]) vanishes, and spin up and spin down electrons remain uncoupled. The only effect of the discontinuity of $f_z$ is a spin-dependent potential step of height $\pm\Delta$ for spin up/down electrons. Incoming spin up electrons having longitudinal energy $\epsilon<\Delta/2$ cannot overcome this step and are reflected with probability one. Since the density of conduction channels is independent of the energy for wires having a two-dimensional cross-section, this mechanism blocks a fraction $\Delta/2E_{\rm F}$ of the conduction channels [@wsj_moriond], all of which exhibit perfect transmission in the absence of the domain wall. If one neglects the effect of the potential step on electrons having higher longitudinal energy ($\epsilon>\Delta/2$), this channel blocking mechanism leads to a relative change in conductance $$\frac{\delta g}{g}=-\frac{\Delta}{2E_{\rm F}}\, ,$$ due to the presence of the domain wall. Taking into account the spin conserving reflections for $\epsilon > \Delta/2$ leads [@falloon] (in the limit $E_{\rm F}\gg \Delta$) to an increase of the effect by a factor $4/3$. The precise shape of the actual domain wall present in an experimental measurement (which is very difficult to know) would in principle be needed to determine the scattering states. In addition, it is not possible to find an analytical solution of Eqs. (\[eq:cde\]) for arbitrary domain walls. This is why we introduce various models of a domain wall, and approximate analytical, as well as numerical calculations. In the bulk, when the magnetization always remains parallel to the domain wall, we have the so-called Bloch walls, whose shape was originally calculated by minimizing the total free energy in the thermodynamic limit [@landau]. In this case, $\vec{f}(\vec{r})$ is given by $$\label{eq:bloch_wall} \vec{f}(z)=\left\{ \tanh\left(\frac{z}{\lambda}\right), \mathrm{sech}\left(\frac{z}{\lambda}\right), 0\right\} \, ,$$ where $\lambda$ is the length scale of the domain wall. The lateral confinement present in a nanowire will certainly alter the previous functional form of $\vec{f}$. Moreover, an easy magnetization axis in the direction of the nanowire will result in a Néel wall, modified by the transverse confinement. Such effects have been recently discussed in the literature [@prejbeanu1; @prejbeanu2; @hausmanns]. The variety of possible domain wall structures motivates us to consider different domain wall profiles: linear, trigonometric and extended (defined below), in order to determine the influence of the domain wall shape on the conductance of the wire. As we will see below, while most of the effects are not very sensitive to the details of the wall, the signature of the particular domain wall appears in some regimes. A possible starting point is to assume that the Néel-like confined domain wall has components with the same functional form as in Eq. (\[eq:bloch\_wall\]). In this case we will consider the “extended” domain wall defined by the magnetization direction $$\label{eq:extended_wall} \vec{f}^{(\mathrm{ex})}(z)= \left\{ {\rm sech}\left(\frac{z}{\lambda}\right), 0 , \tanh\left(\frac{z}{\lambda}\right) \right\} \, .$$ As compared with the situation of a Bloch wall, this leads to a permutation of the spatial variables in Eqs. (\[eq:cde\]), and does not change the results for the transmission coefficients. This is why in the limit of high electron energy we can compare our results with the ones of Cabrera and Falicov [@cabrera], who considered Bloch domain walls. In the case of weak coupling between the spin up and down states described by Eq. (\[eq:cde\]) and short domain walls it is reasonable to approximate $f_z$ in the wall profile with a linear function of position $$\label{eq:linear_wall} \vec f^{(\mathrm{lin})}(z)=\left\{ \begin{array}{cl} \left\{\sqrt{1-(z/\lambda)^2},\;0,\; z/\lambda \right\},& {\rm for}\; |z|< \lambda \\ \left\{\;0 \; ,\;\; 0\;\; , \;{\rm sgn}(z)\; \right\}, &{\rm for}\; |z| \ge \lambda . \end{array}\right.$$ The semi-circle form of the coupling term $f_x$ in this “linear” wall ensures that $|\vec{f}(z)|^2=1$. This is an important difference as compared to our previous work [@wsj_moriond], where the conservation of the absolute value of the magnetization was not respected. The above constraint has only quantitative consequences in the short wall limit which play a role when comparing different wall profiles, but becomes crucial for longer domain walls in the adiabatic regime. For an arbitrary domain wall, the extension of the standard recursive Green function method [@lee; @pastawski] to take into account the spin degree of freedom (in a tight binding setup) allows us to calculate the transmission and reflection coefficients $T_{\uparrow\uparrow}$, $T_{\uparrow\downarrow}$, $R_{\uparrow\uparrow}$ and $R_{\uparrow\downarrow}$. The case of a “trigonometric” domain wall $$\label{eq:trigonometric_wall} \vec f^{(\mathrm{tri})}(z)=\left\{ \begin{array}{cl} \left\{\cos\frac{\pi z}{2 \lambda},\;0,\; \sin\frac{\pi z}{2 \lambda}\right\},& {\rm for}\; |z|< \lambda \\ \left\{\;0 \; ,\;\; 0\;\; , \;{\rm sgn}(z)\; \right\}, &{\rm for}\; |z| \ge \lambda \end{array}\right.$$ admits an exact solution for the wave-function inside the domain wall [@brataas] which has recently been used to calculate the torque that is due to a spin-polarized current [@waintal]. We use the exact solution to determine the scattering properties of the domain wall. Details are presented in Appendix \[sec:exact\_trigo\]. By comparing with this exact analytic solution we checked the accuracy of the numerical method, as well as the absence of lattice effects for the parameters that we work with. Weak coupling {#sec:weak_coupling} ============= The problem can be treated at the analytical level when the differential equations (\[eq:cde\]) are only weakly coupled. This is the case when the domain wall (in which the spin-flip terms $f_x$ are non-zero) is very short, or when the longitudinal energy of the electron is very high such that the transmitted electrons spend only a short time inside the domain wall region. Then, we treat the system of coupled differential equations (\[eq:cde\]) iteratively, considering the spin-flip terms on the right-hand-side as a perturbation. Within this approach, the solution to the homogeneous differential equation (\[eq:df1\]) (in which the spin-flip terms induced by $f_x$ are set to zero) is injected in the spin-flip term of the second differential equation (\[eq:df2\]). This method was used in Ref. \[\], where a mechanism of channel blocking by a domain wall was proposed as a source of resistance in short ferromagnetic quantum wires. The starting point of this approach, which we present here for the example of a linear domain wall as described by (\[eq:linear\_wall\]), is an incoming majority (spin-up) electron from the left $\phi_\uparrow^\mathrm{H}$, and $\phi_\downarrow^\mathrm{H}=0$. Outside the domain wall region, $\phi_\uparrow^\mathrm{H}$ reads \[eq:upoutside\] $$\begin{aligned} \phi_{\uparrow}^\mathrm{H}(z)=&e^{ikz}+r_{\uparrow\uparrow}e^{-ikz}&\quad \textrm{for}\quad z<-\lambda\\ \phi_{\uparrow}^\mathrm{H}(z)=& t_{\uparrow\uparrow}e^{ik'z}&\quad \textrm{for}\quad z>\lambda \, ,\end{aligned}$$ with the wave-numbers $$\begin{aligned} k&=&\sqrt{\frac{2m}{\hbar^2}\left(\epsilon+\frac{\Delta}{2}\right)}\\ k'&=&\sqrt{\frac{2m}{\hbar^2}\left(\epsilon-\frac{\Delta}{2}\right)}\, .\end{aligned}$$ For $-\lambda<z<\lambda$, the homogeneous solution of Eq. (\[eq:df1\]) is $$\begin{aligned} \label{eq:phiH} \phi_{\uparrow}^\mathrm{H}(z) &=& \alpha \, \mathrm{Ai}\left[p^{2/3}\left(-\frac{2\epsilon}{\Delta} + \frac{z}{\lambda} \right)\right]\nonumber \\ &+& \beta \, \mathrm{Bi}\left[p^{2/3}\left(-\frac{2\epsilon}{\Delta} + \frac{z}{\lambda} \right)\right] \end{aligned}$$ with the usual Airy functions Ai and Bi, and the dimensionless parameter $p$ defined by $$\label{p} p=\left( \frac{m}{\hbar^2}\Delta \right)^{1/2} \lambda = \left( \frac{\Delta}{2E_{\textrm{F}}}\right)^{1/2} k_{\textrm{F}}\lambda \, .$$ The coefficients $\alpha$ and $\beta$, as well as $t_{\uparrow\uparrow}$ and $r_{\uparrow\uparrow}$, are obtained from the matching of the expressions (\[eq:upoutside\]) and (\[eq:phiH\]) at $z=\pm \lambda$. For $\epsilon < \Delta/2$, we have imaginary $k'$. The transmission without spin-flip is zero and $|r_{\uparrow\uparrow}|=1$. For $\epsilon > \Delta/2$, the transmission without spin-flip is finite. The first-order correction from the spin-conserving scattering is then obtained by injecting $\phi_\uparrow^\mathrm{H}$ as the inhomogeneous term in the differential equation (\[eq:df2\]) for $\phi_\downarrow$. Since we do not have incoming spin-down electrons, outside the domain wall region we take the outgoing plane waves \[eq:downoutside\] $$\begin{aligned} \phi_{\downarrow}^{(1)}(z)=&r_{\uparrow\downarrow}e^{-ik'z}&\quad \textrm{for}\quad z<-\lambda\\ \phi_{\downarrow}^{(1)}(z)=&t_{\uparrow\downarrow}e^{ikz}&\quad \textrm{for}\quad z>\lambda \, .\end{aligned}$$ For $-\lambda<z<\lambda$ the general solution can be written as a linear combination of Airy functions plus a particular solution $\phi^\textrm{p}(z)$ $$\begin{aligned} \label{eq:phi1} \phi_{\downarrow}^{(1)}(z) &=& \alpha_1 \, \mathrm{Ai}\left[-p^{2/3}\left(\frac{2\epsilon}{\Delta} +\frac{z}{\lambda}\right)\right] \nonumber \\ &+& \beta_1 \, \mathrm{Bi}\left[-p^{2/3}\left(\frac{2\epsilon}{\Delta} +\frac{z}{\lambda}\right)\right] + \phi^\textrm{p}(z) \, .\end{aligned}$$ Spin-flip processes are now included in the description, and electrons undergoing a spin-flip can be transmitted even for energies $\epsilon<\Delta/2$. The parameters $\alpha_1$ and $\beta_1$ and the coefficients $t_{\uparrow\downarrow}$ and $r_{\uparrow\downarrow}$ are determined from the matching of (\[eq:downoutside\]) and (\[eq:phi1\]). Long wavelength limit {#sec:longwl} ---------------------  Perturbative results for $T_{\uparrow\downarrow}$ from Eqs. (\[eq:tupdown\_below\]) and (\[eq:tupdown\_above\]) (solid lines) for three values of $p$, compared with the corresponding full numerical results (circles, squares and diamonds) for the linear domain wall. (b) The total transmission $T$ and $T_{\uparrow\uparrow}$ (diamonds and triangles, respectively) for $p=0.20$.](fig2a-GWJS.eps "fig:"){width="\columnwidth"}\  Perturbative results for $T_{\uparrow\downarrow}$ from Eqs. (\[eq:tupdown\_below\]) and (\[eq:tupdown\_above\]) (solid lines) for three values of $p$, compared with the corresponding full numerical results (circles, squares and diamonds) for the linear domain wall. (b) The total transmission $T$ and $T_{\uparrow\uparrow}$ (diamonds and triangles, respectively) for $p=0.20$.](fig2b-GWJS.eps "fig:"){width="\columnwidth"} In the limit of a short wall, when the wavelength of the incoming electron is much longer than the domain wall, the linear approximation of the Airy functions allows us to write for $\epsilon > \Delta/2$ $$\begin{aligned} T_{\uparrow \uparrow}&=& |t_{\uparrow \uparrow}|^2=\frac{4k k'}{(k + k')^2 + 4(\lambda k k')^2}\\ R_{\uparrow \uparrow}&=&|r_{\uparrow \uparrow}|^2= \frac{(k-k')^2+4(\lambda k k')^2}{(k + k')^2 +4(\lambda k k')^2} \, .\end{aligned}$$ Obviously, for $\lambda\to 0$ we recover the well-known results for a step potential [@merzbacher]. For $\epsilon < \Delta/2$, the transmission probability with spin-flip is given by $$\label{eq:tupdown_below} T_{\uparrow\downarrow}=|t_{\uparrow\downarrow}|^2 =C^2 p^2\left( 1+\frac{2\epsilon}{\Delta} \right) \, .$$ For $\epsilon > \Delta/2$, $$\label{eq:tupdown_above} T_{\uparrow\downarrow}=4C^2p^4\frac{4(x^2-1)+ p^2(x+1)(\frac{2}{3}(x-1)-1/p^2)^2}{\left((\sqrt{x+1}+\sqrt{x-1})^2+ 4p^2(x^2-1)\right)^2}$$ is obtained, with $x=2\epsilon/\Delta$ and the prefactor $C$ defined by $$C=\frac{1}{\lambda}\int_{-\infty}^{\infty}\textrm{d}z\, f_x(z)\, .$$ In Fig. \[fig:perturbative\] (a) we show $T_{\uparrow\downarrow}$ from Eqs. (\[eq:tupdown\_below\]) and (\[eq:tupdown\_above\]) as a function of $2\epsilon/\Delta$, together with numerical calculations for three different values of $p$. We can see an excellent agreement for the smallest values of $p$. When the value of $p$ is increased, deviations appear first at energies close to $\Delta/2$. These features are consistent with the fact that the linear approximation of the Airy functions is justified in the small $p$ limit, and becomes increasingly better for small energies. The total transmission $T=T_{\uparrow\downarrow}+T_{\uparrow\uparrow}$ (Fig. \[fig:perturbative\] (b)) is dominated by the large transmission without spin-flip for $\epsilon>\Delta/2$. This feature justifies the “channel blocking” picture proposed for short domain walls in Ref. \[\], where the presence of the wall suppresses almost completely the transmission at energies $\epsilon<\Delta/2$ ($T_{\uparrow\downarrow}$ is only a small correction for $\epsilon<\Delta/2$ and negligible for $\epsilon\gg \Delta/2$). Neglecting what happens for $\epsilon > \Delta/2$, we have $\delta g/g=(-1+(Cp)^2)\Delta/2E_{\rm F}$. ![\[fig:universality\]The transmission $T_{\uparrow\downarrow}$ divided by the coupling strength $(Cp)^2$ for the linear, trigonometric and extended domain walls (solid line, up and down triangles, respectively). The inset shows the results for $T_{\uparrow\downarrow}$ for the same value of $p=0.09$, before dividing by the corresponding value of $(Cp)^2$.](fig3-GWJS.eps){width="\columnwidth"} In the limit of short domain walls, we have found that the dependence of the transmission coefficients on the shape of the wall is only through the integral over $f_x$ which enters in the prefactor $C$. For the linear, trigonometric and extended domain walls, $C$ takes the values $\pi/2$, $4/\pi$ and $\pi$, respectively. Such a scaling is shown in Fig. \[fig:universality\], where the transmissions $T_{\uparrow\downarrow}$ divided by the coupling strength $(Cp)^2$ for the different domain wall shapes coincide for all energies, except those close to $\Delta/2$. ![\[fig:oscillations\]The coefficients $T_{\uparrow\downarrow}$ and $R_{\uparrow\uparrow}$ as a function of $Cp$ for the three different domain wall shapes, at $\epsilon=0.95\Delta/2$. An oscillatory behavior for the linear and trigonometric walls is found as a consequence of the edges at the connection to the leads.](fig4-GWJS.eps){width="\columnwidth"} Short wavelength limit ---------------------- The perturbative approach is not only applicable for short walls and low energies (as in section \[sec:longwl\]), but for general domain wall parameters as well, provided that $\epsilon\gg (\Delta\lambda/\hbar)^2/m$. That is, when the time that the electron spends inside the domain wall is much shorter than the spin precession period, and therefore the spin-flips are very unlikely. In this limit, the WKB approximation of the scattering wave-functions for the linear domain wall model (Eq. (\[eq:linear\_wall\])) yields the reflection and transmission coefficients $$\begin{aligned} R_{\uparrow\uparrow}&=& \left(\frac{\Delta}{4\epsilon}\right)^2 \frac{\sin^2(2k\lambda)}{(2k\lambda)^2}\\ R_{\uparrow\downarrow}&=& \left(\frac{C\Delta}{8\epsilon}\right)^2 \sin^2(2k\lambda)\\ T_{\uparrow\downarrow}&=& \left(\frac{C\Delta}{8\epsilon}\right)^2 (2k\lambda)^2\\ T_{\uparrow\uparrow}&=& 1-R_{\uparrow\uparrow}-R_{\uparrow\downarrow}- T_{\uparrow\downarrow} \, .\end{aligned}$$ Thus, for energies $\epsilon\gg\Delta$ all scattering coefficients, except the transmission without spin-flip, are very small. Therefore, in first approximation we can neglect the effect of the domain wall for electrons with high longitudinal energies. The conductance associated with the domain wall is then determined by the low-energy electrons [@wsj_moriond]. The algebraic decay in $\Delta/\epsilon$ is less pronounced than the exponential suppression obtained by Cabrera and Falicov [@cabrera]. Such a difference arises from the sharp edges at $z=\pm\lambda$ in the linear domain wall model we used for this calculation. Strong coupling {#sec:strong_coupling} =============== ![\[fig:intermediate\]Transmission and reflection coefficients in the intermediate regime ($p=1$) for the extended domain wall geometry (\[eq:extended\_wall\]). A large transmission with spin-flip (diamonds) is found for energies $\epsilon<\Delta/2$ in this regime, where the transport is adiabatic.](fig5-GWJS.eps){width="\columnwidth"} If we are interested in energies $\epsilon\simeq\Delta/2$ and not necessarily short walls, the previous picture has to be modified. The linear approximation and the perturbative treatment (involving only one spin-flip) in the wall region are no longer justified. Beyond the perturbative regime, the detailed shape of the domain wall might become relevant. In Fig. \[fig:oscillations\], we present $T_{\uparrow\downarrow}$ and $R_{\uparrow\uparrow}$ as a function of the coupling strength for different domain wall shapes and an energy of the order of $\Delta/2$. As discussed in the previous section, it is for these energies that a dependence on the detailed shape of the domain wall appears first when departing from the weak coupling limit ($Cp\ll 1$). We can see from Fig. \[fig:oscillations\] that the transmissions (reflections) for the different domain wall shapes coincide for small values of $Cp$. Even for stronger couplings, the different models do not show very important differences in their behaviors. The only apparent difference are oscillations of the transmission coefficients as a function of $p$, which occur at intermediate $p$ for linear and trigonometric domain walls. The origin of these oscillations is due to the edges of the domain wall region leading to Fabry-Perot like interferences. For a smooth domain wall structure such as the extended domain wall, the oscillations are absent. On the other hand, and as expected, $T_{\uparrow\downarrow}\to 1$ for all shapes in the limit of large $p$. It is the limit of infinite domain wall length where the spin follows adiabatically the orientation of the local magnetization [@stern], corresponding to a rotation from spin up to spin down in the external basis of fixed spin orientations. Electrons are transmitted with probability one through the wall, therefore $T_{\uparrow\downarrow}=1$ and $T_{\uparrow\uparrow}=R_{\uparrow\uparrow}=R_{\uparrow\downarrow}=0$. In this limit the detailed shape of a domain wall, having slow spatial spin rotation, is irrelevant. The adjustment of the spin to the direction of the local magnetization requires an infinite number of spin-flips (in the fixed basis), and obviously cannot be described by taking into account a small number of spin-flips as in the perturbative approach used for short domain walls. The condition for the local adjustment is that the Larmor precession of the spin around the local magnetization is much faster than the rotation of the local magnetization viewed by the traveling electron [@stern]. This condition of adiabaticity translates into $\Delta\gg(h/\lambda)\sqrt{\epsilon/m}$. ![\[fig:integrated\] The relative change in conductance $\delta g/g$ caused by the presence of a domain wall (extended shape), for $E_{\rm F}=2\Delta$.](fig6-GWJS.eps){width="\columnwidth"} We then see that the adiabatic condition strongly depends on the longitudinal kinetic energy of the electrons. In a quantum wire, at a finite value of $\lambda$, electrons with low longitudinal velocity are essentially adiabatic, while the channels with low transverse quantum numbers can be highly non-adiabatic. In calculating the conductance of a ferromagnetic quantum wire, we have to take into account the co-existence of adiabatic (low longitudinal energy) and non-adiabatic (high longitudinal energy) electrons. It then seems important to work out the crossover between the short wall and adiabatic limits, for different shapes of the domain wall. For an intermediate value of $p$ in the case of an extended domain wall, Fig. \[fig:intermediate\] shows that the behavior for $\epsilon<\Delta/2$ is radically different from the weak coupling case of section \[sec:weak\_coupling\]. The weak coupling result of Eq. (\[eq:tupdown\_below\]) is only valid at extremely low energies, and the $T_{\uparrow\downarrow}$ approaches one (adiabatic behavior) for longitudinal energies considerably lower than the step height. Above $\Delta/2$, $T_{\uparrow\downarrow}$ decreases monotonously with energy, returning to the weak coupling regime in the limit of large $\epsilon$. At the same time, $T_{\uparrow\uparrow}$ increases towards one and $T$ remains very close to perfect transmission for all energies. Thus, almost all of the electrons with energy $\epsilon > \Delta/2$ are transmitted. However, while the spin of the transmitted electrons is changed by the domain wall for low $\epsilon$ and high $p$, the spin of electrons having high $\epsilon$ in domain walls of low $p$ remains unaffected by the wall (see also Fig. \[fig:perturbative\] (a)). Therefore, in calculating the effect of the domain wall on the quantum conductance the modes with longitudinal energies in the interval $(-\Delta/2,\Delta/2)$ are most relevant [@wsj_moriond]. The conductance for the ideal ballistic case given in Eq.(\[eq:landauer\]) is obtained by summing over all conductance channels. Fig. \[fig:integrated\] shows an example of the resulting behavior for the difference in conductance between the cases without and with domain wall (normalized to the conductance without domain wall), as a function of the domain wall parameter $p$. We can see that the channel blocking effect due to the presence of the domain wall is rapidly suppressed upon increase of the coupling. Similar results have recently been obtained using a different numerical approach [@falloon]. Summary and Conclusions ======================= The effect of a single domain wall on the electronic transport in a ferromagnetic nanowire has been studied systematically in various parameter regimes. The domain wall leads to a coupling of spin up and spin down electrons in the conduction channels, which is proportional to the exchange energy of the conduction electrons and the length of the wall. For an abrupt domain wall the step in the effective potential felt by the conduction electrons blocks the transmission of channels with low longitudinal energy. In the weak coupling limit, a perturbative approach is possible, leading to the lowest order correction to perfect channel blocking. In this case, the detailed shape of the domain wall is not relevant, and the transmission coefficients scale with the coupling strength. For a very long domain wall the spin of the electrons follows adiabatically the local effective magnetization and the conductance is unaffected by the domain wall, independently of its shape. The intermediate coupling regime is most relevant for the domain walls that can be investigated experimentally. We have shown that the degree of adiabaticity of electrons at the Fermi energy strongly depends on their longitudinal kinetic energy. While the spin of electrons with low longitudinal energy essentially behaves adiabatically, the spin of electrons with high longitudinal energy is practically unaffected by the domain wall. The crossover between these two behaviors, as a function of the longitudinal energy of the electrons, has to be taken into account in calculating the conductance of the quantum wire. Our analysis has been based on coherent scattering at the domain wall, which is connected to perfect leads. However, in realistic situations, the domain wall is not connected to scattering-free regions. The imperfections and impurities at both sides of the wall give rise to elastic scattering, which may be different for the two spin directions of the electrons. Though these coherent effects can in principle be taken into account in a scattering approach, such a coherent picture is not sufficient for wires which are longer than the phase coherence length. Since this is the case in typical experiments, we need in addition to take into account inelastic processes (like electron-phonon or spin-magnon scattering). The length of the leads over which the spin of the electrons is conserved can then be described phenomenologically by classical spin-dependent resistors [@imry]. In this situation, the important part of the electrons which do not undergo spin-flip processes leads to an increase of the resistance due to the GMR mechanism. This picture is likely to be representative of the experimental situations \[\]. More experimental and theoretical work concerning the various relaxation rates will be necessary to establish a complete quantitative understanding of the phenomenon. We thank P. Falloon, H. Pastawski and X. Waintal for very useful discussions. In addition, we are grateful to H. Pastawski for crucial help in the implementation of the numerical method, and to X. Waintal for drawing our attention to the exact solution for the spin spiral of Ref. . This work received financial support from the European Union within the RTN program (Contract No. HPRN-CT-2000-00144). V.G. thanks the French Ministère délégué à la recherche et aux nouvelles technologies and the Center for Functional Nanostructures of the Deutsche Forschungsgemeinschaft (project B2.10) for support. Exact solution for the trigonometric domain wall {#sec:exact_trigo} ================================================ A particularly instructive case is that of a trigonometric domain wall (Eq. (\[eq:trigonometric\_wall\])) since an exact solution for the wave function inside the domain wall region can be obtained [@brataas]. Here we extend this approach to a scattering situation by matching the inner solutions with plane waves, which allows us to calculate the transmission and reflection amplitudes. In addition to the external spin basis $\{|z,\uparrow\rangle , \, |z,\downarrow\rangle\}$, it is useful to introduce a local spin basis $\{|z,\uparrow^{\rm L}\rangle , |z,\downarrow^{\rm L}\rangle\}$, which corresponds to spin orientations parallel and anti-parallel to the (rotating) local magnetization direction $\vec{f}(z)$, leading to $$\left(\begin{array}{c}|z,\uparrow^{\rm L}\rangle\\ |z,\downarrow^{\rm L}\rangle \end{array}\right) =R(z)\left(\begin{array}{c} |z,\uparrow\rangle \\ |z,\downarrow\rangle \end{array}\right)$$ where the spin rotation matrix is given by $$R(z)=\left(\begin{array}{cc} \cos(az+\pi/4)&\sin(az+\pi/4)\\ -\sin(az+\pi/4)&\cos(az+\pi/4) \end{array}\right)$$ with $a=\pi/4\lambda$. For $z=-\lambda$, $R$ is simply the identity matrix (the local rotating basis coincides with the fixed one), and putting $z=\lambda$ corresponds to exchanging the spin directions between the local and fixed bases. Inserting the spinor $$|\psi(z)\rangle=\phi^{\rm L}_\uparrow(z)|z,\uparrow^{\rm L}\rangle +\phi^{\rm L}_\downarrow(z)|z,\downarrow^{\rm L}\rangle$$ into the Schrödinger equation corresponding to the Hamiltonian (\[eq:H\]), we obtain \[eq:cde\_in\_local\_basis\] $$\label{eq:df1_local} \left[\frac{d^2}{dz^2}-a^2\right]\phi^{\rm L}_\uparrow+ \frac{2m}{\hbar^2}\left(\epsilon-\frac{\Delta}{2} \right) \phi^{\rm L}_\uparrow =2a\frac{d}{dz} \phi^{\rm L}_\downarrow$$ $$\label{eq:df2_local} \left[\frac{d^2}{dz^2}-a^2\right]\phi^{\rm L}_\downarrow+ \frac{2m}{\hbar^2}\left(\epsilon+\frac{\Delta}{2} \right) \phi^{\rm L}_\downarrow =-2a\frac{d}{dz} \phi^{\rm L}_\uparrow$$ which is, in fact, Eq. (\[eq:cde\]) expressed in the local spin basis, for the case of a trigonometric domain wall. This system of coupled differential equations can be reduced to a $2\times 2$ eigenvalue problem with the ansatz $$\left(\begin{array}{c}\phi_\uparrow^{\rm L}(z)\\ \phi_\downarrow^{\rm L}(z) \end{array}\right) =\exp(i\tilde{k}z)\left(\begin{array}{c} C_\uparrow \\ C_\downarrow \end{array}\right)\, ,$$ such that the solutions $(C_\uparrow,C_\downarrow)$ and $2m\epsilon/\hbar^2$ are the eigenvectors and eigenvalues, respectively, of the matrix $$M=\left(\begin{array}{cc}\tilde{k}^2+a^2+m\Delta/\hbar^2& 2i\tilde{k}a\\ -2i\tilde{k}a& \tilde{k}^2+a^2-m\Delta/\hbar^2 \end{array}\right)\, .$$ The secular equation of $M$ leads to the dispersion relations $$\epsilon_{1,2}=\frac{\hbar^2}{2m}\left(\tilde{k}^2+a^2\pm\sqrt{4a^2\tilde{k}^2+\frac{2m}{\hbar^2}\left(\frac{\Delta}{2}\right)^2}\right)$$ that is, the eigenenergies of the infinite spin spiral [@brataas]. For a closed spiral, the periodic boundary conditions would lead [@stern] to quantized values of $\tilde{k}$. However, we are interested in a scattering problem, where the region in which the magnetization turns is connected to homogeneous ferromagnetic leads. We therefore express the general solution $(\phi_\uparrow^{\rm L},\phi_\downarrow^{\rm L})$ for a given energy as a linear combination of the four corresponding eigenstates of the spiral, and use the matching conditions between the domain wall region and the perfect leads at $z=\pm \lambda$. Taking into account the rotation of the local basis, and using the expressions given in Eqs.(\[eq:upoutside\]) and (\[eq:downoutside\]) for the wave-function outside the wall region, we get $$\begin{aligned} e^{-ik\lambda}+r_{\uparrow\uparrow}e^{ik\lambda} &=&\phi_\uparrow^{\rm L}(-\lambda) \nonumber \\ t_{\uparrow\uparrow}e^{ik'\lambda} &=&-\phi_\downarrow^{\rm L}(\lambda) \nonumber \\ r_{\uparrow\downarrow}e^{ik'\lambda} &=&\phi_\downarrow^{\rm L}(-\lambda) \nonumber \\ t_{\uparrow\downarrow}e^{ik\lambda} &=&\phi_\uparrow^{\rm L}(\lambda) \nonumber \\ ik\left(e^{-ik\lambda}-r_{\uparrow\uparrow}e^{ik\lambda}\right) &=&\frac{\rm d}{{\rm d}z}\phi_\uparrow^{\rm L}(-\lambda) -a\phi_\downarrow^{\rm L}(-\lambda) \nonumber \\ ik' t_{\uparrow\uparrow}e^{ik'\lambda} &=&-\frac{\rm d}{{\rm d}z}\phi_\downarrow^{\rm L}(\lambda) -a\phi_\uparrow^{\rm L}(\lambda) \nonumber \\ -ik' r_{\uparrow\downarrow}e^{ik'\lambda} &=&\frac{\rm d}{{\rm d}z}\phi_\downarrow^{\rm L}(-\lambda) +a\phi_\uparrow^{\rm L}(-\lambda) \nonumber \\ ik t_{\uparrow\downarrow}e^{ik\lambda} &=&\frac{\rm d}{{\rm d}z}\phi_\uparrow^{\rm L}(\lambda) -a\phi_\downarrow^{\rm L}(\lambda)\, . \nonumber\end{aligned}$$ These conditions allow us to extract the amplitudes $t_{\uparrow\uparrow}$, $r_{\uparrow\uparrow}$, $t_{\uparrow\downarrow}$ and $r_{\uparrow\downarrow}$, as well as the precise form of the wave-function inside the domain wall. The resulting transmission coefficients are presented and discussed in Sec. \[sec:strong\_coupling\]. [50]{} M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, J. Chazelas, Phys. Rev. Lett. **61**, 2472 (1988). G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Phys. Rev. B **39**, 4828 (1989). J.F. Gregg, W. Allen, K. Ounadjela, M. Viret, M. Hehn, S.M. Thompson, J.M.D. Coey, Phys. Rev. Lett. **77**, 1580 (1996); M. Viret, D. Vignoles, D. Cole, J.M.D. Coey, W. Allen, D.S. Daniel, J.F. Gregg, Phys. Rev. B **53**, 8464 (1996). U. Ebels, A. Radulescu, Y. Henry, L. Piraux, K. Ounadjela, Phys. Rev. Lett. **84**, 983 (2000). G. Dumpich, T.P. Krome, B. Hausmanns, J. Magn. Magn. Mater. **248**, 241 (2002). T. Valet, A. Fert, Phys. Rev. B **48**, 7099 (1993). A. Stern, Phys. Rev. Lett. **68**, 1022 (1992). P.M. Levy, S. Zhang, Phys. Rev. Lett. **79**, 5110 (1997). B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P. Kouwenhoven, D. van der Marel, C.T. Foxon, Phys. Rev. Lett. **60**, 848 (1988). D.A. Wharam, T.J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie, G.A.C. Jones, J. Phys. C: Solid State Phys. **21**, L209 (1988). K. Nakanishi, Y.O. Nakamura, Phys. Rev. B **61**, 11278 (2000). H. Imamura, N. Kobayashi, S. Takahashi, S. Maekawa, Phys. Rev. Lett. **84**, 1003 (2000). J.B.A.N. van Hoof, K.M. Schep, A. Brataas, G.E.W. Bauer, P.J. Kelly, Phys. Rev. B **59**, 138 (1999). P. Bruno, Phys. Rev. Lett. **83**, 2425 (1999). I.L. Prejbeanu, L.D. Buda, U. Ebels, K. Ounadjela, Appl. Phys. Lett. **77**, 3066 (2000). I.L. Prejbeanu, M. Viret, L.D. Buda, U. Ebels, K. Ounadjela, J. Magn. Magn. Mater. **240**, 27 (2002). B. Hausmanns, T.P. Krome, G. Dumpich, E.F. Wassermann, D. Hinzke, U. Nowak, K.D. Usadel, J. Magn. Magn. Mater. **240**, 297 (2002). X. Waintal, M. Viret, arXiv:cond-mat/0301293. D. Weinmann, R.L. Stamps, R.A. Jalabert in *Electronic correlations: from meso- to nano-physics*, ed. by T. Martin, G. Montambaux, J. Trân Thanh Vân (EDP Sciences 2001). P.E. Falloon, R.L. Stamps, unpublished. L.D. Landau, E.M. Lifshitz, [*Course of Theoretical Physics*]{}, Vol. 8, 1 Ed. Pergamon Press (1960). G.G. Cabrera, L.M. Falicov, Phys. Stat. Sol. (b), **61**, 539 (1974); *ibid* **62**, 217 (1974). P.A. Lee, D.S. Fisher, Phys. Rev. Lett. **47**, 882 (1981). H.M. Pastawski, E. Medina, Rev. Mex. Fis. **47** S1, 1 (2001), and references therein, also available as arXiv:cond-mat/0103219. A. Brataas, G. Tatara, G.E.W. Bauer, Phys. Rev. B **60**, 3406 (1999). E. Merzbacher, [*Quantum Mechanics*]{}, John Wiley & Sons, 2 Ed. (1970). Y. Imry, [*Introduction to Mesoscopic Physics*]{}, Oxford University Press (New York 1997).

--- abstract: 'Associated Legendre functions arise in many problems of mathematical physics. By using the generalized Abel-Plana formula, in this paper we derive a summation formula for the series over the zeros of the associated Legendre function of the first kind with respect to the degree. The summation formula for the series over the zeros of the Bessel function, previously discussed in the literature, is obtained as a limiting case. The Wightman function for a scalar field with general curvature coupling parameter is considered inside a spherical boundary on background of constant negative curvature space. The corresponding mode sum contains series over the zeros of the associated Legendre function. The application of the summation formula allows us to present the Wightman function in the form of the sum of two integrals. The first one corresponds to the Wightman function for the bulk geometry without boundaries and the second one is induced by the presence of the spherical shell. For points away from the boundary the latter is finite in the coincidence limit. In this way the renormalization of the vacuum expectation value of the field squared is reduced to that for the boundary-free part.' author: - | A. A. Saharian[^1]\ *Department of Physics, Yerevan State University,*\ *1 Alex Manogian Street, 0025 Yerevan, Armenia*\ *and*\ *Departamento de Física, Universidade Federal da Paraíba,*\ * 58.059-970, Caixa Postal 5.008, João Pessoa, PB, Brazil* title: Summation formula over the zeros of the associated Legendre function with a physical application --- PACS numbers: 02.30.Gp, 03.70.+k, 04.62.+v Introduction {#sec:Introd} ============ In a number of problems in mathematical physics we need to sum over the values of a certain function at integer points, and then subtract the corresponding integral. In particular, in quantum field theory the expectation values for physical observables induced by the presence of boundaries are presented in the form of this difference. The corresponding sum and integral, taken separately, diverge and some physically motivated procedure to handle the finite result, is needed. For a number of boundary geometries one of the most convenient methods to obtain such renormalized values is based on the use of the Abel-Plana summation formula [Hard91,Henr74]{} (for different forms of this formula discussed in the literature see also [@Saha07Rev]). Applications of the Abel-Plana formula in physical problems related to the Casimir effect for flat boundary geometries and topologically non-trivial spaces with corresponding references can be found in [@Grib94; @Most97]. The use of this formula allows to extract in a cutoff independent way the Minkowski vacuum part and to obtain for the renormalized part rapidly convergent integrals useful, in particular, for numerical calculations. However, the applications of the Abel-Plana formula in its standard form are restricted to the problems where the normal modes are explicitly known. In [@Sah1] we have considered a generalization of this formula, which essentially enlarges the application range and allows to include problems where the eigenmodes are given implicitly as zeros of a given function. Well known examples of this kind are the boundary-value problems with spherical and cylindrical boundaries. The generalized Abel-Plana formula contains two meromorphic functions and by specifying one of them the Abel-Plana formula is obtained (for other generalizations of the Abel-Plana formula see [Most97,Bart80,Zaya88]{}). Applying the generalized formula to Bessel functions, in [@Sah1; @Sahdis] summation formulae are obtained for the series over the zeros of various combinations of these functions (for a review with physical applications see also Ref. [Saha07Rev,Saha00Rev,Saha06PoS]{}). The summation formulae derived from the generalized Abel-Plana formula have been applied for the evaluation of the vacuum expectation values of local physical observables in the Casimir effect (for the Casimir effect see [Grib94,Most97,Plun86]{}) for plane boundaries with Robin or non-local boundary conditions [@Rome02], for spherical boundaries in Minkowski and global monopole bulks [@Saha01] and for cylindrical boundaries in Minkowski and cosmic string bulks [@Rome01]. By making use of the generalized Abel-Plana formula, the vacuum expectation values of the field squared and the energy-momentum tensor in closely related but more complicated geometry of a wedge with cylindrical boundary are investigated in [@Reza02] for both scalar and electromagnetic fields. As in the case of the Abel-Plana formula, the use of the generalized formula in these problems allows to extract the contribution of the unbounded space and to present the boundary-induced parts in terms of exponentially converging integrals. In [@SahaRind1] summation formulae for the series over the zeroes of the modified Bessel functions with an imaginary order are derived by using the generalized Abel-Plana formula. This type of series arise in the evaluation of the vacuum expectation values induced by plane boundaries uniformly accelerated through the Fulling-Rindler vacuum. Another class of problems where the application of the generalized Abel-Plana formula provides an efficient way for the evaluation of the vacuum expectation values is considered in [@Saha05b]. In these papers braneworld models with two parallel branes on anti-de Sitter bulk are discussed. The corresponding mode-sums for physical observables bilinear in the field contain series over the zeroes of cylinder functions which are summarized by using the generalized Abel-Plana formula. The geometry of spherical branes in Rindler-like spacetimes is considered in [@Saha07RindBr]. In [Saha07Helic]{} from the generalized Abel-Plana formula a summation formula is derived over the eigenmodes of a dielectric cylinder and this formula is applied for the evaluation of the radiation intensity from a point charge orbiting along a helical trajectory inside the cylinder. The physical importance of the Bessel functions is related to the fact that they appear as solutions of the field theory equations in various situations. In particular, in spherical and cylindrical coordinates the radial parts of the solutions for the scalar, fermionic, and electromagnetic wave equations on background of the Minkowski spacetime are expressed in terms of these functions. Another important class of special functions is the so-called Legendre associated functions (see, for instance, [Erde53a,Abra72]{}). These functions can be considered as generalizations of the Bessel functions: in the limit of large values of the degree when the argument is close to unity they reduce to the Bessel functions. The associated Legendre functions arise naturally in many mathematical and physical applications. In particular, they appear as solutions of physical field equations on background of constant curvature spaces (see, for instance, [@Grib94; @Most97; @Birr82]) and the above-mentioned limit corresponds to the limit when the curvature radius of the bulk goes to infinity. The eigenfunctions in braneworld models with de Sitter and anti-de Sitter branes are also expressed in terms of the Legendre functions (see [@Noji00]). Motivated by this, in the present paper, by making use of the generalized Abel-Plana formula, we obtain a summation formula for the series over the zeros of the associated Legendre function of the first kind with respect to the degree. In particular, this type of series appear in the evaluation of expectation values for physical observables bilinear in the operator of a quantum field on background of constant curvature spaces in the presence of boundaries. As in the case of the other Abel-Plana-type formulae, previously considered in the literature, the formula discussed here presents the sum of the series over the zeros of the associated Legendre function in the form of the sum of two integrals. In boundary-value problems the first one corresponds to the situation when the boundary is absent and the second one presents the part induced by the boundary. For a large class of functions the latter is rapidly convergent and, in particular, is useful for the numerical evaluations of the corresponding physical characteristics. We have organized the paper as follows. In the next section, by specifying the functions in the generalized Abel-Plana formula we derive a formula for the summation of series over zeros of the associated Legendre function with respect to the degree. In section \[sec:Special\], special cases of this summation formula are considered. First, as a partial check we show that as a special case the standard Abel-Plana formula is obtained. Then we show that from the summation formula discussed in section \[sec:SumForm\], as a limiting case the formula is obtained for the summation of the series over the zeros of the Bessel function, previously derived in [@Sah1]. A physical application is given in section \[sec:Phys\], where the positive frequency Wightman function for a scalar field is evaluated inside a spherical boundary on background of a negative constant curvature space. It is assumed that the field obeys Dirichlet boundary condition on the spherical shell. The use of the summation formula from section [sec:SumForm]{} allows us to extract from the vacuum expectation value the part corresponding to the geometry without boundaries and to present the part induced by the spherical shell in terms of an integral, which is rapidly convergent in the coincidence limit for points away from the boundary. The main results of the paper are summarized in section [sec:Conclus]{}. In appendix \[sec:Zeros\] we show that the zeros of the associated Legendre function of the first kind with respect to the degree are simple and real, and the asymptotic form for large zeros is discussed. In appendix \[sec:LegAsymp\] asymptotic formulae for the associated Legendre functions are considered for large values of the degree. These formulae are used in section \[sec:SumForm\] to obtain the constraints imposed on the function appearing in the summation formula. Summation formula {#sec:SumForm} ================= In this section we derive a summation formula for the series over zeros of the associated Legendre function of the first kind, $P_{iz-1/2}^{\mu }(u)$, with respect to the degree, assuming that $u>1$ and $\mu \leqslant 0$ (in this paper the definition of the associated Legendre functions follows that given in [@Abra72]). For given values $u$ and $\mu $ this function has an infinity of real zeros. We will denote the positive zeros arranged in ascending order of magnitude as $z_{k}$:$$P_{iz_{k}-1/2}^{\mu }(u)=0,\;k=1,2,\ldots . \label{Pzk0}$$These zeros are functions of the parameters $u$ and $\mu $: $% z_{k}=z_{k}(u,\mu )$. Note that one has $P_{iz-1/2}^{\mu }(u)=P_{-iz-1/2}^{\mu }(u)$ and, hence, $-z_{k}$ are zeros of the function $% P_{iz-1/2}^{\mu }(u)$ as well. In appendix \[sec:Zeros\] we show that the zeros $z_{k}$ are simple and under the conditions specified above the function $P_{iz-1/2}^{\mu }(u)$ has no zeros which are not real. A summation formula for the series over $z_{k}$ can be obtained by making use of the generalized Abel-Plana formula [@Sah1] (see also, [Saha07Rev]{}):$$\lim_{b\rightarrow \infty }\left\{ {\mathrm{p.v.}}\!\int_{a}^{b}dx% \,f(x)-R[f(z),g(z)]\right\} =\frac{1}{2}\int_{a-i\infty }^{a+i\infty }dz\,% \left[ g(z)+{\sigma (z)}f(z)\right] , \label{GAPF}$$where ${\sigma (z)\equiv \mathrm{sgn}}({{\mathrm{Im\,}}}z)$, the functions $% f(z)$ and $g(z)$ are meromorphic for $a\leqslant x\leqslant b$ in the complex plane $z=x+iy$ and p.v. stands for the principal value of the integral. In formula (\[GAPF\]) we have defined $$R[f(z),g(z)]=\pi i\bigg[\sum_{k}\underset{z=z_{g,k}}{\mathrm{Res}}% g(z)+\sum_{k,{{\mathrm{Im\,}}}z_{f,k}\neq 0}\sigma (z_{f,k})\underset{z={% \mathrm{\,}}z_{f,k}}{\mathrm{Res}}f(z)\bigg], \label{Rfg}$$with $z_{f,k}$ and $z_{g,k}$ being the positions of the poles of the functions $f(z)$ and $g(z)$ in the strip $a<x<b$. The functions $f(z)$ and $g(z)$ in formula (\[GAPF\]) we choose in the form$$\begin{aligned} f(z) &=&\sinh (\pi z)h(z), \notag \\ g(z) &=&\frac{e^{-i\mu \pi }h(z)}{\pi iP_{iz-1/2}^{\mu }(u)}\left\{ \cos [\pi (\mu +iz)]Q_{iz-1/2}^{\mu }(u)+\cos [\pi (\mu -iz)]Q_{-iz-1/2}^{\mu }(u)\right\} , \label{gz}\end{aligned}$$where $Q_{iz-1/2}^{\mu }(u)$ is the associated Legendre function of the second kind and the function $h(z)$ is meromorphic for $a\leqslant {{\mathrm{% Re}}}\,z\leqslant b$. By using relation (\[RelPQ\]) between the associated Legendre functions given in appendix \[sec:LegAsymp\], for the combination appearing on the left hand-side of formula (\[GAPF\]) one finds$$g(z)\pm f(z)=\frac{2e^{-i\mu \pi }h(z)}{\pi iP_{iz-1/2}^{\mu }(u)}\cos [\pi (\mu \mp iz)]Q_{\mp iz-1/2}^{\mu }(u). \label{gzplmin}$$With the functions (\[gz\]) the expression for $R[f(z),g(z)]$ takes the form$$R[f(z),g(z)]=2\sum_{k}\frac{e^{-i\mu \pi }Q_{iz-1/2}^{\mu }(u)}{\partial _{z}P_{iz-1/2}^{\mu }(u)}\cos [\pi (\mu +iz)]h(z)\bigg|_{z=z_{k}}+2e^{-i\mu \pi }r[h(z)], \label{Rfg2}$$with the notation$$\begin{aligned} r[h(z)] &=&\sum_{k,{{\mathrm{Im\,}}}z_{h,k}\neq 0}\underset{z=z_{h,k}}{% \mathrm{Res}}\bigg\{\frac{Q_{-\sigma (z)iz-1/2}^{\mu }(u)}{P_{iz-1/2}^{\mu }(u)}\cos [\pi (\mu -\sigma (z)iz)]h(z)\bigg\} \notag \\ &&+\frac{1}{2}\sum_{k,{{\mathrm{Im\,}}}z_{h,k}=0}\underset{z=z_{h,k}}{% \mathrm{Res}}\bigg\{\frac{h(z)}{P_{iz-1/2}^{\mu }(u)}\sum_{l=\pm }\cos [\pi (\mu +liz)]Q_{liz-1/2}^{\mu }(u)\bigg\}. \label{rhz}\end{aligned}$$In formula (\[rhz\]), $z_{h,k}$ are the positions of the poles for the function $h(z)$. In terms of the function $h(z)$ the conditions for the generalized Abel-Plana formula (\[GAPF\]) to be valid take the form$$\begin{aligned} \lim_{w\rightarrow \infty }\int_{a\pm iw}^{b\pm iw}dz\frac{Q_{\mp i z-1/2}^{\mu }(u)}{P_{iz-1/2}^{\mu }(u)}\cos [\pi (\mu \mp i z)]h(z) &=&0, \notag \\ \lim_{b\rightarrow \infty }\int_{b}^{b\pm i\infty }dz\frac{Q_{\mp i z-1/2}^{\mu }(u)}{P_{iz-1/2}^{\mu }(u)}\cos [\pi (\mu \mp iz)]h(z) &=&0. \label{Cond1}\end{aligned}$$By using the asymptotic formulae for the associated Legendre functions given in appendix \[sec:LegAsymp\], it can be seen that these conditions are satisfied if the function $h(z)$ is restricted to the constraint$$|h(z)|<\varepsilon (x)e^{c\eta y},\;z=x+iy,\;|z|\rightarrow \infty , \label{Cond2}$$uniformly in any finite interval of  $x$, where $c<2$, $\varepsilon (x)e^{\pi x}\rightarrow 0$ for $x\rightarrow +\infty $, and $\eta $ is defined by the relation$$u=\cosh \eta . \label{ueta}$$ Substituting the functions (\[gz\]) into formula (\[GAPF\]) and by taking into account relations (\[gzplmin\]), (\[Rfg2\]), we obtain that for a function $h(z)$ meromorphic in the half-plane ${{\mathrm{Re}}}% \,z\geqslant a$ and satisfying condition (\[Cond2\]), the following formula takes place$$\begin{aligned} &&\lim_{b\rightarrow \infty }\bigg\{\sum_{k=m}^{n}T_{\mu }(z_{k},u)h(z_{k})-% \frac{e^{i\mu \pi }}{2}{\mathrm{p.v.}}\!\int_{a}^{b}dx\,\sinh (\pi x)h(x)+r[h(z)]\bigg\} \notag \\ &&\qquad =\frac{i}{2\pi }\int_{a-i\infty }^{a+i\infty }dz\,\frac{Q_{-\sigma (z)iz-1/2}^{\mu }(u)}{P_{iz-1/2}^{\mu }(u)}\cos [\pi (\mu -\sigma (z)iz)]h(z), \label{SumForm0}\end{aligned}$$where and in what follows the notation$$T_{\mu }(z,u)=\frac{Q_{iz-1/2}^{\mu }(u)}{\partial _{z}P_{iz-1/2}^{\mu }(u)}% \cos [\pi (\mu +iz)] \label{Tmu}$$is used. On the left-hand side of formula (\[SumForm0\]), $z_{m-1}<a<z_{m}$, $z_{n}<b<z_{n+1}$ and in the definition of $r[h(z)]$ the summation goes over the poles $z_{h,k}$ in the strip $a<{{\mathrm{Re}}}\,z<b$. Note that from the Wronskian relation for the associated Legendre functions one has$$Q_{iz-1/2}^{\mu }(u)=\frac{e^{i\mu \pi }\Gamma (iz+\mu +1/2)}{% (u^{2}-1)\Gamma (iz-\mu +1/2)\partial _{u}P_{iz-1/2}^{\mu }(u)},\;z=z_{k}. \label{QWrons}$$Now, by taking into account the formula$$\frac{\Gamma (iz+\mu +1/2)}{\Gamma (iz-\mu +1/2)}=\pi \frac{|\Gamma (iz-\mu +1/2)|^{-2}}{\cos [\pi (\mu +iz)]}, \label{GammaRel}$$for the gamma function, the factor $T_{\mu }(z_{k},u)$ in (\[SumForm0\]) can also be written in the form $$T_{\mu }(z_{k},u)=\frac{\pi e^{i\mu \pi }|\Gamma (iz-\mu +1/2)|^{-2}}{% (u^{2}-1)\partial _{u}P_{iz-1/2}^{\mu }(u)\partial _{z}P_{iz-1/2}^{\mu }(u)}% \bigg|_{z=z_{k}}.$$ Taking the limit $a\rightarrow 0$, from (\[SumForm0\]) one obtains that for a function $h(z)$ meromorphic in the half-plane ${{\mathrm{Re}}}% \,z\geqslant 0$ and satisfying the condition (\[Cond2\]) the following formula takes place $$\begin{aligned} \sum_{k=1}^{\infty }T_{\mu }(z_{k},u)h(z_{k}) &=&\frac{e^{i\mu \pi }}{2}% \mathrm{p.v.}\int_{0}^{\infty }dx\,\sinh (\pi x)h(x)-r[h(z)] \notag \\ &&-\frac{1}{2\pi }\int_{0}^{\infty }dx\,\frac{Q_{x-1/2}^{\mu }(u)}{% P_{x-1/2}^{\mu }(u)}\cos [\pi (\mu +x)][h(xe^{\pi i/2})+h(xe^{-\pi i/2})]. \label{SumFormula}\end{aligned}$$If the function $h(z)$ has poles on the positive real axis, it is assumed that the first integral on the right-hand side converges in the sense of the principal value. From the derivation of (\[SumFormula\]) it follows that this formula may be extended to the case of some functions $h(z)$ having branch-points on the imaginary axis, for example, having the form $% h(z)=h_{1}(z)/(z^{2}+c^{2})^{1/2}$, where $h_{1}(z)$ is a meromorphic function. This type of function appears in the physical example discussed in section \[sec:Phys\]. Special cases of formula (\[SumFormula\]) with examples are considered in the next section. Formula (\[SumFormula\]) can be generalized for a class of functions $h(z)$ having purely imaginary poles at the points $z=\pm iy_{k}$, $y_{k}>0$, $% k=1,2,\ldots $, and at the origin $z=y_{0}=0$. Let function $h(z)$ satisfy the condition$$h(z)=-h(ze^{-\pi i})+o((z-\sigma _{k})^{-1}),\;z\rightarrow \sigma _{k},\;\sigma _{k}=0,iy_{k}. \label{Impolecond}$$Now, in the limit $a\rightarrow 0$ the right hand side of (\[SumForm0\]) can be presented in the form$$\frac{i}{2\pi }\sum_{\alpha =+,-}\bigg(\int_{\gamma _{\rho }^{\alpha }}dz+\sum_{\sigma _{k}=\alpha iy_{k}}\int_{C_{\rho }(\sigma _{k})}dz\bigg)\,% \frac{Q_{-\alpha iz-1/2}^{\mu }(u)}{P_{iz-1/2}^{\mu }(u)}\cos [\pi (\mu -\alpha iz)]h(z), \label{Impoles1}$$plus the sum of the integrals along the straight segments $(\pm i(y_{k-1}+\rho ),\pm i(y_{k}-\rho ))$ of the imaginary axis between the poles. In (\[Impoles1\]), $C_{\rho }(\sigma _{k})$ denotes the right half of the circle with radius $\rho $ and with the center at the point $\sigma _{k}$, described in the positive direction. Similarly, $\gamma _{\rho }^{+}$ and $\gamma _{\rho }^{-}$ are upper and lower halves of the semicircle in the right half-plane with radius $\rho $ and with the center at the point $% z=0$, described in the positive direction with respect to this point. In the limit $\rho \rightarrow 0$ the sum of the integrals along the straight segments of the imaginary axis gives the principal value of the last integral on the right-hand side of (\[SumFormula\]). Further, in the terms of (\[Impoles1\]) with $\alpha =-$ we introduce a new integration variable $z^{\prime }=ze^{\pi i}$. By using the relation (\[Impolecond\]) the expression (\[Impoles1\]) is presented in the form$$-\sum_{\sigma _{k}=0,iy_{k}}(1-\delta _{0\sigma _{k}}/2)\underset{z=\sigma _{k}}{\mathrm{Res}}\bigg\{\frac{Q_{-iz-1/2}^{\mu }(u)}{P_{iz-1/2}^{\mu }(u)}% \cos [\pi (\mu -iz)]h(z)\bigg\} \label{Impoles2}$$plus the part which vanishes in the limit $\rho \rightarrow 0$. As a result, formula (\[SumFormula\]) is extended for functions having purely imaginary poles and satisfying condition (\[Impolecond\]). For this, on the right-hand side of (\[SumFormula\]) we have to add the sum of residues (\[Impoles2\]) at these poles and take the principal value of the second integral on the right-hand side. The latter exists due to condition ([Impolecond]{}). Note that for functions having the form $% h(z)=F(z)P_{iz-1/2}^{\mu }(u)$ the left-hand side of (\[SumFormula\]) is zero and from this formula we obtain a formula relating the integrals involving the Legendre associated functions. Special cases {#sec:Special} ============= Here we will consider special cases of the summation formula ([SumFormula]{}). First let us consider the case $\mu =-1/2$. The corresponding associated Legendre functions have the form$$P_{z-1/2}^{-1/2}(\cosh \eta )=\sqrt{\frac{2}{\pi }}\frac{\sinh (z\eta )}{z% \sqrt{\sinh \eta }},\;Q_{z-1/2}^{-1/2}(\cosh \eta )=-i\sqrt{\frac{\pi }{2}}% \frac{e^{-z\eta }}{z\sqrt{\sinh \eta }}. \label{SpCase1}$$In this case one has $z_{k}=\pi k/\eta $. Introducing a new function $F(x)$ in accordance with the relation $F(\eta x/\pi )=\sinh (\pi x)h(x)$, and assuming that this function is analytic in the right half-plane, from formula (\[SumFormula\]) we find the Abel-Plana formula in the standard form:$$\sum_{k=1}^{\infty }F(k)=-\frac{1}{2}F(0)+\int_{0}^{\infty }dx\,F(x)+i\int_{0}^{\infty }dx\frac{F(ix)-F(-ix)}{e^{2\pi x}-1}. \label{AP1}$$Note that the first term on the right-hand side of this formula comes from the residue term with $\sigma _{k}=0$ in (\[Impoles2\]). In the case $\mu =1/2$ for the corresponding associated Legendre functions we have the expressions$$P_{z-1/2}^{1/2}(\cosh \eta )=\sqrt{\frac{2}{\pi }}\frac{\cosh (z\eta )}{% \sqrt{\sinh \eta }},\;Q_{z-1/2}^{1/2}(\cosh \eta )=i\sqrt{\frac{\pi }{2}}% \frac{e^{-z\eta }}{\sqrt{\sinh \eta }}. \label{SpCase2}$$The zeros $z_{k}$ now have the form $z_{k}=\pi (k+1/2)/\eta $ and for functions $F(z)$ analytic in the right half-plane from formula ([SumFormula]{}) we obtain the Abel-Plana formula in the form useful for fermionic field calculations (see, for instance, [@Grib94; @Most97]):$$\sum_{k=1}^{\infty }F(k+1/2)=\int_{0}^{\infty }dx\,F(x)-i\int_{0}^{\infty }dx% \frac{F(ix)-F(-ix)}{e^{2\pi x}+1}. \label{AP2}$$ As a next special case let us consider the formula for the summation over the zeros of the function $P_{isz-1/2}^{-\mu }(\cosh (\eta /s))$ in the limit when $s\rightarrow \infty $. By taking into account the relation (see appendix \[sec:LegAsymp\])$$\lim_{\nu \rightarrow +\infty }\nu ^{\mu }P_{i\nu -1/2}^{-\mu }(\cosh (\eta /\nu ))=J_{\mu }(\eta ), \label{Plim}$$with $J_{\mu }(\eta )$ being the Bessel function of the first kind, in this limit from (\[SumFormula\]) we obtain the summation formula for the series over zeros $\eta =j_{\mu ,k}$, $k=1,2,\ldots ,$ of the Bessel function. In order to take this limit we also will need the formulae (\[Qasymp\]) from appendix \[sec:LegAsymp\] and the formulae [@Erde53b]$$\begin{aligned} \lim_{\nu \rightarrow \infty }\nu ^{\mu }P_{\nu }^{-\mu }[\cosh (x/\nu )] &=&I_{\mu }(x), \notag \\ \lim_{\nu \rightarrow \infty }\nu ^{\mu }Q_{\nu }^{-\mu }[\cosh (x/\nu )] &=&e^{-i\mu \pi }K_{\mu }(x), \label{PQlim}\end{aligned}$$with $I_{\mu }(x)$, $K_{\mu }(x)$ being the modified Bessel functions. First we rewrite formula (\[SumFormula\]) making the replacements $% z\rightarrow sz$, $x\rightarrow sx$, $\mu \rightarrow -\mu $, in both sides of this formula including the terms in $r[h(z)]$, and we take $u=\cosh (\eta /s)$. In order to take the limit $s\rightarrow \infty $ for the second integral on the right-hand side of the resulting formula, we note that, as it follows from the derivation of (\[SumFormula\]), the integrand of this integral (with the replacements described above) should be understood as the limit $$\cos [\pi (sx-\mu )]\sum_{l=+,-}h(sxe^{l\pi i/2})=\lim_{\epsilon \rightarrow +0}\sum_{l=+,-}\cos [\pi (sx-\mu -lis\epsilon )]h(sxe^{l\pi i/2}). \label{note1}$$Taking the limit $s\rightarrow \infty $ with the help of formulae (\[Plim\]),(\[PQlim\]),(\[Qasymp\]), we find the following summation formula over the zeros of the Bessel function$$\begin{aligned} \sum_{k=1}^{\infty }\frac{2f(j_{\mu ,k})}{j_{\mu ,k}J_{\mu }^{\prime 2}(j_{\mu ,k})} &=&\mathrm{p.v.}\int_{0}^{\infty }dx\,f(x)-r_{1}[f(z)] \notag \\ &&-\frac{1}{\pi }\int_{0}^{\infty }dx\,\frac{K_{\mu }(x)}{I_{\mu }(x)}\left[ e^{i\pi \mu }f(xe^{\pi i/2})+e^{-i\pi \mu }f(xe^{-\pi i/2})\right] , \label{SumBess}\end{aligned}$$where $f(z)=\lim_{s\rightarrow \infty }e^{sz/\eta }h(sz/\eta )$, and $$\begin{aligned} r_{J}[f(z)] &=&\pi i\sum_{k}\underset{{{\mathrm{Im\,}}}z_{h,k}>0}{\mathrm{Res% }}\bigg[\frac{H_{\mu }^{(1)}(z)}{J_{\mu }(z)}f(z)\bigg]-\pi i\sum_{k}% \underset{{{\mathrm{Im\,}}}z_{h,k}<0}{\mathrm{Res}}\bigg[\frac{H_{\mu }^{(2)}(z)}{J_{\mu }(z)}f(z)\bigg] \notag \\ &&-\pi \sum_{k}\underset{{{\mathrm{Im\,}}}z_{h,k}=0}{\mathrm{Res}}\bigg[% \frac{Y_{\mu }(z)}{J_{\mu }(z)}f(z)\bigg]-\frac{\pi }{2}\underset{z=0}{% \mathrm{Res}}\bigg[\frac{Y_{\mu }(z)}{J_{\mu }(z)}f(z)\bigg]. \label{r1fz}\end{aligned}$$This formula is a special case of the result derived in [@Sah1; @Sahdis] (see also, [@Saha07Rev]). Now let us consider two important special cases of (\[SumFormula\]) corresponding to $\mu =-l$, $h(z)=H(z)/\cosh (\pi z)$ and $\mu =-l-1/2$, $% h(z)=H(z)/\sinh (\pi z)$ with $l=0,1,2,\ldots $. The associated Legendre functions with these values of the order appear as radial solutions of the equations for various fields on background of constant curvature spaces in cylindrical and spherical coordinates. Let the function $H(z)$ is meromorphic in the half-plane ${{\mathrm{Re}}}\,z\geqslant 0$ and satisfy the condition $$|H(z)|<\varepsilon _{H}(x)e^{c\eta y},\;z=x+iy,\;|z|\rightarrow \infty , \label{condhc}$$uniformly in any finite interval of  $x>0$, where $c<2$, $\varepsilon _{H}(x)\rightarrow 0$ for $x\rightarrow +\infty $. Then from the results of section \[sec:SumForm\] it follows that the formula $$\begin{aligned} \sum_{k=1}^{\infty }\frac{(-1)^{\delta }Q_{iz_{k}-1/2}^{-l-\delta /2}(u)H(z_{k})}{\partial _{z}P_{iz-1/2}^{-l-\delta /2}(u)|_{z=z_{k}}} &=&% \frac{1}{2}\mathrm{p.v.}\int_{0}^{\infty }dx\,\tanh ^{1-\delta }(\pi x)H(x)-r_{\delta }[H(z)] \notag \\ &-&\frac{1}{2\pi }\int_{0}^{\infty }dx\,\frac{Q_{x-1/2}^{-l-\delta /2}(u)}{% P_{x-1/2}^{-l-\delta /2}(u)}[H(xe^{\pi i/2})+(-1)^{\delta }H(xe^{-\pi i/2})], \label{SumFormlcs}\end{aligned}$$takes place, where $\delta =0,1$. In this formula we have introduced the notation$$\begin{aligned} r_{\delta }[H(z)] &=&\sum_{k,{{\mathrm{Im\,}}}z_{h,k}\neq 0}\underset{% z=z_{h,k}}{\mathrm{Res}}\bigg[\sigma ^{\delta }(z)\frac{Q_{-\sigma (z)iz-1/2}^{-l-\delta /2}(u)}{P_{iz-1/2}^{-l-\delta /2}(u)}H(z)\bigg] \notag \\ &&+\sum_{k,{{\mathrm{Im\,}}}z_{h,k}=0}\underset{z=z_{h,k}}{\mathrm{Res}}% \bigg[\frac{Q_{-iz-1/2}^{-l-\delta /2}(u)+(-1)^{\delta }Q_{iz-1/2}^{-l-\delta /2}(u)}{2P_{iz-1/2}^{-l-\delta /2}(u)}H(z)\bigg]\,. \label{rdelta}\end{aligned}$$Adding to the right-hand side of formula (\[SumFormlcs\]) the term $$-(-1)^{\delta }\sum_{\sigma _{k}=0,iy_{k}}(1-\delta _{0\sigma _{k}}/2)% \underset{z=\sigma _{k}}{\mathrm{Res}}\bigg[\frac{Q_{-iz-1/2}^{-l-\delta /2}(u)}{P_{iz-1/2}^{-l-\delta /2}(u)}H(z)\bigg]\,, \label{ImagPolesls}$$with $H(z)$ obeying the condition $H(z)=-(-1)^{\delta }H(ze^{-\pi i})+o((z-\sigma _{k})^{-1})$ for$\;z\rightarrow \sigma _{k}$, we obtain the extension of this formula to the case when the function $H(z)$ has poles at the points $0$, $\pm y_{k}$. From (\[SumFormlcs\]), as an example when the series is summarized in closed form one has$$\begin{aligned} \sum_{k=1}^{\infty }\frac{Q_{iz-1/2}^{-l-1/2}(u)}{\partial _{z}P_{iz-1/2}^{-l-1/2}(u)}\frac{z^{2n}\cos (\alpha z)}{(z^{2}+c^{2})^{m+1}}% \bigg|_{z=z_{k}} &=&\frac{(-1)^{m+n}}{m!}\bigg\{-\frac{\pi }{2^{m+2}}\left( \frac{\partial }{c\partial c}\right) ^{m}(c^{2n-1}e^{-\alpha c}) \notag \\ &&-i\frac{\partial ^{m}}{\partial x^{m}}\bigg[\frac{Q_{x-1/2}^{-l-1/2}(u)}{% P_{x-1/2}^{-l-1/2}(u)}\frac{x^{2n}\cosh (\alpha x)}{(x+c)^{m+1}}\bigg]_{x=c}% \bigg\}\,, \label{Example1}\end{aligned}$$where $\alpha <2\eta $,$\;c>0$, with $m\geqslant 0$ and $0\leqslant n\leqslant m$ being integers. The last term on the right-hand side of this formula comes from the residue at the pole $\sigma _{k}=ic$. As a next example, we take in formula (\[SumFormlcs\]) with $\delta =1$ the function$$H(z)=z^{2n-\nu }J_{\nu }(az)\frac{J_{\alpha }(b\sqrt{z^{2}+c^{2}})}{% (z^{2}+c^{2})^{\alpha /2}}, \label{Example2}$$where $a$, $b$, $c$ are positive constants and $n$ is a non-negative integer. This function is analytic in the right half-plane and satisfies the condition (\[condhc\]) if $a+b<2\eta $, $2n<\alpha +\nu $. By taking into account that (\[Example2\]) is an even function of $z$, from ([SumFormlcs]{}) we find$$\sum_{k=1}^{\infty }J_{v}(az_{k})\frac{J_{\alpha }(b\sqrt{z_{k}^{2}+c^{2}})}{% (z_{k}^{2}+c^{2})^{\alpha /2}}\frac{z_{k}^{\nu +2n}Q_{iz_{k}-1/2}^{-l-1/2}(u)% }{\partial _{z}P_{iz-1/2}^{-l-1/2}(u)|_{z=z_{k}}}=-\frac{1}{2}% \int_{0}^{\infty }dx\,x^{2n-\nu }J_{\nu }(ax)\frac{J_{\alpha }(b\sqrt{% x^{2}+c^{2}})}{(x^{2}+c^{2})^{\alpha /2}}. \label{Example2n}$$ Wightman function inside a spherical boundary in a constant curvature space {#sec:Phys} =========================================================================== In this section we consider a physical application of the summation formula derived in section \[sec:SumForm\]. Namely, we will evaluate the positive frequency Wightman function for a scalar field and the vacuum expectation value of the field squared inside a spherical shell in a constant negative curvature space assuming that the field obeys the Dirichlet boundary condition on the shell (for quantum effects on background of constant curvature spaces see, for instance, [@Grib94; @Most97; @Birr82]  and references therein). Consider a quantum scalar field $\varphi (x)$ with the curvature coupling parameter $\xi $ on background of the space with constant negative curvature described by the line element $$ds^{2}=dt^{2}-a^{2}\left[ dr^{2}+\sinh ^{2}r(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\right] , \label{metric}$$where $a$ is a constant which is related to the non-zero components of the Ricci tensor and the Ricci scalar by the relations$$R_{1}^{1}=R_{2}^{2}=R_{3}^{3}=-\frac{2}{a^{2}},\;R=-\frac{6}{a^{2}}. \label{Rii}$$The field equation has the form$$\left( \nabla _{l}\nabla ^{l}+M^{2}+\xi R\right) \varphi (x)=0, \label{FieldEq}$$where $M$ is the mass of the field quanta. We are interested in quantum effects induced by the presence of a spherical shell with radius $r=r_{0}$, on which the field obeys Dirichlet boundary condition: $\varphi (x)|_{r=r_{0}}=0$. This boundary condition modifies the spectrum of the zero-point fluctuations compared with the case of free space and changes the physical properties of the vacuum. Among the most important characteristics of the vacuum are the expectation values of quantities bilinear in the field operator such as the field squared and the energy-momentum tensor. These expectation values are obtained from two-point functions in the coincidence limit. As a two-point function here we will consider the positive frequency Wightman function $W(x,x^{\prime })=\langle 0|\varphi (x)\varphi (x^{\prime })|0\rangle $, where $|0\rangle $ is the amplitude for the vacuum state. This function also determines the response of Unruh-De Witt type particle detectors [@Birr82]. Expanding the field operator over the complete set $\{\varphi _{\alpha }(x),\varphi _{\alpha }^{\ast }(x)\}$ of classical solutions to the field equation satisfying the boundary condition, the Wightman function is presented as the mode-sum$$W(x,x^{\prime })=\sum_{\alpha }\varphi _{\alpha }(x)\varphi _{\alpha }^{\ast }(x^{\prime }), \label{WFsum}$$where $\alpha $ is a set of quantum numbers specifying the solution. By the symmetry of the problem under consideration, the eigenfunctions for the scalar field can be presented in the form$$\varphi _{\alpha }(x)=Z(r)Y_{lm}(\theta ,\phi )e^{-i\omega t}, \label{eigfunc1}$$where $Y_{lm}(\theta ,\phi )$ are standard spherical harmonics, $% l=0,1,2,\ldots $, $-l\leqslant m\leqslant l$. From the field equation ([FieldEq]{}) we obtain the equation for the radial function $Z(r)$:$$\frac{1}{\sinh ^{2}r}\frac{d}{dr}\left( \sinh ^{2}r\frac{dZ}{dr}\right) +% \left[ (\omega ^{2}-m_{\mathrm{eff}}^{2})a^{2}-\frac{l(l+1)}{\sinh ^{2}r}% \right] Z=0, \label{Zeq}$$where we have introduced the effective mass defined by $$m_{\mathrm{eff}}^{2}=M^{2}-6\xi /a^{2}. \label{meff}$$In the region inside the spherical shell the solution of equation (\[Zeq\]), finite at $r=0$, is expressed in terms of the associated Legendre function of the first kind and the eigenfunctions have the form$$\varphi _{\alpha }(x)=C_{\alpha }\frac{P_{iz-1/2}^{-l-1/2}(\cosh r)}{\sqrt{% \sinh r}}Y_{lm}(\theta ,\phi )e^{-i\omega t}, \label{eigfunc2}$$with the notation$$z^{2}=(\omega ^{2}-m_{\mathrm{eff}}^{2})a^{2}-1. \label{lambda}$$ From the boundary condition on the spherical shell we find that the eigenvalues for $z$ are solutions of the equation$$P_{iz-1/2}^{-l-1/2}(\cosh r_{0})=0, \label{Eigvalues}$$and, hence, $z=z_{k}$, $k=1,2,\ldots $, in the notations of section [sec:SumForm]{}. The corresponding eigenfrequencies are found to be$$\omega _{k}^{2}=\omega ^{2}(z_{k})=(z_{k}^{2}+1-6\xi )/a^{2}+M^{2}. \label{eigfreq}$$Hence, the set $\alpha $ of the quantum numbers is specified to $\alpha =(l,m,k)$. The coefficient $C_{\alpha }$ in (\[eigfunc2\]) is determined from the orthonormalization condition$$\int d^{3}x\,\sqrt{|g|}\varphi _{\alpha }(x)\varphi _{\alpha ^{\prime }}^{\ast }(x)=\frac{\delta _{\alpha \alpha ^{\prime }}}{2\omega }, \label{normcond}$$where the integration goes over the region inside the spherical shell. Substituting the eigenfunctions (\[eigfunc2\]) into (\[normcond\]), by taking into account the integration formula (\[int3\]) and the boundary condition, one finds$$C_{\alpha }^{-2}=a^{3}\frac{\omega (z)}{z}(u_{0}^{2}-1)\partial _{z}P_{iz-1/2}^{-l-1/2}(u_{0})\partial _{u}P_{iz-1/2}^{-l-1/2}(u)|_{z=z_{k},u=u_{0}}, \label{Cnorm}$$where and in the discussion below we use the notations$$u=\cosh r,\;u_{0}=\cosh r_{0}. \label{uu0}$$By using the Wronskian relation (\[QWrons\]), the formula for the normalization coefficient is written as$$C_{\alpha }^{2}=\frac{z_{k}\Gamma (iz_{k}+l+1)Q_{iz_{k}-1/2}^{-l-1/2}(u_{0})e^{i(l+1/2)\pi }}{a^{3}\omega (z_{k})\Gamma (iz_{k}-l)\partial _{z}P_{iz-1/2}^{-l-1/2}(u_{0})|_{z=z_{k}}}. \label{Cnorm2}$$Note that the ratio of the gamma functions in this formula can also be presented in the form$$\frac{\Gamma (iz_{k}+l+1)}{\Gamma (iz_{k}-l)}=|\Gamma (iz_{k}+l+1)|^{2}\frac{% \cos [\pi (iz-l-1/2)]}{\pi }. \label{GamRatio}$$ Substituting the eigenfunctions into the mode-sum formula (\[WFsum\]) and using the addition theorem for the spherical harmonics, for the Wightman function one finds$$\begin{aligned} W(x,x^{\prime }) &=&\frac{1}{4\pi ^{2}a^{3}}\sum_{l=0}^{\infty }\frac{% (2l+1)P_{l}(\cos \gamma )}{\sqrt{\sinh r\sinh r^{\prime }}}e^{i(l+1/2)\pi }\sum_{k=1}^{\infty }z_{k}|\Gamma (iz_{k}+l+1)|^{2} \notag \\ &&\times T_{-l-1/2}(z_{k},u_{0})P_{iz_{k}-1/2}^{-l-1/2}(\cosh r)P_{iz_{k}-1/2}^{-l-1/2}(\cosh r^{\prime })\frac{e^{-i\omega (z_{k})\Delta t}}{\omega (z_{k})}, \label{WF1}\end{aligned}$$where $\Delta t=t-t^{\prime }$ and $T_{\mu }(z,u)$ is defined by relation (\[Tmu\]). In (\[WF1\]), $P_{l}(\cos \gamma )$ is the Legendre polynomial and $$\cos \gamma =\cos \theta \cos \theta ^{\prime }+\sin \theta \sin \theta ^{\prime }\cos (\phi -\phi ^{\prime }). \label{cosgam}$$As the expressions for the zeros $z_{k}$ are not explicitly known, formula (\[WF1\]) for the Wightman function is not convenient. In addition, the terms in the sum are highly oscillatory for large values of quantum numbers. For the further evaluation of the Wightman function we apply to the series over $k$ the summation formula (\[SumFormula\]) taking in this formula$$h(z)=z|\Gamma (iz+l+1)|^{2}P_{iz-1/2}^{-l-1/2}(\cosh r)P_{iz-1/2}^{-l-1/2}(\cosh r^{\prime })\frac{e^{-i\omega (z)\Delta t}}{% \omega (z)}. \label{hz}$$The corresponding conditions are met if $r+r^{\prime }+\Delta t/a<2r_{0}$. In particular, this is the case in the coincidence limit $t=t^{\prime }$ for the region under consideration. For the function (\[hz\]) the part of the integral on the right-hand side of formula (\[SumFormula\]) over the region $(0,x_{M})$ vanishes and for the Wightman function one finds$$\begin{aligned} W(x,x^{\prime }) &=&W_{0}(x,x^{\prime })-\frac{1}{4\pi ^{2}a^{2}}% \sum_{l=0}^{\infty }\frac{(2l+1)P_{l}(\cos \gamma )}{\sqrt{\sinh r\sinh r^{\prime }}}e^{i(l+1/2)\pi }\int_{x_{M}}^{\infty }dx\,x\frac{\Gamma (x+l+1)% }{\Gamma (x-l)} \notag \\ &&\times \frac{Q_{x-1/2}^{-l-1/2}(u_{0})}{P_{x-1/2}^{-l-1/2}(u_{0})}% P_{x-1/2}^{-l-1/2}(\cosh r)P_{x-1/2}^{-l-1/2}(\cosh r^{\prime })\frac{\cosh (% \sqrt{x^{2}-x_{M}^{2}}\Delta t/a)}{\sqrt{x^{2}-x_{M}^{2}}}, \label{WF2}\end{aligned}$$where we have defined$$x_{M}=\sqrt{M^{2}a^{2}+1-6\xi }. \label{xM}$$ In formula (\[WF2\]), the first term on the right-hand side is given by$$\begin{aligned} W_{0}(x,x^{\prime }) &=&\frac{1}{8\pi ^{2}a^{3}}\sum_{l=0}^{\infty }\frac{% (2l+1)P_{l}(\cos \gamma )}{\sqrt{\sinh r\sinh r^{\prime }}}e^{-i(l+1/2)\pi }\int_{0}^{\infty }dx\,x\sinh (\pi x) \notag \\ &&\times |\Gamma (ix+l+1)|^{2}P_{ix-1/2}^{-l-1/2}(\cosh r)P_{ix-1/2}^{-l-1/2}(\cosh r^{\prime })\frac{e^{-i\omega (x)\Delta t}}{% \omega (x)}. \label{WF0}\end{aligned}$$This function does not depend on the sphere radius and is the Wightman function for a scalar field in background spacetime described by the line element (\[metric\]) when boundaries are absent. This can also be seen by the direct evaluation. Indeed, when boundaries are absent the eigenfunctions are still given by formula (\[eigfunc2\]), where now the spectrum for $z$ is continuous. In this case the corresponding part on the right of the orthonormalization condition (\[normcond\]) should be understood as the Dirac delta function. In the case $z=z^{\prime }$ the normalization integral diverges and, hence, the main contribution comes from large values $r$. By using the asymptotic formulae for the associated Legendre functions for large values of the argument, we can see that$$\int_{1}^{\infty }du\,P_{iz-1/2}^{-l-1/2}(u)P_{iz^{\prime }-1/2}^{-l-1/2}(u)=\left\vert \frac{\Gamma (iz)}{\Gamma (l+1+iz)}\right\vert ^{2}\delta (z-z^{\prime }). \label{NormInt0}$$By using this result for the normalization coefficient in the case when boundaries are absent one finds$$C_{\alpha }=\frac{1}{\sqrt{2\omega a^{3}}}\left\vert \frac{\Gamma (l+1+iz)}{% \Gamma (iz)}\right\vert , \label{Calfa}$$and the eigenfunctions have the form (see also, [@Grib94; @Grib74])$$\varphi _{\alpha }(x)=\left\vert \frac{\Gamma (l+1+iz)}{\Gamma (iz)}% \right\vert \frac{P_{iz-1/2}^{-l-1/2}(\cosh r)}{\sqrt{2\omega a^{3}\sinh r}}% Y_{lm}(\theta ,\phi )e^{-i\omega t}. \label{phialfa0}$$Substituting these eigenfunctions into the mode-sum (\[WFsum\]), for the corresponding Wightman function we find the formula which coincides with (\[WF0\]). The case of a spherical boundary in the Minkowski spacetime is obtained in the limit $a\rightarrow \infty $, with fixed $ar=R$. In this limit one has $% x_{M}=aM$. Introducing a new integration variable $y=x/a$, using the formulae (\[PQlim\]) and the asymptotic formula for the gamma function for large values of the argument, we find$$\begin{aligned} W^{\mathrm{(M)}}(x,x^{\prime }) &=&W_{0}^{\mathrm{(M)}}(x,x^{\prime })-\sum_{l=0}^{\infty }\frac{(2l+1)P_{l}(\cos \gamma )}{4\pi ^{2}\sqrt{% RR^{\prime }}}\int_{M}^{\infty }dy\,y \notag \\ &&\times I_{l+1/2}(Ry)I_{l+1/2}(R^{\prime }y)\frac{K_{l+1/2}(R_{0}y)}{% I_{l+1/2}(R_{0}y)}\frac{\cosh (\sqrt{y^{2}-M^{2}}\Delta t)}{\sqrt{y^{2}-M^{2}% }}. \label{WMink}\end{aligned}$$This formula gives the the positive frequency Wightman function inside a spherical shell with radius $R_{0}$ in the Minkowski bulk and is a special case of the general formula given in the first paper of [@Saha01] for a scalar field with Robin boundary conditions in arbitrary number of spatial dimensions. Having the Wightman function (\[WF2\]), we can evaluate the vacuum expectation value of the field squared taking the coincidence limit of the argument. Of course, this limit is divergent and some renormalization procedure is necessary. Here the important point is that for points outside the spherical shell the local geometry is the same as for the case of without boundaries and, hence, the structure of the divergences is the same as well. This is also directly seen from formula (\[WF2\]), where the second term on the right-hand side is finite in the coincidence limit. Since in formula (\[WF2\]) we have already explicitly subtracted the boundary-free part, the renormalization is reduced to that for the geometry without boundaries. In this way for the renormalized vacuum expectation value of the field squared one has$$\begin{aligned} \langle \varphi ^{2}\rangle _{\mathrm{ren}} &=&\langle \varphi ^{2}\rangle _{0,\mathrm{ren}}-\sum_{l=0}^{\infty }\frac{e^{i(l+1/2)\pi }}{4\pi ^{2}a^{2}}% \frac{(2l+1)}{\sinh r}\int_{x_{M}}^{\infty }dx\,x \notag \\ &&\times \frac{\Gamma (x+l+1)}{\Gamma (x-l)}\frac{Q_{x-1/2}^{-l-1/2}(\cosh r_{0})}{P_{x-1/2}^{-l-1/2}(\cosh r_{0})}\frac{\left[ P_{x-1/2}^{-l-1/2}(% \cosh r)\right] ^{2}}{\sqrt{x^{2}-x_{M}^{2}}}, \label{phi2}\end{aligned}$$where the first term on the right-hand side is the corresponding quantity in the constant negative curvature space without boundaries and the second one is induced by the presence of the spherical shell. For large values $x$, the integrand in (\[phi2\]) behaves as $e^{-(r_{0}-r)x}/(2x\sinh r)$ and the integral is exponentially convergent at the upper limit for strictly interior points. For $r\rightarrow 0$ one has $P_{x-1/2}^{-l-1/2}(\cosh r)\approx (r/2)^{l+1/2}/\Gamma (l+3/2)$, and in the boundary induced part at the sphere center the $l=0$ term contributes only:$$\langle \varphi ^{2}\rangle _{\mathrm{ren}}=\langle \varphi ^{2}\rangle _{0,% \mathrm{ren}}-\frac{1}{2\pi ^{2}a^{2}}\int_{x_{M}}^{\infty }dx\,\frac{% x^{2}(x^{2}-x_{M}^{2})^{-1/2}}{e^{2xr_{0}}-1},\quad r=0. \label{phi2Cent}$$where we have used formulae (\[SpCase1\]). Note that for a conformally coupled field the boundary induced part in (\[phi2Cent\]) coincides with the corresponding quantity for the sphere with radius $ar_{0}$ in the Minkowski bulk. Conclusion {#sec:Conclus} ========== The associated Legendre functions are an important class of special functions that appear in a wide range of problems of mathematical physics. In the present paper, specifying the functions in the generalized Abel-Plana formula in the form (\[gz\]), we have derived summation formula ([SumFormula]{}) for the series over the zeros of the associated Legendre function $P_{iz-1/2}^{\mu }(u)$ with respect to the degree. This formula is valid for functions $h(z)$ meromorphic in the right half-plane and obeying condition (\[Cond2\]). Using formula (\[SumFormula\]), the difference between the sum over the zeros of the associated Legendre function and the corresponding integral is presented in terms of an integral involving the Legendre associated functions with real values of the degree plus residue terms. For a large class of functions $h(z)$ this integral converges exponentially fast and, in particular, is useful for numerical calculations. Frequently used two standard forms of the Abel-Plana formula are obtained as special cases of formula (\[SumFormula\]) with $\mu =-1/2$ and $\mu =1/2$ and for an analytic function $h(z)$. Applying the summation formula for the series over the zeros of the function $P_{iz-1/2}^{\mu }(u\cosh (\eta /s))$ and taking the limit $s\rightarrow \infty $ we have obtained formula ([SumBess]{}) for the summation of the series over zeros of the Bessel function. The latter is a special case of the formula, previously derived in [@Sah1]. Further, we specify the summation formula for two special cases of the order $\mu =-l$ and $\mu =-l-1/2$ with $l$ being a non-negative integer and give examples of the application of this formula. The associated Legendre functions with these values of the order arise as solutions of the wave equation on background of constant curvature spaces in cylindrical and spherical coordinates. In section \[sec:Phys\] we consider a physical application of the summation formula. Namely, for a quantum scalar field we evaluate the positive frequency Wightman function and the vacuum expectation value of the field squared inside a spherical shell in a constant negative curvature space assuming that the field obeys the Dirichlet boundary condition on the shell. In spherical coordinates the radial part of the corresponding eigenfunctions contains the function $P_{iz-1/2}^{-l-1/2}(\cosh r)$ and the eigenfrequencies are expressed in terms of the zeros $z_{k}$ by relation (\[eigfreq\]). As a result, the mode-sum for the Wightman function includes the summation over these zeros. For the evaluation of the corresponding series we apply summation formula (\[SumFormula\]) with the function $h(z)$ given by (\[hz\]). The term with the first integral on the right-hand side of formula (\[SumFormula\]) corresponds to the Wightman function for the constant curvature space without boundaries and the term with the second integral is induced by the spherical boundary. For points away from the shell the latter is finite in the coincidence limit and can be directly used for the evaluation of the boundary induced part in the vacuum expectation value of the field squared. The latter is given by the second term on the right-hand side of formula (\[phi2\]). The renormalization is necessary for the boundary-free part only and this procedure is the same as that in quantum field theory without boundaries. On the physical example considered we have demonstrated the advantages for the application of the Abel-Plana-type formulae in the evaluation of the expectation values of local physical observables in the presence of boundaries. For the summation of the corresponding mode-sums the explicit form of the eigenmodes is not necessary and the part corresponding to the boundary-free space is explicitly extracted. Further, the boundary induced part is presented in the form of an integral which rapidly converges and is finite in the coincidence limit for points away from the boundary. In this way the renormalization procedure for local physical observables is reduced to that in quantum field theory without boundaries. Note that methods for the evaluation of global characteristics of the vacuum, such as total Casimir energy, in problems where the eigenmodes are given implicitly as zeros of a given function, are described in references [@Eliz94]. Acknowledgements {#acknowledgements .unnumbered} ================ The work was supported by the Armenian Ministry of Education and Science Grant No. 119 and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil). On zeros of the function $P_{iz-1/2}^{\protect\mu }(u)$ {#sec:Zeros} ======================================================= In this appendix we show that the zeros $z=z_{k}$ are simple and real. By making use of the differential equation for the associated Legendre functions it can be seen that the following integration formula takes place$$\int du\,P_{\nu ^{\prime }}^{\mu }(u)P_{\nu }^{\mu }(u)=(1-u^{2})\frac{% P_{\nu ^{\prime }}^{\mu }(u)\partial _{u}P_{\nu }^{\mu }(u)-P_{\nu }^{\mu }(u)\partial _{u}P_{\nu ^{\prime }}^{\mu }(u)}{\nu ^{\prime }(\nu ^{\prime }+1)-\nu (\nu +1)}+\mathrm{const}. \label{int1}$$Taking the limit $\nu ^{\prime }\rightarrow \nu $ and applying the Lopital’s rule for the right-hand side, from this formula we find$$\int du\,[P_{\nu }^{\mu }(u)]^{2}=(1-u^{2})\frac{[\partial _{\nu }P_{\nu }^{\mu }(u)]\partial _{u}P_{\nu }^{\mu }(u)-P_{\nu }^{\mu }(u)\partial _{\nu }\partial _{u}P_{\nu }^{\mu }(u)}{2\nu +1}+\mathrm{const}. \label{int2}$$By taking into account the relation $P_{-iz-1/2}^{\mu }(u)=P_{iz-1/2}^{\mu }(u)$, we see that for real $z$ one has $[P_{iz-1/2}^{\mu }(u)]^{2}=|P_{iz-1/2}^{\mu }(u)|^{2}$. Hence, from formula (\[int2\]) we find$$\int_{1}^{u}dv\,|P_{iz-1/2}^{\mu }(v)|^{2}=\frac{u^{2}-1}{2z}\left\{ [\partial _{z}P_{iz-1/2}^{\mu }(u)]\partial _{u}P_{iz-1/2}^{\mu }(u)-P_{iz-1/2}^{\mu }(u)\partial _{z}\partial _{u}P_{iz-1/2}^{\mu }(u)\right\} . \label{int3}$$Here we have taken into account that for $u\rightarrow 1$ one has $% P_{iz-1/2}^{\mu }(u)\sim (u-1)^{-\mu }$ and, hence, $\lim_{u\rightarrow 1}P_{iz-1/2}^{\mu }(u)=0$ for $\mu <0$. From formula (\[int3\]) it follows that $[\partial _{z}P_{iz-1/2}^{\mu }(u)]_{z=z_{k}}\neq 0$, and, hence, the zeros $z_{k}$ are simple. Now let us show that under the conditions $u>1$ and $\mu \leqslant 0$ all zeros of the function $P_{iz-1/2}^{\mu }(u)$ are real. Suppose that $% z=\lambda $ is a zero of $P_{iz-1/2}^{\mu }(u)$ which is not real. As the function $P_{z-1/2}^{\mu }(u)$ has no real zeros (see, for instance, [Grad]{}), $\lambda $ is not a pure imaginary. If $\lambda ^{\ast }$ is the complex conjugate to $\lambda $, then it is also a zero of $P_{iz-1/2}^{\mu }(u)$, because $P_{i\lambda ^{\ast }-1/2}^{\mu }(v)=[P_{i\lambda -1/2}^{\mu }(v)]^{\ast }$. As a result, from formula (\[int1\]) we find$$\int_{1}^{u}dv\,P_{i\lambda ^{\ast }-1/2}^{\mu }(v)P_{i\lambda -1/2}^{\mu }(v)=0. \label{int4}$$We have obtained a contradiction, since the integrand on the left hand-side is positive. Hence the number $\lambda $ cannot exist and the function $% P_{iz-1/2}^{\mu }(u)$ has no zeros which are not real. From the asymptotic formula (\[largey\]) for the function $P_{iz-1/2}^{\mu }(u)$ (see appendix \[sec:LegAsymp\] below) we obtain the asymptotic expression for large zeros:$$z_{k}\sim (\pi k-\pi \mu /2-\pi /4)/\eta . \label{zkAsymp}$$Note that this result can also be obtained by taking into account that for large values $z$ from (\[Plim\]) one has $P_{iz-1/2}^{-\mu }(\cosh (\eta ))\approx z^{-\mu }J_{\mu }(\eta z)$ and using the asymptotic form for the zeros of the Bessel function  (see, for instance, [@Abra72]). Asymptotics of the associated Legendre functions {#sec:LegAsymp} ================================================ In this appendix we consider asymptotic expressions for the associated Legendre functions for large values of the degree. As a starting point we use the formula $$Q_{z-1/2}^{\mu }(\cosh \eta )=\sqrt{\pi }e^{i\mu \pi }\frac{\Gamma (1/2+z+\mu )}{\Gamma (1+z)}\frac{(1-e^{-2\eta })^{\mu }}{e^{(z+1/2)\eta }}% F(1/2+\mu ,1/2+z+\mu ;1+z;e^{-2\eta }). \label{Qform1}$$Using the linear transformation formula 15.3.4 from [@Abra72] for the hypergeometric function, the expression for the function $Q_{z-1/2}^{\mu }(\cosh \eta )$ is presented in the form$$Q_{z-1/2}^{\mu }(\cosh \eta )=\sqrt{\pi }e^{i\mu \pi }\frac{\Gamma (1/2+z+\mu )}{\Gamma (1+z)}\frac{e^{-z\eta }}{\sqrt{2\sinh \eta }}F(1/2+\mu ,1/2-\mu ;1+z;1/(1-e^{2\eta })). \label{Qform2}$$Now, by using the result that for large $|c|$ one has $F(a,b;c;z)=1+O(1/|c|)$, from (\[Qform2\]) the asymptotic formula for the function $% Q_{z-1/2}^{\mu }(\cosh \eta )$ is obtained for large values $|z|$. The corresponding formula for the function $P_{z-1/2}^{\mu }(\cosh \eta )$ is obtained by using the relation$$\pi e^{i\mu \pi }\sin (\pi z)P_{z-1/2}^{\mu }(\cosh \eta )=\cos [\pi (z-\mu )]Q_{-z-1/2}^{\mu }(\cosh \eta )-\cos [\pi (z+\mu )]Q_{z-1/2}^{\mu }(\cosh \eta ). \label{RelPQ}$$In this way we obtain the following formulae$$\begin{aligned} P_{z-1/2}^{\mu }(\cosh \eta ) &\sim &\sqrt{\frac{2}{\pi }}\frac{y^{\mu -1/2}% }{\sqrt{\sinh \eta }}\sin (\eta y-i\eta x+\pi \mu /2+\pi /4), \notag \\ Q_{z-1/2}^{\mu }(\cosh \eta ) &\sim &\sqrt{\frac{\pi }{2}}\frac{y^{\mu -1/2}% }{\sqrt{\sinh \eta }}\exp [-\eta x-i(\eta y-\pi \mu /2-\pi /4)], \label{largey}\end{aligned}$$in the limit $y\rightarrow +\infty $, $z=x+iy$, and the formulae$$\begin{aligned} P_{z-1/2}^{\mu }(\cosh \eta ) &\sim &\frac{x^{\mu -1/2}}{\sqrt{2\pi \sinh \eta }}e^{\eta x+i\eta y}, \notag \\ Q_{z-1/2}^{\mu }(\cosh \eta ) &\sim &\sqrt{\frac{\pi }{2}}e^{i\mu \pi }\frac{% x^{\mu -1/2}}{\sqrt{\sinh \eta }}e^{-\eta x-i\eta y}, \label{largex}\end{aligned}$$in the limit $x\rightarrow +\infty $. Now let us consider the asymptotics of the functions $P_{i\nu -1/2}^{-\mu }(\cosh (\eta /\nu ))$, $Q_{\pm i\nu -1/2}^{-\mu }(\cosh (\eta /\nu ))$ as $% \nu \rightarrow +\infty $. These asymptotics are obtained in the way similar to that used in [@Erde53b] for formulae (\[PQlim\]). Our starting point is the formula $$P_{i\nu -1/2}^{-\mu }(\cosh (\eta /\nu ))=\frac{\tanh ^{\mu }(\eta /2\nu )}{% \Gamma (1+\mu )}F(1/2-i\nu ,1/2+i\nu ;1+\mu ;-\sinh ^{2}(\eta /2\nu )), \label{PtoF}$$relating the associated Legendre function to the hypergeometric function. From the definition of the hypergeometric function it is not difficult to see that $$\lim_{\nu \rightarrow +\infty }F(1/2-i\nu ,1/2+i\nu ;1+\mu ;-\sinh ^{2}(\eta /2\nu ))=\Gamma (1+\mu )(2/\eta )^{\mu }J_{\mu }(\eta ). \label{Flim}$$Combining (\[PtoF\]) and (\[Flim\]) we obtain formula (\[Plim\]). The corresponding formula for the functions $Q_{\pm i\nu -1/2}^{-\mu }(\cosh (\eta /\nu ))$ are obtained by making use of the relation$$\frac{2}{\pi }\sin (\mu \pi )e^{i\mu \pi }Q_{\pm i\nu -1/2}^{-\mu }(u)=\frac{% \Gamma (\pm i\nu -\mu +1/2)}{\Gamma (\pm i\nu +\mu +1/2)}P_{i\nu -1/2}^{\mu }(u)-P_{i\nu -1/2}^{-\mu }(u), \label{QPrel}$$and formula (\[Plim\]). In this way we find $$\lim_{\nu \rightarrow +\infty }\nu ^{\mu }e^{i\mu \pi }Q_{\pm i\nu -1/2}^{-\mu }(\cosh (\eta /\nu ))=\pi \frac{e^{\mp i\mu \pi }J_{-\mu }(\eta )-J_{\mu }(\eta )}{2\sin (\mu \pi )}, \label{Qlim}$$ or in the equivalent form $$\begin{aligned} \lim_{\nu \rightarrow +\infty }\nu ^{\mu }e^{i\mu \pi }Q_{i\nu -1/2}^{-\mu }(\cosh (\eta /\nu )) &=&-\frac{\pi i}{2}e^{-i\mu \pi }H_{\mu }^{(2)}(\eta ), \notag \\ \lim_{\nu \rightarrow +\infty }\nu ^{\mu }e^{i\mu \pi }Q_{-i\nu -1/2}^{-\mu }(\cosh (\eta /\nu )) &=&\frac{\pi i}{2}e^{i\mu \pi }H_{\mu }^{(1)}(\eta ), \label{Qasymp}\end{aligned}$$where $H_{\mu }^{(1,2)}(\eta )$ are the Hankel functions. [99]{} G.H. Hardy, *Divergent Series* (Chelsea Publishing Company, New York, 1991). P. Henrici, *Applied and Computational Complex Analysis,* Vol. 1 (Wiley, New York, 1974). A.A. Saharian, “The generalized Abel-Plana formula with applications to Bessel functions and Casimir effect,” Preprint ICTP/2007/082; arXiv: 0708.1187. A.A. Grib, S.G. Mamayev, and V.M. Mostepanenko, *Vacuum Quantum Effects in Strong Fields* (Friedmann Laboratory Publishing, St. Petersburg, 1994). V.M. Mostepanenko and N.N. Trunov, *The Casimir Effect and Its Applications* (Oxford University Press, Oxford, 1997). A.A. Saharian, Izv. AN Arm. SSR. Matematika **22**, 166 (1987) \[Sov. J. Contemp. Math. Analysis, **22**, 70 (1987)\]. G. Barton, J. Phys. A **14**, 1009 (1981); G. Barton, J. Phys. A **15**, 323 (1982). Yu.B Zayaev and I.Yu. Sokolov, In Abstracts of the 3rd All-Union Conference on “Quantum Metrology and Fundamental Physical Constants” (VNIIM “D.I. Mendeleev”, Leningrad, 1988 (in Russian)). A.A. Saharian, “Electromagnetic vacuum effects in presence of macroscopic bodies,” PhD thesis (Yerevan, 1987, in Russian). A.A. Saharian, “The generalized Abel-Plana formula. Applications to Bessel functions and Casimir effect,” Preprint IC/2000/14; hep-th/0002239. A.A. Saharian, Generalized Abel-Plana formula as a renormalization tool in quantum field theory with boundaries. Proceedings of the Fifth International Conference on Mathematical Methods in Physics, 24-28 April, 2006, Rio de Janeiro, Brazil; PoS(IC2006)019. G. Plunien, B. Muller, and W.Greiner, Phys. Rept. **134**, 87 (1986); M. Bordag, U. Mohidden, and V.M. Mostepanenko, Phys. Rept. **353**, 1 (2001); K.A. Milton, *The Casimir effect: Physical Manifestation of Zero-Point Energy* (World Scientific, Singapore, 2002). A. Romeo and A.A. Saharian, J. Phys. A **35**, 1297 (2002); A.A. Saharian and G. Esposito, J. Phys. A **39**, 5233 (2006). A.A. Saharian, Phys. Rev. D **63**, 125007 (2001); A.A. Saharian and M.R. Setare, Class. Quantum Grav. **20**, 3765 (2003); A.A. Saharian, Astrophys. **47**, 303 (2004); A.A. Saharian and M.R. Setare, Int. J. Mod. Phys. A **19**, 4301 (2004); A.A. Saharian and E.R. Bezerra de Mello, J. Phys. A **37**, 3543 (2004); A. A. Saharian and E. R. Bezerra de Mello, Int. J. Mod. Phys. A **20**, 2380 (2005); E.R. Bezerra de Mello and A.A. Saharian, Class. Quantum Grav. **23**, 4673 (2006). A. Romeo and A.A. Saharian, Phys. Rev. D **63**, 105019 (2001); A.A. Saharian and A.S. Tarloyan, J. Phys. A **39**, 13371 (2006); E.R. Bezerra de Mello, V.B. Bezerra, A.A. Saharian, and A.S. Tarloyan, Phys. Rev. D **74**, 025017 (2006); E.R. Bezerra de Mello, V.B. Bezerra, and A.A. Saharian, Phys. Lett. B **645**, 245 (2007). A.H. Rezaeian and A.A. Saharian, Class. Quantum Grav. **19**, 3625 (2002); A.A. Saharian and A.S. Tarloyan, J. Phys. A **38**, 8763 (2005); A.A. Saharian, Eur. Phys. J. C **52**, 721 (2007); A.A. Saharian and A.S. Tarloyan, Ann. Phys., in press, arXiv:0708.2970 \[hep-th\]. A.A. Saharian, Class. Quantum Grav. **19**, 5039 (2002); R.M. Avagyan, A.A. Saharian, and A.H. Yeranyan, Phys. Rev. D **66**, 085023 (2002); A.A. Saharian, R.M. Avagyan, and R.S. Davtyan, Int. J. Mod. Phys. A **21**, 2353 (2006). A.A. Saharian, Nucl. Phys. B **712**, 196 (2005); A.A. Saharian, Phys. Rev. D **73**, 044012 (2006); A.A. Saharian, Phys. Rev. D **73**, 064019 (2006). A.A. Saharian and M.R. Setare, JHEP **0702**, 089 (2007). A.A. Saharian, A.S. Kotanjyan, and M.L. Grigoryan, J. Phys. A **40**, 1405 (2007). A. Erdélyi *et al*, *Higher Transcendental Functions*, Vol. 1 (McGraw Hill, New York, 1953). *Handbook of Mathematical Functions*, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1972). N.D. Birrell and P.C.W. Davis, *Quantum Fields in Curved Space* (Cambridge University Press, Cambridge, England, 1982). S. Nojiri, S. Odintsov, and S. Zerbini, Class. Quantum Grav. **17**, 4855 (2000); W. Naylor and M. Sasaki, Phys. Lett. B **542**, 289 (2002); E. Elizalde, S. Nojiri, S.D. Odintsov, and S. Ogushi, Phys. Rev. D **67**, 063515 (2003); I.G. Moss, W. Naylor, W. Santiago-Germán, M. Sasaki, Phys. Rev. D **67**, 125010 (2003); A. Flachi, A. Knapman, W. Naylor, and M. Sasaki, Phys. Rev. D **70**, 124011 (2004); J.P. Norman, Phys.Rev. D **69**, 125015 (2004); W. Naylor and M. Sasaki, Prog. Theor. Phys. **113**, 535 (2005); M. Minamitsuji, W. Naylor, and M. Sasaki, Nucl. Phys. B **737**, 121 (2006). A. Erdélyi *et al*, *Higher Transcendental Functions*, Vol. 2 (McGraw Hill, New York, 1953). A.A. Grib, B.A. Levitskii, and V.M. Mostepanenko, Theor. Math. Phys. **19**, 349 (1974). E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, and S. Zerbini, Zeta regularization techniques with applications (World Scientific, Singapore, 1994); E. Elizalde, S. Leseduarte, and A. Romeo, J. Phys. A **26**, 2409 (1993); S. Leseduarte and A. Romeo, J. Phys. A **27**, 2483 (1994); M. Bordag, J. Phys. A **28**, 755 (1995); M. Bordag and K. Kirsten, Phys. Rev. D **53**, 5753 (1996); M. Bordag, E. Elizalde, and K. Kirsten, J. Math. Phys. **37**, 895 (1996); S. Leseduarte and A. Romeo, Ann. Phys. **250**, 448 (1996); M. Bordag, K. Kirsten, and J.S. Dowker, Commun. Math. Phys. **182**, 371 (1996); M. Bordag, E. Elizalde, K. Kirsten, and S. Leseduarte, Phys. Rev. D **56**, 4896 (1997); E. Elizalde, M. Bordag, and K. Kirsten, J. Phys. A **31**, 1743 (1998); V.V. Nesterenko and I.G. Pirozhenko, Phys. Rev. D 57, 1284 (1998). I.S. Gradshtein and I.M. Ryzhyk, *Tables of Integrals, Series and Products* (Academic Press, New York, 1980). [^1]: E-mail: saharian@ictp.it

--- abstract: 'We propose an optical read-out scheme allowing a demonstration of principle of information extraction below the diffraction limit. This technique, which could lead to improvement in data read-out density onto optical discs, is independent from the wavelength and numerical aperture of the reading apparatus, and involves a multi-pixel array detector. Furthermore, we show how to use non classical light in order to perform bit discrimination beyond the quantum noise limit.' author: - 'V. Delaubert' - 'N. Treps' - 'G. Bo' - 'C. Fabre' title: 'Optical storage of high density information beyond the diffraction limit : a quantum study' --- Introduction {#introduction .unnumbered} ============ The reconstruction of an object from its image beyond the diffraction limit, typically of the order of the wavelength, is a hot field of research, though a very old one, as Bethe already dealt with the theory of diffraction by sub-wavelength holes in 1944 [@Bethe]. More recently, theory has been developed to be applied to the optical storage problem, in order to study the influence of very small variations of pit width or depth relative to the wavelength [@Bethe; @Marx1; @Marx2; @Wang; @Liu; @Brok]. To date, only a few super-resolution techniques [@Kolobov] include a quantum treatment of the noise in the measurement, but to our knowledge, none has been applied to the optical data storage problem. Optical discs are now reaching their third generation, and have improved their data capacity from 0.65 GB for compact discs (using a wavelength of 780 nm), to 4.7 GB for DVDs ($\lambda$ = 650 nm), and eventually to 25 GB for the Blu-Ray discs (using a wavelength of 405 nm). In addition to new coding techniques, this has been achieved by reducing the spot size of the diffraction-limited focused laser beam onto the disc, involving higher numerical apertures and shorter wavelengths. Several further developments are now in progress, such as the use of volume holography, 266 nm reading lasers, immersion lenses, near field systems, multi-depths pits [@Hsu], or information encoding on angle positions of asymmetrical pits [@TorokCD]. These new techniques rely on bit discrimination using small variations of the measured signals. Therefore, the noise is an important issue, and ultimately, quantum noise will be the limiting factor. In this paper, we investigate an alternative and complementary way to increase the capacity of optical storage, involving the retrieval of information encoded on a scale smaller than the wavelength of the optical reading device. We investigate a way to optimize the detection of sub-wavelength structures using multi-pixel array. An attempt to a full treatment of the optical disc problem being far too complex, we have chosen to illustrate our proposal on a very simple example, leaving aside most technical constraints and complications, but still involving all the essence of the overall problem. We first explain how the use of an array detector can lead to an improvement of the detection and distinction of sub-wavelength structures present in the focal spot of a laser beam. We then focus on information extraction from an optical disc with a simple but illustrative example, considering that only a few bits are burnt on the dimensions of the focal spot of the reading laser, and show how the information is encoded from the disc to the light beam, propagated to the detector, and finally detected. We explain the gain configuration of the array detector that has to be chosen in order to improve the signal-to-noise ratio (SNR) of the detection. Moreover, as quantum noise is experimentally accessible, and will be a limiting factor for further improvements, we perform a quantum calculation of the noise in the detection process. Indeed, we present how this detection can be optimized to perform simultaneous measurements below the quantum noise limit, using non classical light. Proposed scheme for bit sequence recognition in optical discs ============================================================= ![Color online [**Scheme for information extraction from optical disc, using an array detector.**]{}[]{data-label="general_scheme"}](Fig1-general_scheme){width="7cm"} We propose a novel optical read-out scheme shown on figure (\[general\_scheme\]) allowing information extraction from optical discs beyond the diffraction limit, based on multi-pixel detection. Bits, coded as pits and holes on the optical disc, induce phase flips in the electric field transverse profile of the incident beam at reflection. The reflected beam is imaged in the far field of the disc plane, where the detector stands. In the far field, the phase profile induced by the disc is converted into an intensity profile, that the multi-pixel detectors can, at least partly, reconstruct. Taking into account that a lot of a priori information is available - i.e. only a finite number of intensity profiles is possible - we propose to use a detector with a limited number of pixels $D_{k}$ whose gains can independently be varied depending on which bit sequence one wants to detect. The signal is then given by $$\begin{aligned} \label{signal} S = \sum_{k}\sigma_{k}N_{k}\end{aligned}$$ where $N_{k}$ is the mean photon number detected on pixel $D_{k}$, and $\sigma_{k}$ is the electronic gain of the same pixel. Ideally, to each bit sequence present on the disc corresponds a set of gains chosen so that the value of the measurement is zero, thus cancelling noise from the mean field. Measuring the signal for a given time interval $T$ around the centered position of a bit sequence in the focal spot, and testing, in parallel, all the pre-define sets of gain in the remaining time, allows to deduce which bit sequence is present on the disc. We will first show that this improvement in density of information encoded on an optical disc is already possible using classical resources. Moreover, as the measurement is made around a zero mean value, the classical noise is mostly cancelled. Hence, we reach regimes where the quantum noise can be the limiting factor. We will demonstrate how to perform measurements beyond the quantum noise limit, using previous results on quantum noise analysis in multi-pixel detection developed in reference [@Treps]. Encoding information from a disc onto a light beam ================================================== We have explained the general principle of reading-out sub-wavelength bit sequences encoded on an optical disc, and now focus on the information transfer from the optical disc to the laser beam, through an illustrative example. Let us recall that bits are encoded by pits and holes on the disc surface: a step change from hole to pit (or either from pit to hole) encodes bit $1$, whereas no depth change on the surface encodes bit $0$, as represented on figure (\[bits\]). A hole depth of $\lambda/4$ insures a $\pi$ phase shift between the fields reflected by a pit and a hole. In this section, we compute the incident field distribution on the optical disc affected by the presence of a bit sequence in the focal spot, and finally analyze the intensity back reflected in the far field, in the detection plane, as sketched on figure (\[general\_scheme\]). ![Color online [**Example of bit sequence on an optical disc. The spacing between the bits is smaller than the wavelength, the minimum waist of the incident laser beam being of the order of $\lambda$. A hole depth of $\lambda/4$ insures a $\pi$ phase shift between fields reflected on a pit and a hole.**]{}[]{data-label="bits"}](Fig2-bits){width="7cm"} Beam focalization ----------------- Current optical disc read-out devices involve a linearly polarized beam strongly focused on the disc surface to point out details whose size is of the order of the laser wavelength. The numerical aperture ($NA$) of the focusing lens can be large ($0.47$ for CDs, $0.6$ for DVDs, and $0.85$ for BLU RAY discs), and the exact calculation of the field cannot be done in the paraxial and scalar approximation. Thus, the vectorial theory of diffraction has to be taken into account. The structure of the electromagnetic field in the focal plane of a strongly focused beam has been investigated for decades now [@VanNie], as its applications include areas such as microscopy, laser micro-fabrication, micromanipulation, and optical storage [@Landesman; @Rodriguez; @Ulanowski; @Lax; @Seshadri; @Ciattoni; @Cao; @Nieminen]. In our case of interest, we can restrict the field calculation to the focal plane, which is the disc plane. Thus Richards and Wolf integrals [@Richards], which are not suitable for a general propagation of the field, but which can provide the field profile in the focal plane for any type of polarization of the incoming beam as long as the focusing length is much larger than the wavelength, can be used to achieve this calculation. These integrals have already been used in many publications dealing with tight focusing processes [@Quabis1; @Novotny; @Dorn; @Quabis2; @Sheppard; @Torok; @Youngworth; @Zhan]. As highlighted in these references, the importance of the vectorial aspect of the field can easily be understood when a linearly polarized beam is strongly focused, as the polarization of the wave after the lens is not perpendicular to the propagation axis anymore and has thus components along this axis. In order to estimate the limit of validity of the paraxial approximation, we computed focused spot sizes of linearly polarized beam in the focal plane for different numerical apertures, first in the paraxial approximation, and then calculated with Richards and Wolf integrals. The results are compared on figure (\[waistcomparison\]) for an incident plane wave in air medium with $\lambda=780nm$, where the spot size is defined as the diameter which contains $86\%$ of the focused energy, as in reference [@Siegman]. ![Color online[**Evolution of the focused spot size of an incident plane wave with the numerical aperture (for $\lambda$ = 780 nm in air medium). The spot size is limited to the order of the wavelength in the non-paraxial case (o), whereas it goes to zero for very high numerical apertures in the paraxial case($\ast$).**]{}[]{data-label="waistcomparison"}](Fig3-waist_NA){width="9cm"} We see that when the numerical aperture exceeds $0.6$, a good prediction requires a non-paraxial treatment. Moreover, whereas there is no theoretical limit to focalization in the paraxial case, we see that non paraxial effects prevent us to reach a waist smaller than the order of the wavelength. Note that this limit is not fundamental and can be overcome by modifying the polarization of the incoming beam. Quabis [*et al.*]{} have indeed managed to reduce the spot area to about $0.1~\lambda^{2}$ using an incident radially polarized doughnut beam [@Quabis1; @Quabis2]. As our aim is to present a demonstration of principle and not a full treatment of the optical disc problem, the following calculations will be done using the physical parameters of the actual Compact Discs ($\lambda=780nm$ and $NA=0.47$, corresponding to a focalization angle of $27$ degrees in air medium). In this case, the paraxial and scalar approximations are still valid. Indeed, figure (\[fieldcomponents\]), giving the transverse profile of the three field components and the resultant intensity in the focal plane using the former parameters, shows that although the field is not strictly linearly polarized as foreseen before, $E_{y}\ll E_{z}\ll E_{x}$, and we can thus consider that only $E_{x}$ is different from zero with a good approximation. Note that the exact expression would not intrinsically change the problem, as our scheme can be adapted to any field profile discrimination. ![Color online[**Norm of the different field components and resultant intensity in the focal plane with a linearly polarized incident field along the $x$ axis, focused with a $0.47$ numerical aperture.**]{}[]{data-label="fieldcomponents"}](Fig4-components){width="7cm"} Reflection onto the disc ------------------------ In order to compute the reflected field, we simply assume that bumps and holes are generated in such a way that they induce a $\pi$ phase shift between them at reflection on the field profile. Note that the holes depth is usually $\lambda/4$, but precise calculations would be required to give the exact shape of the pits, as they are supposed to be burnt below the wavelength size, and as the field penetration in those holes is not trivial [@Wang; @Brok; @Liu]. As we have shown that only one vectorial component of the field was relevant in the focal plane, we can directly apply this phase shift to the amplitude profile of this component. We first envision a scheme with only three bits in the focal spot, which means that $2^{3}$ different bit sequences, i.e. a byte, have to be distinguished from each other, using the information extracted from the reflected field. Note that we neglect the influence of other bits in the neighborhood. A more complete calculation involving this effect with more bits will be considered in a further approach. The amplitude profiles obtained when the incident beam is centered on a bit of the CD are presented on figure (\[summary\]), for a particular bit sequence. Note that we have chosen the space between two bits on the disc equal to the waist size of the reading beam. The first three curves respectively show the field amplitude profile incident on the disc, an example of a bit sequence, and the corresponding profile just after reflection onto the disc. We see that binary information is encoded from bumps and holes on the CD to phase flips in the reflected field. ![Color online[**Modifications of the transverse amplitude field profile trough propagation, in the case of a $111$ bit sequence in the focal spot : a) incoming beam profile, b) 111 bit sequence, c) corresponding reflected field in the disc plane, d) far field profile in the detector plane .**]{}[]{data-label="summary"}](Fig5-summary){width="9cm"} Back propagation to the detector plane -------------------------------------- In order to extract the information encoded in the transverse amplitude profile of the beam, the field has to be back propagated to the detector plane. A circulator, composed of a polarizing beam splitter and a Faraday rotator, ensures that the linearly polarized reflected beam reaches the array detector, as shown on figure (\[general\_scheme\]). Assuming that the detector is positioned just behind the lens plane, the expression of the detected field is given by the far field of the disc plane, apertured by the diameter of the focusing lens. As the focal length and the diameter of the lens are large compared to the wavelength, we use Rayleigh Sommerfeld integral to compute the field in the lens plane [@Born]. As an example, the calculated far field profile when the bit sequence $111$ is present in the focal spot is shown on the fourth graph of figure (\[summary\]). The presence of the lens provides a limited aperture for the beam and cuts the high spatial frequencies of the field, which can be a source of information loss, as the difference between each bit sequence can rely on those high frequencies. However, we will see that enough information remains in the low frequency part of the spatial spectrum, so that the $8$ bits can be distinguished. This is due to the fact that we have in this problem a lot of a priori information on the possible configurations to distinguish. We see on figure (\[far\_field\]) that, with the physical parameters used in compact disc read out devices, 6 over 8 profiles in the detector plane are still different enough to be distinguished. ![Color online[**Field profiles in the array detector plane, for each of the $8$ bit sequence configuration. Note that they are clearly distinguishable, except for the bit sequences $100$ and $001$, and $011$ and $110$, which have the same profile because of the symmetry of the bit sequence relative to the position of the incident laser beam.** ]{}[]{data-label="far_field"}](Fig6-far_field.eps){width="9cm"} At this stage, we are nevertheless unable to discriminate between symmetric configurations, because they give rise to the same far field profile. Therefore, 100 and 001, and 110 and 011, cannot be distinguished. Note that this problem can be solved thanks to the rotation of the disc. Indeed, an asymmetry is created when the position of the disc relative to the laser beam is shifted, thus modifying differently the two previously indistinguishable profiles. As shown on figure (\[non\_centered\]), where the far field profiles are represented after a shift of $w_{0}/6$ in the position of the disc, the degeneracy has been removed. Moreover, it is important to notice that the other profiles experience a small shape modification. This redundant information is very useful in order to remove ambiguities while the disc is rotating. ![Color online[**Field profiles in the array detector plane, for each of the $8$ bit sequence configuration, when the position of the disc has been shifted of $w_{0}/6$ relative to the incident beam. The profile degeneracy for $100$ and $001$, and $011$ and $110$ is raised. Note that the other profiles have experienced a much smaller shape modification between the two positions of the disc.** ]{}[]{data-label="non_centered"}](Fig7-FF_non_centered.eps){width="9cm"} Information extraction for bit sequence recognition =================================================== In this section, we describe the detection,present some illustrative results, and the way they can be used to increase the read-out precision of information encoded on optical discs. We show here that a pixellised detector with a very small number of pixels is enough to distinguish between the $8$ bit sequences. Note that for technical and computing time reasons, it is not realistic to use a CCD camera to record the reflected images, as such cameras cannot yet combine good quantum efficiency and high speed. Detected profiles ----------------- For simplicity reason, we limit our calculation to a $5$ pixels array detector $D_{1}..D_{5}$, each of whom has an electronic gain $\sigma_{1}..\sigma_{5}$, as shown on figure (\[FF\_detection\]). The size of each detector has been chosen without a systematic optimization, which will be done in a further approach. Gain values are adapted to detect a mean signal equal to zero for each bit configuration present in the focal spot, in order to cancel the common mode classical noise present in the mean field [@Treps]. It means that for each bit sequence $i$, gains are chosen to satisfy the following relation $$\label{gaindef} \sum_{k=1}^{5} \sigma_{k}(i)N_{k}(i) =0$$ where $N_{k}(i)$ is the mean photon number detected on pixel $D_{k}$ when bit $i$ is present in the focal spot on the disc $$\label{Idef} N_{k}(i) = \int_{D_{k}}n_{i}(x)dx$$ where $n_{i}(x)$ is the number of photon incident on the array detector, at position $x$, when bit sequence $i$ is present in the focal spot. As all profiles are symmetrical when the incident beam is centered on a bit, we have set $\sigma_{1}=\sigma_{5}$ and $\sigma_{2}=\sigma_{4}$. In addition, we have chosen $\sigma_{3}=-\frac{\sigma_{1}}{2}$. Using these relations and equation (\[gaindef\]), we compute gain values adapted to the recognition of each bit sequence. ![Color online[**Far field profiles in detection the plane for each bit configuration, and array detector geometry. The 5 detectors $D_{1}..D_{5}$ have electronic gains $\sigma_{1}(i)..\sigma_{5}(i)$ according to the bit sequence $i$ which is present in the focal spot.**]{}[]{data-label="FF_detection"}](Fig8-FF_detection.eps){width="8cm"} Note that the calculation of each gain configuration requires a priori information on the far field profiles, or at least an experimental calibration using a well-known sample. Now that these gain configurations are set, we can investigate for a bit sequence on the optical disc. Classical results ----------------- The expression of the detected signal $S_{i}(j)$ is given by $$\begin{aligned} \label{signal} S_{i}(j) = \sum_{k=1}^{5}\sigma_{k}(j)N_{k}(i)\end{aligned}$$ where $i$ refers to the bit sequence effectively present in the focal spot, and $j$ to the gain set adapted to the detection of the bit sequence $j$. It merely corresponds to the intensity weighted by the electronic gains. Note that for $i=j$ - and only in this case if the detector is well chosen - the mean value of the signal $S_{i}(i)$ is equal to zero, according to equation (\[gaindef\]). All possible values of $S_{i}(j)$ are presented for a total number of incident photons $N_{inc}=25$, in table (\[table\]) where $i$ is read vertically, and corresponds to the bit sequence on the disc, whereas $j$ is read horizontally and refers to the gain set adapted to the detection of bit $j$. In order not to have redundant information, we have gathered results corresponding to identical far field profiles. A zero value is obtained for only one gain configuration, allowing an identification of the bit sequence present in the focal spot. 000 001/100 010 011/110 101 111 --------- ----- --------- ------ --------- ----- ------ 000 0 -34 -204 -254 -77 -303 001/100 15 0 -76 -99 -19 -121 010 23 20 0 -6 16 -13 011/110 24 22 5 0 19 -5 101 19 11 -36 -50 0 -63 111 24 23 9 5 20 0 : [**Detected signals $S_{i}(j)$ where $i$ is read vertically and corresponds to the bit sequence on the disc, whereas $j$ is read horizontally and refers to the gain set adapted to the detection of bit $j$. A zero value means that the tested gain configuration is adapted to the bit sequence.**]{}[]{data-label="table"} The reading process to determine which bit sequence is lit on the disc follows these few steps : - the time dependent intensity is first measured on each of the five detectors with all electronic gains set to one. - these intensities are integrated for a time $T$. - the signal is then calculated, using the different gain configurations $j$ - the bit sequence effectively present in the focal spot is determined by the only signal yielding a zero value. Note that the second step just corresponds to the $N_{k}$ measurements. The integration time $T$ is chosen as the time interval during which the signal leads to the determination of a unique bit sequence. The third step corresponds to the simple calculation of a line in table \[table\]. This can be done in parallel thanks to the speed of data processing on dedicated processors, and the reading rate will thus not be affected compared to current devices. Finally, note that the last step requires a good choice of the parameters in order to be able to distinguish all bit sequences. It means that the noise level has to be smaller than the difference between the two closest values from $0$, in order to get a zero mean value for only one bit sequence. Indeed, there must be no overlap between the expectation values when we take into account the noise and thus the uncertainty relative to each measurement. Note that using the zero value as the discriminating factor could be combined with the use of all the calculated values, as each line of table (\[table\]) is distinct. We just need to know how to weight each data point according to the noise related to its obtention. Noise calculation ================= The shot noise limit -------------------- To include the noise in our calculation, we separate classical and quantum noise contributions. The classical noise comprises residual noise of the laser diode, mechanical and thermal vibrations. The major part of this noise is directly proportional to the signal, i.e. to the number of detected photons. For a detection of the total number of photons $N_{inc}$ in the whole beam during the integration time of the detector, the classical noise contribution $\sqrt{\langle\delta N_{inc}^{2}\rangle}$ would thus be written as $$\label{noise1} \sqrt{\langle\delta N_{inc}^{2}\rangle}=\beta N_{inc}$$ where $\beta$ is a constant factor. And the individual noise variable $\delta N_{i}(k)$ arising from detection on pixel $D_{k}$ is given by $$\label{noise2} \delta N_{i}(k)= \frac{N_{i}(k)}{N_{inc}}\delta N_{inc}$$ Using equations \[signal\], \[noise1\] and \[noise2\], a simple calculation yields the variance of the signal arising from the classical noise $$\langle \delta \hat{S}^{2}_{i}(j)\rangle_{Cl}=\frac{B {S}^{2}_{i}(j)}{N_{inc}}$$ where the constant $B=N_{inc}\beta^{2}$ is the classical noise factor, and is chosen so that, when $B=1$ and when all the intensity is detected by one detector, the classical noise term is equal to the shot noise term. Note that classical noise does not deteriorate measurements having a zero mean value. For this reason, we have chosen to discriminate bit sequences by choosing gains such as $S_{i}(i)=0$, as mentioned earlier. ![Color online[**Classical noise ($10~dB$ of excess noise) represented as error bars, for $\lambda=0.78 \mu m$, $NA=0.47$, and $25$ detected photons. Each inset corresponds to the $6$ signals obtained for the different gain configurations, when one particular bit sequence is present in the focal spot. Each bit sequence present in the focal spot can be clearly identified as only one gain configuration can give a zero value for each inset.**]{}[]{data-label="classical"}](Fig9-noise_classical.eps){width="8cm"} The calculation of the quantum contribution requires the use of quantum field operators, describing the quantum fluctuations in all transverse modes of the field. By changing the gain configuration of the array detector, not only the signal $S_{i}(j)$ is modified, but also the related quantum noise denoted $\langle \delta\hat{S}^{2}_{i}(j)\rangle_{Qu}$, as different gain configurations are sensitive to noise in different modes of the field. We have shown in reference [@Treps] that for a multi-pixel detection of an optical image, the measurement noise arises from only one mode component of the field, referred to as the [*detection mode*]{}, or [*noise-mode*]{} [@Del]. The expression of the quantum noise is then : $$\begin{aligned} \label{qnoise} \langle \delta \hat{S}^{2}_{i}(j)\rangle_{Qu} = f_{i,j}^{2} N_{inc}\langle\delta \hat{X}^{2}_{w_{i,j}}\rangle\end{aligned}$$ where $\delta \hat{X}_{w_{i,j}}$ is the quantum noise contribution of the noise-mode $w_{i,j}(x)$ which is defined for one set of gain $j$, when the bit sequence $i$ is present in the focal spot, as $$\label{wdef} \forall x \in D_{k} \qquad w_{i,j}(x) = \frac{\sigma_{k}(j)n_{i}(x)}{f_{i,j}}$$ and where $f_{i,j}$ is a normalization factor, which expression is $$\label{fdef} f^{2}_{i,j} = \frac{\sum_{k=1}^{5}\sigma^{2}_{k}(j)N_{k}(i)dx}{N_{inc}}$$ The noise-mode corresponds in fact to the incident field profile weighted by the gains. The shot noise level corresponds to $\langle\delta \hat{X}^{2}_{w_{i,j}}\rangle=1$. The variance of the signal can eventually be written as: $$\begin{aligned} \label{noisedef} \langle \delta \hat{S}^{2}_{i}(j)\rangle = f_{i,j}^{2} N_{inc}\langle\delta \hat{X}^{2}_{w_{i,j}}\rangle + \frac{B {S}^{2}_{i}(j)}{N_{inc}}\end{aligned}$$ ![Color online[**Shot noise represented as error bars, for $\lambda=0.78 \mu m$, $NA=0.47$, $25$ detected photons. Some bit sequences cannot be determined without ambiguity because of the noise level.**]{}[]{data-label="shot"}](Fig10-noise_shot.eps){width="8cm"} We have first represented the classical noise with an excess noise of $10~dB$, as error bars for each result $S_{i}(j)$, on figures (\[classical\]). We have chosen a representation with a number of detected photons of only $25$. Each of the $6$ insets refers to the measurement obtained for a particular bit sequence in the focal spot. The $6$ data points and associated error bars refer to the results obtained when the $6$ gain configurations are tested. One inset thus corresponds to one line in table (\[table\]). We can see that with this choice of parameters, the bit sequence effectively present in the focal spot can be determined without ambiguity by the only zero value. The sequence corresponds to the one for which the gains were optimized. We see that bit sequence discrimination can be achieved even with a very low number of photons. The relative immunity to classical noise of our scheme arises from the fact that measurements are performed around a zero mean value. Thus, given this limit in the minimum necessary photon number and the flux of photons one can calculate the maximum data rate, which is found to be $2.10^{7} Mbits/s$ (this estimation takes into account an integration time $T$ corresponding to $1/10$ of the delay between the read-out process of two adjacent bits with a $1mW$ laser). This very high value shows that classical noise should not be a limit for data rate in such a scheme. The effect of quantum noise is very small, but becomes a limiting factor for such a small number of detected photons, or for a large number of bits encoded on the disc in the wavelength size. In order to see independently the effect of each contribution to the noise, we have thus represented on figure (\[shot\]) the shot noise also for $25$ detected photons, appearing as the threshold under which it is impossible to distinguish bit sequences because of the quantum noise. Note that for the represented case, the shot noise is the most important contribution, and that it prevents a bit sequence discrimination, as a zero value for the signal can be obtained for several gain configurations in the same inset. Beyond the shot noise limit --------------------------- When the shot noise is the limiting factor, non classical light can be used to perform measurements beyond the quantum noise limit. We have shown in reference [@Treps] that squeezing the noise-mode of the incident field was a necessary and sufficient condition to a perfect measurement. What we are interested in is improving the measurements that yield a zero value, which are obtained when the gain configuration matches the bit sequence in the focal spot, as $S_{i}(i)=0$. Using equation (\[noisedef\]), we see that $w_{i,i}$ has to be squeezed. As no information on the bit present in the focal spot is available before the measurement, in order to improve simultaneously all the bit sequences detections, the $6$ noise-modes have to be squeezed at the same time in the incident field. These $6$ transverse modes are not necessarily orthogonal, but one can show that squeezing the subspace that can generate all of them is enough to induce the same amount of squeezing. ![Color online[**Quantum detection noise represented as error bars, for $\lambda=0.78 \mu m$, $NA=0.47$, $25$ detected photons and $-10~dB$ of simultaneous squeezing for all the flipped modes. The ambiguity in presence of shot noise has been removed and each bit sequence can be identified.**]{}[]{data-label="SQZ"}](Fig11-noise_sqz.eps){width="8cm"} The quantum noise with $10~dB$ of squeezing on the sub space generated by the $w_{i,i}$ is represented as error bars on figure (\[SQZ\]). The noise of each noise-mode $w_{i,j}$ is computed using its overlap integrals with the generator modes of the squeezed sub space, assuming that all modes orthogonal to the squeezed subspace are filled with coherent noise. In this case, the effect of squeezing, reducing the quantum noise on the measurements, and especially on the measurement for which the gains have been optimized, is enough to discriminate bit sequences that were masked by quantum noise. Conclusion ========== We have proposed a novel way of information extraction from optical discs, based on multi-pixel detection. We have first demonstrated, using only classical resources, that this detection could allow large data storage capacity, by burning several bits in the spot size of the reading laser. We have presented a demonstration of principle through a simple example which will be refined in further studies. We have also shown that in shot noise limited measurements, using squeezed light in appropriate modes of the incident laser beam can lead to improvement in bit sequence discrimination. The next steps are to study in details how to extract the redundant information when the disc is spinning, and to systematically optimize the number of bits in the focal spot, the number and size of pixels in the array detector. Such a regime involving a large number of bits in the focal spot will ultimately be limited by the shot noise, and will require the quantum noise calculations presented in this paper. Acknowledgment ============== We thank Magnus Hsu, Ping Koy Lam and Hans Bachor for fruitful discussions. Laboratoire Kastler Brossel, of the Ecole Normale Superieure and University Pierre et Marie Curie, is associated to CNRS. This work has been supported by the European Union in the frame of the QUANTIM network (contract IST 2000-26019). H. A. Bethe, Phys. Rev. 66 [**7 and 8**]{}, 163 (1944) D. S. Marx, and P. Psaltis, J. Opt. Soc. Am. A 14 [**6**]{}, 1268 (1997) D. S. Marx, and P. Psaltis, Appl. Opt. 36 [**25**]{}, 6434 (1997) X. Wang, J. Mason, M. Latta, T. C. Strand, D. S. Marx, and P. Psaltis, J. Opt. Soc. Am. A 18 [**3**]{} 565 (2001) W.-C. Liu, and M. W. Kowarz, Appl. Opt. 38 [**17**]{}, 3787 (1999) J. M. Brok, and H. P. Urbach, J. Opt. Soc. Am. A 20 [**2**]{}, 256 (2003) M. Kolobov [*et al*]{}, Opt. Let. [**85**]{}, 3789 (2000) M.T.L. Hsu, W.P.Bowen, C.Fabre, H.-A. Bachor and P.K. Lam, “Two-Pixel Phase Coding of Optical Beams Using Squeezed Light” (to be submitted) P. Török, Proceeding AOI conference (2005). N. Treps, V. Delaubert, A. Maître, J.M. Courty and C. Fabre, Phys. Rev. A [**71**]{}, 013820 (2005) A. G. van Nie, Rigorous calculation of the electromagnetic field of a wave beam, Philips Res. Rep. 19, 378?394 (1964). M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975). S. R. Seshadri, J. Opt. Soc. Am. 19, 2134 (2002) and references therein. A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000). Q. Cao and X. Deng, J. Opt. Soc. Am. A 15, 1144 (1998). B. T. Landesman and H.H. Barrett, J. Opt. Soc. Am. A 5, 1610 (1988). G. Rodriguez-Morales and S. Chavez-Cerda, Opt. Let. 5, 430 (2004). Z.Ulanowski and I. K. Ludlow, Opt. Let. 25 1792 (2000). T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, submitted to Elsevier. B. Richards, E.Wolf, Electromagnetic diffraction in optical systems, Proc. R. Soc. A 253, 359 (1959). S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001). L. Novotny, R. D. Grober, and K. Karrai, Ot. Let. 26, 789 (2001). R. Dorn, S. Quabis, and G. Leuchs, preprint : arxiv/physics/0310007. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Com. 179, 1 (2000). C. J. R. Sheppard, J. Opt. Soc. Am. A 18, 1579 (2001). P. Török, P. Varga, Z. Laczik, and G. R. Booker, J. Opt. Soc. Am. A 12, 325 (1995). K. S. Youngworth, and T. G. Brown, Opt. Expr. 7, 77 (2000). Q. Zhan, and J. R. Leger, Opt. Expr. 10, 324 (2002). A. E. Siegman, [*Lasers*]{}, Mill Valley, CA: University Science (1986) M. Born, E. Wolf, [*Principle of optics*]{}, Cambridge University Press (1959) V. Delaubert, N. Treps, C. C. Harb, P. K. Lam and H-A. Bachor, [*Surpassing the quantum noise limit on displacement and tilt measurement of a gaussian beam* ]{}, submitted in 2005

--- abstract: 'We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf-Witten theory. The appropriate initial data for the construction are certain three object categories with coefficients satisfying a partially degenerate cocycle condition.' author: - 'I. J. Lee and D. N. Yetter' title: 'Dijkgraaf-Witten Type Invariants of Seifert Surfaces in 3-Manifolds' --- Introduction ============ Throughout we work exclusively in the PL category, all manifolds are compact, and we write composition in diagrammatic order. The goal of this paper is to construct state-sum invariants of triples $K \subset \Sigma \subset M$, in which $M$ is an oriented 3-manifold, $K$ an oriented knot or link in $M$ and $\Sigma$ an oriented surface with boundary embedded in $M$ with $\partial \Sigma = K$, extending and generalizing the state-sum construction of finite-gauge group Dijkgraaf-Witten theory given by Wakui [@W]. To avoid confusion between link in the knot-theoretic sense and the link of a simplex in the sense of combinatorial topology, we will always refer to $K$ as “the knot”, even though it might be a link in the knot-theoretic sense. The arrangement $K \subset \Sigma \subset M$ is a very simple stratified space of the sort Crane and Yetter [@CY] called “starkly stratified ”. However, as was the case in two dimensions (cf. [@DPY]), our constructions will require us to restrict our attention to flag-like triangulations. The first task will, thus, be to show that the instances of extended bistellar moves which preserve flag-likeness suffice to give all PL homeomorphisms between stratified PL spaces of the form $K \subset \Sigma \subset M$. From [@CY] recall A [*starkly stratified space*]{} is a PL space $X$ equipped with a filtration $$X_0 \subset X_1 \subset \ldots \subset X_{n-1} \subset X_n = X$$ satisfying 1. There is a triangulation $\mathcal T$ of $X$ in which each $X_k$ is a subcomplex. 2. For each $k = 1,\ldots n$ $X_k \setminus X_{k-1}$ is a(n open) $k$-manifold. 3. If $C$ is a connected component of $X_k \setminus X_{k-1}$, then $\mathcal T$ restricted to $\bar{C}$ gives $\bar{C}$ the structure of a combinatorial manifold with boundary. 4. For each combinatorial ball $B_k$ with $\accentset{\circ}B_k \subset X_k \setminus X_{k-1}$, $\accentset{\circ}B_k$ admits a closed neighborhood $N$ given inductively as a cell complex as follows (although we require $B_k$ to be a combinatorial ball, the triangulation is then ignored): $N = N_n$, where $N_m$ for $k \leq m \leq n$ is given inductively by $$N_k = B_k$$ and $$N_{\ell+1} = N_\ell \cup \bigcup_{v \in S_\ell} L(v) \ast v$$ for $S_\ell$ a finite set of points in $X\setminus X_\ell$, andl $L(\cdot)$ a function on $S_\ell$ valued in $$\{ L | L\; \mbox{\rm is a combinatorial ball and}\; B_k \subset L \subset N_\ell \}$$ We will call such a neighborhood of the interior of a combinatorial ball lying in a stratum of the same dimension a [*stark neighborhood*]{}. It is quite easy to show that for a knot or link $K$ with a Seifert surface $\Sigma$ in a 3-manifold $M$, the filtered space $\emptyset \subset K \subset \Sigma \subset M$ is a starkly stratified space – the stark neighborhoods of open simplexes $\Sigma$ are simply the joins of their closure with two points of $M \setminus \Sigma$, one on each side of $\Sigma$, $\Sigma$ being, of course, two-sided, while those of open simplexes in $K$ are iterated joins, first with a point in $\Sigma$ and a point in $M \setminus \Sigma$ and then of with two points of $M \setminus \Sigma$, one on either side of $\Sigma$ (or more properly the union of $\Sigma$ with the first join, which is easily seen to also be an oriented surface with boundary). Throughout, despite allowing the possibility that is is actually a link with more than one component, we will refer to $K$ as “the knot” to avoid the ambiguity of the word “link” in the context of PL topology, as we will have cause to mention the link of a simplex in the PL sense fairly often in our exposition. We will also need A triangulation $\mathcal T$ of a stratified PL space $$X_0 \subset X_1 \subset \ldots \subset X_{n-1} \subset X_n = X$$ is [*flag-like*]{} if each of the $X_i$ is a subcomplex, and moreover for each simplex $\sigma$ of $\mathcal T$, the restriction of the filtration to the simplex, that is the distinct non-empty intersections in the sequence $$X_0\cap \sigma \subset X_1 \cap \sigma \subset \ldots \subset X_{n-1} \cap \sigma \subset \sigma$$ form a (possibly incomplete) simplicial flag. Observe that in any starkly stratified space, it follows from the iterative construction of a stark neighborhood, that the restriction of the filtration to any simplex of a stark neighborhood is a simplical flag. Moreover, in general, flag-like triangulations are plentiful. In particular, it is easy to show If $\mathcal T$ is a triangulation of a stratified PL space $$X_0 \subset X_1 \subset \ldots \subset X_{n-1} \subset X_n = X$$ for which each $X_i$ is a subcomplex, then its barycentric subdivision $\beta X$ is flag-like. As our construction will require a flag-like triangulation, the most direct way to show topological invariance will be show that combinatorial moves which preserve flag-likeness suffice to give all PL homeomorphisms between our stratified spaces $K \subset \Sigma \subset M$. An Alexander move (stellar subdivision or stellar weld), a Pachner move (bistellar move), or an extended Pachner move (extended bistellar move in the sense of [@CY]) on a triangulation of a stratified space is [*flag-like*]{} if the restriction of the filtration to each simplex of both its initial and its final state is a simplicial flag. From Alexander Moves to Extended Pachner Moves ============================================== As observed in [@DPY], any Alexander subdivision is flag-like, while there exist non-flag-like Alexander welds, but the theorem of [@DPY] that flag-like Alexander moves suffice to characterize PL homeorphism of stratified PL spaced equipped with flag-like triangulations hold in full generality: If $\mathcal T$ and ${\mathcal T}^\prime$ are flag-like triangulations of a stratified PL spaces $$X_0 \subset X_1 \subset \ldots \subset X_{n-1} \subset X_n = X$$ and $$X_0^\prime \subset X_1^\prime \subset \ldots \subset X_{n-1}^\prime \subset X_n^\prime = X^\prime ,$$ respectively, then there is a stratification-preserving PL homeomorphism from $X$ to $X^\prime$ if and only if there is a sequence of flag-like Alexander moves starting with $\mathcal T$ and resulting in a triangulation of $X$ which is combinatorially equivalent to the triangulation ${\mathcal T}^\prime$ of $X^\prime$. The proof is identical to that given in the special case in [@DPY]: Sufficiency is clear, necessity follows from the full force of Alexander’s theorem – that two complexes are PL homoemorphic if and only if there is a common subdivision by Alexander subdivisions – and the observation that subdivisions of flag-like triangulations are always flag-like: starting at $\mathcal T$ and ${\mathcal T}^\prime$, apply Alexander subdivisions (necessarily flag-like) until a common combinatorial type of triangulation is reached. The sequence in the theorem is then the sequence of subdivsions beginning at $\mathscr T$ followed by the welds reversing the subdivisions starting at ${\mathscr T}^\prime$ in reverse order. Working with Alexander moves in the context of state-sum models is difficult, since in dimensions three and greater there are infintely many combinatorial types of Alexander move, and what is more, the number of simplexes in the subdivided triangulation is quite large. We therefore wish to reduce PL homeorphism of our triples $K \subset \Sigma \subset M$, equipped with flag-like triangulations to flag-like instances of the extended Pachner moves of [@CY]. \[sufficient\_moves\] Every flag-like Alexander move relating triangulations of a knot, Seifert surface, 3-manifold triple $K \subset \Sigma \subset M$ can be accomplished by a sequence of flag-like extended Pachner moves. Every flag-like Alexander move relating triangulation of a knot, Seifert surface, 3-manifold triple $K \subset \Sigma \subset M$ can be accomplished by a sequence of flag-like moves of the following types: Pachner moves on tetrahedra, extended Pachner moves on triangles of the Seifert surface, and Alexander subdivision of an edge of the knot whose link has exactly three edges (a 3-6 move). It suffices to show that flag-like Alexander subdivisions can be accomplished by sequences of flag-like extended Pachner moves. For subdivisions of tetrahedra, subdivisions of triangles lying in $\Sigma$ and of edges of $K$ with regular neighborhoods as depicted below, (1.9,-0.5) –(1.5,-1.8) – (1.5,1.8); (0,0) –(1.9,-0.5) – (3,0.5) – (1.5,1.8)– (0,0); (0,0) –(1.5,-1.8)– (3,0.5); (1.9,-0.5)– (1.5,-1.8); (1.5,1.8)– (1.9,-0.5); (0,0) – (1.2,1.0) – (3,0.5); (1.5,1.8) – (1.2,1.0) – (1.5,-1.8); (1.5,1.8) – (1.5,-0.2); (1.5,-0.6) – (1.5,-1.8); (1.5,1.8)–(1.9,-0.5)–(1.5,-1.8); (1.5,1.8) circle (1mm); (1.9,-0.5) circle (1mm); (1.5,-1.8) circle (1mm); the first statement is immediate, since the Alexander move itself is an extended Pachner move. Likewise the second statement is immediate for subdivisions of tetrahedra, triangles lying in $\Sigma$ and edges of $K$ with regular neighborhoods as depicted here: (1.9,-0.5) –(1.5,-1.8) – (1.5,1.8); (0,0) –(1.9,-0.5) – (3,0.5) – (1.5,1.8)– (0,0); (0, 0) – (3,0.5); (0,0) –(1.5,-1.8)– (3,0.5); (1.9,-0.5)– (1.5,-1.8); (1.5,1.8)– (1.9,-0.5); (1.5,1.8) – (1.5,-0.2); (1.5,-0.6) – (1.5,-1.8); (1.5,1.8)–(1.9,-0.5)–(1.5,-1.8); (1.5,1.8) circle(1mm); (1.9,-0.5) circle(1mm); (1.5,-1.8) circle(1mm); For subdivisions of edges in $\Sigma$, the result follows by extending the Pachner moves in the proof of the corresponding result in [@DPY] to a stark neighborhood formed by taking the join with a point on either side of the surface. Subdivisions of triangles with interior in $M \setminus \Sigma$ are easily seen to be accomplished by a sequence of two flag-like Pachner moves, regardless of how the filtration restricts to the pair of tetrahedra sharing the triangular face: first perform a 1-4 Pachner move in one of the tetrahedra, as a subdivision, of flag-like triangulation this move is necessarily flag-like. The resulting complex than has the face to be subdivided shared by the other of the original tetrahedra and one of those from the subdivsion, whose interesection with lower-dimensional strata necessarily lies in the boundary of triangle being subdivided. The 2-3 Pachner move on this pair of tetrahedra is necessarily flag-like since the intersection with the lower dimensional strata lies entirely in the original tetrahedron, and performing it accomplished the desired subdivision. The main difficulty in the proof of each statement comes from handling subdivisions of edges lying on the knot, the link of which has a different number of edges than that included in the set of moves, and subdivisions of edges with interior in the top dimensional stratum. For the latter, we proceed as follows: first, observe that the link of any given edge is a polygon, and the star of the edge consists of tetrahedra including an edge of the link and the given edge (as one of its pairs of disjoint edges) and triangles shared by two of the tetrahedra, having one vertex in the link and the given edge as their opposites side, and the faces of these. As the edge has interior in the top dimensional stratum, and the triangulation is flag-like, the given edge has at most one vertex in a lower dimensional stratum (the knot or the Seifert surface), and the higher Now, perform a 1-4 Pachner move on each tetrahedron in the star, as Alexander subdivisions of a flag-like triangulation, these moves are necessarily flag-like. Having done this, consider the triangles which were shared by the original tetrahedra of the given edge. They are now each shared by two tetrahedra which intersect the lower-dimensional strata only in the intersection of the triangle with the lower-dimensional – by flag-likeness this must be empty, a single vertex (in $\Sigma$ or in $K$) or an edge with its end-points (all in $\Sigma$, all in $K$, or in $\Sigma$, except for a vertex in $K$) – and thus the entire star of each of the triangles in the new triangulation intersects the lower-dimensional strata only in the empty set or the same vertex or edge. It is immediate then that the 2-3 Pachner move on the star of each triangle is flag-like. Now, perform all of the 2-3 Pachner moves on the stars of the triangles shared by the original tetrahedra. The link of the given edge in this new triangulation has as its vertices exactly the new vertices introduced by the 1-4 moves of the first step, and as its edges precisely the new edges introduced by the 2-3 moves. It is easy to see that the intersection of the star of the given edge in this new triangulation with the lower-dimensional strata is exactly the intersection of (the closure of) the given edge, that is, either empty, or a single vertex in either $\Sigma$ or $K$. In any event, any Pachner moves performed within the star of the given edge in the last triangulation are necessarily flag-like. By Cassali’s [@C] improvement of Pachner’s result, we can reach any triangulation of the star (as a manifold with boundary) with the same triangulation of the boundary by a squence of Pachner moves on the interior of the star, in particular we can perform the Alexander subdivision of the edge by a sequence of such moves. Having subdivided the given edge in the modified triangulation, we must now apply more flag-like Pachner moves to obtain the Alexander subdivision of the given edge [*in the original triangulation*]{}. Now, consider the cells in the most recently constructed triangulation within which we had performed the 2-3 Pachner moves. These cells are still cells of the latest triangulation, and still intersect the lower-dimensional strata in such a way that any moves performed in their interiors are necessarily flag-like, and are still each the star of and edge introduced by one of the 2-3 moves, but this star now consists of four closed tetrahedra all sharing the edge – being in particular the join of a pair of triangles sharing the edge with two vertices. By Cassali [@C] we can perform the join with the two vertices of a 2-dimensional 2-2 Pachner move removing the edge introduced by the 2-3 move. Now, observe that the last constructed triangulation is a proper subdivision the triangulation we had after applying the 1-4 moves (and thus of the original triangulation), and, moreover, in it, the triangle shared by the tetrahedra of the original star have been divided into two triangles, exactly as in the result of the desired Alexander move. Applying Cassali’s main theorem [@C] to the subdivision of each of the original tetrahedra (any moves within which much be flag-like) then obtains the desired triangulation by a sequence of flag-like Pachner moves. Finally, we must reduce all Alexander subdivisions of an edge of the knot to flag-like extended Pachner moves on simplexes not on the knot, together with the subdivion of an edge of the knot with a four (resp. three) edge link to establish the first (resp. second) statement. For the second statement, consider first the case of reducing the Alexander subdivision of the edge in the first picture above to flag-like extended Pachner moves on simplexes not on the knot, together with the subdivion of an edge of the knot with a four three edge link. The star of the edge to be subdivided is show below. (2,0)–(1.7,3)–(1.7,-2); (A) at (0,0); (B) at (2,0); (C) at (3,1.2); (D) at (1,1.2); (E) at (1.7,3); (F) at (1.7,-2); (G) at (1.7,0.3); (A)–(B)–(C); (A)–(D)–(C); (A)–(E)–(B); (E)–(C); (D)–(E); (A)–(F)–(B); (F)–(C); (D)–(F); (E)–(G); (1.7,-0.2)–(F); (E) circle (3pt); (F) circle (3pt); (E)–(B)–(F); First, apply a 2-3 Pachner move to two tetrahedra $AFED$ and $DFCE$, sharing the triangle $DEF$. Observe that this move is flag-like. Giving the triangulation given within the original star below. (2,0)–(1.7,3)–(1.7,-2); (A) at (0,0); (B) at (2,0); (C) at (3,1.2); (D) at (1,1.2); (E) at (1.7,3); (F) at (1.7,-2); (G) at (1.7,0.3); (A)–(B)–(C); (A)–(D)–(C); (A)–(C); (A)–(E)–(B); (E)–(C); (D)–(E); (A)–(F)–(B); (F)–(C); (D)–(F); (E)–(G); (1.7,-0.2)–(F); (E) circle (3pt); (F) circle (3pt); (E)–(B)–(F); Then we can apply 3-6 move to the new star of the edge EF. (2,0)–(1.7,3)–(1.7,-2); (A) at (0,0); (B) at (2,0); (C) at (3,1.2); (D) at (1,1.2); (E) at (1.7,3); (F) at (1.7,-2); (G) at (1.7,0.3); (A)–(B)–(C); (A)–(D)–(C); (A)–(C); (A)–(E)–(B); (E)–(C); (D)–(E); (A)–(F)–(B); (F)–(C); (D)–(F); (E)–(G); (1.7,-0.2)–(F); (A)–(G); (B)–(G); (C)–(G); (E) circle (3pt); (F) circle (3pt); (E)–(B)–(F); Subdividing the edge AC using flag-like Pachner moves as in the argument above, then reversing a like sequence of flag-like Pachner moves to introduce an edge DG then gives the desired final configuration shown below. (2,0)–(1.7,3)–(1.7,-2); (A) at (0,0); (B) at (2,0); (C) at (3,1.2); (D) at (1,1.2); (E) at (1.7,3); (F) at (1.7,-2); (G) at (1.7,0.53); (A)–(B)–(C); (A)–(D)–(C); (A)–(E)–(B); (E)–(C); (D)–(E); (A)–(F)–(B); (F)–(C); (D)–(F); (E)–(G); (1.7,-0.2)–(F); (A)–(G); (B)–(G); (C)–(G); (D)–(G); (E) circle (3pt); (F) circle (3pt); (E)–(B)–(F); Now for edges of the knot with links of greater than four edges, the number of edges in the link can be reduced by a 2-3 move, as was done in the initial step of the construction just completed, and after the application of the 3-6 move, a sequence of pairs of a subdivision of an edge introduced by a 2-3 move, followed by a reverse subdivision to introduce a radial edge from the subdivision point introduced by the 3-6 move, will accomplish the desired Alexander subdivision. Thus we establish the second statement. The same argument will also reduce Alexander subdivisions of edges on the knot with links of more than four edges to sequences of flag-like extended Pachner moves. To establish the first statement, it thus remains only to show that the 3-6 move can be accomplished by a sequence of flag-like extended Pachner moves. Consider an edge of the knot with a three edge link. The link consists of two edges incident with the Seifert surface and one not incident with the Seifert surface. As shown above, the edge not incident with the Seifert surface can be subdivided by a sequence of flag-like Pachner moves. Having done this, the edge of the knot now has a 4 edge link, and can be subdivided by the extended Pachner move (4-8 move). It remains then to weld two edges which had been the edge of the link not incident with the Seifert surface. This, however, is the reverse of a subdivision of an edge not incident with either the knot or the Seifert surface, so as shown above can be done by a sequence of flag-like Pachner moves, thus establishing the result. Initial Data: Untwisted Case ============================ The initial data describing local states in Wakui’s construction [@W] of state-sum for finite-gauge group Dikjkgraaf-Witten theory is a finite group, $G$, together with a 3-cocycle $\alpha:G^3 \rightarrow K^\times$ valued in the multiplicative group of a field $K$. In [@DPY], 2-dimensional Dijgraaf-Witten theory was extended to pairs $C\subset \Sigma$ of a closed curve lying in a surface, using local states given by a triple $(H,X,G)$ of two groups $H$ and $G$ and a set $X$ with commuting actions of the groups on the left and right, respectively, together with $K^\times$-valued coefficients extending a 2-cocycle on $G$ and satisfying 2-cocycle-like conditions. For now we defer consideration of the analogue of a cocycle and describe the initial data sufficient for the “untwisted” construction, which will turn out to be equivalent to the construction involving an analogue of a cocycle which is identically $1$. Brief consideration of the conditions on the triple $(H,X,G)$ show that the data can be regarded as a two-object category in which the elements of $H$ are the endomorphisms of one object, which we denote 1, the elements of $G$ the endomorphisms of the other, which we denote 2, there are no arrows from 2 to 1 and the elements of $X$ are the arrow from 1 to 2 – the commutativity of the actions being part of the associativity of composition in the category. A $\mathcal P$-[*parcel*]{} is a small category $\mathcal C$ equipped with a surjective conservative functor $U:{\mathcal C}\rightarrow {\mathcal P}$, which is injective on objects, to a poset $\mathcal P$. A finite $\mathsf 3$-parcel, where $\mathsf 3$ is the three-element chain $1 < 2 < 3$, will be the initial data needed for the untwisted construction, and part of the data needed for the general construction. Throughout we will denote an object of a $\mathsf 3$-parcel by the number which is its image under the functor $U$. The State-Sum Construction: Untwisted Case ========================================== We now describe a state-sum invariant of knot, Seifert surface, 3-manifold triples (which by general principles, cf. [@Y], can be seen as arising from a generalized TQFT given by a functor on cobordisms of surfaces marked with curves with boundary), using only a finite $\mathsf 3$-parcel as initial data. Let $\mathcal T$ be a flag-like triangulation of $K \subset \Sigma \subset M$ for a knot (or link) $K$ with a Seifert surface $\Sigma$ in a 3-manifold $M$. The $\Gamma({\mathcal T})$, [*directed graph*]{} of $\mathcal T$, is the directed graph with ${\mathcal T}_0$, the set of vertices of $\mathcal T$ as vertices, and an edge $(v,w)$ whenever $v$ and $w$ are vertices connected by an edge of $\mathcal T$ and $\dim(v) \leq \dim(w)$, where the dimension of a vertex is the dimension of the stratum in which it lies. ${\mathcal C}({\mathcal T})$, the category of $\mathcal T$, is then the quotient of the path category of $\Gamma({\mathcal T})$ by the relations $(v,w)(w,v) = Id_v$ whenever there are edges in both directions between two vertices, and $(v,w)(w,x) = (v,x)$ whenever there is a 2-simplex of $\mathcal T$ with vertices $v,w,$ and $x$. ${\mathcal C}({\mathcal T})$ then has a surjective conservative functor $D$ to $\mathsf 3$, given on object by mapping each vertex to the dimension of the stratum in which it lies. Given a $\mathsf 3$-parcel, $U:{\mathcal C}\rightarrow {\mathsf 3}$, a $\mathcal C$-coloring of a flag-like triangulation $\mathcal T$ of a knot-Seifert surface-3-manifold triple $K \subset \Sigma \subset M$ is a functor $\sigma:{\mathcal C}({\mathcal T}) \rightarrow {\mathcal C}$ such that $U(\sigma) = D$. \[untwisted\] Let $K \subset \Sigma \subset M$ be a knot, Seifert surface, 3-manifold triple, $\mathcal T$ a flag-like triangulation, and $U:{\mathcal C}\rightarrow {\mathsf 3}$ a $\mathsf 3$-parcel. Let $[\mathcal T, \mathcal C]$ denote the set of $\mathcal C$-clorings of $\mathcal T$, and $G_i = {\mathcal C}(i,i)$ for $i = 1, 2,3$, denote the endomorphism groups of $\mathcal C$. And finally let $\mathcal T_0^i$ for $i = 1,2,3$ denote the set of vertices of $\mathcal T$ lying in the stratum of dimension $i$. Then, using $\#$ to denote cardinality, the quantity $$\frac{ \#[\mathcal T, \mathcal C] } { \prod_{i=1}^3 \#G_i^{\#T_0^i} }$$ is independent of the triangulation and thus gives a topological invariant of the triple. We prove this theorem using Alexander moves. The extended Pachner moves of section 2 will become important for the twisted case. Whenever an Alexander subdivision is performed, a single new vertex is introduced. Chosing an edge from the new vertex to an adjacent vertex in the same stratum induces an equivalence between the category of the old triangulation and the category of the new triangulation, by functors commuting with the underlying functors to $\mathsf 3$. A $\mathcal C$-coloring of the old triangulation determines most of a $\mathcal C$-coloring of the new – all that remains to be specified are the images of the arrows named by edges incident with the new vertex. A brief consideration shows that the choice of an element of $G_i$ (where $i$ is the dimension of the stratum in which the new vertex lies) to be the image of the edge chosen to induce the equivalence of categories completely determines a unique $\mathcal G$-coloring of the new triangulation, and thus whenever a simplex in the $i$-dimensional stratum is subdivided, the number of $\mathcal G$-colorings of the new triangulation is exactly $\#G_i$ times the number of $\mathcal G$-colorings of the old triangulation, thus establishing the result. As with the invariants of $n$-manifolds given by untwisted Dikjgraaf-Witten theory, which for connected $n$-manifolds count group homomorphisms from the fundamental group to the guage group, and more generally count groupoid homomorphisms from any skeleton of the fundamental groupoid to the guage group, the invariants given by Theorem \[untwisted\] admit a counting interpretation. Observe that the relation $x \preceq y$ on points of $M$ given by $x \preceq y$ if $\dim(x) \leq \dim(y)$, where the dimension of a point is the dimension of the stratum in which it lies, is a preorder, whose restriction to each stratum is chaotic. Taking the family of directed paths in $M$ to be the continuous maps $p:[0,1]\rightarrow M$ which are non-decreasing with respect to the usual order on $[0,1]$ and the preorder on $M$, $M$ is given the structure of a directed topological space, or $d$-space in the sense of Grandis [@Gr1; @Gr2]. It is easy to establish The fundamental category $\uparrow\!\!\Pi_1(X)$ with respect to the $d$-space structure of the previous paragraph has an underlying functor to $\mathsf 3$ given on objects by $x \mapsto \dim(x)$, and $\uparrow\!\!\pi_1(X)$ is any skelton of $\uparrow\!\!\Pi_1(X)$, the invariant of Theorem \[untwisted\] counts the number of functors from $\uparrow\!\!\pi_1(X)$ to $\mathcal C$, which commute with the underlying functors to $\mathsf 3$. Partial Cocycles ================ We now turn to the matter of introducing local coefficients which generalize the group 3-cocycle in Wakui’s construction [@W]. The obvious generalization of a $K^\times$-valued group 3-cocycle to a category – a function from composable triples of arrows in $\mathcal C$ to $K^\times$, satisfying the group 3-cocycle condition [*mutatis mutandis*]{} for every composable quadruple of arrows – will give an instance of the rest of the necessary data. However, a moment’s thought reveals that the values of a cocycle on composable triples for which the target of the composition is the object $1$ or $2$ will be irrelevant to the construction, since they cannot occur as labels of a tetrahedron, and, in fact weaker conditions on the values on triples whose composition has target $3$ suffice. The analogue of Wakui’s 3-cocycles in our construction is given by a function on the composable triples of arrows which can occur as labels of the long oriented path in a tetrahedron, satisfying equations corresponding to the combinatorial moves that suffice to ensure topological invariance: A [*$K^\times$-valued partial 3-cocycle*]{} on a $\mathsf 3$-parcel $U:{\mathcal C}\rightarrow {\mathsf 3}$ is a function $$\alpha: \{ (f,g,h) | f,g,h \in Arr({\mathcal C}), \; fgh \; \mbox{\rm is defined,}\; t(h) = 3, \; \mbox{\rm and } s(h) \neq 1\; \} \rightarrow K^\times$$ for some field $K$, satisfying 1. $\alpha_{j,k,3,3}(g,h,k)\cdot\alpha_{i,k,3,3}^{-1}(fg,h,k)\cdot\alpha_{i,j,3,3}(f,gh,k)\cdot$\ $\alpha_{i,j,k,3}^{-1}(f,g,hk)\cdot\alpha_{i,j,k,3}(f,g,h) = 1$, whenever $k \geq 2$, and\ $f,g,h,k$ is composable. 2. $ \alpha_{i , j , 2 , 3} ( a , b c , d ) \cdot \alpha_{j , 2 , 2 , 3} ( b , c , d ) \cdot \alpha^{-1}_{i , j , 2 , 3} ( a , b , c d ) \cdot \alpha^{-1}_{i , 2 , 2 , 3} ( a b , c , d ) = $\ $ \alpha_{i , j , 2 , 3} ( a , b c , e ) \cdot \alpha_{j , 2 , 2 , 3} ( b , c , e ) \cdot \alpha^{-1}_{i , j , 2 , 3} ( a , b , c e ) \cdot \alpha^{-1}_{i , 2 , 2 , 3} ( a b , c , e ), $\ whenever $a,b,c,d$ and $a,b,c,e$ are both composable. 3. $\alpha_{1 , 1 , 2 , 3} ( a , b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( a , b c , d ) = $\ $\alpha_{1 , 1 , 2 , 3} ( f , g b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( f , g b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( f , g b c , d ) \cdot $\ $\alpha_{1 , 1 , 2 , 3} ( g , b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( g , b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( g , b c , d ) $, whenever\ $a,b,c,d$ is composable, $e$ is any arrow such that $be = bcd$,\ and $fg = a$. where $\alpha_{i,j,k,3}$ denotes the restriction of $\alpha$ to $\{ (f,g,h) | s(f) = i, t(f) = j = s(g), t(g) = k = s(h), t(h) = 3 \}$ for $1 \leq i \leq j \leq k \leq3$, and the quadruples of maps in the condition have sources and targets agreeing with those specified by the subscripts on instances of $\alpha$. If $\beta:\{(f,g,h) | f,g,h \in Arr({\mathcal C}) \; \mbox{\rm and } fgh \;\mbox{\rm is defined} \}\rightarrow K^\times$ is a 3-cocycle on the category $\mathcal C$ of a $\mathsf 3$-parcel, then the restriction of $\beta$ to $\{ (f,g,h) | f,g,h \in Arr({\mathcal C}), \; fgh \; \mbox{\rm is defined,}\; t(h) = 3 \; \mbox{\rm and } s(h) \neq 1\}$ is a partial 3-cocycle. The first condition in the definition of a partial cocycle follows trivially from the cocycle condition, being simply instances of the cocycle condition. The second follows since in the presence of the cocycle condition each side equals $\beta(a, b, c)$. The third requires a little work. Solving the equation to move all factors to the right hand side, gives the equivalent condition $$\begin{aligned} 1 & = & \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c ) \cdot \alpha_{1 , 1 , 2 , 3} ( a , b , e ) \cdot \alpha^{-1}_{1 , 1 , 3 , 3} ( a , b c , d ) t\\ & & \alpha_{1 , 1 , 2 , 3} ( f , g b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( f , g b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( f , g b c , d ) \cdot \\ & &\alpha_{1 , 1 , 2 , 3} ( g , b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( g , b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( g , b c , d ). \end{aligned}$$ Now, suppressing the subscripts denoting sources and targets, and substituting the cocycle $\beta$ gives $$\begin{aligned} 1 & = & \beta^{-1} ( a , b , c ) \cdot \beta ( a , b , e ) \cdot \beta^{-1}( a , b c , d ) \cdot\\ & & \beta( f , g b , c ) \cdot \beta^{-1} ( f , g b , e ) \cdot \beta ( f , g b c , d ) \cdot \\ & &\beta( g , b , c ) \cdot \beta^{-1} ( g , b , e ) \cdot \beta ( g , b c , d ). \end{aligned}$$ Now, recalling the $fg = a$, the factors aligned in each column in the preceding expression are three factors of an instance of the coboundary of $\beta$, for the composable quadruples $f,g,b,c$; $f,g,b,e$ and $f,g,bc,d$, respectively. Replacing the factors in each column with the product of two factors equal to them by the cocycle condition gives $$\begin{aligned} 1 & = & \beta( f , g , bc ) \cdot \beta^{-1} ( f , g , be ) \cdot \beta ( f , g , bcd ) \cdot \\ & & \beta^{-1} ( f, g, b ) \cdot \beta ( f, g, b ) \cdot \beta^{-1}(f, g, bc ) \end{aligned}$$ the factors on the right hand side of which cancel in pairs, when the condition that $be = bcd$ is recalled, thus completing the proof. As usual, to twist by coefficients, we need to specify orientations on the edges of the triangulation, and in the proof of invariance show not only invariance under the combinatorial moves giving PL homoemorphism, for us the moves of the second statement of Theorem \[sufficient\_moves\], but under changes to the edge-orientations. The orientation on the knot and the non-invertibility conditions in the definition of the category ${\mathcal C}({\mathcal T})$ remove some of the need to make choices, but not all – the edges of the knot are oriented in agreement with the orientation of the knot, and edges incident with the knot, but not lying in it, are all oriented away from the knot, and edges incident with the Seifert surface, but not lying in it, are all oriented away from the surface. To orient the other edges, as usual chose a linear ordering of the vertices (in this case it suffices to chose separate linear orderings of the vertices in the interior of the Seifert surface and of the vertices in the complement of the closed Seifert surface) and orient the remaing edges from the earlier vertex to the later vertex. Once the edges are oriented, each tetrahedron in the triangulation has a longest oriented path of edges, and a combinatorial orientation induced by the ordering of the vertices along the longest path. We can now state our main theorem: \[main\] Let $U:{\mathcal C} \rightarrow {\mathsf 3}$ be a $\mathsf 3$-parcel, and $\alpha$ be a partial 3-cocycle on it. Then for any knot, Seifert surface, 3-manifold triple $K \subset \Sigma \subset M$, the quantity given by $$\sum_{\lambda:{\mathcal C}({\mathcal T})\rightarrow {\mathcal C}} \frac{\prod_{\sigma \in {\mathcal T}_3} \alpha^{\epsilon(\sigma)}(\lambda(\sigma))} { \prod_{i=1}^3 \#G_i^{\#T_0^i} }$$ where the summation ranges over all $\mathcal C$-colorings of $\mathcal T$, $\lambda(\sigma)$ denotes the composable of arrows $\lambda$ assigns to the longest path of the tetrahedron $\sigma$ and $\epsilon(\sigma)$ is $+1$ if the combinatorial orientation of $\sigma$ agrees with the orientation of $M$, and $-1$ otherwise, for any flag-like triangulation $\mathcal T$ and a choice of ordering of the vertices off the knot, is independent of the triangulation and choice of ordering, and thus a topological invariant of the triple. As in the case of Theorem \[untwisted\], moves which introduce a new vertex involve a choice of a label for one of the edges incident with the new vertex and lying in the same stratum as the new vertex, which suffices, in the presence of labels from the remaining edges not subdivided by the move to determine uniquely a $\mathcal C$-coloring of the new triangulation. The denominator is thus multiplied by the number of summands in the numerator for the new triangulation. It thus suffices to show that for any move which does not introduce a new vertex (the 2-3 Pachner moves and the 4-4 extended Pachner move) for any $\mathcal C$-coloring of the initial triangulation, there is a unique $\mathcal C$-coloring of the triangulation resulting from the move, agreeing with the given coloring on all edges not modified by the move and that the summand in the numerator for these colorings are equal, and similarly for moves which do introduce a new vertex (the 1-4 Pachner move, the 2-6 extended Pachner move and the 3-6 move on an edge of the knot) that given a $\mathcal C$-coloring of the initial triangulation and a choice of label for one of the new edges incident with the new vertex and lying in the same stratum, there is a unique $\mathcal C$-coloring of the new triangulation agreeding with the given coloring on all edges not modified by the move, having the chosen label on the new edge, and, moreover, that the summand in the numerator for the new triangulation for this coloring equals the summand in the numerator for the old triangulation of the given coloring. Condition (3) is exactly the condition needed to ensure that each summand corresponding to labelings of the six-tetrahedon state in the 3-6 move are equal to the summand corresponding to the labeling of the three-tetrahedron state, as illustrated in Figures \[three\_tetrahedra\] and \[six\_tetrahedra\]. --------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[thick][scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \begin{scope} \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$ d $}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, left]{$ c d $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b $}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ a b c d $\,}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$ a $}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw [dashed,thick] (1.5,-0.6) -- (1.5,-1.8); \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \fill[red, opacity=.5] (A) circle (3pt); \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$ a $}; \fill[red, opacity=.5] (E) circle (3pt); \draw [dashed,thick] (1.5,-0.6) -- (1.5,-1.8); \fill[blue, opacity=.5] (C) circle (3pt); \fill[red, opacity=.5] (A) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(E); \fill[red, opacity=.5] (E) circle (3pt); \draw[thick,blue,blue, opacity=.5] (A)--(C); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue, opacity=.5] (E)--(C); \draw[thick,red,red, opacity=.5] (A)--(E); \end{scope} \draw[thick,blue,blue, opacity=.5] (A)--(C); \end{tikzpicture} \draw[thick,blue,blue, opacity=.5] (E)--(C); \end{scope} & \alpha_{1 , 1 , 2 , 3} ( a , b , c ) \end{tikzpicture}$ \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, left]{$ c d $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ a b c d $\,}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ a $}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(E); \draw[thick,blue,blue, opacity=.5] (A)--(C); \draw[thick,blue,blue, opacity=.5] (E)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$ d $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ a b c d $\,}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, right]{$ a $}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(E); \end{scope} \end{tikzpicture} & \alpha_{1 , 1 , 3 , 3} ( a , b c , d ) \\ \end{array} \right.$ --------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[thick][scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \fill[blue,blue, opacity=.5] (1.5,0)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,0){}; \begin{scope} \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$ d $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ ( a b c = ) f g b c $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b ( = f g b )$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c d $}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,0) node[font=\tiny, midway, left]{$ f $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b $}; \fill[red, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ a b c d $\,}; \fill[blue, opacity=.5] (C) circle (3pt); \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw[thick,red,red, opacity=.5] (A)--(1.5,0); \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw[thick,blue,blue, opacity=.5] (A)--(C); \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [dashed, thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, above]{$ g b c $}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$ f $}; \draw [dashed, thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, left]{$ g b $}; \draw [dashed,thick, midarrow={>}] (1.5,-0.6) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ g $}; \draw[thick,blue,blue, opacity=.5] (F)--(C); \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (F) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \end{scope} \fill[blue, opacity=.5] (C) circle (3pt); \end{tikzpicture} \draw[thick,red,red, opacity=.5] (A)--(E); \draw[thick,blue,blue, opacity=.5] (A)--(C); & \alpha_{1 , 1 , 2 , 3} ( f , g b , c ) \draw[thick,blue,blue, opacity=.5] (E)--(C); \draw [thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, above]{$ g b c $}; \\ \draw [thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, left]{$ g b $}; \\ \draw [thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, above]{$ g b c d $}; \end{scope} \begin{tikzpicture}%[thick][scale=1.5] \end{tikzpicture}$ \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$ d $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ ( a b c = ) f g b c $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ f g b c d ( = a b c d ) $\,}; \draw [thick, midarrow={>}] (1.5,1.8) -- (F) node[font=\tiny, midway, left]{$ f $}; \fill[red, opacity=.5] (A) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(F); \draw [thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, below]{$ g b c $}; \draw [thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, below]{$ g b c d $}; \fill[red, opacity=.5] (F) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha_{1 , 1 , 3 , 3} ( f , g b c , d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-0.1)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{$ c d $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ f g b(=\!a b ) $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ f g b c d ( = a b c d ) $\,}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$ f $}; \fill[red, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(F); \draw[thick,blue,blue, opacity=.5] (A)--(C); \draw [thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, left]{$ g b $}; \draw [dashed, thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, right]{$ g b c d $}; \draw[thick,blue,blue, opacity=.5] (F)--(C); \fill[red, opacity=.5] (F) circle (3pt); \end{scope} \end{tikzpicture} & \alpha^{-1}_{1 , 1 , 2 , 3} ( f , g b , c d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,0){}; \begin{scope} \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw [dashed,thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ g $}; \fill[red, opacity=.5] (E) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue, opacity=.5] (E)--(C); \draw [thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, above]{$ g b c $}; \draw [thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, above]{\,\,\,\,$ g b $}; \fill[red, opacity=.5] (F) circle (3pt); \draw[thick,red,red, opacity=.5] (F)--(E); \draw[thick,blue,blue, opacity=.5] (F)--(C); \end{scope} \end{tikzpicture} & \alpha_{1 , 1 , 2 , 3} ( g , b , c ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, above]{$ d $}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ g $}; \fill[red, opacity=.5] (F) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \draw[thick,red,red, opacity=.5] (F)--(E); \draw [thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, below]{$ g b c $}; \draw [thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, below]{$ g b c d $}; \end{scope} \end{tikzpicture} & \alpha_{1 , 1 , 3 , 3} ( g , b c , d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,-0.1); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c d $}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {\!\!$ b $\,}; \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ g $}; \fill[red, opacity=.5] (F) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (F)--(E); \draw[thick,blue,blue, opacity=.5] (E)--(C); \draw[thick,blue,blue, opacity=.5] (F)--(C); \draw [thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, below]{$ g b $}; \draw [thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, above]{$ g b c d $}; \end{scope} \end{tikzpicture} & \alpha^{-1}_{1 , 1 , 2 , 3} ( g , b , c d ) \\ \end{array} \right.$ ---------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------- Conditions (1) and (2) each ensure invariance under two types of moves – the 1-4 and 2-3 Pachner moves and the 2-6 and 4-4 extended Pachner moves, respectively. In each case, the differently indexed instances of the conditions correspond to the ways in which a flag-like triangulation of the state with fewer tetrahedra can intersect the strata. For instance, the cases of the 4-4 move and the 2-6 move in which the tetrahedra being modified by the moves are not incident with the knot are show in Figures \[four\_before\] and \[four\_after\] and Figures \[two\_extended\] and \[six\_no\_knot\], respectively. The latter depends on the choice of the label $f$, which then induces the labels $g$ and $h$ such that there are factorizations of $c$ and $d$ as $c = fg$ and $d = fh$, respectively. ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(3.5,1.2)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \begin{tikzpicture}%[scale=1.5] \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \begin{scope} \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [dashed,thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [dashed,midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw[thick,blue,blue] (A)--(B); \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw[dashed,thick,blue,blue] (D)--(A); \draw [dashed,midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \draw[dashed,thick,blue,blue] (D)--(B); \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \fill[blue, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \fill[blue, opacity=.5] (B) circle (3pt); \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \fill[blue, opacity=.5] (D) circle (3pt); \draw [dashed,midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \end{scope} \draw[thick,blue,blue] (A)--(B)--(C); \end{tikzpicture} \draw[dashed,thick,blue,blue] (C)--(D)--(A); \fill[blue, opacity=.5] (A) circle (3pt); & \alpha_{2 , 2 , 2 , 3} ( a , b c , d ) \fill[blue, opacity=.5] (B) circle (3pt); \\ \fill[blue, opacity=.5] (C) circle (3pt); \\ \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \begin{tikzpicture}%[scale=1.5] \end{tikzpicture}$ \fill[blue,blue, opacity=.5] (2.5,0)--(3.5,1.2)--(1,1.2); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \draw[thick,blue,blue] (B)--(C); \draw[thick,blue,blue] (B)--(D); \draw[dashed,thick,blue,blue] (C)--(D); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha_{2 , 2 , 2 , 3} ( b , c , d ) \\ \\ \begin{tikzpicture}%[scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \draw [dashed,midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \draw[thick,blue,blue] (A)--(B); \draw[thick,blue,blue] (D)--(A); \draw[thick,blue,blue] (D)--(B); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( a , b c , e ) \\ \\ \begin{tikzpicture}%[scale=1.5] \fill[blue,blue, opacity=.5] (2.5,0)--(3.5,1.2)--(1,1.2); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \draw [midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \draw[thick,blue,blue] (B)--(C); \draw[thick,blue,blue] (C)--(D); \draw[thick,blue,blue] (B)--(D); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( b , c , e ) \\ \end{array} \right.$ ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(3.5,1.2)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \begin{tikzpicture}%[scale=1.5] \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(3.5,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \begin{scope} \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [dashed,thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw[thick,blue,blue] (A)--(B)--(C); \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw[dashed,thick,blue,blue] (C)--(A); \draw [dashed,midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \fill[blue, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \fill[blue, opacity=.5] (B) circle (3pt); \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \fill[blue, opacity=.5] (C) circle (3pt); \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \end{scope} \draw [dashed,midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \end{tikzpicture} \draw[thick,blue,blue] (A)--(B)--(C); \draw[dashed,thick,blue,blue] (C)--(D)--(A); & \alpha_{2 , 2 , 2 , 3} ( a , b , c d ) \fill[blue, opacity=.5] (A) circle (3pt); \\ \fill[blue, opacity=.5] (B) circle (3pt); \\ \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \begin{tikzpicture}%[scale=1.5] \end{scope} \fill[blue,blue, opacity=.5] (0,0)--(3.5,1.2)--(1,1.2)--(0,0); \end{tikzpicture}$ \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \begin{scope} \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw [dashed,midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \draw[thick,blue,blue] (A)--(C); \draw[dashed,thick,blue,blue] (C)--(D)--(A); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha_{2 , 2 , 2 , 3} ( a b , c , d ) \\ \\ \begin{tikzpicture}%[scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(3.5,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \draw[thick,blue,blue] (A)--(B)--(C); \draw[thick,blue,blue] (C)--(A); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( a , b , c e ) \\ \\ \begin{tikzpicture}%[scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(3.5,1.2)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \begin{scope} \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \draw [dashed,midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \draw[thick,blue,blue] (A)--(C); \draw[thick,blue,blue] (C)--(D)--(A); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( a b , c , e ) \\ \end{array} \right.$ ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------ $\displaystyle $ \left\{ \begin{tikzpicture}%[thick][scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \begin{scope} \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [dashed,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ a b $\,\,\,}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, right]{$ b $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ a b c $}; \draw [dashed,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ a b $\,\,\,}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ b c $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ b c $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ a b d $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left] {$ b d $}; \fill[blue, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \draw[thick,blue,blue] (A)--(B); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue] (B)--(C); \draw[thick,blue,blue] (A)--(B); \draw[dashed,thick,blue,blue] (A)--(C); \draw[thick,blue,blue] (B)--(C); \end{scope} \draw[dashed,thick,blue,blue] (A)--(C); \end{tikzpicture} \end{scope} \end{tikzpicture}$ & \alpha_{2 , 2 , 2 , 3} ( a , b , c ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, above]{$ a b $\,\,\,}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ a b d $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left] {$ b d $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue] (A)--(B); \draw[thick,blue,blue] (B)--(C); \draw[thick,blue,blue] (A)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( a , b , d ) \\ \end{array} \right.$ ------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[thick][scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(1.5,0.1)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,0.1){}; \begin{scope} \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{\!$ b $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ abc $}; \draw [dashed,midarrow={>}] (A)--(C) node[font=\tiny, midway, above]{$ a b $\,\,\,\,\,\,\,}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ abc $}; \draw [dashed,thick, midarrow={>}] (1.5,0.1) -- (1.5,1.8) node[font=\tiny, midway, left]{$ g $}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \fill[blue, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \fill[blue, opacity=.5] (B) circle (3pt); \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abd $}; \fill[blue, opacity=.5] (F) circle (3pt); \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right] {$ bd $}; \draw[thick,blue,blue] (A)--(B); \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \draw [dashed,thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, above]{$ abf $}; \draw [dashed,thick, midarrow={>}] (1.5,-0.1) -- (1.5,1.8) node[font=\tiny, midway, left]{$ g $}; \draw [dashed,thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, right]{\,\,$ bf $}; \draw [dashed,thick, midarrow={>}] (1.5,-0.25) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ h $}; \draw[dashed,thick,blue,blue] (A)--(F); \fill[blue, opacity=.5] (A) circle (3pt); \draw[dashed,thick,blue,blue] (F)--(B); \fill[blue, opacity=.5] (B) circle (3pt); \end{scope} \fill[blue, opacity=.5] (C) circle (3pt); \end{tikzpicture} %\draw[thick,red,red, opacity=.5] (A)--(E); \draw[thick,blue,blue] (A)--(B); & \alpha_{2 , 2 , 2 , 3} ( a , b f , g ) \draw[thick,blue,blue] (B)--(C); \draw[dashed,thick,blue,blue] (A)--(C); \\ \draw [thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, above]{$ abf $}; \\ \draw [thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, below]{$ bf $\,\,}; \draw [thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, below]{\!\!\!\!\!$ f $}; \begin{tikzpicture}%[thick][scale=1.5] \end{scope} \fill[blue,blue, opacity=.5] (0,0)--(1.5,-0.1)--(3,0.5)--(0,0); \end{tikzpicture}$ \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [dashed,midarrow={>}] (A)--(C) node[font=\tiny, midway, above]{$ a b $\,\,\,\,\,\,\,}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ abc $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \draw [thick, midarrow={>}] (1.5,-0.1) -- (1.5,1.8) node[font=\tiny, midway, right]{$ g $}; \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[dashed,thick,blue,blue] (A)--(C); \draw [thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, below]{$ abf $}; \draw [thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, below]{$ f $}; \draw[thick,blue,blue] (F)--(A); \draw[thick,blue,blue] (F)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( ab , f , g ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-0.1)--(1.9,-0.5)--(3,0.5)--(1.5,-0.1); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{\!$ b $}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \draw [thick, midarrow={>}] (1.5,-0.1) -- (1.5,1.8) node[font=\tiny, midway, left]{$ g $}; \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[thick,blue,blue] (B)--(C); \draw [thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, below]{$ bf $\,\,\,}; \draw [dashed,thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, above]{$ f $}; \draw[thick,blue,blue] (F)--(B); \draw[dashed,thick,blue,blue] (F)--(C); \end{scope} \end{tikzpicture} & \alpha_{2 , 2 , 2 , 3} ( b , f , g ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(1.5,0.1)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,0.1){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abd $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right] {$ bd $}; \draw [dashed,thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ h $}; \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[thick,blue,blue] (A)--(B); \draw [thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, above]{$ abf $}; \draw [thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, right]{$ bf $}; \draw[thick,blue,blue] (F)--(B); \draw[thick,blue,blue] (F)--(A); \end{scope} \end{tikzpicture} & \alpha^{-1}_{2 , 2 , 2 , 3} ( a , b f , h ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(1.5,-0.1)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, above]{$ a b $\,\,\,\,\,\,\,}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \draw [thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ h $}; \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[dashed,thick,blue,blue] (A)--(C); \draw [thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, below]{$ abf $}; \draw [thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, below]{\!\!\!\!\!$ f $}; \draw[thick,blue,blue] (F)--(C); \draw[thick,blue,blue] (F)--(A); \end{scope} \end{tikzpicture} & \alpha_{2 , 2 , 2 , 3} ( a b , f , h ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-0.1)--(1.9,-0.5)--(3,0.5)--(1.5,-0.1); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{\!$ b $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right] {$ bd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \draw [thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ h $}; \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[thick,blue,blue] (B)--(C); \draw [thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, below]{$ bf $\,\,}; \draw [thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, below]{\!\!\!\!\!$ f $}; \draw[thick,blue,blue] (F)--(C); \draw[thick,blue,blue] (F)--(B); \end{scope} \end{tikzpicture} & \alpha^{-1}_{2 , 2 , 2 , 3} ( b , f , h ) \\ \end{array} \right.$ ----------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- In each case, separating the factors involving $d$ (resp. $g$) from those involving $e$ (resp. $h$) gives an instance of condition (2) in the definition of partial cocycle, which is thus seen to ensure invariance under the illustrated instance of the 4-4 extended Pachner move and the 2-6 extended Pachner move. Invariance under moves of these forms with a vertex (resp. an edge) of the polygon in the Seifert surface being modified by the move lying in the knot is, by an identical argument, given by the instances of condition (2) with $i=1$ and $j =2$ (resp. $i = j = 1$). In the case illustrated of the 2-6 extended Pachner move, the new vertex is added to the ordering after the vertices of the triangle lying in the Seifert surface. However, if it is inserted anywhere else in the ordering, an undefying calculation shows that an equation of the same form ensures invariance under the 2-6 move. The case of 1-4 and 2-3 Pachner moves is similar. Invariance under each is ensured by condition (1). We illustrate this with the instances of the moves in which the boundary contains an edge of the knot bounding a triangle of the Seifert surface. The before and after states of the 2-3 move are shown in Figure \[2-3\_with\_knot\_and\_surface\] [ll]{} $\displaystyle \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,1.8)--(0,0)--(1.9,-0.5); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, below]{$b \cdot c$}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$a \cdot b$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$a \cdot b \cdot c$\,}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$b \cdot c \cdot d$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, left] {$c \cdot d$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \draw[thick,blue,blue, opacity=.5] (B)--(C)--(A); \end{scope} \end{tikzpicture}$ & $ \left\{ \begin{array}{ll} \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,1.8)--(0,0)--(1.9,-0.5); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, below]{$b \cdot c$}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$a \cdot b$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$a \cdot b \cdot c$\,}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \draw[thick,blue,blue, opacity=.5] (B)--(C)--(A); \end{scope} \end{tikzpicture} & \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, below]{$b \cdot c$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$b \cdot c \cdot d$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, left] {$c \cdot d$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue, opacity=.5] (B)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{1 , 2 , 3 , 3} ( b , c , d ) \\ \end{array} \right.$ \ \ $\displaystyle \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,1.8)--(0,0)--(1.9,-0.5); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$bc$}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ab$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$abc$\,}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$bcd$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right] {$cd$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$abcd$}; \draw [dashed,thick] (1.5,-0.6) -- (1.5,-1.8); \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \draw[thick,blue,blue, opacity=.5] (B)--(C)--(A); \end{scope} \end{tikzpicture}$ & $ \left\{ \begin{array}{ll} \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,1.8)--(0,0)--(1.9,-0.5); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ab$}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$bcd$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right] {$cd$\,}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$abcd$}; \draw [dashed,thick] (1.5,-0.6) -- (1.5,-1.8); \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \draw[thick,blue,blue, opacity=.5] (B)--(C)--(A); \end{scope} \end{tikzpicture} & \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ab$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$abc$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right] {$cd$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$abcd$}; \fill[red, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue, opacity=.5] (A)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{1 , 2 , 3 , 3} ( a b , c , d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$bc$}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$abc$\,}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$bcd$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, right]{$abcd$}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \end{scope} \end{tikzpicture} & \alpha_{1 , 1 , 3 , 3} ( a , b c, d ) \\ \end{array} \right.$ Plainly invariance under this move is equivalent to the relation $$\begin{aligned} \lefteqn{ \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c ) \, \cdot \, \alpha^{-1}_{1 , 2 , 3 , 3} ( b , c , d ) =} \\ & & \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c d ) \, \cdot \, \alpha^{-1}_{1 , 2 , 3 , 3} ( a b , c , d ) \, \cdot \, \alpha_{1 , 1 , 3 , 3} ( a , b c, d ) \end{aligned}$$ which solves to give the instance of condition (1) with indices 1,1,2,3,3. A little effort shows that the same relation suffices to give invariance under the other instance of a 2-3 move with an triangle on the boundary lying in the Seifert surface, with an edge on the knot – in which the edge of the knot lies on the equatorial triangle, and under a 1-4 move in which the tetrahedron subdivided similarly has a triangle of its boundary lying in the Seifert surface with an edge on the knot, regardless of whether the new vertex is inserted into the ordering before or after the original vertex lying in the 3-dimensional stratum. Likewise the other instances of condition (1) with indices 3,3,3,3,3 (resp. 2,3,3,3,3; 1,3,3,3,3; 2,2,3,3,3; 1,2,3,3,3; 1,1,3,3,3; 2,2,2,3,3; 1,2,2,3,3) give invariance under 2-3 and 1-4 moves in which the boundary of the cell modified is not incident with either the knot or Seifert sufrace (resp. intersect the Seifert surface in a single vertex not on the knot; intersect the knot in a single vertex; is not incident with the knot, but intersects the Seifert surface in an edge; intersects the knot in a single vertex and the Seifert surface in an edge ending at that vertex; intersects the knot in a single edge; is not incident with the knot, but intersects the Seifert surface in a triangle; intersect the Seifert surface in a triangle with a single vertex on the knot). Again, inserting the new vertex in 1-4 moves anywhere in the ordering gives rise to different instances of the same relation. Thus the value of the expression is independent of both the triangulation and the ordering(s) of the vertices. As with the untwisted invariant, which is the special case of the preceding in which $\alpha \equiv 1$, by general principles explained in [@Y], this invariant is the restriction to the endomorphisms of the monoidal identity, the empty triple of spaces, of a generalized TQFT, a monoidal functor from (3,2,1)-[**COBORD**]{}, the category whose objects are triples of an oriented surface containing an oriented curve with boundary, with disjoint union as the monoidal product, and whose arrows are PL-homeomorphism classes of cobordisms between these, to $k$-[**v.s.**]{}. Examples of Initial Data ======================== Finite $\mathsf 3$-parcels are easy to construct. Given any three finite groups $G_i$ $i = 1, 2,3$ and finite sets $X_{1,2}$ and $X_{2,3}$, with $X_{i,j}$ equipped with a left $G_i$-action and a right $G_j$ action, which commute with each other, there is a $\mathsf 3$-parcel $\mathcal C$ with ${\mathcal C}(i,i) = G_i$, ${\mathcal C}(i,j) = X_{i,j}$ for $(i,j) = (1,2)$ or $(2,3)$ and $${\mathcal C}(1,3) = X_{1,2} \times X_{2,3} / \sim$$ where $\sim$ is the equivalence relation induced by $$(xg,y) \sim (x,gy)$$ whenever $x\in X_{1,2}$, $g\in G_2$ and $y\in X_{2,3}$, where the actions of the group are denoted by the null infix. Composition is given by the group multiplications, group actions, and the map $(x,y) \mapsto [x,y]$ where $[x,y]$ is the equivalence class of the pair under $\sim$ for $x\in X_{1,2}$ and $y\in X_{2,3}$, as appropriate. The identity arrows for the object $i$ is the group identity of $G_i$. It is trivial to verify associativity. If we add to the data a finite set $X_{1,3}$ with a commuting left $G_1$ action and a right $G_3$ action, together with an biequivariant map $\varphi: X_{1,2} \times X_{2,3} / \sim \rightarrow X_{1,3}$, and let the composition arrows $x:1\rightarrow 2$ and $y:2\rightarrow 3$ be given by $\varphi([x,y])$, we obtain a complete description of all finite $\mathsf 3$-parcels. Among these are $\mathsf 3$-parcels constructed from subgroups and subsets of a groups with suitable closure properties after the manner of [@DPY] Example 5.2, on which it is, moreover, easy to find examples of partial 3-cocycles using ordinary group cohomology: Fix a group $G$. Let $G_i$ $i = 1,2,3$ be subgroups of $G$ and $X_{1,2}$ (resp. $X_{2,3}$) be a subset of $G$ closed under left multiplication by elements of $G_1$ (resp. $G_2$) and right multiplication by elements of $G_2$ (resp. $G_3$), and let $X_{1,3} := \{ x\xi \; |\; x \in X_{1,2}\; \&\; \xi \in X_{2,3} \}$. The disjoint union of the $G_i$ for $i = 1,2,3$ and the $X_{i,j}$ for $(i,j) = (1,2), (2,3), (1,3)$ is easily seen to be a $\mathsf 3$-parcel $\mathcal C$ under a composition induced by the group law on $G$ with the obvious underlying functor to $\mathsf 3$. It is also easy to see that given any $K^\times$-valued 3-cocycle $\alpha$ on $G$, the map induced on $Arr({\mathcal C})$ by composing the canonical maps to $G^3$, induced by the inclusions of the homsets into $G$, with $\alpha$ is a 3-cocycle on $\mathcal C$, and thus a partial 3-cocycle. [1]{} Alexander, J., “The Combinatorial Theory of Complexes,” [*Ann. of Math.*]{} [**31**]{} (2) (1930) 292-320. Casali, Maria Rita, “A note about bistellar operations on PL-manifolds with boundary,” [*Geometria Dedicata*]{} [**56**]{} (1995) 257-262. Crane, L. and Yetter, D.N., “Moves on Filtered PL Manifolds and Stratified PL Spaces,” \#arXiv:1404.3142. Dougherty, Aria L., Park, Hwajin, and Yetter, D.N., “On 2-dimensional Dikjgraaf-Witten Theory with Defects,” (2014) Grandis, Marco, “Directed Homotopy Theory, I. The fundamental category”, [*Cah. Topol. Géom. Diff. Catég.*]{} [**44**]{} (2003), 281-316. Grandis, Marco, [*Directed Algebraic Topology: Models of non-reversible worlds*]{}, Cambridge University Press, (2009). Wakui, Michihisa, “On Dijkgraaf-Witten Invariant for 3-manifolds,” [*Osaka J. Math.*]{} [**29**]{} (4) (1992) 675-696. Yetter, D.N., “Triangulations and TQFTs,” in [*Conference Proceedings on Quantum Topology*]{}, (Randy Baadhio, ed.), World Scientific (1993) 354-370.

--- author: - | \ HEP Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA\ E-mail: - | J. B. Kogut\ Department of Energy, Division of High Energy Physics, Washington, DC 20585, USA\ and\ Dept. of Physics – TQHN, Univ. of Maryland, 82 Regents Dr., College Park, MD 20742, USA\ E-mail: title: Searching for the elusive critical endpoint at finite temperature and isospin density --- Introduction ============ Direct simulations of QCD at finite baryon/quark number density are made difficult if not impossible because at finite quark-number chemical potential $\mu$ the fermion determinant is complex. At small chemical potentials, close to the finite temperature transition, various methods have been devised to circumvent this difficulty, series expansions in $\mu$ [@Allton:2002zi; @Allton:2005gk; @Gavai:2003mf], analytic continuation from imaginary $\mu$ [@deForcrand:2002ci; @deForcrand:2006pv; @Azcoiti:2004ri; @Azcoiti:2005tv; @D'Elia:2002gd; @D'Elia:2004at], reweighting methods [@Fodor:2001au; @Fodor:2004nz] and canonical ensemble techniques [@Kratochvila:2005mk; @Alexandru:2005ix]. We adopt a different strategy, and simulate using the magnitude of the fermion determinant and ignoring the phase [@Kogut:2002zg; @Kogut:2005yu]. This can be thought of as considering all quarks to be in isodoublets and introducing a finite chemical potential $\mu_I$ for isospin. In the region of small $\mu/\mu_I$, where the phase is expected to be less important one can argue that the finite $\mu$ and $\mu_I$ transitions might be identical. Since our fermion determinant is positive (or at least non-negative), we can use standard hybrid molecular-dynamics HMD(R) simulations [@Gottlieb:1987mq]. However, for this algorithm, the Binder cumulants used to determine the nature of the finite temperature transition turn out to be strongly dependent on the updating increment $dt$. For this reason we now simulate using the rational hybrid monte-carlo (RHMC) algorithm [@Kennedy:1998cu; @Clark:2005sq], which is exact in the sense of having no $dt$ dependence for observables. In the low chemical potential domain, the most interesting feature expected in the phase diagram is the critical endpoint, where the finite temperature transition changes from a crossover to a first-order transition as chemical potential is increased. The critical endpoint is expected to lie in the universality class of the 3-dimensional Ising model. For 3 flavours it had been expected that the critical point at zero chemical potentials, where the transition changes from a first order transition to a crossover as mass is increased, would move to higher masses as the chemical potential increases, thereby becoming the critical endpoint. Our preliminary results indicate that this does not happen. In section 2 we give the fermion action and make a few comments on the RHMC implementation. Section 3 gives our preliminary results. Our conclusions occupy section 4. QCD at finite isospin density and the RHMC ========================================== The pseudo-fermion action for QCD at finite $\mu_I$, used for the implementation of the RHMC algorithm is $$S_{pf}=p_\psi^\dag {\cal M}^{-N_f/8} p_\psi \label{eqn:action}$$ where $p_\psi$ is the momentum conjugate to the pseudo-fermion field $\psi$. $${\cal M} = [D\!\!\!\!/(\frac{1}{2}\mu_I)+m]^\dag [D\!\!\!\!/(\frac{1}{2}\mu_I)+m] + \lambda^2$$ is the quadratic Dirac operator, and we set $\lambda=0$ for our $\mu_I < m_\pi$ simulations. To implement the RHMC method we need to know positive upper and lower bounds to the spectrum of ${\cal M}$. $25$ exceeds the upper bound for the $\mu_I$ range of interest. We use a speculative lower bound of $10^{-4}$ since the actual lower bound of the spectrum is unknown. This is justified by varying the choice of lower bounds and comparing the results [@Kogut:2006jg]. For $N_f=3$ we use a $(20,20)$ rational approximation to ${\cal M}^{(\pm 3/16)}$ at the ends of each trajectory, and a $(10,10)$ rational approximation to ${\cal M}^{(-3/8)}$ for the updating. Simulations and Results ======================= We are simulating lattice QCD with staggered fermions and $N_f=3$ at quark masses close to $m_c$, the critical mass for $\mu=\mu_I=0$ on $8^3 \times 4$, $12^3 \times 4$ and $16^3 \times 4$ lattices. $m=0.02$, $0.025$, $0.03$, $0.035$, and $\mu_I=0.0,\,0.2,\,0.3$. For our $12^3 \times 4$ simulations we use runs of 300,000 trajectories at each of 4 $\beta$ values close to $\beta_c$, for each $m$ and $\mu_I$. We mostly use $dt=0.05$ for which length-1 trajectories give acceptances of $\sim 70\%$ for the RHMC algorithm. To determine the nature of the transition, we use 4-th order Binder cumulants [@Binder:1981sa] for the chiral condensate. For any observable $X$ this cumulant is defined by $$B_4(X) = {\langle(X-\langle X \rangle)^4\rangle \over \langle(X-\langle X \rangle)^2\rangle^2}$$ where the $X$s are lattice averaged quantities. For infinite volumes, $B_4=3$ for a crossover, $B_4=1$ for a first-order transition and $B_4=1.604(1)$ for the 3-dimensional Ising model. Thus, if there is a critical endpoint we would expect $B_4$ to decrease with increasing $\mu_I$, passing through a value close to the Ising value at the critical $\mu_I$. Figure \[fig:b4mass\] shows our preliminary measurements of the Binder cumulant for the chiral condensate as a function of mass at $\mu_I=\mu=0$ from our $12^3 \times 4$ simulations. Taking the point where the straight-line fit passes through the Ising value as our estimate for the critical mass yields $m_c=0.0264(3)$. Each of the points in this graph were obtained by averaging the Binder cumulants taken from several $\beta$ values close to the transition, and extrapolated to $\beta_c$ which minimizes these cumulants, using Ferrenberg-Swendsen rewieghting [@Ferrenberg:1988yz]. The $\mu_I$ dependence of this Binder cumulant at $\beta_c(\mu_I)$ is shown in figure \[fig:b4m0.03\], for $m=0.03$, a little above $m_c$. It is clear that, rather than decreasing with increasing $\mu_I$, it actually increases slowly. Since $\beta_c$ and hence $T_c$ decrease with increasing $\mu_I$, in physical units $m$ is actually decreasing with increasing $\mu_I$ meaning that at fixed physical $m$ the rise would be even more pronounced. The behaviour at $m=0.035$ is very similar. Figure \[fig:beta\_c\] shows the dependence of the transition $\beta$, $\beta_c$, on $\mu_I$. As mentioned above, $\beta_c$ and hence the transition temperature $T_c$ fall (slowly) with increasing $\mu_I$ as expected. The fits shown to this preliminary ‘data’ are: $$\begin{aligned} \beta_c &=& 5.15326(10) - 0.173(2) \mu_I^2 \hspace{0.5in} m=0.035\nonumber \\ \beta_c &=& 5.14386(\:\,8) - 0.172(1) \mu_I^2 \hspace{0.5in} m=0.030\nonumber \\ \beta_c &=& 5.13426(12) - 0.179(4) \mu_I^2 \hspace{0.5in} m=0.025\nonumber \\\end{aligned}$$ which is in reasonable agreement with the results of de Forcrand and Philipsen for the $\mu$ dependence of the transition temperature, obtained from analytic continuation from imaginary $\mu$ if we make the identification $\mu_I=2\mu$. Figure \[fig:rhmc&hmdr\] shows the $dt$ dependence of the Binder cumulants at the transition for $m=0.035$, $\mu_I=0.2$ in the HMD(R) simulations. The exact RHMC result, which has no $dt$ dependence, is plotted on this graph at $dt=0$. It is clear that the RHMC result is consistent with the $dt \rightarrow 0$ limit of the HMD(R) results. The actual value of $dt$ used in the RHMC simulations was $dt=0.05$, the value of $dt$ for the rightmost point on this graph, showing one advantage of using this new algorithm. Conclusions =========== We simulate lattice QCD with 3 flavours of staggered quarks with a small chemical potential $\mu_I < m_\pi$ for isospin, in the neighbourhood of the finite temperature transition from hadronic matter to a quark-gluon plasma. Fourth order Binder cumulants are used to probe the nature of this transition and search for the critical endpoint for masses slightly above the critical mass for zero chemical potentials. Earlier simulations using the HMD(R) algorithm were plagued by large finite $dt$ errors [@Kogut:2005yu]. We now use the RHMC algorithm which is exact in the sense of having no finite $dt$ errors. We measure the critical mass to be $m_c=0.0264(3)$ for $N_t=4$, in agreement with the recent results of de Forcrand and Philipsen [@deForcrand:2006pv], but considerably below earlier measurements which found values close to $m_c=0.033$ [@Karsch:2001nf; @Christ:2003jk; @deForcrand:2003hx]. These higher values were due to using the HMD(R) algorithm with $dt$ large enough to produce large systematic errors. For masses greater than $m_c$ we found that the Binder cumulant for the chiral condensate increases with increasing $\mu_I$ and thus shows no evidence for a critical endpoint, contrary to earlier expectations. This also agrees with the observations of de Forcrand and Philipsen [@deForcrand:2006pv] for finite $\mu$, emphasizing the similarities between finite $\mu$ and finite $\mu_I$ for small $\mu$,$\mu_I$, near the finite temperature transition. On these relatively small lattices ($12^3 \times 4$), we really should minimize the Binder cumulant of linear combinations of the chiral condensate, the plaquette and the isospin density to obtain the desired eigenfield of the renormalization group equations, to draw reliable conclusions [@Karsch:2001nf] [^1]. We end with the observation that we have used RHMC simulations where we do not know a positive lower bound for the spectrum of the quadratic Dirac operator. This is done by choosing a speculative lower bound and justifying our choice a postiori. We refer the reader to our recent paper on this subject [@Kogut:2006jg]. Acknowledgements {#acknowledgements .unnumbered} ================ JBK is supported in part by a National Science Foundation grant NSF PHY03-04252. DKS is supported by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38. We thank Philippe de Forcrand and Owe Philipsen for helpful discussions. Simulations are performed on Jazz at Argonne, Tungsten and Cobalt at NCSA, Bassi and Jacquard at NERSC, DataStar at SDSC and PC’s in Argonne HEP. [99]{} C. R. Allton [*et al.*]{}, Phys. Rev. D [**66**]{}, 074507 (2002) \[arXiv:hep-lat/0204010\]. C. R. Allton [*et al.*]{}, Phys. Rev. D [**71**]{}, 054508 (2005) \[arXiv:hep-lat/0501030\]. R. V. Gavai and S. Gupta, Phys. Rev. D [**68**]{}, 034506 (2003) \[arXiv:hep-lat/0303013\]. P. de Forcrand and O. Philipsen, Nucl. Phys. B [**642**]{}, 290 (2002) \[arXiv:hep-lat/0205016\]. P. de Forcrand and O. Philipsen, arXiv:hep-lat/0607017. V. Azcoiti, G. Di Carlo, A. Galante and V. Laliena, JHEP [**0412**]{}, 010 (2004) \[arXiv:hep-lat/0409157\]. V. Azcoiti, G. Di Carlo, A. Galante and V. Laliena, Nucl. Phys. B [**723**]{}, 77 (2005) \[arXiv:hep-lat/0503010\]. M. D’Elia and M. P. Lombardo, Phys. Rev. D [**67**]{}, 014505 (2003) \[arXiv:hep-lat/0209146\]. M. D’Elia and M. P. Lombardo, Phys. Rev. D [**70**]{}, 074509 (2004) \[arXiv:hep-lat/0406012\]. Z. Fodor and S. D. Katz, Phys. Lett. B [**534**]{}, 87 (2002) \[arXiv:hep-lat/0104001\]. Z. Fodor and S. D. Katz, JHEP [**0404**]{}, 050 (2004) \[arXiv:hep-lat/0402006\]. S. Kratochvila and P. de Forcrand, PoS [**LAT2005**]{}, 167 (2006) \[arXiv:hep-lat/0509143\]. A. Alexandru, M. Faber, I. Horvath and K. F. Liu, Phys. Rev. D [**72**]{}, 114513 (2005) \[arXiv:hep-lat/0507020\]. J. B. Kogut and D. K. Sinclair, Phys. Rev. D [**66**]{}, 034505 (2002) \[arXiv:hep-lat/0202028\]. J. B. Kogut and D. K. Sinclair, arXiv:hep-lat/0509095. S. A. Gottlieb, W. Liu, D. Toussaint, R. L. Renken and R. L. Sugar, Phys. Rev. D [**35**]{}, 2531 (1987). A. D. Kennedy, I. Horvath and S. Sint, Nucl. Phys. Proc. Suppl.  [**73**]{}, 834 (1999) \[arXiv:hep-lat/9809092\]. M. A. Clark, P. de Forcrand and A. D. Kennedy, PoS [**LAT2005**]{}, 115 (2006) \[arXiv:hep-lat/0510004\]. J. B. Kogut and D. K. Sinclair, arXiv:hep-lat/0608017. K. Binder, Z. Phys. B [**43**]{}, 119 (1981). A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett.  [**61**]{}, 2635 (1988). F. Karsch, E. Laermann and C. Schmidt, Phys. Lett. B [**520**]{}, 41 (2001) \[arXiv:hep-lat/0107020\]. N. H. Christ and X. Liao, Nucl. Phys. Proc. Suppl.  [**119**]{}, 514 (2003). P. de Forcrand and O. Philipsen, Nucl. Phys. B [**673**]{}, 170 (2003) \[arXiv:hep-lat/0307020\]. [^1]: We thank Frithjof Karsch for reminding us of this fact

--- author: - | Hisashi <span style="font-variant:small-caps;">Kotegawa</span>$^{1,3}$[^1], Yudai <span style="font-variant:small-caps;">Hara</span>$^{1}$, Hiroki <span style="font-variant:small-caps;">Nohara</span>$^{1}$, Hideki <span style="font-variant:small-caps;">Tou</span>$^{1,3}$,\ Yoshikazu <span style="font-variant:small-caps;">Mizuguchi</span>$^{2,3}$, Hiroyuki <span style="font-variant:small-caps;">Takeya</span>$^{2,3}$, and Yoshihiko <span style="font-variant:small-caps;">Takano</span>$^{2,3}$ title: | Possible Superconducting Symmetry and Magnetic Correlations in K$_{0.8}$Fe$_2$Se$_2$:\ A $^{77}$Se–NMR Study --- The recent discovery of superconductivity in K$_{0.8}$Fe$_2$Se$_2$ is accelerating the research on related materials.[@Guo; @Krzton] The research will spread widely because of the relatively high transition temperature $T_c$ of $\sim32$ K and absence of arsenic. Another interesting point is whether superconductivity occurs in the same framework as that of other Fe-based superconductors. Most Fe-based superconductors possess both hole-like bands and electron-like bands. The nesting between those bands is considered to yield multigap superconductivity characterized by $s^{\pm}$ symmetry.[@Mazin] The band calculations suggest that the electronic state of stoichiometric KFe$_2$Se$_2$ is far from that of other Fe-based superconductors, and the appearance of the common band topology to other systems depends on the amount of hole doping, which is probably induced by the deficiency of K and Fe.[@Shein; @Cao; @Nekrasov] Recent angle-resolved photoemission spectroscopy (ARPES) studies suggest that K$_{0.8}$Fe$_2$Se$_2$ is a heavily electron-doped system compared with other Fe-based superconductors, and the hole-like Fermi surface disappears.[@Qian; @Zhang] If superconductivity is realized under such a situation, $s^{\pm}$ symmetry should be excluded. In this letter, we report $^{77}$Se-NMR results in single-crystalline K$_{0.8}$Fe$_2$Se$_2$ to investigate the superconducting (SC) symmetry and magnetic correlations. A single-crystalline sample with $T_c=32$ K was prepared as described elsewhere.[@Mizuguchi] $^{77}$Se-NMR measurement using a standard spin-echo method was performed under a magnetic field of $\sim8.995$ T along the $ab$ plane in both the normal state and the SC state. $^{77}$Se possesses the nuclear spin of $I=1/2$, which corresponds to one NMR transition. $T_c=31$ K was estimated under $\sim8.995$ T from the onset of diamagnetism. The Knight shift was obtained using the gyromagnetic ratio of $\gamma_n=8.13$ MHz/T. The nuclear spin-lattice relaxation rate $1/T_1$ was obtained by a nice fitting of the recovery curve to the single exponential function in the normal state. In the SC state, we omitted a small amount of fast relaxation arising from the vortex core in the fitting. {width="0.9\linewidth"} Figure 1(a) shows the $^{77}$Se-NMR spectra measured by applying $\sim8.955$ T along the $ab$ plane. The line width was 25 kHz at $32$ K. The spectral shape at high temperatures is not the Lorenz type but rather close to a rectangle, suggesting that slight inhomogeneity exists in the sample. However, we confirmed that the electronic state is homogeneous, within the accuracy of $1/T_1$, from the frequency dependence of $1/T_1$. Knight shift was estimated from a central position of the spectrum in the normal state. The spectrum shows a slight and asymmetric broadening owing to the existence of a vortex in the SC state, where Knight shift was obtained from the peak of the spectrum. The spectrum at 29 K just below $T_c$ shifts from that at 32 K without obvious broadening, ensuring that $T_c$ is almost homogenous in the sample. Figure 1(b) shows the temperature dependence of Knight shift. In the normal state, Knight shift shows a strong temperature dependence. This reduction in Knight shift toward low temperatures originates in the high density of states (DOS) near the Fermi level,[@Ikeda] which is often seen in the electron-doped systems among Fe-based superconductors.[@Nakai; @Grafe; @Ning; @Imai] In K$_{0.8}$Fe$_2$Se$_2$, Knight shift continues to decrease down to $T_c$ without the Fermi liquid region, as shown in the inset. This indicates that the high DOS is located in the vicinity of the Fermi level, and the spin susceptibility of $q=0$ is suppressed with decreasing temperature. Knight shift markedly decreases below $T_c=31$ K and approaches $\sim0.02$% at zero temperature. Generally, Knight shift consists of the temperature-dependent spin part $K_s$ and temperature-independent orbital part (or chemical shift) $K_{orb}$. This suggests that the spin part of superconducting symmetry is a singlet and $K_{orb} \sim 0.02$%, because field-induced DOS is not expected at the present magnetic field sufficiently lower than $H_{c2}$.[@Mizuguchi] {width="0.8\linewidth"} Figure 2 shows temperature dependences of $1/T_1T$ for K$_{0.8}$Fe$_2$Se$_2$, FeSe, and non superconducting (Fe$_{0.9}$Co$_{0.1}$)Se. In K$_{0.8}$Fe$_2$Se$_2$, $1/T_1T$ also decreases with decreasing temperature owing to the band structure effect. The marked difference from Knight shift is that $1/T_1T$ is almost constant below $\sim60$ K, while Knight shift continues to decrease toward low temperatures. This disagreement suggests that the spin fluctuation of $q\neq0$, which is most likely an antiferromagnetic (AF) one, develops at least in this temperature region, even though it is not so strong. A strong temperature dependence of DOS toward low temperatures likely masks the development of spin fluctuations. We compare $1/T_1T$ with those in FeSe and electron-doped (Fe$_{0.9}$Co$_{0.1}$)Se. $1/T_1T$ in FeSe was measured using a high-quality sample made with high-temperature annealing,[@Mizuguchi_review] and the NMR line width of which is almost the same as that of stoichiometric FeSe.[@Imai] Even in this sample, $1/T_1T$ separated into two components below 40 K, so that we plotted the short component of $\sim75$% in the volume fraction, which is considered to be intrinsic. In FeSe, an obvious increase in $1/T_1T$ is seen below $\sim40$ K owing to the development of a low-energy part of AF spin fluctuations. The development of $1/T_1T$ is enhanced under pressure with a close relationship with $T_c$.[@Imai; @Masaki] In the high-temperature region, the temperature dependence of $1/T_1T$ in K$_{0.8}$Fe$_2$Se$_2$ is markedly stronger than that in FeSe, and it is weak in (Fe$_{0.9}$Co$_{0.1}$)Se. The reduction of $1/T_1T$ in (Fe$_{0.9}$Co$_{0.1}$)Se upon electron doping indicates that the high DOS, which is located below the Fermi level in FeSe, is removed from the Fermi level by electron doping. This doping suppresses the AF spin fluctuations and superconductivity. By contrast, strong temperature dependence in $1/T_1T$ above $\sim60$ K in K$_{0.8}$Fe$_2$Se$_2$ suggests that high DOS is located closer to the Fermi level than in the case of FeSe. However, the development of AF spin fluctuations at low temperatures is not drastically enhanced in K$_{0.8}$Fe$_2$Se$_2$. This indicates that the character of the band near the Fermi level is different between FeSe and K$_{0.8}$Fe$_2$Se$_2$. {width="0.8\linewidth"} Figure 3 shows Knight shift vs $(1/T_1T)^{1/2}$ in the normal state. The Korringa ratio `K` is given as follows. $$\texttt{K}=\frac{1}{T_1TK_s^2}\frac{\hbar}{4\pi k_B} \frac{\gamma_e^2}{\gamma_n^2}$$ Here, $\gamma_e$ is the electron gyromagnetic ratio. `K` corresponds to the character of spin correlations, for example, $\texttt{K}<<1$ corresponds to ferromagnetic correlations and $\texttt{K}>>1$ corresponds to AF correlations. The observed strong temperature dependences of Knight shift and $1/T_1T$ are mainly attributed to the band structure effect, but Knight shift vs $(1/T_1T)^{1/2}$ does not show a linear relationship, indicating that the Korringa relation of $\texttt{K}=const.$ is not realized. The relationship of upward convex suggests that `K` increases with decreasing temperature, because the slope in the figure is proportional to $\texttt{K}^{\ 1/2}$. If we assume $K_{orb}=0.02$% as $K_s(T=0)=0$, the temperature dependence of `K` is as indicated in the inset. `K` is not very far from 1, indicating that spin correlations are not strong in K$_{0.8}$Fe$_2$Se$_2$, but the obvious increase in `K` toward low temperatures suggests the development of AF spin correlations. {width="0.7\linewidth"} Next we focus on the symmetry of the superconducting gap in K$_{0.8}$Fe$_2$Se$_2$. Figure 4 shows the temperature dependence of $1/T_1$. The steep decrease in $1/T_1$ was observed below $T_c$ without any sign of a coherence peak. The power is $T^6$-like just below $T_c$, approaches $T^3$-like below $\sim T_c/2$, and then starts to deviate from $T^3$ below $\sim T_c/5$. Such power-law behavior in $1/T_1$ is often seen in most Fe-based superconductors. The absence of the coherence peak and non exponential temperature dependence of $1/T_1$ clearly excludes the possibility of a single isotropic gap pointed out from the ARPES measurement.[@Zhang] The green dashed curve is a calculation using a single isotropic gap of $\Delta_0=4k_BT_c$ ignoring the coherence effect. This curve completely disagrees with the data for the low-temperature region. Non exponential behavior suggests that the multigap is realized with different magnitudes of gap size or that the anisotropic superconductivity is realized with a finite DOS in the SC gap. The red solid curve is the results of a calculation based on the $s^{\pm}$ isotropic gap.[@Nagai] This model has been adapted for other Fe-based systems such as LaFeAsO, (Ba,K)Fe$_2$As$_2$, and LiFeAs.[@Matano; @Yashima; @Li] We can reproduce the data well using $\Delta_1=4.2 k_BT_c$, $\Delta_2=1.7 k_BT_c$, $N_1:N_2=0.7:0.3$, and $\eta=0.048\Delta_1$. Here, $\Delta_1$ and $\Delta_2$ are the magnitudes of the SC gap in the respective bands, and $N_1$ and $N_2$ are respective DOS at the Fermi level. $\eta$ is the smearing factor due to the impurity effect, and induces the residual DOS at the Fermi level. $\Delta_1=4.2 k_BT_c$ indicates the superconductivity in the strong-coupling regime, and is comparable to that of (Ba$_{0.6}$K$_{0.4}$)Fe$_2$As$_2$.[@Yashima] The value of $\eta$ is also comparable to those of other Fe-based superconductors with $s^{\pm}$ symmetry.[@Matano; @Yashima; @Li] The multigap feature is not obvious in the Knight shift probably due to the relatively large $N_1$, which is similar to the case of (Ba$_{0.6}$K$_{0.4}$)Fe$_2$As$_2$.[@Yashima] The third model is a two-dimensional $d$-wave model ($\Delta(\theta)=\Delta_0 \cos(2\theta)$) with line nodes that produce $T^3$ behavior at low temperatures. As shown by the blue dotted curve, this model also can reproduce the data roughly using $\Delta_0 = 5 k_BT_c$ in the strong-coupling regime and the residual DOS of 3%, which is generally induced by impurity scattering. The model of $d$-wave symmetry is not sufficiently far from the data to exclude the possibility that $d$-wave symmetry is realized in this system. We tried the two-gap $d$-wave model, but it did not improve the fitted result. The $1/T_1$ at the lowest temperature gives the upper limit of the residual DOS of $\sim7$%. The small residual DOS of $\sim3-7$% indicates that impurity scattering is small in spite of the system with a deficiency of ions. The $s^{\pm}$ symmetry is more plausible, but for its identification, measurement at lower temperatures will be required, as well as under lower field to avoid the contribution in the relaxation process from the vortex core. The ARPES measurement suggests the isotropic gap opens only in the electron pocket.[@Zhang] The interpretation based on our data seems to be inconsistent with this observation. Another ARPES measurement also shows that this system possesses only an electron pocket.[@Qian] If the nesting between the electron pockets plays a crucial role in superconductivity, the $d$-wave symmetry is likely to be realized. The present NMR result cannot exclude this possibility. From the results of our study, we can claim that the single isotropic gap is excluded, and the $s^{\pm}$-wave or $d$-wave with a strong-coupling regime are possible candidates. Finally, we mention earlier NMR reports presented by Yu and co-workers quite recently.[@Yu; @Ma] The obtained data in this study are similar to their data. The difference in the absolute value of Knight shift seems to originate in the different value of $\gamma_n$. Yu and co-workers claim that AF spin correlations are absent in this system from the Korringa ratio. Note that we defined the reciprocal of their Korringa ratio as `K`. The `K` is sensitive to the estimation of $K_{orb}$, and they estimated $K_{orb}$ considering that Korringa ratio is constant against temperature. (they use the notation of $K_{c}$). However, our measurement at higher temperatures shows that the Korringa ratio depends on temperature. If we can obtain the susceptibility data removing the contribution from impurity, it will help us to estimate the exact $K_{orb}$. As for the symmetry of the SC gap, they tried the fitting of $1/T_1$ with an isotropic gap and discussed the possibility of the multigap. We clearly excluded a single isotropic gap and suggest the multigap or the anisotropic gap from the fitting of $1/T_1$. In summary, we performed NMR measurements in K$_{0.8}$Fe$_2$Se$_2$ using a single-crystalline sample. The strong temperature dependences in Knight shift and $1/T_1T$ at high temperatures suggest that the high DOS is located in the vicinity of the Fermi level. This feature is much stronger than that in FeSe. The breakdown of the Korringa relation suggests the AF spin correlations develop toward low temperatures, although it is not obvious compared with FeSe. The temperature dependence of $1/T_1$ in the SC state can exclude the single isotropic gap. It can be reproduced well by the $s^{\pm}$-wave model, but the possibility of $d$-wave symmetry still exists. This work has been partly supported by Grants-in-Aid for Scientific Research (Nos. 19105006, 20740197, 20102005, 22013011, and 22710231) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan. [99]{} J. Guo, S. Jin, G.Wang, S.Wang, K. Zhu, T. Zhou, M. He, and X. Chen: Phys. Rev. B [**82**]{} (2010)180520(R). A. Krzton-Maziopa, Z. Shermadini, E. Pomjakushina, V. Pomjakushin, M. Bendele, A. Amato, R. Khasanov, H. Luetkens, and K. Conder: J. Phys.: Condens. Matter [**23**]{} (2011) 052203. I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du: Phys. Rev. Lett. [**101**]{} (2008) 057003. I. R. Shein and A. L. Ivanovskii: arXiv:1012.5164 (2010). C. Cao and J. Dai: arXiv:1012.5621 (2010). I. A. Nekrasov and M. V. Sadovskii: Pisma ZhETF [**93**]{} (2011) 182. Y. Zhang, L. X. Yang, M. Xu, Z. R. Ye, F. Chen, C. He, J. Jiang, B. P. Xie, J. J. Ying, X. F. Wang, X. H. Chen, J. P. Hu, and D. L. Feng: arXiv:1012.5980 (2010). T. Qian, X.-P. Wang, W.-C. Jin, P. Zhang, P. Richard, G. Xu, X. Dai, Z. Fang, J.-G. Guo, X.-L. Chen, and H. Ding: arXiv:1012.6017 (2010). Y. Mizuguchi, H. Takeya, Y. Kawasaki, T. Ozaki, S. Tsuda, T. Yamaguchi and Y. Takano: Appl. Phys. Lett. [**98**]{} (2011) 042511. H. Ikeda: J. Phys. Soc. Jpn. [**77**]{} (2008) 123707. Y. Nakai, K. Ishida, Y. Kamihara, M. Hirano, and H. Hosono: J. Phys. Soc. Jpn. [**77**]{} (2008) 073701. H.-J. Grafe, D. Paar, G. Lang, N. J. Curro, G. Behr, J. Werner, J. Hamann-Borrero, C. Hess, N. Leps, R. Klingeler, and B. Büchner: Phys. Rev. Lett. [**101**]{} (2008) 047003. F. Ning, K. Ahilan, T. Imai, A. S. Sefat, R. Jin, M. A. Mcguire, B. C. Sales, D. Mandrus: J. Phys. Soc. Jpn. [**77**]{} (2008) 103705. T. Imai, K. Ahilan, F. L. Ning, T. M. McQueen, and R. J. Cava: Phys. Rev. Lett. [**102**]{} (2009) 177005. Y. Mizuguchi and Y. Takano. J. Phys. Soc. Jpn. [**79**]{} (2010) 102001. S. Masaki, H. Kotegawa, Y. Hara, H. Tou, K. Murata, Y. Mizuguchi, and Y. Takano: J. Phys. Soc. Jpn. [**78**]{} (2009) 063704. Y. Nagai, N. Hayashi, N. Nakai, H. Nakamura, M. Okumura, and M. Machida: New J. Phys. [**10**]{} (2008) 103026 K. Matano, Z. Li, G. L. Sun, D. L. Sun, C. T. Lin, M. Ichioka, and G. -q. Zheng: EPL [**87**]{} (2009) 27012. M. Yashima, H. Nishimura, H. Mukuda, Y. Kitaoka, K. Miyazawa, P. M. Shirage, K. Kihou, H. Kito, H. Eisaki, and A. Iyo; J. Phys. Soc. Jpn. [**78**]{} (2009) 103702. Z. Li, Y. Ooe, X.-C. Wang, Q.-Q. Liu, C.-Q. Jin, M. Ichioka, and G.-q. Zheng: J. Phys. Soc. Jpn. [**79**]{} (2010) 083702. W. Yu, L. Ma, J. B. He, D. M. Wang, T.-L. Xia, and G. F. Chen: arXiv:1101.1017 (2011). L. Ma, J. B. He, D. M. Wang, G. F. Chen, and W. Yu: arXiv:1101.3687 (2011). [^1]: E-mail address: kotegawa@crystal.kobe-u.ac.jp

--- abstract: 'Bio-medical ontologies can contain a large number of concepts. Often many of these concepts are very similar to each other, and similar or identical to concepts found in other bio-medical databases. This presents both a challenge and opportunity: maintaining many similar concepts is tedious and fastidious work, which could be substantially reduced if the data could be derived from pre-existing knowledge sources. In this paper, we describe how we have achieved this for an ontology of the mitochondria using our novel ontology development environment, the [Tawny-OWL]{}library.' address: 'School of Computing Science, Newcastle University, Newcastle-upon-Tyne, UK' author: - 'Jennifer D. Warrender and Phillip Lord[^1]' bibliography: - '2015\_scaffolding\_pwl\_jw.bib' title: Scaffolding the Mitochondrial Disease Ontology from extant knowledge sources --- Introduction ============ Bio-medical ontologies vary in size, with largest containing millions of concepts. Building ontologies of this size is complex, time-consuming and expensive and just as challenging to maintain and update. Ontologies are only one of many mechanisms for the computational representation of knowledge. In some cases, ontologies are created where many of the needed concepts will be available elsewhere as terms in different structured representations. Being able to reuse these representations as a *scaffold* for the rest of an ontology might be able to reduce the cost and work-load of producing ontologies. This is evidenced by, for instance, SIO [@sio] which contains a list of all the chemical elements. Or the Gene Ontology (GO) [@go], which contains many terms related to chemical homeostasis, each of which need to relate to a specific chemical described in ChEBI [@chebi]. In addition to being described elsewhere, these concepts are often highly similar to each other. In extreme cases such as the amino acid ontology [@greycite9379], ontologies can consist of only related concepts, and “support” concepts that are used to describe them. One solution to this is the use of patterns. A pattern is an abstract specification of an ontology axiomatisation with a number of “variables”. The pattern is instantiated by providing values for these variables, which are then expanded into the full axiomatisation providing one or more concepts. Patterns have been implemented by a number of different tools, which differ in how the patterns are specified, and how and when the values are provided for the variables. For example, *termgenie* is a website which allows submission to GO (and others) [@Dietze_2014]. Variable values are entered through a form which then generates axioms, definitions and cross-references. For instance, this is the axiomatisation from termgenie when defining the term “cytosine homeostasis” is_a: GO:0048878 {is_inferred="true"} ! chemical homeostasis intersection_of: GO:0048878 ! chemical homeostasis intersection_of: regulates_levels_of CHEBI:16040 ! cytosine relationship: regulates_levels_of CHEBI:16040 {is_inferred="true"} ! cytosine As well as the axiomatisation, termgenie also generates a number of different annotations including a definition, submitter information, and status. With termgenie, patterns are specified through the use of JavaScript functions. In addition to termgenie, other systems also allow patterns. For example, both the desktop and web version of [Protégé]{}contain forms, which grant users the ability to customise the GUI and specify several axioms at once. In this case, patterns are declaratively defined (implicitly, with a GUI design) in XML [@tudorach_icd_webprotege]. Applications like Populous [@Jupp_Wolstencroft_Stevens_2011] and Rightfield [@rightfield] use spreadsheets or spreadsheet-like interfaces to enter data, which is then transformed into a set of OWL axioms based on a pattern. In the case of these two, the patterns are specified in OPPL, a pattern language for OWL which can also be used independently [@aranguren_Stevens_Antezana_2009]. Finally, the Brain API allows programmatic construction of ontologies in an easy to use manner using Java [@croset2013]. While these systems are all aimed at somewhat different use-cases, they all address the same problem; how to produce a large number of concepts all of which are similar, and to do so with a high-degree of repeatability. However, the use of this form of patternised ontology tool presents a number of problems. These tools provide a mechanism for adding many axioms at once, but not removing them again[^2]. If the knowledge changes, then this is a problem as the axioms added from a given pattern need to be removed or updated. Furthermore, if the knowledge engineering changes i.e. the pattern is updated, then all axioms added from any use of the pattern must also be updated. In this paper, we describe how we have addressed these problems with the Mitochondrial Disease Ontology (MDO), through the use of the [Tawny-OWL]{}environment, which is a fully programmatic environment for ontology development. With [Tawny-OWL]{}, we can use a *pattern-first* ontology development process, building with patterns and data from extant knowledge sources from the start. This has allowed us to generate a *scaffold* which we can then populate further with hand-crafted links between parts of this scaffold where the knowledge exists. As a result, it is possible to update both the knowledge and the patterns by simply regenerating the ontology. This process promises to aid in both the construction and maintenance of ontologies. The MDO is available from <https://github.com/jaydchan/tawny-mitochondria>. [Tawny-OWL]{}is available from <https://github.com/phillord/tawny-owl>. The Mitochondria Disease Ontology (MDO) {#sec:mitoch-dise-ontol} ======================================= Mitochondria are complex organelles found in most eukaryotic cells. Their key function is to enable the production of ATP through oxidative phosphorylation, providing usable energy for the rest of the cell. The mitochondria carry their own small genome containing 37 genes in human. Many other genes are involved in producing proteins involved in mitochondrial function, but these are encoded in the nuclear genome. A number of mitochondrial genes are associated with diseases; the first identified of these is the MELAS [@melas], which is most commonly caused by a point mutation in a tRNA found in the mitochondrial genome. As with many areas of biology, mitochondrial research is a large, knowledge-rich discipline. Our purpose with the MDO is to attempt to formalise this knowledge, using an incremental or “pay-as-you-go” data integration approach. The ontology here serves as a tool for reasoning and knowledge exploration, rather than to form as a reference ontology [@handbook2]. This is an approach we have previously found useful in classifying phosphatases [@wolstencroftetal2006]. The hope is that we can incorporate new knowledge as it is released, checking it for consistency and cross-linking it with existing knowledge. [Tawny-OWL]{} {#sec:tawny} ============= In this section, we give a brief description of [Tawny-OWL]{} [@tawny] and how it supports pattern-first development. [Tawny-OWL]{}is a library written in Clojure, a dialect of lisp. It wraps the OWL API [@owlapi] and allows the fully programmatic constructions of ontologies. It has a simple syntax which was modelled on the Manchester Syntax [@ms2], modified to integrate well with Clojure. It can be used to make simple statements in OWL: (defclass A :super (some r B)) which makes defines a new class $A$ such that $A\sqsubseteq~\exists~r~B$. Although this is similar to the equivalent Manchester Syntax statements, [Tawny-OWL]{}provides a feature called “broadcasting” which is, essentially a form of pattern. So this following statement: (some r B C) is equivalent to the two statements $\exists~r~B$ and $\exists~r~C$. We apply the first two arguments (|some| and |r|) to the remaining ones consecutively. It also provides simple patterns, such as the covering axiom, so: (some-only r B C) is equivalent to three statements $\exists~r~B$, $\exists~r~C$ and $\forall~r~(B~\sqcup~C)$. While the patterns shown here are provided by [Tawny-OWL]{}, end ontology developers are using the same programmatic environment. Patterns are encoded as functions and instantiated with function calls. For instance, we could define |some-only| as follows: (defn some-only \[property & classes\] (list (some property classes) (only property (or classes)))) Here |defn| introduces a new function, |property & classes| are the arguments, and |list| packages the return values as a list. |some|, |only| and |or| are defined by [Tawny-OWL]{}as the appropriate OWL class constructors. It is, therefore, possible to build *localised patterns* — custom patterns for use predominately with the current ontology [@warrender_thesis_2015]. Patterns can call each other and can be of arbitrary complexity. The use of [Tawny-OWL]{}, therefore, inverts the usual style of ontology development. Non-patternised classes are just trivial instantiations of patterns. Building a Mitochondrial Scaffold {#sec:build-mitoch-scaff} ================================= Following a requirements gathering phase for MDO, it was clear from our competency questions (for example “What are all the genes/proteins that are associated with a specific syndrome?”) that we needed many concepts which were heavily repetitive, and further which have comprehensive and curated lists available. We describe these parts of the domain knowledge as the *scaffold*. For example, there are around 761 genes whose products are involved in mitochondrial function. Classes representing these genes do not, in the first instance, require complex descriptions, and are defined within MDO as follows: (defclass Gene) (defn gene-class \[name\] (owl-class name :label name :super Gene)) This pattern is then populated using a simple text file, with the 761 gene names present. The gene pattern is an extremely simple pattern, as these concepts are self-standing. Other parts of the ontology are even simpler; for instance, for describing mitochondrial anatomy, the classes have similar complexity to the genes, but there are only 15. In this case, classes are defined with a pattern and a list “hard-coded” into the MDO source code, rather than using an external text file. Other patterns are more complex. For instance, the subclasses of |Disease| are defined as follows: (defn disease-class \[name omim lname\] (let \[disease (owl-class name :label name :super Disease)\] (if-not (nil? omim) (refine disease :annotation (see-also (str “OMIMID:” omim)))) (if-not (nil? lname) (refine disease :label (str “Long name:” lname))))) This function adds two annotations to each disease class, if they are available. This function also demonstrates the use of conditionals (|if|), predicates (|nil?|) and string concatenation (|str|); these are not provided by [Tawny-OWL]{}, but by Clojure and demonstrate the value of building [Tawny-OWL]{}inside a fully programmatic environment. Fitting out the Scaffold {#sec:what-we-have} ======================== The top-level of the MDO is shown in Figure \[fig:mtoplevel\]. Of these classes, “Paper” and “Term” are described later. ![The top-level structure of Mitochondrial Disease Ontology. Classes that are a part of the scaffold are coloured in orange, while classes that are built on top of the scaffold are coloured in green.[]{data-label="fig:mtoplevel"}](mstructure.png){width="\columnwidth"} The remaining classes define the scaffold, which now has a total of 1357 classes; a break-down of these classes and their sources is shown in Table \[tab:genericStats\]. For the next stage of the process, we are now building on top of this scaffold, using hand-crafted and bespoke knowledge. This is being achieved by manual extraction of knowledge from papers about mitochondria. Our initial process is to find references in papers to the terms that are represented by classes we have built in the scaffold, and draw explicit relationships between these papers and the scaffolded knowledge that they describe. Currently, these classes also use a patternised approach; the raw data is held in a bespoke (but human readable) syntax[^3], which is then parsed and used to instantiate patterns. In total, there are now 2174 classes created from this approach from around 30 papers. These terms currently are not defined beyond their name and the source paper from which they were identified. We do not consider them directly as part of the scaffold, as they are not from an extant knowledge source, but one that we have created; they are the first layer build on top of our scaffold. We expect future layers to use the [Tawny-OWL]{}syntax directly, as the knowledge increases in complexity and decreases in regularity. Resiliance to Change {#sec:resiliance-change} ==================== One key feature of our development process is that the OWL which defines the MDO is no longer *source code* but generated. Rather it is generated from patterns defined in [Tawny-OWL]{}and text files which are used to instantiate these patterns. The in-memory OWL classes and associated OWL files are generated on-demand, by *evaluating* the patterns. Effectively, we regenerate the ontology every time we restart the environment. In this section, we consider the types of changes that can happen, and how these changes impact on MDO. The scaffold of MDO is sensitive to changes in its dependency knowledge sources. First, new terms can be entered into extant sources, which will necessitate the addition of new classes. For the MDO, this simply necessitates re-importing the knowledge. The addition of equivalent new classes will then happen automatically according to the patterns already defined; no other changes should be necessary for the MDO, although we may wish to refer to the new classes in other parts of the ontology. Second, terms may be removed from dependencies; so, for example, a disease may be redefined by the UMDF. In many cases, for the MDO, this is not problematic – the equivalent classes will simply disappear from the ontology. [Tawny-OWL]{}provides two features to help with changes to terms in the scaffold when these terms are also referred to outside of the scaffold. [Tawny-OWL]{}uses a “declare-before-use” semantics, so removal of classes from the scaffold will cause fail-fast behaviour when they are used elsewhere. The Brain environment uses the same semantics for similar reasons [@croset2013]. In addition, [Tawny-OWL]{}provides a “deprecation” facility which allows the developer to continue refer to terms from the scaffold which have been removed, but to receive warnings about this use; this is rather like obsolescence, but happens automatically[^4]. Third, the MDO scaffold can also cope straight-forwardly with changes to patterns. As with the addition or removal of terms from dependencies, pattern changes will simply take place by re-evaluating the ontology. Finally, the MDO is resilient to changes in ontology engineering conventions. For example, MDO does not use OBO style numeric identifiers, nor provide stable IRIs for integration with linked data sources since these are not critical at the current time[^5]. They, however, could be added easily to all existing (and future) terms in a few lines of code, using an existing facility within [Tawny-OWL]{}for minting and persisting numeric identifiers in an automatic, yet managed, way. This change would just alter IRIs and would have no impact on references between concepts inside or outside of the scaffold. In conclusion, as well as enabling rapid construction of the MDO, we believe that the pattern-first scaffolding approach should also allow easy maintenance of the ontology. Discussion {#sec:discussion} ========== In this paper, we have described how we have used a number of extant knowledge sources, combined with patterns defined using the [Tawny-OWL]{}library to rapidly, reliably and repeatedly construct a scaffold for MDO. We have previously used a related patternised methodology to construct a complex ontology describing human chromosome rearrangements (i.e. The Karyotype Ontology (KO) [@warrender-karyotype]). However, unlike KO, the mitochondrial knowledge we want to encapsulate is found in numerous independent sources (e.g. published papers and online databases) and in a variety of formats (e.g. “free text” and CSV); the use of several patterns to form a scaffold is unique to MDO. Conversely, the axiomatisation of MDO from these sources is simple; this cannot be said for KO, most of which is generated from a single large pattern [@warrender-pattern]. In addition, while our knowledge of the karyotype is constrained and is essentially finished, the community’s understanding of mitochondria and mitochondrial disease is incomplete and will grow in response to the demands of changing knowledge. This methodology is extremely attractive for a number of reasons. First of all, it allows a very rapid way of scaffolding an ontology for a complex area of knowledge. At this stage, most of the classes created are simple and self-standing, although in some cases do have relationships to other entities in the scaffold. At this point, we have built the ontological equivalent of a data warehouse: terms have been taken from elsewhere and have undergone a form of schema reconciliation into ontological classes. One key feature of the MDO is that it has been built using tools designed for software development; these tools are relatively advanced and well-maintained[^6] [@tawny]. Moreover, recreating the MDO ontology from our original [Tawny-OWL]{}source code is an intrinsic part of the development process; there is no complex release process and any ontology developer can recreate the OWL file with a single command. While, the system as it stands has a high-degree of replicability, the design decisions implicit in the source code are not necessarily apparent. For the basic scaffold this is, perhaps, not a major issue, however as MDO is developed outside of its scaffold , we expect to integrate more documentation into the source code itself, using *lentic*, a recently developed tool for literate programming [@greycite23590]. We believe that the engineering process that we have used to build the scaffold is resilient to change, as described in Section \[sec:resiliance-change\]. Despite this resilience, our use of external sources of knowledge does bring with it new dependencies, with all the issues that this entails for change management. We believe that we can manage this by borrowing best practice from software engineering. Importing knowledge into the scaffold can, in many cases, happens entirely automatically from our extant knowledge sources. Considering just the gene lists, we can either import from a local, fixed copy of this list, or take the current version live from the NCBI portal. In software engineering terms, the former is a *release dependency* and provides stability, while the latter is a *snapshot dependency* which will fail-fast, allowing rapid incorporation of new knowledge. The latter is particularly useful within a continuous integration environment which are used with other ontologies [@greycite2899], and are also fully supported by [Tawny-OWL]{}[@tawny]. Although we have not described its usage here, with the MDO we are not forced to use [Tawny-OWL]{}for all development. It would be possible to combine predominately hand-crafted development using [Protégé]{}, for instance, with some patternised classes; for example, the OBI uses this approach [@2041-1480-1-s1-s7]. For, the MDO, in fact almost all terms other than the top-level has been created from other syntaxes, generally a flat-file. For larger projects, we envisage that most ontology developers would not need to use the programmatic nature of [Tawny-OWL]{}. While we appreciate the value of a single environment, a tool should not force all users into it. In this paper, we have described our approach to building the MDO using a patternised scaffold based around existing knowledge sources. While the work described in this paper allows us to integrate structured data into an ontology, we are now investigating new ways of integrating unstructured literate-based knowledge into our ontology; while we have started the process of formalising, this new knowledge is far from finished. As described in this paper, though, a pattern-first, scaffolded approach to ontology development has enabled us to make significant advances with the MDO. We believe that this approach is likely to be applicable to many other domains also. [^1]: To whom correspondence should be addressed: phillip.lord@newcastle.ac.uk [^2]: OPPL can remove axioms as well as add them but this is not automatic. [^3]: In this case *EDN* which is a text representation of Clojure data structures; it looks rather like JSON. [^4]: [Tawny-OWL]{}is implemented in a Lisp and so is homoiconic; this makes it particularly straight-forward to automate code updates if we choose. [^5]: Our initial intention was to use PURLS from [www.purl.org](www.purl.org) but have found practical problems with generating these. [^6]: And, usefully, not dependent on academic developers for future maintenance.

--- abstract: 'We study the phase diagram of director structures in cholesteric liquid crystals of negative dielectric anisotropy in homeotropic cells of thickness $d$ which is smaller than the cholesteric pitch $p$. The basic control parameters are the frustration ratio $d/p$ and the applied voltage $U$. Upon increasing $U$, the direct transition from completely unwound homeotropic structure to the translationally invariant configuration ($TIC$) with uniform in-plane twist is observed at small $d/p\lessapprox 0.5$. Cholesteric fingers that can be either isolated or arranged periodically occur at $0.5\lessapprox d/p<1$ and at the intermediate $U$ between the homeotropic unwound and $TIC$ structures. The phase boundaries are also shifted by (1) rubbing of homeotropic substrates that produces small deviations from the vertical alignment; (2) particles that become nucleation centers for cholesteric fingers; (3) voltage driving schemes. A novel re-entrant behavior of $TIC$ is observed in the rubbed cells with frustration ratios $0.6\lessapprox d/p\lessapprox 0.75,$ which disappears with adding nucleation sites or using modulated voltages. In addition, Fluorescence Confocal Polarising Microscopy (FCPM) allows us to directly and unambiguously determine the 3-D director structures. For the cells with strictly vertical alignment, FCPM confirms the director models of the vertical cross-sections of four types of fingers previously either obtained by computer simulations or proposed using symmetry considerations. For rubbed homeotropic substrates, only two types of fingers are observed, which tend to align along the rubbing direction. Finally, the new means of control are of importance for potential applications of the cholesteric structures, such as switchable gratings based on periodically arranged fingers and eyewear with tunable transparency based on $TIC$.' address: - | Liquid Crystal Institute and Chemical Physics\ Interdisciplinary Program, Kent State University, Kent, Ohio 44242 - | Department of Mathematical Sciences, Kent State University, Kent,\ Ohio 44242 - 'AlphaMicron Inc., Kent, Ohio 44240.' author: - | I. I. Smalyukh ($\thanks{% Author for correspondence (e-mail: smalyukh@lci.kent.edu)}$), B. I. Senyuk, P. Palffy-Muhoray, and O. D. Lavrentovich - 'H. Huang and E. C. Gartland, Jr.' - 'V. H. Bodnar, T. Kosa, and B. Taheri' title: 'Electric-field-induced nematic-cholesteric transition and 3-D director structures in homeotropic cells' --- Introduction ============ The unique electro-optic and photonic properties of cholesteric liquid crystals (CLCs) make them attractive for applications in displays, switchable diffraction gratings, eyeglasses with voltage-controlled transparency, for temperature visualization, for mirrorless lasing, in beam steering and beam shaping devices, and many others [Blinov-Ch-book,McManamon,NatureChColor,DKYang,EyeLC1,Patel,ChGrating1,Bistable-chiral,ChLaser1,ChLaserAPL,ChLaser3,de Gennes-book,KlemanLavrentovichBook,ChiralityInLC]{}. In nearly all of these applications, CLCs are confined between flat glass substrates treated to set the orientation of molecules at the liquid crystal (LC)-glass interface along some well-defined direction (called easy axis) and an electric field is often used to switch between different textures. In the confined CLCs, the magnitude of the free energy terms associated with elasticity, surface anchoring, and coupling to the applied field are frequently comparable; their competition results in a rich variety of director structures that can be obtained by appropriate surface treatment, material properties of CLCs, and applied voltage. Understanding these structures and transitions between them is of great practical interest and of fundamental importance[@de; @Gennes-book; @KlemanLavrentovichBook; @ChiralityInLC]. CLCs have a twisted helicoidal director field in the ground state. The axis of molecular twist is called the helical axis and the spatial period over which the liquid crystal molecules twist through $2\pi $ is called the cholesteric pitch $p$. CLCs can be composed of a single compound or of mixtures of a nematic host and one or more chiral additives. Cholesterics usually have the equilibrium pitch $p$ in the range $100nm-100\mu m$; the pitch $p$ can be easily modified by additives. When CLCs are confined in the cells with different boundary conditions or subjected to electric or magnetic fields, one often observes complex three-dimensional (3-D) structures. The cholesteric helix can be distorted or even completely unwound by confining CLCs between two substrates treated to produce homeotropic boundary conditions [@Oswald-review00]. Interest in this subject was initiated by Cladis and Kleman [@CladisKleman], subsequently a rich variety of spatially periodic and uniform structures have been reported [Oswald-review00,LGil,ourreview,PressArrott,MCLC-PressArrott,Flow-PressArrott,Gil-PRL98,Baudry-PRE99,CHnegativeDeltaE,Tarasov,Oswald2004,InvWalls,T-Junktions,FLequeux,Ishikawa-fingerprint]{}.  These structures can be controlled by varying the cell gap thickness $d$, pitch $p$, applied voltage $U$, and the dielectric and elastic properties of the used CLC. The complexity of many LC structures usually does not allow simple analytic descriptions of the director configuration. Since the pioneering work of Press and Arrott[@PressArrott; @MCLC-PressArrott], a great progress has been made in computer simulations of static director patterns in CLCs confined into homeotropic cells (see, for example [Oswald-review00,LGil,ourreview,PressArrott,Flow-PressArrott,Gil-PRL98,Baudry-PRE99]{}), which brought much of the current understanding of these structures. The first goal of this work is to study phase diagram and director structures that appear because of geometrical frustration of CLCs in the cells with either strictly vertical or slightly tilted ($<2^{\circ }$) easy axis at the confining substrates. We start with the phase diagram in the plane of $\rho =d/p$ and $U$ similar to the one reported in Refs.[Oswald-review00,CHnegativeDeltaE]{} and then proceed by studying influence of such extra parameters as rubbing, introducing nucleation sites, and voltage driving schemes. We use CLCs with negative dielectric anisotropy; the applied voltages are sufficiently low and the frequencies are sufficiently high to avoid hydrodynamic instabilities [@de; @Gennes-book]. Cell gap thicknesses $d$ are smaller than $p$ and the phase diagrams are explored for frustration ratios $\rho =0-1$ and $U=(0-4)Vrms$. For small $\rho $ and $U$, the boundary conditions force the LC molecules throughout the sample to orient perpendicular to the glass plates. Above the critical values of $\rho $ and/or $U$, cholesteric twisting of the director takes place [Oswald-review00]{}. Depending on $\rho ,$ $U,$ and other conditions, the twisted director structures can be either uniform or spatially periodic, with wave vector in the plane of the cell. Upon increasing $U$ for $\rho <0.5 $, the direct transition from completely unwound homeotropic structure to the translationally invariant configuration ($TIC$) [Oswald-review00,PressArrott,MCLC-PressArrott]{} with uniform in-plane twist is observed. Cholesteric fingers ($CF$s) of different types that can be either isolated or arranged periodically are observed for $0.5\lessapprox \rho <1$ and intermediate $U$ between the homeotropic unwound and $TIC$ structures. The phase diagrams change if the homeotropic alignment layers are rubbed, if particles that become the nucleation centers for $CF$s are present, and if different driving voltage schemes are used. Upon increasing U in rubbed homeotropic cells with $0.6<\rho <0.75$, we observe a re-entrant behavior of $TIC$ and the following transition sequence: (1) homeotropic untwisted state, (2) translationally invariant twisted state, (3) periodic fingers structure (4) translationally invariant twisted state with larger in-plane twist. This sequence has not been observed in our own and in the previously reported[@Oswald-review00] studies of unrubbed homeotropic cells. The second goal of this work is to unambiguously reconstruct director field of $CF$s and other observed structures in the phase diagram. For this we use the Fluorescence Confocal Polarizing Microscopy (FCPM) [@FCPMCPL]. Although the fingers of different types look very similar under the polarizing microscope (which may explain some confusion in the literature [@Baudry-PRE99]), FCPM allows clear differentiation of $CF$s, as well as other structures. We directly visualize the $TIC$ with the total director twist ranging from $0$ to $2\pi $, depending on $\rho $ and $U,$ and rubbing. We reconstruct director structure in the vertical cross-sections of four different types of $CF$s that are observed for cells with strictly vertical alignment. We unambiguously prove the models described recently by Oswald et al. [@Oswald-review00] while disproving some of the other models that were proposed in the early literature (see, for example[Baudry-PRE99]{}). Only two types of $CF$ structures are observed in CLCs confined to cells with slightly tilted easy axes at the substrates. The third goal of our work stems from the importance of the studied structures for practical applications. The spatially-uniform $TIC$ and homeotropic-to-$TIC$ transition are used in the electrically driven light shutters, intensity modulators, eyewear with tunable transparency, and displays [@Blinov-Ch-book; @EyeLC1; @Patel; @Bistable-chiral]. In these applications, it is often advantageous to work in the regime of high $\rho $ but fingers are not desirable since they scatter light. In our study, we therefore focus on obtaining maximum effect of different factors on the phase diagrams. We demonstrate that the combination of rubbing and low-frequency voltage modulation can stabilize the uniformly twisted structures up to $\rho \approx 0.75$, much larger than $\rho \approx 0.5$ reported previously [@Oswald-review00]. The presence of nucleation centers, such as particles used to set the cell thickness, tends to destroy homogeneously twisted cholesteric structure even at relatively low $\rho \approx 0.5$ confinement ratios; this information is important for the optimal design of the finger-free devices. On the other hand, periodic finger patterns with well controlled periodicity and orientation may be used as voltage-switchable diffraction gratings. Our finding, which enables the very possibility of such application, is that rubbing can set the unidirectional orientation of periodically arranged fingers. The article is organized as follows. We describe materials, cell preparation, and experimental techniques in section II. The phase diagrams are described in section III.A and the reconstructed director structures in section III.B. Section IV gives an analytical description of the transition from homeotropic to a twisted state as well as a brief discussion of other structures and transitions along with their potential applications. The conclusions are drawn in section V. Experiment ========== Materials and cell preparation ------------------------------ The cells with homeotropic boundary conditions were assembled using glass plates coated with transparent ITO electrodes and the polyimide JALS-204 (purchased from JSR, Japan) as an alignment layer. JALS-204 provides strong homeotropic anchoring; anchoring extrapolation length, defined as the ratio of the elastic constant to the anchoring strength, is estimated to be in the submicron range. Some of the substrates with thin layers of JALS-204 were unidirectionally buffed (5 times using a piece of velvet cloth) in order to produce an easy axis at a small angle $\gamma $ to the normal to the cell substrates. $\gamma $ was measured by conoscopy and magnetic null methods [@MagneticNull].  The value of $\gamma $ weakly depends on the rubbing strength, but in all cases it was small, $\gamma <2^{\circ }$. The cell gap thickness was set using either the glass micro-sphere spacers uniformly distributed within the area of a cell (one spacer per approximately $100\mu m\times 100\mu m$ area) or strips of mylar film placed along the cell edges. The cell gap thickness $d$ was measured after cell assembly using the interference method [@BornWolf] with a LAMBDA18 (Perkin Elmer) spectrophotometer. In order to study textures as a function of the confinement ratio $\rho =d/p$, we constructed a series of cells, with identical thickness, but filled with CLCs of different pitch $p$. To minimize spherical aberrations in the FCPM, observations were made with immersion oil objectives, using glass substrates of thickness $0.15mm$ with refractive index $1.52$ [@FCPMCPL].  Regular ($1mm$) and thick ($3mm$) substrates were used to construct cells for polarizing microscopy (PM) observations. Cholesteric mixtures were prepared using the nematic host AMLC-0010 (obtained from AlphaMicron Inc., Kent, OH) and the chiral additive ZLI-811 (purchased from EM Industries). The helical twisting power $HTP=10.47\mu m^{-1}$ of the additive ZLI-811 in the AMLC-0010 nematic host was determined using the method of Grandjean-Cano wedge [@SmalyukPRE; @TKosaMCLC].  The obtained mixtures had pitch $2<p<500\mu m$ as calculated from $% p=1/(c_{chiral}\cdot HTP)$ where $c_{chiral}$ is the weight concentration of the chiral agent, and verified by the Grandjean-Cano wedge method [SmalyukPRE,TKosaMCLC,SmalyukhPRL]{}. The low frequency dielectric anisotropy of the AMLC-0010 host$\ $is $\Delta \varepsilon =-3.7$ ($\varepsilon _{\parallel }=3.4,$ $\varepsilon _{\perp }=7.1$) as determined from capacitance measurements for homeotropic and planar cells using an SI-1260 Impedance/Gain-phase analyzer (Schlumberger) [@Blinov-Ch-book; @deJeu]. The birefringence of AMLC-0010 is $\Delta n=0.078$ as measured with an Abbe refractometer.  The elastic constants describing the splay, twist, and bend deformations of the director in AMLC-0010 are $K_{11}=17.2pN$, $% K_{22}=7.51pN $, $K_{33}=17.9pN$ as determined from the thresholds of electric and magnetic Freedericksz transition in different geometries [Blinov-Ch-book,deJeu]{}.  The cholesteric mixtures were doped with a small amount of fluorescent dye n,n’-bis(2,5-di-tert-butylphenyl)-3,4,9,10-perylenedicarboximide (BTBP) [FCPMCPL]{} for the FCPM studies. Small quantities ( $0.01$wt. % ) of BTBP dye were added to the samples; at these concentrations, the dye is not expected to affect properties of the CLCs used in our studies. Constant amplitude and modulated amplitude signals were applied to the cells using a DS345 generator (Stanford Research Systems) and a Model 7602 Wide-band Amplifier (Krohn-Hite) which made possible the use of a wide range of carrier and modulation frequencies ($10-100000$) $Hz$. The transitions from the homeotropic untwisted to a variety of twisted structures were monitored via capacitance measurements and by measuring the light transmittance of the cell between crossed polarizers. The transitions between different director structures and textures were characterized with PM and FCPM [@FCPMCPL] as described below. Polarizing Microscopy and Fluorescence Confocal Polarizing Microscopy --------------------------------------------------------------------- Polarizing Microscopy (PM) observations were performed using the Nikon Eclipse E600 POL microscope with the Hitachi HV-C20 CCD camera. The PMstudies were also performed using a BX-50 Olympus microscope in the PM mode. In order to directly reconstruct the vertical cross-sections of the cholesteric structures, we performed further studies in the FCPM mode of the very same modified BX-50 microscope[@FCPMCPL] as described below. The PM and FCPM techniques are used in parallel and provide complementary information. The FCPM set-up was based on a modified BX-50 fluorescence confocal microscope [@FCPMCPL]. The excitation beam ($\lambda =488nm$, from an Ar laser) is focused by an objective onto a small submicron volume within the CLC cell. The fluorescent light from this volume is detected by a photomultiplier in the spectral region $510-550nm$. A pinhole is used to discriminate against emission from the regions above and below the selected volume. The pinhole diameter $D$ is adjusted according to magnification and numerical aperture ${\rm NA}$ of the objective; $D=$ $100\,\mu m$ for an immersion oil $60\times $ objective with ${\rm NA}=1.4$. A very same polarizer is used to determine the polarization of both the excitation beam and the detected fluorescent light collected in the epifluorescence mode. The relatively [low birefringence (]{}${\Delta n\approx 0.0}78$) [of the AMLC-0010 nematic host ]{}mitigates two problems that one encounters in FCPM imaging of CLCs: (1) defocussing of the extraordinary modes relative to the ordinary modes [@FCPMCPL] and (2) the Mauguin effect, where polarization follows the twisting director field [@ourreview; @Yehbook]. The used BTBP dye has both absorption and emission transition dipoles parallel to the long axis of the molecule [SmalyukPRE,FordKamat,SmalyukhPRL]{}. The FCPM signal, resulting from a sequence of absorption and emission events, strongly depends on the angle $% \beta $ between the transition dipole moment of the dye (assumed to be parallel to the local director ${\bf \hat{n}}$) and the polarization ${\bf \hat{P}}$.$\ $The intensity scales as $I_{FCPM}\sim $ $\cos ^{4}\beta $, [@FCPMCPL] as both the absorption and emission are proportional to $\cos ^{2}\beta $[**.**]{} The strongest FCPM signal corresponds to ${\bf \hat{n}% \parallel \hat{P}}$, where $\beta =0$, and sharply decreases when $\beta $ becomes non-zero [@ourreview; @FCPMCPL; @SmalyukPRE; @SmalyukhPRL]. By obtaining the FCPM images for different ${\bf \hat{P}}$, we reconstruct director structures in both in-plane and vertical cross-sections of the cell from which then the entire 3-D director pattern is reconstructed. We note that in the FCPM images the registered fluorescence signal from the bottom of the cell can be somewhat weaker than from the top, as a result of light absorption, light scattering caused by director fluctuations, depolarization, and defocussing. To mitigate these experimental artefacts and to maintain both axial and radial resolution within $1\mu m$, we used relatively shallow ($\leq $ $20$ $\mu m$) scanning depths [ourreview,FCPMCPL]{}. The other artefacts, such as light depolarization by a high NA objective, are neglected as they are of minor importance [ourreview,FCPMCPL,SmalyukPRE,SmalyukhPRL]{}. Results ======= Phase diagrams of textures and structures ----------------------------------------- We start with an experimental phase diagram of cholesteric structures in the homeotropic cells similar to the one reported in [Oswald-review00,CHnegativeDeltaE]{} and then explore how this diagram is affected by rubbing of homeotropic substrates, using different voltage driving schemes, and introducing nucleation sites. We note that for pitch $% p\gtrapprox 5\mu m$ and the cell gap $d\gtrapprox 5\mu m$ much larger than the anchoring extrapolation length ($<1\mu m,$ describing polar anchoring at the interface of CLC and JALS-204 layer), the observed structures depend on $% \rho =d/p$ but not explicitly on $d$ and $p$. We therefore construct the diagrams of structures in the plane of the applied voltage $U$ and the frustration ratio $\rho $; to describe the phase diagram we adopt the terminology introduced in Ref. [@Oswald-review00]. The diagrams display director structures (phases) of homeotropic untwisted state, isolated $CF$s and periodically arranged $CF$s, the $TIC$ and the modulated (undulating) $% TIC$. The phase boundary lines are denoted as $V0-V3$, $V01$, $V02$, Fig.1, similarly to Refs. [@Oswald-review00; @CHnegativeDeltaE] (for comparison with the phase diagrams reported for other LCs). As we show below, the phase diagram can be modified to satisfy requirements for several electro-optic applications of the CLC structures. ### Cells with unrubbed homeotropic substrates The diagram for unrubbed homeotropic cells is shown in Fig.1. The completely unwound homeotropic texture is observed at small $U$ and $\rho $, Fig.1. At high $U$ above $V0$, $V01$, and $V02$, the $TIC$ with some amount of director twist (up to $2\pi $, helical axis along the cell normal) is observed; the twist in $TIC$ is accompanied by splay and bend deformations. The $TIC$ texture is homogeneous within the plane of a cell except that it often contains the so-called umbilics, defects in direction of the tilt [de Gennes-book]{}, Fig.2f. Periodically arranged $CF$s are observed for voltages $U\approx 1.5-3.5Vrms$ and for $0.5<\rho <1$, Fig.1 and Fig.2b-e. If the values of $U$ and $\rho $ are between the $V0$ and $V01$ boundary lines, Fig.1, a transient $TIC$ appears first but then it is replaced (within $0.1-10s$ after voltage pulse, depending on $\rho $ and $U$) by a periodic pattern of $CF$s, which also undergoes slow relaxation; equilibrium is reached only in $3-50s$, Fig.2d. The isolated $CF$s coexisting with the homeotropic state are observed at $U\lessapprox 1.8$ and for $0.75<\rho <1$, Fig.1 and Fig.2a,b. For $\rho $ and $U$ between $V0$ and $V1$, the isolated fingers start growing from nucleation sites such as spacers, Fig.2b, or from already existing fingers. In both cases the $CF$s separated by homeotropic regions split in order to fill in the entire space with a periodic texture of period $\sim p$, similar to the one shown in Fig.2c. In the region between $V1$ and $V2$, isolated $CF$s nucleate and grow but they do not split and do not fill in the whole sample; fingers in this part of diagram coexist with homeotropic untwisted structure, Fig.2a. Hysteresis is observed between $V2$ and $V3$ lines: a homeotropic texture is observed if the voltage is increased, but isolated fingers coexisting with untwisted homeotropic structure can be found if $U$ is decreased from the initial high values. Even though the neighboring $CF$s in the fingers pattern are locally parallel to each other, Fig.2c,d, there is no preferential orientation of the fingers in the plane of the cell on the scales $\gtrapprox 10mm$. Finally, the periodic structure observed between $V01$ and $V02$ does not contain interspersed homeotropic regions, Fig.2e. The director field of $CF$s as well as other structures of the diagram will be revealed by FCPM below, see Sec. III.B. The behavior of the voltage-driven transitions between untwisted homeotropic and different types of twisted structures is reminiscent of conventional temperature-driven phase transitions with voltage playing a role similar to temperature. The phase diagram of structures has a Landau tricritical point $% \rho =\rho _{tricritical}$ at which $V2$ and $V0$ meet. The order of the transition changes from the second order (continuous) at $\rho <\rho _{tricritical}$ to the first order (discontinuous, proceeding via nucleation) at $\rho >\rho _{tricritical}$, Fig.1. The phase diagram also has a triple point at $\rho =\rho _{triple},$ where $V0$ and $V01$ meet. At the triple point, the untwisted homeotropic texture coexists with two different twisted structures, spatially uniform $TIC$ and periodic fingers pattern. The phase diagram and transitions in homeotropic cells with perpendicular easy axes at the substrates are qualitatively similar to those reported by Oswald et al. [@Oswald-review00; @CHnegativeDeltaE] for other materials; both qualitative and quantitative differences are observed when the homeotropic substrates are rubbed to produce slightly tilted easy axes at the confining substrates as discussed below. ### Effects of rubbing and nucleation centers Rubbing of the homeotropic alignment layers induces small pretilt angle from the vertical axis, $\gamma <2^{\circ }$. The azimuthal degeneracy is therefore broken, and the projection of easy axis defines a unique direction in the plane of a cell. Therefore, even rubbing of only one of the cell substrates has a strong effect on the CLC structures: (1) no umbilics are observed in the $TIC$, Fig.3a; (2) $CF$s preferentially align along the rubbing direction, Fig.3b. In addition, the homeotropic-$TIC$ transition, which is sharp in cells with vertical alignment, becomes somewhat blurred for rubbed homeotropic substrates with small $\gamma $, Fig.3c,d. In principle, one can set opposite rubbing directions on the substrates; we report a phase diagram of structures for such anti-parallel rubbing in Fig.4. The cells used to obtain the diagram were constructed from thick $3mm$ glass plates and only the mylar spacers at the cell edges were used to set the cell gap thickness. Compared to phase diagrams of structures with unrubbed substrates, dramatic changes are observed at $\rho \gtrapprox 0.5$. The direct homeotropic to $TIC$ transition is observed up to $\rho \approx 0.75$. The experimental triple and tricritical points are closer to each other than for unrubbed cells (compare  Fig.1 and Fig.4). Interestingly, within the range $0.6<\rho <0.75$ and upon increasing $U$, one first observes a homogeneous $TIC$, Fig.5b, which is then replaced by a periodic fingers pattern at higher $U$, Fig.5c, and again a uniform $TIC$ at even higher $U>3-3.5V$, Fig.5d. The same sequence, $TIC$-fingers-$TIC$-homeotropic, is also observed upon decreasing $U$ from initial high values. Pursuing the analogy with temperature-driven phase transitions, the $TIC$ texture between the fingers pattern and homeotropic texture can be considered as a re-entrant $TIC$ phase. As compared to unrubbed cells, the antiparallel rubbing has little effect on $V0$, but shifts the other boundary lines towards increasing $\rho $. The effects of anti-parallel rubbing on the phase diagram can be explained as follows. At $\rho \sim 0.5$ anti-parallel rubbing matches the director twist of $TIC,$ which at high $U$ is $\approx \pi $. Therefore, $TIC$ is stabilized by anti-parallel rubbing and $CF$s do not appear until higher $\rho $, Fig.4. The transient $TIC$ disappears if large quantities of spacers ($>100/mm^{2}$) or other nucleation sites for fingers are present in the cells with anti-parallel rubbing; in this case the phase diagram is closer to the one shown in Fig.1. The spacers with perpendicular surface anchoring produce director distortions in their close vicinity even in the part of diagram corresponding to homeotropic unwound state, Fig.6a. In the vicinity of the homeotropic-$TIC$ transition, Fig.6b, the director realignment starts in the vicinity of spacers. Similar to the observations in Refs. [InvWalls,PECladis]{}, particles with perpendicular surface anchoring cause inversion walls ($IW$s) and disclinations. The $TIC$ with $0.5<\rho <0.75$, Fig.6c, is eventually replaced by fingers, which are facilitated by the particles, Fig.6d. Moreover, even at high $U$, $TIC$ remains spatially non-uniform and contains different types of $IW$s and disclination lines [@InvWalls; @PECladis], which are caused by the boundary conditions at the surfaces of the particles. ### Phase diagrams for different voltage driving schemes The phase diagrams of structures shown in Figs.1 and 4 were obtained with constant amplitude sinusoidal voltages applied to the cells. The diagram changes dramatically if the applied voltage is modulated. The effect is especially strong in the cells with rubbed homeotropic substrates, for which we present results in Fig.7a-c; somewhat weaker effect is also observed for unrubbed substrates. We explored modulation with rectangular-type, triangular, and sinusoidal signals of different duration and modulation depth. The strongest effect is observed with $100\%$ modulation depth and sinusoidal modulation signal at frequencies (10-200)$Hz$. The fingers patterns are shifted towards increasing $\rho $, Fig.7. At the same time, the $rms$ voltage values of homeotropic-$TIC$ transition are practically the same for different voltage driving schemes, Figs.1,4,7. We assume that the effect of amplitude modulation is related to the very slow dynamics of some of the structures (see Secs. III.A.1, III.A.2), such as $CF$s; the corresponding parts of the diagram are the most sensitive to voltage driving schemes. The substantial combined effect of rubbing and voltage driving schemes is important for practical applications of the homeotropic-$TIC$ transition when it is important to have strongly twisted but finger-free field-on state [@EyeLC1; @Bistable-chiral]. We therefore present only the diagrams corresponding to the largest $\rho $ values at which fingers do not appear for given surface rubbing conditions, Fig.7. On the other hand, voltage modulation could be a way to study the stability of different parts of the diagram in the $\rho ,U$ plane and deserves to be explored in more details; we leave this for forthcoming publications. Finally, to understand the diagrams and transitions explored in this section it is important to know the director fields that are behind different textures; this will be explored in the following section. Director structures ------------------- ### Spatially-homogeneous twisted structures, umbilics, and inversion walls In this section, we take advantage of the FCPM and study the director field $% {\bf \hat{n}}\left( x,y,z\right) $ in the vertical cross-sections (i.e., along the $z$, normal to the cell substrates) of the cholesteric structures. This is important as, for example, in the $TIC$ ${\bf \hat{n}}$ varies only along $z$ and not in the plane of a cell. The $TIC$, observed above the $V0$ and $V02$ lines in the phase diagrams, Figs.1,4,7, can be visualized as having ${\bf \hat{n}}$ rotating with distance from the cell wall on a cone whose axis is along $z$; the half angle of this cone varies from $\theta =0$ at the substrates to $\theta _{\max }$ in the middle plane of a cell ($% \theta _{\max }<\pi /2$), Fig.8. FCPM reveals that the in-plane twist of the director in the $TIC$ depends on $\rho $. For small $\rho \approx 0,$ the $% TIC$ contains practically no in-plane twist. When $\rho \approx 1/2$, the in-plane twist at high $U$ reaches $\pi $, Fig.8a,c. Finally, when $\rho \approx 1$, the twist of the $TIC$ structure at high $U$ can reach $2\pi $, Fig.8b,d. The maximum in-plane twist at high $U$ is $\approx 2\pi \rho $; we stress that the twist of $TIC$ depends not only on $\rho $, but also on $U.$ In addition, for the cells with rubbed homeotropic plates, the in-plane twist is affected by the rubbing direction. For example, the re-entrant $TIC$ in the rubbed cells of $0.5<\rho <0.75$ has twist $\approx \pi $ at small $U$ just above the $V0$ and the twist close to $2\pi $ at high voltages $U>4Vrms$. If both of the homeotropic substrates are rubbed, the natural twist of the $TIC$ structure may or may not be compatible with the tilted easy axes at the confined substrates. Since the amount of twist in the $TIC$ depends both on $\rho $ and $U$, it is impossible to match tilted homeotropic boundary conditions to a broad range of $\rho $ and $U$. However, since the in-plane anchoring is weak, the effect of rubbing on the twist in $TIC$ is not as strong as in the case of planar cells. The FCPM also allows us to probe the defects that appear in $TIC$. We confirm that the defects with four brushes, Fig.2f, are umbilics of strength $\pm 1$[@de; @Gennes-book] rather than disclinations with singular cores. We also verify that the umbilics are caused by degeneracy of director tilt when $U$ is applied; such degeneracy is eliminated by rubbing, Fig.3a. Within $TIC$, we also observe $IW$s[@InvWalls; @PECladis]. The appearance of these walls was previously attributed to a variety of factors, such as flow of liquid crystal, hydrodynamic instabilities, alignment induced at the edges of the sample, and others [@InvWalls; @PECladis]. FCPM observations indicate that in the presence of spacers with perpendicular anchoring, the $% IW$s appear at the particles when $U$ above the threshold for homeotropic-$% TIC$ transition is applied. This is believed to be caused by director distortions in the vicinity of the particles [@InvWalls; @PECladis]. When the confinement ratio is $0.5<\rho <0.75$, the distorted $TIC$ with umbilics, disclinations, and $IW$s is replaced by $CF$s with the spacers serving as nucleation sites for the fingers, Fig.6. ### Fingers structures; nonsingular fingers of $CF1$-type Fingers structures have translational invariance along their axes ($y$-direction) and can be observed as isolated between $V3$ and $V0$ or periodically arranged between $V0$ and $V01$ boundary lines, Figs.1,4,7 (see Sec.III.A for details). We again take advantage of FCPM by visualizing the vertical cross-sections and then reconstruct ${\bf \hat{n}}\left( x,y,z\right) $ of the fingers directly from the experimental data. To describe the results, we use the $CFs$ classification of Oswald et al. [Oswald-review00]{}. The finger of $CF1$ type is the most frequently observed in cells with vertical as well as slightly tilted alignment, Fig.9. $CF1$ is isolated and co-existing with the homeotropic untwisted structure between $% V3 $ and $V0$ and is a part of the spatially periodic pattern between $V0$ and $V01$ lines, Figs.1,4,7. The reason for abundance of $CF1$, is that it can form from $TIC$ by continuous transformation of director field above the $V0$ line and it also can easily nucleate from homeotropic untwisted structure below the $V0$ boundary line, Figs.1,4,7[@Oswald-review00]. The director structure of $CF1$ reconstructed from the FCPM vertical cross-section, Fig.9, is in a good agreement with the results of computer simulations [Oswald-review00,LGil,ourreview,PressArrott,MCLC-PressArrott]{}. In the $CF1$, the axis of cholesteric twist is tilted away from the cell normal $z$, Fig.9; the in-plane twist in direction perpendicular to the finger in the middle of a cell is $2\pi $. The isolated $CF1s$ that are separated from each other by large regions of homeotropic texture, Fig.2a, assume random tilt directions. The width of an isolated $CF1$ is somewhat larger than $d$; this is in a good agreement with computer simulations of L. Gil [@LGil]. $CF1$ is nonsingular in ${\bf \hat{n}}$ (i.e., the spatial changes of ${\bf \hat{n}}$ are continuous and ${\bf \hat{n}}$ can be defined everywhere within the structure) as the twist is accompanied with escape of ${\bf \hat{n% }}$ into the third dimension along its center line. An isolated $CF1$ can be represented as a quadrupole of the non-singular $\lambda $-disclinations, two of strength $+1/2$ and two of strength $-1/2$, as shown in Fig.9b. The $% \lambda $-disclinations, with core size of the order of $p$, cost much less energy than the disclinations with singular cores [KlemanLavrentovichBook]{}. The pair of disclinations $\lambda ^{+1/2}\lambda ^{-1/2}$ introduces $2\pi $-twist at one homeotropic substrate; this $2\pi $-twist is then terminated by introducing another $\lambda ^{+1/2}\lambda ^{-1/2}$ pair in order to satisfy the homeotropic boundary conditions at another substrate, Fig.9b. A segment of an isolated $CF1$ has different ends; one is rounded while the another is pointed. Behavior of these ends is different during growth; the pointed end remains stable, while the rounded end continuously splits, as also discussed in Ref. [@Oswald2004]. The FCPM vertical cross-section, Fig.10, reveals details of $CF1$ tiling into periodically arranged structures that are observed above $V0$ line, Figs.1,4,7. When $U$ or $\rho $ are relatively large, the $CF1$ fingers are close to each other so that the homeotropic regions in between cannot be clearly distinguished. The tilt of the helical axis in the periodic $CF$ structures is usually in the same direction, Fig.10. A possible explanation is that the elastic free energy is minimized since the structure of unidirectionally tilted $CFs$ is essentially space-filling. On rare occasions, the tilt direction of neighboring $CF1s$ is opposite. Upon increasing $U$, the width of fingers originally separated by homeotropic regions, Fig.11a, gradually increases, Fig.11b-e; the fingers then merge to form a periodically modulated $TIC$, Fig.11f. Finally, at high applied voltages, the transition to uniform $TIC$ is observed, Fig.11g. The details of transformation of periodically arranged fingers into the in-plane homogeneous $TIC$ via the modulated (undulating) twisted structure were not known before and would be difficult to grasp without FCPM. $TIC$ can also be formed by expanding one of the $CF1$s; structure of coexisting fingers and $% TIC$ contains only $\lambda $-disclinations nonsingular in ${\bf \hat{n}}$ again demonstrating the natural tendency to avoid singularities, Fig.12. Periodically arranged $CF1$s slowly (depending on rubbing, $U$ and $\rho $; usually up to $1s$) appear from $TIC$ if $U$ is between $V0$ and $V01,$ and quickly disappear (in less than $50ms$) if $U$ is increased above $V02$. This allowed us to use the amplitude-modulated voltage driving schemes in combination with rubbing and obtain finger-free $TIC$ up to $\rho \approx 0.8 $ (Sec. III.A.3), as needed for applications of $TIC$ in the electrically driven light shutters, intensity modulators, eyewear with tunable transparency, displays, etc. [Blinov-Ch-book,EyeLC1,Patel,Bistable-chiral]{} ### Fingers of $CF2$, $CF3$, and $CF4$ types containing defects; T-junctions of fingers Another type of fingers is $CF2$, Fig.13, which is observed for vertical as well as slightly tilted alignment in the same parts of the diagram as $CF1$, Figs.1,4,7. However, in contrast to the case of nonsingular $CF1$, a segment of $CF2$ has point defects at its ends. Because of this, $CF2$ does not nucleate from the homeotropic or $TIC$ structures as easily as $CF1$ and normally dust particles, spacers, or irregularities are responsible for its appearance. Therefore, fingers of $CF2$-type, Fig.13, are found less frequently than $CF1$. Using FCPM, we reconstruct the director structure in the vertical cross-section of $CF2$, Fig.13; the experimental result resembles the one obtained in computer simulations by Gil and Gilli [Gil-PRL98]{}, proving the latest model of $CF2$ [Oswald-review00,Gil-PRL98]{} and disproving the earlier ones [Baudry-PRE99]{}. Within the $CF2$ structure, one can distinguish the non-singular $\lambda ^{+1}$ disclination in the central part of the cell and two half-integer $\lambda ^{-1/2}$ disclinations in the vicinity of the opposite homeotropic substrates. The total topological charge of the $CF2$ is conserved, similarly to the case of $CF1$. Polarizing microscopic observations show that unlike in $CF1$-type fingers, the ends of $CF2$ segments have similar appearance. FCPM reveals that the point defects (of strength $1$) at the two ends have different locations being closer to the opposite substrates of a cell. Unlike the $CF1$ structure, $CF2$ is not invariant by $\pi $-rotation around the $y$-axis along the finger, as also can be seen from the FCPM cross-section, Fig.13. Different symmetries of $% CF1 $ and $CF2$ are responsible for their different dynamics under electric field [@Oswald-review00; @Gil-PRL98]. This, along with computer simulations, allowed Gil and Gilli [@Gil-PRL98] to propose the model of $% CF2$; our direct imaging using FCPM unambiguously proves that this model is correct, Fig.13b. The isolated $CF2$ fingers (coexisting with homeotropic state) expand, when $U\gtrapprox 2.1Vrms$ is applied, Fig.13c,d. The structures with non-singular $\lambda $-disclinations often separate the parts of a cell with different twist, Fig.14; they resemble the structures of thick lines that are observed in Grandjean-Cano wedges with planar surface anchoring [@SmalyukPRE; @SmalyukhPRL]. The appearance of these lines in homeotropic cells is facilitated by sample thickness variations and spacers. The width of $CF2$ coexisting with the homeotropic state is usually the same or somewhat (up to 30%) smaller than $CF1$; this can be seen in Fig.15 showing a $T$-junction of the $CF1$ and $CF2$ fingers. Even though $CF1$ and $CF2$ have similar appearance under a polarizing microscope [@Oswald-review00], FCPM allows one to clearly distinguish these structures. Note also the tendency to avoid singularities in ${\bf \hat{n}}$ evidenced by the reconstructed structure of the $T$-junction, Fig.15b. The metastable cholesteric fingers of $CF3$-type, Fig.16, occur even less frequently than $CF2s$. The director structure of this finger was originally proposed by Cladis and Kleman [@CladisKleman]. In polarizing microscopy observations, the width of $CF3$ fingers is about half of that in $CF1$ and $% CF2$. The reconstructed FCPM structure of $CF3$ indicates that the director ${\bf \hat{n}}$ rotates through only $\pi $ along the axis perpendicular to the finger ($x$-axis). This differs from both $CF1$ and $CF2$, which both show a rotation of ${\bf \hat{n}}$ through $2\pi $.  Two twist disclinations of opposite signs near the substrates allow the cholesteric $% \pi $-twist in the bulk to match the homeotropic boundary conditions, Fig.16. The structure of $CF3$ is singular in ${\bf \hat{n}}$; the disclinations are energetically costly and this explains why $CF3$ is observed rarely even in the cells with vertical alignment. In cells with rubbed homeotropic substrates $CF3$ was never observed. This is likely due to the easy axis having the same tilt on both sides of the finger on a rubbed substrate, whereas the $\pi $-twisted configuration of  $CF3$ requires director tilt in opposite directions. The $CF4$-type metastable finger shown in Fig.17 is also singular, and is usually somewhat wider than the other $CFs$. It can be found in all regions of existence of $CF1$, Fig.1, but is very rare and usually is formed after cooling the sample from isotropic phase. $CF4$ contains two singular disclinations at the same substrate. In the plane of a cell, the director $% {\bf \hat{n}}$ rotates by $2\pi $ with the twist axis being along ${\bf x}$ and perpendicular to the finger. Using the direct FCPM imaging, we reconstruct the director structure of $CF4$, Fig.17b, which is in a good agreement with the model of Baudry et al. [@T-Junktions]. The bottom part of this finger, Fig.17, is nonsingular, and is similar in this respect to $CF1$ and $CF2$; its top part, however, contains two singular twist disclinations. The $CF4$ structure is observed only in cells with no rubbing. Similar to the case of $CF3$, the structure of $CF4$ is not compatible with uniform tilt produced by rubbing of homeotropic substrates. Of the four different fingers structures, $CF4$ might be the least favorable energetically, since it usually rapidly transforms into $CF1$ or $CF2$; less frequently, it also splits into two $CF3$ fingers. Transformation of other fingers into $CF4$ was never observed. Discussion ========== The system that we study is fairly rich and complicated; some of the structures and transitions can be described analytically while the other require numerical modeling. Below we first restrict ourselves to translationally uniform structures (i.e., homeotropic and $TIC$) which can be described analytically. We then discuss the other experimentally observed structures and transitions comparing them to the analytical as well as numerical results available in literature [Oswald-review00,CHnegativeDeltaE]{}, as well as our own numerical study of the phase diagram that will be published elsewhere[@Garlandetal]. Finally, we discuss the practical importance of the obtained results on the phase diagrams of director structures. Translationally uniform homeotropic and $TIC$ structures -------------------------------------------------------- We represent the director ${\bf \hat{n}}$ in terms of the polar angle $% \theta $ (between the director and the $z$-axis) and the azimuthal angle $% \phi $ (the twist angle); $\psi $ is electric potential. For the $TIC$ configurations, these fields are functions of $z$ only, and the Oseen-Frank free-energy density takes the form $$\begin{aligned} 2f &=&(K_{11}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta )\theta _{z}^{2}+(K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta )\sin ^{2}\!\theta \,\phi _{z}^{2} \nonumber \\ &&{}-K_{22}\frac{4\pi }{p}\sin ^{2}\!\theta \,\phi _{z}+K_{22}\frac{4\pi ^{2}% }{p^{2}}-(\varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta )\psi _{z}^{2}, \label{eqn:fe}\end{aligned}$$where, $K_{11}$, $K_{22}$, and $K_{33}$ are the splay, twist, and bend elastic constants, respectively; $\varepsilon _{\parallel }$,$\varepsilon _{\perp }$ are the dielectric constants parallel and perpendicular to ${\bf \hat{n},}$ respectively; $\theta _{z}=d\theta /dz,$ $\phi _{z}=d\phi /dz,$ and $\psi _{z}=d\psi /dz$. The associated coupled Euler-Lagrange equations are $$\begin{aligned} \lefteqn{\frac{d}{dz}\left[ \left( K_{11}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta \right) \theta _{z}\right] =\sin \theta \cos \theta \Bigl\{(K_{11}-K_{33})\theta _{z}^{2}} \nonumber \\ &&{}+\left[ (2K_{22}-K_{33})\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta % \right] \phi _{z}^{2}-K_{22}\frac{4\pi }{p}\phi _{z}-\Delta \varepsilon \psi _{z}^{2}\Bigr\}, \label{eqn:ELa}\end{aligned}$$ $$\frac{d}{dz}\left\{ \sin ^{2}\!\theta \left[ (K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta )\phi _{z}-K_{22}\frac{2\pi }{p}\right] \right\} =0, \label{eqn:ELb}$$ $$\frac{d}{dz}\left[ \left( \varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta \right) \psi _{z}\right] =0, \label{eqn:ELc}$$ with associated boundary conditions $\theta (0)=\theta (d)=0$; $\phi (0),\phi (d)$ undefined; and $\psi (0)=0$, $\psi (d)=U$. Dielectric anisotropy is negative for the studied material, $\Delta \varepsilon =\varepsilon _{\parallel }-\varepsilon _{\perp }<0$. Representative solutions of these equations are plotted in Fig.18 and describe how $\theta (z),$ $\phi (z),$ and $\psi (z)$ vary across the cell. Equations (\[eqn:ELb\]) and (\[eqn:ELc\]) above admit first integrals, $$\phi _{z}=\frac{K_{22}}{K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta }% \frac{2\pi }{p} \label{eqn:phiz}$$and $$\left( \varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta \right) \psi _{z}=\frac{U}{\displaystyle\int_{0}^{d}\frac{1}{% \varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta }dz},$$which allow to express the free energy in terms of the tilt angle $\theta $ only: $$\begin{aligned} {\cal F}[\theta ] &=&\frac{1}{2}\int_{0}^{d}\left[ (K_{11}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta )\theta _{z}^{2}+\frac{K_{2}K_{3}\cos ^{2}\!\theta }{K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta }\frac{4\pi ^{2}}{p^{2}}% \right] dz \nonumber \\ &&{}-\frac{1}{2}U^{2}\left[ \int_{0}^{d}\frac{1}{\varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta }dz\right] ^{-1}.\end{aligned}$$This is similar to Eq. (3.221) of Ref. [@stewart:04] (see p. 91), where the splay Freedericksz transition with a coupled electric field is discussed. We expand the free energy in terms of $\theta (z)$ about the undistorted $\theta =0\,$homeotropic configuration to obtain$${\cal F}[\theta ]=\frac{1}{2}\left[ \frac{4\pi ^{2}dK_{22}}{p^{2}}-\frac{% \varepsilon _{\parallel }U^{2}}{d}\right] +\frac{1}{2}\int_{0}^{d}\left[ K_{33}\theta _{z}^{2}-\left( \frac{\Delta \varepsilon U^{2}}{d^{2}}+\frac{% 4\pi ^{2}K_{22}^{2}}{p^{2}K_{33}}\right) \theta ^{2}\right] dz+O(\theta ^{4}).$$The first term is the free energy of the uniform homeoptropic configuration. The second-order term is positive definite if $U$ and $\rho $ are sufficiently small. Ignoring higher order terms, we find that the loss of stability occurs when $$\frac{4K_{22}^{2}}{K_{33}^{2}}\rho ^{2}+\frac{\Delta \varepsilon }{K_{33}\pi ^{2}}U^{2}=1. \label{eqn:ellipse}$$Eq. (\[eqn:ellipse\]) is the spinodal ellipse. The homeotropic configuration is metastable with respect to translationally homogeneous perturbations for the $\rho $ and $U$ parameters inside the ellipse ([eqn:ellipse]{}), which corresponds to the boundary line $V0$, Figs.1,4,7. Eq. (\[eqn:ellipse\]) gives the threshold voltage for transition between homeotropic and $TIC$ structures: $$U_{th}=\pi \sqrt{K_{33}/\Delta \varepsilon }\cdot \sqrt{1-4\rho ^{2}K_{22}^{2}/K_{33}^{2}}. \label{U0-threshold}$$ Eq. (\[U0-threshold\]) is in a good agreement with our experimental results described in Sec. III above and with Ref. [@Rosenblatt]. The experimental data for boundary line $V0$ are well described by Eq. ([eqn:ellipse]{}) for rubbed and unrubbed homeotropic cells, Figs.1,4,7. According to the linear stability analysis above, the ellipse in the $\rho $-$U$ plane (\[U0-threshold\]) defines the limit of metastability of the homeotropic phase: for $\rho $ and $U$ inside this ellipse, the uniform homeotropic configuration is metastable, while outside the ellipse, it is a locally unstable equilibrium. In an idealized cell with infinitely strong homeotropic anchoring and no pretilt, the transition from homeotropic to $% TIC $ is a forward pitchfork bifurcation, that is, a second-order transition. For voltages $U$ below the $V0$ line (inside the ellipse), the $% TIC$ configuration does not exist. On the other hand, when there is a slight tilt of the easy axis, the reflection symmetry is broken. The pitchfork is unfolded into a smoother transition, i.e., the transition becomes supercritical and the precise transition threshold is not well defined. The experimentally observable artefact of this is the somewhat blurred transition, which is described in Sec. III.A.2, for the cells with rubbing and resembles a similar effect in planar cells with small pretilt[Yehbook]{}. The above analysis allows one to understand the dependence of the total in-plane twist of $TIC$ on $\rho $ and $U$ that was described in Sec. III.B.1. The first integral (\[eqn:phiz\]) gives the tilt-dependence of the local twist rate. The total twist across a cell of thickness $d$ is $$\Delta \phi =\frac{2\pi }{p}\int_{0}^{d}\frac{K_{22}}{K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta }\,dz. \label{eqn:Dphi}$$For the AMLC-0010 material with $K_{22}/K_{33}\approx 0.42$ for given $\rho $ the total twist $\Delta \phi $ can be varied $$0.42\ast 2\pi \rho <\Delta \phi <2\pi \rho , \label{eqn:DphiLimits}$$by changing $U$. $\Delta \phi $ approaches the lower limit for relatively small $U$ that are just above $U_{th}$ and $\theta \approx 0$ and the upper limit for $U\gg U_{th}$ and $\theta \approx \pi /2$ . This analysis is in a good agreement with the FCPM images of the vertical cross-sections of $TIC$ for different $\rho $, as described in Sec. III. Finally, knowledge of $% \Delta \phi $ variation with changing $U$ is important for the practical applications of $TIC$ as it will be discussed below. Other structures and transitions of the phase diagram ----------------------------------------------------- Modeling of transitions associated with $CFs,$ in which ${\bf \hat{n}}$ is a function of two coordinates, is more complicated than in the case of $TIC$. Ribière, Pirkl, and Oswald[@CHnegativeDeltaE] obtained complete phase diagram in calculations assuming a simplified model of a cholesteric finger. This theoretical diagram qualitatively resembles our experimental result for the cells with vertical alignment, Fig.1. We explored the phase diagram using 2-D numerical modeling in which the equilibrium structure of the $CFs$ and equilibrium period of periodically arranged fingers are determined from energy minimization in a self-consistent way and the nonlocal field effects are taken into account[@Garlandetal]. The numerical phase diagrams show a good quantitative agreement with the experimental results presented here, predicting even the re-entrant behavior of $TIC$ that we experimentally obtain for cells with rubbed substrates (Sec.III.A.2). Presentation of these results requires detailed description of numerical modeling and will be published elsewhere[@Garlandetal]. Therefore, we only briefly discuss the qualitative features of the phase diagrams shown in Figs.1,4,7 in the light of the previous theoretical studies[@Oswald-review00; @CHnegativeDeltaE] and also summarize the new experimental results below. The important feature of the studied diagram is that the nematic-cholesteric transition changes order: it is second order for $0<\rho <\rho _{tricritical} $ and first order for $\rho >\rho _{tricritical}$, Fig.1, in agreement with Refs.[@Oswald-review00; @CHnegativeDeltaE] The phase diagram has a triple point at $\rho =\rho _{triple}$, where $V0$ and $V01$ meet and the untwisted homeotropic texture coexists with two twisted structures, $TIC$ and periodically arranged $CFs$, Figs1,4,7. For vertical alignment, the direct voltage-driven homeotropic-$TIC$ transition is observed at small $\rho \lessapprox 0.5$. Structures of isolated $CFs$ and periodically arranged $CFs $ occur for $0.5\lessapprox \rho <1$ and intermediate $U$ between the homeotropic state and $TIC$. The theoretical analysis of Ref. [@CHnegativeDeltaE] allows one to determine $\rho $ corresponding to the tricritical and triple points in the phase diagrams. Solving the equations given in Ref.[@CHnegativeDeltaE] numerically[error]{}  and using the material parameters of the AMLC-0010 host doped with the chiral agent ZLI-811, we find $\rho $ corresponding to the triple and tricritical points: $\rho _{triple}=0.816$ and $\rho _{tricritical}=0.861.$ These values are somewhat larger than $\rho _{triple}$ and $\rho _{tricritical}$ determined experimentally for the cells with vertical alignment, Figs.1,4,7, as also observed in [@CHnegativeDeltaE] for other CLCs. The calculated $\rho _{triple}$ and $\rho _{tricritical}$ are closer to the experimental ones in the case of rubbed substrates; this may indicate the possible role of umbilics and $IWs$ in the $TIC,$ which were not taken into account in the model [@CHnegativeDeltaE] (umbilics and $IWs$ are nucleation sites for fingers and may also increase elastic energy of $TIC$). Agreement is improved when phase diagrams are obtained using 2-D numerical modeling [@Garlandetal]. An interesting new finding revealed by the FCPM is that upon increasing $U$ the periodically arranged fingers merge with each other forming modulated (undulating) $TIC$ that is observed in a narrow voltage range between the structures of $TIC$ and periodically arranged $CFs$, Figs.1,4,7. We also find that the phase boundaries can be shifted in a controlled way by rubbing-induced tilt ($<2^{\circ }$) of easy axis from the vertical direction, by introducing particles that become nucleation sites for $CFs,$ as well as by using different amplitude-modulated voltage schemes. A novel and unexpected result is the re-entrant behavior of $TIC$ in the rubbed cells with $0.6\lessapprox \rho \lessapprox 0.75,$ which, however, disappears if nucleation sites are present. FCPM allows us to directly and unambiguously determine the 3-D director structures corresponding to different parts of the phase diagram. In particular, we unambiguously reconstruct the structures of four types of $CFs$. In all parts of diagrams corresponding to stability or metastability of $CFs$, the fingers of $CF1$-type are the most frequently observed. $CF2$ fingers are less frequent; the metastable $CF3$ and $CF4$ are very rare. Such findings indicate that fingers of $CF1$-type have the lowest free energy out of four fingers; this is consistent with the reconstruction of the structure of $CF1,$ which is nonsingular in ${\bf \hat{n}}$. It is also natural that $CF2$ with singular point defects and especially metastable $CF3$ and $CF4$ with singular line defects are less frequently observed. In the case of rubbed homeotropic substrates, only $CF1$ and $CF2$ are observed whereas $CF3$ and $CF4$ newer appear because the rubbing-induced tilting of the easy axis at one or both substrates contradicts with their symmetry. Control of phase diagrams to enable practical applications ---------------------------------------------------------- The combination of rubbing and amplitude-modulated voltage driving allows one to suppress appearance of fingers up to high $\rho \approx 0.75$, compare Fig.1 and Fig.7. This is a valuable finding for many practical applications such as eyeglasses with voltage-tunable transparency and light shutters,[@EyeLC1] bistable[@Bistable-chiral] and inverse twisted nematic displays,[@Patel] etc. In these applications of the homeotropic-$% TIC$ transition, it is important to have a broad range of well controlled total twist $\Delta \phi $ in the finger-free $TIC$. The broad range of voltage-tuned $\Delta \phi $ allows one to control optical phase retardation in the displays and electro-optic devices[@Patel; @Bistable-chiral] as well as light absorption when the dye-doped CLC is used in the tunable eyeglasses and light shutters[@EyeLC1]. A very subtle tilt of easy axis from the vertical direction not only makes the director twist in $TIC$ vary in a controlled way but also suppresses the appearance of fingers, $IW$s, and umbilics, Figs.3-7. Slow appearance of fingers from $TIC$ and untwisted homeotropic states allowed us to magnify the effect of rubbing via using amplitude-modulated voltage schemes and suppress appearance of fingers up to even higher $\rho $, Fig.7. For example, we can control $\Delta \phi =55^{\circ }-270^{\circ }$ in the finger-free $TIC$. Even stronger effects of rubbing and voltage modulation can be expected if the tilt of the easy axis is increased. This might be implemented by using the approach recently developed by Huang and Rosenblatt [@Homeotrop-tilt] in which case a tilt of easy axis up to $30^{\circ }$ could be achieved. On the other hand, when constructing cells for all of the above applications of tightly-twisted $TIC$, it is important to remember about the effect of particles, which become nucleation sites for fingers and can cause their appearance at lower $\rho $. Such particles are often used as spacers to set cell thickness and it is, therefore, important to either avoid their usage or limit (optimize) their concentration in order to obtain finger-free $TIC$. The finding that fingers align along the rubbing direction, Sec. III.A, may enable the use of periodically-arranged $CF$s in switchable diffraction gratings with the diffraction pattern corresponding to the field-on state. The spatial periodicity and the diffraction properties of such gratings can be easily controlled by selecting proper pitch $p$ and cell gap $d$, which can be varied from sub-micron to tens of microns. Our preliminary study shows that the grating periodicity can be changed in range $1-50\mu m$. More detailed studies of the cholesteric diffraction gratings based on voltage-induced well-oriented pattern of $CF$s will be published elsewhere. Conclusions =========== The major findings of this work are threefold: (1) we obtained phase diagram of CLC structures as a function of confinement ratio $\rho =d/p$ and voltage $U$ for different extra parameters such as rubbing, voltage driving, presence of nucleation sites; (2) we enabled new applications of finger-free tightly twisted $TIC$ and well-oriented fingers; (3) we unambiguously deciphered 3-D director fields associated with different structures and transitions in CLCs using FCPM. In the phase diagram, the direct homeotropic-$TIC$ transition upon increasing $U$ was observed for $\rho \lessapprox 0.5$; the analytical model of this transition is in a very good agreement with the experiment. Structures of isolated and periodically arranged fingers were found at $0.5\lessapprox \rho <1$ and intermediate $U$ between the homeotropic and $TIC$ phases. We observed the re-entrant behavior of $TIC$ in the rubbed cells of $0.6\lessapprox \rho \lessapprox 0.75$ for which the following sequence of transitions has been observed upon increasing $U$: (1) homeotropic untwisted - (2) $TIC$ - (3) periodically arranged fingers - (4) $% TIC$ with larger in-plane twist. The re-entrant behavior of $TIC$ is also observed in our numerical study of the phase diagram that will be published elsewhere[@Garlandetal]. The re-entrant $TIC$ disappears if nucleation sites are present or amplitude-modulated driving schemes are used. Rubbing also eliminates non-uniform in-plane structures such as umbilics and inversion walls that otherwise are often observed in $TIC$ and also influence the phase boundary lines. The lowest $\rho $ for which periodically arranged fingers start to be observed upon increasing $U$ can be shifted for up to $0.3$ towards higher $\rho $-values by rubbing and/or voltage driving schemes. The FCPM allowed us to unambiguously determine and confirm the latest director models [@Oswald-review00] for the vertical cross-sections of four types of $CFs$ ($CF1-CF4$) while disproving some of the earlier models[@Baudry-PRE99]. The $CF1$-type fingers are observed in all regions of the phase diagrams where the fingers are either stable or metastable; other fingers appear in the same parts of the diagram but less frequently. For the rubbed cells, only two types of $CFs$ ($CF1$ and $CF2$) are observed, which align along the rubbing direction. The new means to control structures in CLCs are of importance for potential applications, such as switchable gratings based on periodically arranged $CF$s and eyewear with tunable transparency based on $TIC$. Acknowledgements ================ The work is part of the AlphaMicron/TAF collaborative project “Liquid Crystal Eyewear”, supported by the State of Ohio and AlphaMicron, Inc. I.I.S. and O.D.L. acknowledge support of the NSF, Grant DMR-0315523. I.I.S. acknowledges Fellowship of the Institute for Complex and Adaptive Matter. E.C.G. acknowledges support under NSF Grant DMS-0107761, as well as the hospitality and support of the Department of Mathematics at the University of Pavia (Italy) and the Institute for Mathematics and its Applications at the University of Minnesota, where part of work was carried out. We are grateful to M. Kleman, L. Longa, Yu. Nastishin, and S. Shiyanovskii for discussions. Figure Captions: FIG.1. Phase diagram of structures in the $U-\rho $ parameter space for CLCs in cells with homeotropic surface anchoring. The boundary lines $V0-V3$, $% V01 $, $V02$ separate different phases (cholesteric structures). The two dashed vertical lines mark $\rho _{triple}=0.816$ and $\rho _{tricritical}=0.861$ as estimated according to Ref. [@CHnegativeDeltaE] for the material parameters of AMLC-0010 - ZLI-811 LC mixture. The solid line $V0$ was calculated using Eq.(1) and parameters of the used CLC; the solid lines $V1-V3$, $V01$, $V02$ connect the experimental points to guide the eye. FIG.2. Polarizing microscope textures observed in different regions of the phase diagram of structures shown in Fig.1: (a) isolated $CFs$ coexisting with the homeotropic untwisted state between the boundary lines $V1$ and $V2$ of Fig.1; (b) dendritic-like growth of $CFs$ (observed between the boundary lines $V0$ and $V1$); (c) branching of $CFs$ with increasing voltage, between the boundary lines $V0$ and $V1$; (d) periodically arranged $CFs$ where the individual $CFs$ are separated by homeotropic narrow stripes, observed between $V0$ and $V01$; (e) $CFs$ merge producing undulating $TIC$, observed between $V01$ and $V02$; (f) $TIC$ with umbilics, observed above the lines $V0$ and $V02$. Picture shown in part (b) was taken about 2 seconds after voltage was applied; it shows an intermediate state in which the circular domains grow from nucleation sites and will eventually fill in the whole area of the cell by the fingers. FIG.3. (a,b) Polarizing microscope textures of (a) the $TIC$ with no umbilics and (b) periodic fingers pattern in a homeotropic cell with one of the substrates rubbed along the black bar. (c,d) Light transmission through the cell with rubbed homeotropic substrates placed between crossed polarizers for (c) $\rho =0$ and (d) $\rho =0.5$. The insets in (c,d) show details of intensity changes in the vicinity of homeotropic-$TIC$ transition; note that the rubbing-induced pretilt makes these dependencies not as sharp as normally observed in non-rubbed homeotropic cells (see, for example, Ref. [@Rosenblatt]). FIG.4. Phase diagram of structures in the $U$ vs. $\rho $ parameter space for CLCs in the cells with homeotropic boundary conditions and substrates rubbed in anti-parallel directions. The cell has mylar spacers at the edges; no spacer particles are present in the bulk. The lines $V0-V3$, $V01$, $V02$ separate different phases and the two dashed vertical lines mark $\rho _{triple}=0.816$ and $\rho _{tricritical}=0.861$ corresponding to the triple and tricritical points, similar to Fig.1. The solid line $V0$ was calculated using Eq.(1) and is the same as in Fig.1; the solid lines $V1-V3$, $V01$, $% V02$ connect the experimental points to guide the eye. FIG.5. Polarizing microscope textures illustrating the transition from (a) homeotropic untwisted state to (b) $TIC$ with no umbilics and total twist $% \Delta \phi \approx \pi $ between the substrates, and then to (c) fingers pattern that slowly ($\sim 1s$) appears from $TIC$, and then to (d) uniform $% TIC$ with $\ \lessapprox 2\pi $ twist. The applied voltages are indicated. The homeotropic cell has substrates rubbed in anti-parallel directions; $% \rho =0.65$. The cell was assembled by using mylar spacers at the cell edges; no particles or other nucleation sites are present in the working area of the cell. FIG.6. Influence of spherical particles with perpendicular surface anchoring on the CLC structures in homeotropic cells: (a) particle-induced director distortions in the homeotropic state; (b) director distortions in the $TIC$ at $U\approx U_{th}$; (c) $TIC$ $10ms$ after $U>U_{th}$ is applied and (d) relaxation of the distortions in $TIC$ via formation of fingers as observed $% \approx 1s$ after $U$ is applied (the particles become nucleation sites for the fingers). FIG.7. Phase diagram of structures in the $U$ vs. $\rho $ parameter space for CLCs in homeotropic cells with rubbed substrates : (a) anti-parallel (i.e., at 180$^{\circ }$); (b) at 90$^{\circ }$; (c) at 270$^{\circ }$. The frequency of the applied voltage is $1kHz$, which is amplitude-modulated with a $50Hz$ sinusoidal signal. The boundary lines $V0-V3$, $V01$, $V02$ separate different phases and the two dashed vertical lines mark $\rho _{triple}=0.816$ and $\rho _{tricritical}=0.861$, similar to Figs.1 and 4. The solid line $V0$ was calculated using Eq.(1) and is the same as in Figs.1,4; the solid lines $V1-V3$, $V01$, $V02$ connect the experimental points to guide the eye. FIG.8. FCPM cross-sections (a,b) and schematic of director structures (c,d) of $TIC$ with twist: (a,c) $\approx \pi $ at $U=5Vrms$ and $\rho =1/2$; (b,d) $\approx 2\pi $ at $U=5Vrms$ and $\rho =1$. The polarization of the probe light in FCPM marked by ”P” is in the $y$-direction, along the normal to the pictures in (a,b). FIG.9. FCPM vertical cross-sections (a) and schematic of director structures (b) of a $CF1$-type isolated finger. The polarization of the probe light in FCPM marked by ”P” is in the $y$-direction, along the normal to the picture in (a). The non-singular $\lambda $-disclinations are marked by circles in (b); the open circles correspond to the $\lambda ^{-1/2} $ and the solid circles correspond to the $\lambda ^{+1/2}$ disclinations. FIG.10. FCPM cross-sections (a) and schematic of director structures (b) of a periodic finger pattern composed of $CF1$s separated by homeotropic stripes. The polarization of the FCPM probe light marked by ”P” is normal to the picture in (a). FIG.11. FCPM vertical cross-section illustrating the voltage-induced transition from (a) isolated fingers coexisting with homeotropic state to (f) periodically-modulated $TIC$ and then to (g) a uniform $TIC$. The fingers gradually widen (b-e) and then merge in order to form the modulated $% TIC$ (f). The polarization of the FCPM probe light marked by ”P” is normal to the pictures in (a). The applied voltages are indicated, the confinement ratio is $\rho =0.9$. FIG.12. FCPM cross-section (a) and reconstructed director structure (b) illustrating the $CF$ expanding into $TIC$. The polarization of the FCPM probe light marked by ”P” is along $y$, normal to the picture in (a). FIG.13. FCPM cross-section (a) and reconstructed director structure (b) of $% CF2$ finger; the $CF2$ can expand in one (c) or two in-plane directions (d) forming $TIC$. The non-singular $\lambda $-disclinations are marked by circles in (b); the open circles correspond to the $\lambda ^{-1/2}$ disclinations and the solid circle corresponds to the $\lambda ^{+1}$ disclination. The FCPM polarization is normal to the picture in (a). FIG.14. FCPM cross-section (a) and reconstructed director structures (b,c)formed between the parts of a cell with different in-plane twist and helical axis along the $z$: (b) $TIC$ with $\approx \pi $ twist, coexisting with homeotropic untwisted state and separated by $CF2$-like structure with two nonsingular $\lambda ^{-1/2}$ disclinations; (c) $TIC$ with $\approx 2\pi $ twist coexisting with $TIC$ with $\approx \pi \,\ $twist. The FCPM polarization marked by ”P” is normal to the picture in (a). FIG.15. FCPM images and director structure of a $T$-junction of $CF1$ and $% CF2$: (a) in-plane FCPM cross-section; (b) perspective view of the 3-D director field of the $T$-junction; (c,d,e) FCPM cross-sections along the lines $c-c$, $d-d$, and $e-e$ in part (a). The FCPM cross-sections in (c,e) correspond to $CF2$ and cross-section (d) corresponds to $CF1$. The FCPM polarization marked by ”P” is normal to the pictures in (c,d,e). FIG.16. FCPM cross-section (a) and reconstructed director structures (b) of $% CF3$ finger. The FCPM polarization marked by ”P” is normal to the picture in (a). Two $CF3$s can be seen in (a); the director structure of only one $% CF3$ is shown in (b). The singular disclinations at the two substrates are marked by circles in (b). FIG.17. FCPM cross-sections (a,c) and reconstructed director structure (b) of the $CF4$ fingers. The FCPM polarization is along $y$ in (a,c). The singular disclinations at one of the substrates are marked by circles in (b). The $CF4$s in (c) have singular disclinations at the same substrate. FIG.18. A representative $TIC$ equilibrium configuration obtained as numerical solution of Eqs. (\[eqn:ELa\]–\[eqn:ELc\])): (a) tilt angle $% \theta $, (b) twist angle $\phi $, (c) electric potential $\psi $. The material parameters used in the calculations were taken for the AMLC-0010host doped with ZLI-811, $d=5\,\mu m$, $\rho =0.5$, $U=3.5Vrms$. L. M. Blinov and V. G. Chigrinov, [*Electrooptic Effects in Liquid Crystal Materials,*]{} (Springer, New York, 1994). P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, [*Proceedings of the IEEE*]{} [**84**]{}, 268 (1996). P. Palffy-Muhoray, [*Nature*]{}[** 391**]{}, 745 (1998). D.-K. Yang, X.-Y. Huang, and Y.-M. Zhu, [*Annu. Rev. Mater. Sci.*]{} [**27**]{}, 117 (1997). P. Palffy-Muhoray, T. Kosa, B. Taheri, Variable light attenuating dichroic dye guest-host device, PCT Int. Appl. WO 9967681 (1999); B. Taheri, P. Palffy-Muhoray, T. Kosa, and D. L. Post., [*Proc. SPIE*]{} [**4021**]{}, 114 (2002). J. S. Patel, G. B. Cohen, [*Appl. Phys. Lett.*]{} [**68**]{}, 3564 (1996). B. I. Senyuk, I. I. Smalyukh, O. D. Lavrentovich, [*Optics Lett.*]{} [**30**]{}, 349 (2005). J. S. Hsu, B.-J. Liang, S.-H. Chen, [*Appl. Phys. Lett. [**85**]{},* ]{}5511(2004). F. A. Munoz, P. Palffy-Muhoray, B. Taheri, [*Optics Lett.*]{} [**26**]{}, 804 (2001). M. F. Moreira, I. C. S. Carvalho, W. Cao, C. Bailey, B. Taheri, and P. Palffy-Muhoray, [*Appl. Phys. Lett.*]{} [**85**]{}, 2691 (2004). J. Schmidtke, W. Stille, and H. Finkelmann, [*Phys. Rev. Lett.*]{} [**90**]{}, 083902 (2003). P. G. de Gennes and J. Prost, [*The Physics of Liquid Crystals*]{} (Clarendon Press, Oxford, 1993). M. Kleman and O. D. Lavrentovich, [*Soft Matter Physics: An Introduction*]{} (Springer, New York, 2003). O. D. Lavrentovich, and M. Kleman, Cholesteric Liquid Crystals: Defects and Topology, Chapter 5 in: [*Chirality in Liquid Crystals*]{}, C. Bahr and H. Kitzerow, (eds.), (Springer, New York, 2001). P. Oswald, J. Baudry, and S. Pirkl, [*Phys. Reports*]{} [**337**]{}, 67 (2000). P. E. Cladis and M. Kleman, [*Mol. Cryst. Liq. Cryst.*]{} [**16**]{}, 1 (1972). L. Gil, [*J. Phys. II France*]{} [**5**]{}, 1819 (1995). S. V. Shiyanovskii, I. I. Smalyukh, and O. D. Lavrentovich, [*Chapter in: Defects in Liquid Crystals: Computer Simulations, Theory and Experiments,*]{} edited by O. D. Lavrentovich, P. Pasini, C. Zannoni, and S. Žumer, NATO Science Series, (Kluwer Academic Publishers, Dordrecht, 2001). M. J. Press, A. S. Arrott, [*J. Phys. France* ]{}[**37**]{}, 387 (1976). M. J. Press, A. S. Arrott, [*Mol. Cryst. Liq. Cryst.*]{} [**37**]{}, 81 (1976). M. J. Press, A. S. Arrott, [*J. Phys. France* ]{}[**39**]{}, 750 (1978). L. Gil and J. M. Gilli, [*Phys. Rev. Lett.*]{} [**80**]{}, 5742 (1998). J. Baudry, S. Pirkl, and P. Oswald, [*Phys. Rev. E*]{} [**57**]{}, 3038 (1998); ibid, [*Phys. Rev. E*]{} [**59**]{}, 5562 (1999). P. Ribiere, S. Pirkl, and P. Oswald, [*Phys. Rev. A* ]{}[**44**]{}, 8198 (1991). O. S. Tarasov, A. P. Krekhov, L. Kramer, [*Phys. Rev. E*]{} [**68** ]{} 031708 (2003). P. Oswald, J. Baudry, and T. Rondepierre, [*Phys. Rev. E*]{} [**70**]{}, 041702 (2004). J. M. Gilli, S. Thiberge, A. Vierheilig, and F. Fried,[* Liq. Cryst.*]{} [**23**]{}, 619 (1997). J. Baudry, S. Pirkl, and P. Oswald, [*Phys. Rev. E*]{} [**57**]{}, 3038 (1998). F. Lequeux, P. Oswald, and J. Bechhoefer, [*Phys. Rev. A*]{} [**40**]{}, 3974 (1989). T. Ishikawa and O. D. Lavrentovich, [*Phys. Rev. E* ]{}[**60**]{}, R5037 (1999). I. I. Smalyukh, S. V. Shiyanovskii, and O. D. Lavrentovich, [*Chem. Phys. Lett.*]{} [**336,**]{} 88 (2001). B. L. van Horn and H. Henning Winter, [*Applied Optics*]{} [**40**]{}, 2089 (2001); T. J. Scheffer and J Nehring, [*J. Appl. Phys.*]{} [**48**]{}, 1783 (1977). M. Born, E. Wolf, [*Principles of Optics*]{} (Cambridge Unversity Press, Cambridge, 1999). I.I. Smalyukh and O. D. Lavrentovich, [*Phys. Rev. E*]{} [**66**]{}, 051703 (2002). T. Kosa, V. H. Bodnar, B. Taheri, and P. Palffy-Muhoray, [*Mol. Cryst. Liq. Cryst.* ]{}[**369**]{}, 129 (2001). W. E. Ford, P. V. Kamat, [*J. Phys. Chem.*]{} [**91,**]{} 6373 (1987). I.I. Smalyukh and O.D. Lavrentovich, [*Phys. Rev. Lett.* ]{}[**90**]{}, 085503 (2003). W. H. de Jeu, [*Physical properties of liquid crystalline materials* ]{}(Gordon and Breach, New York, 1979). P. Yeh, C. Gu, [*Optics of Liquid Crystal Displays*]{} (John Wiley & Sons, Inc., 1999). K. A. Crandall, M. R. Fisch, R. G. Petschek, and C. Rosenblatt, [*Appl. Phys. Lett.*]{} [**64**]{}, 1741 (1994). P. E. Cladis, W. van Saarloos, P. L. Finn, and A. R. Kortan, [*Phys. Rev. Lett. [**58**]{},* ]{}222 (1987). E. C. Gartland, Jr. et al., unpublished. I. W. Stewart, [*The Static and Dynamic Continuum Theory of Liquid Crystals*]{} (Taylor & Francis, 2004). We note that there is a mistake in Eq.(29) of Ref.[Oswald-review00]{} as it is different from Eq.(14) of Ref.[CHnegativeDeltaE]{}. Z. Huang and C. Rosenblatt, [*Appl. Phys. Lett.*]{} [**86**]{}, 011908 (2005).

--- author: - 'Fabien Durand, Dominique Perrin' bibliography: - 'dimensionGroups.bib' title: Dimension groups and dynamical systems ---

--- abstract: 'We study the vibrational and electronic properties of (x)Na$_2$S-(1-x)GeS$_2$ glasses through DFT-based molecular dynamics simulations, at different sodium concentrations ($0<x<0.5$). We compute the vibrational density of states for the different samples in order to determine the contribution of the Na$^+$ ions in the VDOS. With an in-depth analysis of the eigenvectors we determine the spatial and atomic localization of the different modes, and in particular in the zone corresponding to the Boson peak. We also calculate the electronic density of states as well as the partial EDOS, in order to determine the impact of the introduction of the sodium modifiers on the electronic properties of the GeS$_2$ matrix.' address: 'Laboratoire de Physicochimie de la Matière Condensée, Université Montpellier 2, Place E. Bataillon, Case 03, 34095 Montpellier, France' author: - Sébastien Blaineau and Philippe Jund title: Calculated vibrational and electronic properties of various sodium thiogermanate glasses --- PACS numbers: 61.43.Fs,61.43.Bn,71.23.Cq,71.23.-k,71.15.Pd Introduction ============ Sodium thiogermanate (x)Na$_2$S-(1-x)GeS$_2$ glasses are good solid electrolytes, with a high ionic conductivity at room temperature [@robinel; @tranchant]. As in other glassy systems, the ionic transport process has been clearly determined but its microscopic origin is still not well understood. In particular the mechanisms leading to the high conductivity at room temperature are not clearly established: is the conductivity dominated by random back and forth jumps [@funke] or do preferential pathways (“channels”) exist inside the glassy matrix as in amorphous silica [@jund] ? A first step in the comprehension of these mechanisms is to determine the effect of the modifier ions (Na) on the physical properties of the glass (GeS$_2$). To that purpose, theoretical studies, and in particular molecular dynamics (MD) simulations are interesting tools that provide detailed informations at the atomic level on the modifications of the amorphous sample as the concentration of sodium is increased. These modifications have then to be connected to the evolution of the ionic transport if this connection exists.\ In previous works we have studied germanium disulfide glasses (GeS$_2$) through DFT-based MD simulations [@blaineau1; @blaineau2; @blaineau3]. The results provided by our simulations were in very good agreement with the existing experimental data. Furthermore additional informations at the atomic scale provided by these simulations were found to explain some properties observed experimentally at the macroscopic scale especially concerning the vibrational and electronic properties of GeS$_2$ glasses.\ In this paper we aim to analyze Na-Ge-S systems using the same model, in order to study the impact of the Na$^+$ ions in GeS$_2$ glasses. Although similar studies have been performed in other glassy systems, such as SiO$_2$ [@jund], showing the evidence of conduction channels created dynamically by the Na cations inside the glass, no MD simulations have been performed yet in sodium thiogermanate glasses to our knowledge. The aim is [*in fine*]{} to analyze the influence of the glassy matrix on the conduction properties: do the conduction channels exist also in chalcogenide glasses or are they specific to oxide glasses ? Are the modifications introduced by the alkali ions similar in both types of glasses ? To that purpose, we focus here (in a first step) on the vibrational and electronic properties of Na-Ge-S glasses, which should be directly connected to the ionic conduction properties. In order to evaluate the evolution of theses properties with the concentration of alkali ions, we simulate several (x)Na$_2$S-(1-x)GeS$_2$ samples, for $0<x<0.5$, and analyze the vibrational and electronic densities of states for these different Na concentrations. The article is organized as follows : In section II we briefly describe the theoretical foundations of our model, whereas in section III we study the vibrational properties of the amorphous samples through an in-depth analysis of the vibrational eigenvectors. Subsequently we study the electronic properties in section IV, and finally in section V we summarize the major conclusions of our work. Model ===== The code we have used is a first-principles type molecular dynamics program called FIREBALL96, which is based on the local orbital electronic structure method developed by Sankey and Niklewski [@sankey]. The foundations of this model are the Density Functional Theory (DFT) [@dft] within the Local Density Approximation (LDA) [@lda], and the non-local pseudopotential scheme [@pseudo]. The use of the non-self-consistent Harris functional [@harris], with a set of four atomic orbitals (1$s$ and 3$p$) per site that vanish outside a cut-off radius of $5a_0$ (2.645 Å) considerably reduces the CPU time.\ The pseudo-wave function $\Psi$ of the system is given by the following equation: $$\begin{aligned} \Psi_j(\vec{k},\vec{r})=\Sigma_{\mu}C_{\mu}^j(\vec{k})\Phi_{fireball}^{\mu}(\vec{r})\end{aligned}$$ where j is the band index, $\Phi_{fireball}^{\mu}$ is the fireball basis function for orbital $\mu$, and C$_{\mu}^j(\vec{k})$ are the LCAO expansion coefficients. Only the $\Gamma$ point is used to sample the Brillouin zone ($\vec{k}=\vec{0}$).\ This model has given excellent results in many different chalcogenide systems over the last ten years [@blaineau1; @drabold; @junli]. In the present work we melt a crystalline $\alpha$-GeS2 configuration containing 258 particles at 2000K during 60 ps (24000 timesteps) in a cubic box of 19.21 Å, until we obtain an equilibrated liquid. Subsequently we replace randomly GeS$_4$ tetrahedral units by artificial Na$_2$S$_3$ “molecules”, following a procedure similar to the one used in SiO$_2$ glasses [@jund], in order to obtain a given sodium concentration (the total number of atoms, N, is kept constant at 258). We generate thus eight (x)Na$_2$S-(1-x)GeS$_2$ samples at different sodium concentrations (x= 0, 0.015, 0.03, 0.06, 0.11, 0.2, 0.33 and 0.5). The bounding box is rescaled each time so that the density matches its experimental counterpart (from 19.21 Å  for x=0[@boolchand] to 18.3 Å  for x=0.5[@ribes]), in order to limit artificial pressure effects on the system. Then, we melt the resulting system at 2000K during 60 ps so that the system completely loses the memory of the initial artificial configuration, and becomes a homogeneous liquid (x)Na$_2$S-(1-x)GeS$_2$ system. Finally, we quench the liquid structure at a quenching rate of 6.8$\times$10$^{14}$ K/s, decreasing the temperature to 300K through the glass transition temperature T$_g$, and we let our sample relax at 300K during 100 ps. The dynamical matrix and the electronic density of states have been calculated at the end of this relaxation time. It is worth noticing that the results obtained with FIREBALL96 on several Na-Ge-S test samples are almost identical to those obtained with the self-consistent [*ab initio*]{} SIESTA code [@siesta] in which the largest available basis set has been used. This shows the ability of our model to accurately describe sodium thiogermanate glasses at a relatively reduced CPU time cost. Results ======= Vibrational properties ---------------------- First we compute the Vibrational Density of States (VDOS), which can be measured experimentally by inelastic neutron diffraction spectroscopy. The VDOS is calculated through the diagonalization of D, the dynamical matrix of the system given by: $$\begin{aligned} D(\phi_i,\phi_j)=\frac{\partial^2 E(\phi_i,\phi_j)}{\partial\phi_i\partial\phi_j} , ~~\phi=x,y,z\end{aligned}$$ for two particles $i$ and $j$. Fig.1 presents the calculated VDOS for different values of x ( for clarity only the results for x=0, x=0.2 and x=0.5 are shown). The VDOS of GeS$_2$ (x=0) has been studied in detail in a previous work [@blaineau2] in which we have determined the presence of two bands (the acoustic and optic band), separated by a “gap”, which was found to contain a few localized modes caused by bond defects. We can see in Fig. 1 how the introduction of sodium atoms modifies the VDOS, and it appears clearly that the vibrational contribution of the Na atoms takes place between the acoustic and optic band (200 cm$^{-1}$- 300 cm$^{-1}$). In the Na$_2$GeS$_3$ compound (x=0.5) the two bands cannot be distinguished anymore, since the density of states is almost flat over the whole spectrum. It can also be seen that the introduction of sodium cations diminishes the acoustic band especially on the low frequency side, contrarily to the optic band that remains practically unchanged. The diminution of the low frequency modes is counterbalanced by an accumulation of modes in the “gap” zone between 200 cm$^{-1}$ and 300 cm$^{-1}$ when the sodium concentration increases. The low-frequency zone at 35 cm$^{-1}$ has been attributed experimentally to the well-known “Boson peak”, which is a signature of amorphous materials in the VDOS [@philips]. It appears in our simulation that above x=0.2, the density of states in that zone decreases significantly (together with the rest of the acoustic band). Unfortunately, the VDOS of sodium thiogermanate glasses has never been measured experimentally to our knowledge, and therefore we cannot confirm this lack of low frequency modes by experimental data. In order to measure the localization of the vibrational modes, we calculate the participation ratio P$_r$ [@bell2]: $$\begin{aligned} P_r=\frac{(\Sigma_{i=1}^N|\vec{e}_i(\omega)|^2 )^2}{N\Sigma_{i=1}^N|\vec{e}_i(\omega)|^4}\end{aligned}$$ where the summation is done over the N particles of the sample. If the mode corresponding to eigenvalue $\omega$ is delocalized and all atoms vibrate with equal amplitudes, then P$_r$($\omega$) will be close to 1. On the contrary, if the mode is strongly localized, then P$_r$($\omega$) will be close to 0. The results are shown in Fig.2, and it can be seen that the modes that appear in the zone between 200 cm$^{-1}$ and 300 cm$^{-1}$ in the sodium-enriched samples are relatively delocalized. The P$_r$, which is close to zero for GeS$_2$ in that region, becomes higher as the Na concentration increases. An in-depth study of the vibrational eigenvectors in that region shows that these modes are mainly caused by sodium atoms, as it could be deduced from the VDOS (Fig.1). Although a few localized modes appear at the end of the optic band (beyond 480 cm$^{-1}$), it can be said that the contribution of the Na atoms in the VDOS is principally sensitive in delocalized modes (for x=0.5, the maximum of Pr appears at 190 cm$^{-1}$). It should also be noted that in the region attributed to the Boson Peak (35 cm$^{-1}$) [@tanaka], the participation ratio is lower for sodium-enriched systems. Since we have previously seen that this zone showed a lack of modes for x=0.33 and x=0.5 in comparison to the low values of x one can conclude that the remaining “soft modes” become more localized when the Na concentration increases. In order to calculate the spatial localization of the vibrational modes, we calculate the center of gravity $\vec{r}_g(\omega)$ of each mode of eigenvalue $\omega$, and the corresponding “localization” length L [@zotov], as $$\begin{aligned} \vec{r}_g(\omega)=\frac{\Sigma_{i=1}^N\vec{r}_i|\vec{e}_i(\omega)|^2/m_i}{\Sigma_{i=1}^N |\vec{e}_i(\omega)|^2/m_i}\end{aligned}$$ and $$\begin{aligned} L(\omega)=\sqrt{\frac{\Sigma_{i=1}^N|\vec{r}_i-\vec{r}_g(\omega)|^2|\vec{e}_i(\omega)|^2/m_i}{\Sigma_{i=1}^N|\vec{e}_i(\omega)|^2/m_i}}\end{aligned}$$ where $\vec{r}_i$ and $m_i$ are respectively the position and the atomic mass of particle $i$. Periodic boundary conditions must be taken into account in these calculations. The localization length (Fig.3) represents the spatial localization of a given mode, and its maximal value corresponds to the half-size of the box. Beyond this length, the amplitude of the atomic vibrations decreases significantly. It can be seen in Fig.3 that contrarily to the participation ratio, the localization length remains unchanged at low frequencies as the Na concentration increases, and corresponds approximately to the half-size of the box. The low frequency modes appear therefore completely delocalized in space, but the number of particles involved in these vibrations decreases as the sodium concentration becomes higher as shown by the decreasing participation ratio. This means that in pure GeS$_2$ a large amount of connected particles were involved in the low frequency modes whereas in Na-Ge-S glasses, Na breaks these connections and therefore small groups of particles (small participation ratio) scattered in the whole simulation box (large localization length) participate in these modes with the consequence of a decrease of the density of states at low frequency. Electronic properties --------------------- The Sankey-Niklewski scheme that has been described in section II allows the determination of the electronic energy eigenvalues for the different samples. The Electronic Density of States (EDOS), which can be calculated by “binning” these eigenvalues, has been measured experimentally by X-ray Photoelectron Spectroscopy (XPS) in GeS$_2$ and Na$_2$GeS$_3$ (x=0.5) systems [@foix]. In a previous work we have compared in GeS$_2$ our calculated EDOS with its experimental counterpart, and we have analyzed in detail the different features of the valence spectrum [@blaineau3]. Three bands, called A, B and C were clearly distinguished, with good agreement with the XPS spectrum. We show in Fig.4 the valence band of our calculated EDOS for x=0 and x=0.5, whereas Fig.5 presents the calculated and experimental valence spectra of Na$_2$GeS$_3$. We can see in Fig.4. that the impact of the sodium atoms in the EDOS of amorphous GeS$_2$ is negligible (the spectra of the other simulated samples are rather similar to these two graphs). It can however be seen that the width of band B decreases with the introduction of the sodium atoms in the glassy sample and that the density of states in band C increases. In addition, the small peak at the end of band A is slightly shifted in the sodium-enriched sample. We can see in Fig.5 that this calculated spectrum is in good agreement with the experimental data, even though a small energy shift is visible at the end of band C ($\approx$ 1 eV).\ In order to determine which atomic orbitals are responsible of these bands, we must compute the partial EDOS by summing the $|C_{\mu}^j(\vec{k})|^2$ for each element and each orbital. Here the $s$ and $p$ orbitals of germanium and sulfur atoms can be distinguished, as well as the $s$ orbitals of sodium atoms. We have scaled the partial EDOS so that that their sum is equal to the total EDOS. The results are illustrated in Fig.6, where the solid line represents the total EDOS and the dashed area shows the contribution of a given orbital.\ It can be seen that zone A is almost exclusively caused by the $3s$ orbitals of sulfur atoms. The small peak at the end of this band, which appears to be shifted for x=0.5, has been attributed in GeS$_2$ to S-S homopolar bonds[@blaineau3]. We find that in the Na$_2$GeS$_3$ sample these homopolar bonds are also connected to a sodium atom, which changes thus the energy eigenvalues of these orbitals. This explains why the modes at the end of band A have a higher energy ($\approx$ 0.3 eV) for x=0.5 than in the GeS$_2$ compound. The band B, which is sharper for sodium-enriched systems (Fig.4), is mainly caused by the $4s$ orbitals of Ge atoms (here the respective concentrations of each element must be taken into account). This property has been observed experimentally, and has been attributed to the increase of the Ge-Ge interatomic length, limiting thus the height of band B [@foix]. We have analyzed the interatomic distances for Ge-Ge pairs, and we can confirm this experimental prediction. Indeed, the distance between germanium pairs increases in both edge-sharing and corner-sharing units. In GeS$_2$ these lengths were found to be equal respectively to 2.91 Å  and 3.41 Å[@blaineau1], whereas in the Na$_2$GeS$_3$ sample they become equal to 3.01 Å  and 3.67 Å. Therefore this variation could indeed be responsible of the evolution of band B when the sodium concentration increases.\ We see that the contribution of the sodium atoms in the EDOS appears at the beginning of band C. The electronic energy of the $3s$ orbitals of sodium atoms is therefore close to the energy of the $4p$ orbitals of germanium atoms, and both are responsible of the shoulder which can be distinguished at the beginning of this band. It can be seen at the end of band C that the last occupied states before the Fermi Level are caused by the $3p$ orbitals of sulfur atoms as in GeS$_2$, and no contribution of the Na ions could be determined at the top of the valence band.\ The localization of the electronic eigenstates can be evaluated through the inverse participation ratio (IPR), which can be calculated with the Mulliken charges [@mulliken]. It appears that in comparison to the IPR of GeS$_2$, which has been shown in our previous work [@blaineau3], no localized states concerning sodium atoms have been found. Indeed the IPR of Na$_2$GeS$_3$ is very similar to that of GeS$_2$ and therefore it is not shown here.\ Conclusion ========== We have studied the vibrational and electronic properties of sodium thiogermanate glasses through DFT-based molecular dynamics simulations for different sodium concentrations. We find that the vibrational contribution of the sodium ions in the VDOS takes place between the acoustic and optic band. Thus, the modes in that zone are almost exclusively caused by the Na particles of the system, and are due to delocalized motions of the ions. In the same time, the acoustic band decreases, and in particular the well known “soft modes” tend to disappear because of the introduction of the sodium atoms in the system, which limits the collective vibrations connected to the Boson peak in that zone. This effect should be confirmed or infirmed experimentally. The calculation of the EDOS shows that the electronic contribution of Na in (x)Na$_2$S-(1-x)GeS$_2$ glasses is quite negligible. However, several variations could be seen in the three main contributions of the valence spectrum, and in particular in band B, which becomes sharper in the sodium-enriched systems. This change can be attributed to the increase of the Ge-Ge interatomic length, as it has been proposed experimentally. With the calculation of the partial EDOS we find that the contribution of the Na$^+$ ions takes place at the beginning of band C, in the same energy zone that the $p$ orbitals of the germanium atoms, and no localized states could be attributed to the sodium ions at the top of the valence band.\ This work is the first step in the study of the influence of the alkali ions on the properties of GeS$_2$ glasses and has to be continued by the analysis of the structural changes before trying to link all these modifications to the transport properties: this work is currently in progress.\ [**Acknowledgments**]{} Parts of the simulations have been performed on the computers of the “Centre Informatique National de l’Enseignement Supérieur” (CINES) in Montpellier. [1]{} E. Robinel, B. Carette and M. Ribes, J. Non-Cryst. Solids [**57**]{}, 49 (1983). S. Tranchant, S. Peytavin, M. Ribes, H. Dexpert and P. Lagarde, in: Proc. 6th, Riso Int. Symp. on Transport Structure. Relations in Fast Ion and Mixed Conductors, ed. F. W. Poulsen (Riso Nat. Lab., Roskilde, 1985) p.485. K. Funke, Prog. Solid State Chem. [**22**]{}, (1993) E. Sunyer, P. Jund and R. Jullien, Phys. Rev. B [**65**]{}, 214203 (2002). S. Blaineau, P. Jund, and D. A. Drabold, Phys. Rev. B [**67**]{}, 094204 (2003) S. Blaineau and P. Jund, Phys. Rev. B [**69**]{}, 064201 (2004). S. Blaineau and P. Jund, Phys. Rev. B [**70**]{}, 184210 (2004). O. F. Sankey an D. J. Niklewski, Phys. Rev. B, [**40**]{}, 3979 (1989). P. Hohenberg and W. Kohn, Phys. Rev [**136**]{}, B864 (1964). D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. [**45**]{}, 566 (1980). G. B. Bachelet, D. R. Hamman, and M. Schluter, Phys. Rev. B [**26**]{}, 4199 (1982). J. Harris, Phys. Rev. B, [**31**]{}, 1770 (1985). M. Durandurdu, D. A. Drabold, and N. Mousseau, Phys. Rev B, [**62**]{} 15307 (2000). J. Li, D.A. Drabold, Phys Rev. B, [**64**]{} 104206 (2001). P. Boolchand, J. Grothaus, M. Tenhover, M. A. Hazle and R. K. Grasselli, Phys. Rev. B [**33**]{}, 5421 (1986). J. O. Fourcade, M. Ribes, E. Philippot and M. Maurin. C. R. Acad. Sc. Paris [**272**]{} series C, 1964-67 (1971). http://www.uam.es/siesta “Amorphous solids, Low-Temperature Properties”, edited by W.A. Phillips (Springer, Berlin 1981) and references therein. R. J. Bell, Methods Comput. Phys. [**15**]{}, 215 (1976). K. Tanaka, M. Yamaguchi, J. Non-Cryst. Solids [**227-230**]{}, 757 (1988). M. Marinov and N. Zotov, Phys. Rev. B, [**55**]{}, 2938 (1997). D. Foix, Phd. Thesis, Université de Pau, France (2003). A. Szabo and N.S. Ostlund, [*[Modern Quantum Chemistry]{}*]{} (Dover, New York, 1996). 0,5cm ![ Calculated VDOS of (x)Na$_2$S-(1-x)GeS$_2$ for x=0, x=0.2 and x=0.5. []{data-label="fig1"}](fig1.eps){width="11cm"} ![ Participation ratio (see text for definition) of $(a)$ GeS$_2$ (x=0), and $(b)$ Na$_2$GeS$_3$ (x=0.5), as a function of $\omega$. []{data-label="fig2"}](fig2.eps){width="12cm"} ![ Localization length (see text for definition) of GeS$_2$ (x=0) $(a)$, and Na$_2$GeS$_3$ (x=0.5) $(b)$, as a function of $\omega$. []{data-label="fig3"}](fig3.eps){width="12cm"} ![ Calculated EDOS of GeS$_2$ and Na$_2$GeS$_3$ $(a)$. []{data-label="fig4"}](fig4.eps){width="10cm"} ![ Calculated EDOS of Na$_2$GeS$_3$ $(a)$ and experimental valence spectrum obtained by XPS measurements (Ref.18) . []{data-label="fig5"}](fig5.eps){width="9cm"} ![ Partial EDOS of Na$_2$GeS$_3$ (shaded area) and total EDOS (solid line). []{data-label="fig6"}](fig6.eps){width="8cm"}

--- abstract: 'Contextuality is a central property in comparative analysis of classical, quantum, and supercorrelated systems. We examine and compare two well-motivated approaches to contextuality. One approach (“contextuality-by-default”) is based on the idea that one and the same physical property measured under different conditions (contexts) is represented by different random variables. The other approach is based on the idea that while a physical property is represented by a single random variable irrespective of its context, the joint distributions of the random variables describing the system can involve negative (quasi-)probabilities. We show that in the Leggett-Garg and EPR-Bell systems, the two measures essentially coincide.' author: - 'J. Acacio de Barros^1^, Ehtibar N. Dzhafarov^2^, Janne V. Kujala^3^, Gary Oas^4^' title: Measuring Observable Quantum Contextuality --- Introduction ============ Contextuality is a key feature of quantum systems, as no noncontextual hidden-variable theory exists that is consistent with quantum theory. This feature has been at the core of recent research in quantum information, such as attempts to identify the underlying principles for the quantum boundary. Despite its importance, there seem to be no universally accepted measure of contextuality, and it is clear that the many definitions proposed in the literature [@larsson_kochen-specker_2002; @kleinmann_memory_2011; @kurzynski_entropic_2012; @chaves_entropic_2012; @svozil_how_2012; @grudka_quantifying_2014; @dzhafarov_contextuality_2014; @dzhafarov_probabilistic_2014; @dzhafarov_all-possible-couplings_2013; @dzhafarov_random_2013; @dzhafarov_generalizing_2014; @dzhafarov_qualified_2014; @DKL2014; @KDL2014] are not all equivalent. Here, we consider and compare two measures inspired by the idea that contextuality means the impossibility of finding a joint probability distribution (jpd) for different sets of random variables with some elements in common. One measure (related in various ways to Refs. [@larsson_kochen-specker_2002; @dzhafarov_all-possible-couplings_2013; @dzhafarov_qualified_2014; @Simon-Brukner-Zeilinger; @Winter2014; @svozil_how_2012; @DK_PLOS_2014] and in its current form presented in Refs. [@DKL2014; @KDL2014; @KDconjecture; @DKL_overview; @dzhafarov_contextuality_2014; @dzhafarov_generalizing_2014; @dzhafarov_probabilistic_2014]) is based on extended sets of context-indexed random variables and another on negative (quasi-)probabilities (dating back to Dirac, and recently explored in connection to contextuality in Refs. [@spekkens_negativity_2008; @abramsky_sheaf-theoretic_2011; @de_barros_decision_2014; @de_barros_negative_2015; @oas_exploring_2014; @de_barros_negative_2014]). As an example of contextuality, let there be three properties of a system, $P$, $Q$, and $R$, whose measurement outcomes are represented by the random variables $\mathbf{P}$, $\mathbf{Q}$, and $\mathbf{R}$[^1]. Assume we can never observe $P$, $Q$, and $R$ simultaneously, but only in pairwise combinations, $\left(\mathbf{P},\mathbf{Q}\right)$, $\left(\mathbf{P},\mathbf{R}\right)$, or $\left(\mathbf{Q},\mathbf{R}\right)$. We may think of each pair as recorded under a different experimental condition providing a context. The system exhibits contextuality if one cannot find a jpd of $\left(\mathbf{P},\mathbf{Q},\mathbf{R}\right)$ that agrees with the observed distributions of $\left(\mathbf{P},\mathbf{Q}\right)$, $\left(\mathbf{P},\mathbf{R}\right)$, and $\left(\mathbf{Q},\mathbf{R}\right)$ as its marginals. The two approaches to be considered in this paper deal with this situation differently. The negative probabilities (NP) approach relaxes the notion of a jpd by allowing some (unobservable) joint probabilities for $\left(\mathbf{P},\mathbf{Q},\mathbf{R}\right)$ to be negative. The “contextuality-by-default” (CbD) approach treats random variables recorded under different conditions as different “by default”, so that, e.g., property $P$ in the context of experiment $\left(\mathbf{P},\mathbf{Q}\right)$ is represented by some random variable $\mathbf{P}_{A}$, and in the context $\left(\mathbf{P},\mathbf{R}\right)$ by another random variable, $\mathbf{P}_{B}$. Denoting the three contexts by $A,B,C$, this yields three pairs of contextually labeled random variables, $\left(\mathbf{P}_{A},\mathbf{Q}_{A}\right)$, $\left(\mathbf{P}_{B},\mathbf{R}_{B}\right)$, and $\left(\mathbf{Q}_{C},\mathbf{R}_{C}\right)$, and in the CbD approach the joint distribution imposed on them allows, say, $\mathbf{P}_{A}$ and $\mathbf{P}_{B}$ to be unequal with some probability. Here we compare the NP and CdB approaches applied to the simplest contextual case possible, given by three pairwise correlated random variables, and to the standard EPR-Bell experiment. We show that for such examples the two measures of contextuality are the same. Negative Probabilities (NP) ============================ Using our above example, with $\mathbf{P},\mathbf{Q},\mathbf{R}$ observed in pairs, in the NP approach one ascribes to the vector $\left(\mathbf{P},\mathbf{Q},\mathbf{R}\right)$ a joint quasi-distribution by means of assigning to each possible combination $w=\left(p,q,r\right)$ a real number $\mu\left(w\right)$ (possibly negative), such that $$\begin{array}{c} \sum_{r}\mu\left(w\right)=\Pr\left[\mathbf{P}=p,\mathbf{Q}=q\right],\\ \sum_{q}\mu\left(w\right)=\Pr\left[\mathbf{P}=p,\mathbf{R}=r\right],\\ \sum_{p}\mu\left(w\right)=\Pr\left[\mathbf{Q}=q,\mathbf{R}=r\right]. \end{array}\label{eq:NP exmaple}$$ Such $\mu$ exists if and only if the no-signaling condition (built into EPR paradigms with spacelike separation) is satisfied [@al-safi_simulating_2013; @oas_exploring_2014; @abramsky_operational_2014], i.e., the distribution of, say, $\mathbf{P}$ is the same in $\left(\mathbf{P},\mathbf{Q}\right)$ and in $\left(\mathbf{P},\mathbf{R}\right)$.[^2] The numbers $\mu\left(w\right)$ can then be interpreted as quasi-probabilities of events $\left\{ w\right\} $, with the quasi-probability of any other event (subset of $w$ values) being computed by additivity, inducing thereby a signed measure [@halmos_measure_1974] on the set of all events. The quasi-probability of the entire set of $w$ will then be necessarily equal to unity, because, e.g., $$1=\sum_{p,q}\Pr\left[\mathbf{P}=p,\mathbf{Q}=q\right]=\sum_{w}\mu\left(w\right).$$ The function $\mu$ is generally not unique. In our approach [@de_barros_decision_2014; @de_barros_negative_2015] we restrict the class of possible $\mu$ to those as close as possible to a proper jpd by requiring that the L1 norm of the probability distribution, defined by $M=\sum_{w}\left|\mu\left(w\right)\right|,$ be minimized. This ensures that if the class of all possible $\mu$ satisfying (\[eq:NP exmaple\]) contains proper probability distributions, the chosen $\mu$ will have to be one of them. Since in this case $\left|\mu\left(w\right)\right|=\mu\left(w\right)$ for all $w$, the minimum of $M$ is 1. If (and only if) no proper probability distribution exists, then the minimum of $M$ exceeds 1. As a result, the smallest possible value $\Gamma_{\min}$ of $M-1$ can be taken as a measure of contextuality. Contextuality-by-Default (CbD) ============================== A more direct approach to contextuality [@dzhafarov_all-possible-couplings_2013; @dzhafarov_random_2013; @dzhafarov_contextuality_2014; @dzhafarov_generalizing_2014; @dzhafarov_probabilistic_2014; @dzhafarov_qualified_2014; @KDL2014; @DKL2014; @DK_PLOS_2014; @DKL_overview; @KDconjecture] is to posit that the identity of a random variable is determined by all conditions under which it is recorded. Thus, in $\left(\mathbf{P}_{A},\mathbf{Q}_{A}\right)$, $\left(\mathbf{P}_{B},\mathbf{R}_{B}\right)$, and $\left(\mathbf{Q}_{C},\mathbf{R}_{C}\right)$ of our example, any random variable in any of the pairs is *a priori* different from and stochastically unrelated to any random variable in any other pair [@dzhafarov_contextuality_2014; @dzhafarov_qualified_2014], but a jpd can always be imposed on the six random variables. In other words, one can always assign probability masses $\lambda$ to $v=\left(p_{A},p_{B},q_{A},q_{C},r_{B},r_{C}\right)$ in such a way that $$\begin{array}{c} \sum_{p_{B},q_{C},r_{B},r_{C}}\lambda\left(v\right)=\Pr\left[\mathbf{P}_{A}=p_{A},\mathbf{Q}_{A}=q_{A}\right],\\ \sum_{p_{A},q_{A},q_{C},r_{C}}\lambda\left(v\right)=\Pr\left[\mathbf{P}_{B}=p_{B},\mathbf{R}_{B}=r_{B}\right],\\ \sum_{p_{A},p_{B},q_{A},r_{B}}\lambda\left(v\right)=\Pr\left[\mathbf{Q}_{C}=q_{C},\mathbf{R}_{C}=r_{C}\right]. \end{array}\label{eq:CbD exmaple}$$ The noncontextuality hypothesis for $\mathbf{P}_{A},\mathbf{Q}_{A},\mathbf{R}_{B}$ and $\mathbf{P}_{B},\mathbf{Q}_{C},\mathbf{R}_{C}$ is that among these jpds $\lambda$ we can find at least one for which $\Pr\left[\mathbf{P}_{A}\not=\mathbf{P}_{B}\right]=\Pr\left[\mathbf{Q}_{A}\not=\mathbf{Q}_{C}\right]=\Pr\left[\mathbf{R}_{B}\not=\mathbf{R}_{C}\right]=0,$ which is equivalent to $\Delta=\Pr\left[\mathbf{P}_{A}\not=\mathbf{P}_{B}\right]+\Pr\left[\mathbf{Q}_{A}\not=\mathbf{Q}_{C}\right]+\Pr\left[\mathbf{R}_{B}\not=\mathbf{R}_{C}\right]=0.$ Such a jpd need not exist, and then the smallest possible value $\Delta_{\min}$ of $\Delta$ for which a jpd of $\left(\mathbf{P}_{A},\mathbf{P}_{B},\mathbf{Q}_{A},\mathbf{Q}_{C},\mathbf{R}_{B},\mathbf{R}_{C}\right)$ exists can be taken as a measure of contextuality.[^3] The CdB approach has its precursors in the literature: various aspects of the contextual indexation of random variables and probabilities of the kind shown are considered in Refs. [@larsson_kochen-specker_2002; @dzhafarov_qualified_2014; @Simon-Brukner-Zeilinger; @Winter2014; @svozil_how_2012; @dzhafarov_all-possible-couplings_2013; @Khr2005; @Khr2008; @Khr2009]. The principal difference, however, is in the use of minimization of $\Delta$ under the assumption that a jpd exists. This is a well-defined mathematical problem, solvable in principle for any set of distributions observed empirically. We will now compare and interrelate the two approaches, NP and CbD, by applying them to the Leggett-Garg and the EPR-Bell setups. Leggett-Garg ============= Let us consider Leggett and Garg’s $\pm1$-valued random variables, $\mathbf{Q}_{1}$, $\mathbf{Q}_{2}$, and $\mathbf{Q}_{3}$ [@leggett_quantum_1985]. Applying the NP approach, we seek signed probabilities $\mu$ for $\left(\mathbf{Q}_{1},\mathbf{Q}_{2},\mathbf{Q}_{3}\right)$ that are consistent with the observed correlations $\left\langle \mathbf{Q}_{i}\mathbf{Q}_{j}\right\rangle $ and individual expectations $\left\langle \mathbf{Q}_{i}\right\rangle $, with the smallest possible value of the L1 norm $M\equiv\sum_{w}\left|\mu\left(w\right)\right|$, where $w$ denotes all possible combinations of values $\left(q_{1},q_{2},q_{3}\right)$ for $\left(\mathbf{Q}_{1},\mathbf{Q}_{2},\mathbf{Q}_{3}\right)$. Here, we use the standard notation $\left\langle \cdot\right\rangle $ for the expectation operator. This problem can be easily solved, as we only have $2^{3}$ atomic elements $w$: $\left(1,1,1\right)$, $\left(1,1,-1\right)$, ... , $\left(-1,-1,-1\right)$. Thus, for $\mathbf{Q}_{1}$, $\mathbf{Q}_{2}$, and $\mathbf{Q}_{3}$, the minimal L1 norm $1+\Gamma_{\min}$ satisfies $$\begin{aligned} \Gamma_{\min} & =\max & \left\{ 0,-\frac{1}{2}+\frac{1}{2}S_{LG}\right\} ,\label{eq:Mm-LG}\end{aligned}$$ where $S_{LG}$ is defined as $$S_{LG}\equiv\max_{\#^{-}=1,3}\{\pm\left\langle \mathbf{Q}_{1}\mathbf{Q}_{2}\right\rangle \pm\left\langle \mathbf{Q}_{1}\mathbf{Q}_{3}\right\rangle \pm\left\langle \mathbf{Q}_{2}\mathbf{Q}_{3}\right\rangle \},\label{eq:=00003D000023- in LG}$$ where each $\pm$ in the expression should be replaced with $+$ or $-$, and $\#^{-}$ indicates the possible numbers of minuses. Notice that $S_{LG}\leq1$, which is equivalent to $\Gamma_{\min}=0$, is a necessary and sufficient condition for the existence of a proper jpd. Turning now to the CbD approach, we create a set of six random variables $$\mathbf{Q}_{1,2},\mathbf{Q}_{1,3},\mathbf{Q}_{2,1},\mathbf{Q}_{2,3},\mathbf{Q}_{3,1},\mathbf{Q}_{3,2},\label{eq:singles}$$ each indexed by the measurement conditions under which it is recorded: for any two random variables recorded at moments $t_{i}$ and $t_{j}$, with $i<j$, the $\mathbf{Q}_{i,j}$ designates the earlier variable and $\mathbf{Q}_{j,i}$ the later one. We have thus three pairs of variables with known jpds: $$\left(\mathbf{Q}_{1,2},\mathbf{Q}_{2,1}\right),\left(\mathbf{Q}_{1,3},\mathbf{Q}_{3,1}\right),\left(\mathbf{Q}_{2,3},\mathbf{Q}_{3,2}\right).\label{eq:pairs}$$ A jpd can always be constructed for these pairs (e.g., they can always be connected as stochastically independent pairs), but we seek a jpd with the smallest value $\Delta_{\min}$ of $$\begin{array}{r} \Delta=\Pr\left[\mathbf{Q}_{1,2}\neq\mathbf{Q}_{1,3}\right]+\Pr\left[\mathbf{Q}_{2,1}\neq\mathbf{Q}_{2,3}\right]+\Pr\left[\mathbf{Q}_{3,1}\neq\mathbf{Q}_{3,2}\right].\end{array}$$ A classical joint exists for $\mathbf{Q}_{1}$, $\mathbf{Q}_{2}$, and $\mathbf{Q}_{3}$ (no contextuality) if and only if a joint exists for (\[eq:pairs\]) with $\Delta=0$. The more we depart from the classical joint, the larger the minimum value $\Delta_{\min}$. Thus, $\Delta_{\min}$ can serve as a measure of contextuality. Requiring a jpd consistent with (\[eq:pairs\]) means to assign a probability to each of the $2^{6}$ possible values of these random variables, $$\mathbf{Q}_{1,2}=\pm1,\mathbf{Q}_{1,3}=\pm1,\ldots,\mathbf{Q}_{3,2}=\pm1,$$ constrained by being nonnegative and summing to the observed probabilities. For instance, the probabilities assigned to all combinations with $\mathbf{Q}_{1,2}=1$ and $\mathbf{Q}_{2,1}=-1$ should sum to the observed $\Pr\left[\mathbf{Q}_{1,2}=1,\mathbf{Q}_{2,1}=-1\right]$. A computer-assisted Fourier-Motzkin elimination algorithm gives the following analytic expression for the minimum value of $\Delta$ consistent with the observable pairs (\[eq:pairs\]): $$\Delta_{\min}=\max\left\{ 0,-\frac{1}{2}+\frac{1}{2}S_{LG}\right\} .\label{eq:Delta-LG}$$ This is a special case of the result in Ref. [@dzhafarov_generalizing_2014; @DKL2014; @KDL2014]. Comparing the general expressions (\[eq:Mm-LG\]) for $\Gamma_{\min}$ and (\[eq:Delta-LG\]) for $\Delta_{\min}$ we see that the two simply coincide: $$\Delta_{\min}=\Gamma_{\min}.$$ EPR-Bell ======== We now turn to the EPR-Bell case where Alice and Bob have each two distinct settings, $1$ and $2$, corresponding to four observable random variables $\mathbf{A}_{1}$, $\mathbf{A}_{2}$, $\mathbf{B}_{1}$, and $\mathbf{B}_{2}$. This notation implicitly contains the assumption that the identity of Alice’s measurements as random variables does not depend on Bob’s settings, and vice versa. It is well known [@fine_hidden_1982] that under the no-signaling conditions the existence of the jpd is equivalent to the CHSH inequalities being satisfied. Applying the NP approach, the minimal L1 norm of the probability distribution is given by [@oas_exploring_2014] $$\Gamma_{\min}=\max\left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} ,\label{eq:M-inequality-AliceBob}$$ where $$\begin{array}{r} S_{CHSH}=\raisebox{0pt}[0pt][0pt]{\ensuremath{{\displaystyle \max_{\#^{-}=1,3}}}}\{\pm\left\langle \mathbf{A}_{1,1}\mathbf{B}_{1,1}\right\rangle \pm\left\langle \mathbf{A}_{1,2}\mathbf{B}_{1,2}\right\rangle \phantom{\mbox{\}}.}\\ \pm\left\langle \mathbf{A}_{2,1}\mathbf{B}_{2,1}\right\rangle \pm\left\langle \mathbf{A}_{2,2}\mathbf{B}_{2,2}\right\rangle \mbox{\}}. \end{array}$$ Here $\Gamma_{\min}=0$ corresponds to the CHSH inequalities, and $\Gamma_{\min}>0$ to contextuality. Turning now to the CbD approach, we have four pairs of random variables, $$\left(\mathbf{A}_{1,1},\mathbf{B}_{1,1}\right),\left(\mathbf{A}_{1,2},\mathbf{B}_{1,2}\right),\left(\mathbf{A}_{2,1},\mathbf{B}_{2,1}\right),\left(\mathbf{A}_{2,2},\mathbf{B}_{2,2}\right).\label{eq:AB pairs}$$ Here, $\mathbf{A}_{i,j}$ denotes Alice’s measurement under her setting $i=1,2$ when Bob’s setting is $j=1,2$, and analogously for $\mathbf{B}_{i,j}$. We seek a jpd with the smallest value $\Delta_{\min}$ of $$\begin{array}{r} \Pr\left[\mathbf{A}_{1,1}\neq\mathbf{A}_{1,2}\right]+\Pr\left[\mathbf{A}_{2,1}\neq\mathbf{A}_{2,2}\right]+\Pr\left[\mathbf{B}_{1,1}\neq\mathbf{B}_{2,1}\right]+\Pr\left[\mathbf{B}_{1,2}\neq\mathbf{B}_{2,2}\right].\end{array}$$ No contextuality means $\Delta_{\min}=0$. A computer assisted Fourier-Motzkin elimination algorithm yields (this is a special case of the result in Ref. [@dzhafarov_generalizing_2014; @DKL2014; @KDL2014]) $$\Delta_{\min}=\max\left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} .\label{eq:delta-AliceBob}$$ We have the same simple coincidence the two measures as in the case of the Leggett-Garg systems, $$\Delta_{\min}=\Gamma_{\min}.$$ Final remarks ============== We have discussed two ways to measure contextuality. The direct approach, named Contextuality-by-Default (CbD), assigns to each random variable an index related to their context. If a system is noncontextual, a jpd can be imposed on the random variables so that any two of them representing the same property in different contexts always have the same values. If the system is contextual, the minimum value of $\Delta$ in (\[eq:Delta-LG\])-(\[eq:delta-AliceBob\]) across all possible jpds has the interpretation of how close a variable can be in two different contexts: the larger the value the greater contextuality, zero representing a necessary and sufficient condition for no contextuality. The other approach maintains the original set of random variables, but requires negative (quasi-)probabilities. This leads to nonmonotonicity (i.e., a set of outcomes can have a smaller probability than some of its proper subsets), which is a characteristic of quantum interference. The departure from a proper probability distribution is measured by $\Gamma_{\min}$ in the minimum L1 norm $1+\Gamma_{\min}$. Similar to the CbD approach, we use here a minimization principle that gives the closest probability distribution to an ideal (but impossible) jpd. The value of $\Gamma_{\min}$ has the interpretation of how contextual the system is: a necessary and sufficient condition for no contextuality is $\Gamma_{\min}=0$, and the larger the value of $\Gamma_{\min}$, the more contextual the system is. As we have seen, in the case of EPR-Bell and Leggett-Garg systems the two approaches lead to simple coincidence, $\Delta_{\min}=\Gamma_{\min}$. The two measures, $\Gamma_{\min}$ and $\Delta_{\min}$, can be computed, in principle, for any given system. For detailed examples of such computations, see Appendix. Of the two measures of contextuality, $\Gamma_{\min}$ is computationally much simpler, as it involves fewer random variables and a simpler set of conditions (no nonnegativity constraints). However, CbD has the advantage of being more general than NP, as it can include cases where no NP distributions exist due to violations of the no-signaling condition [@dzhafarov_probabilistic_2014; @DKL2014; @KDL2014]. #### Acknowledgments. This work was supported by NSF grant SES-1155956 and AFOSR grant FA9550-14-1-0318. The authors are grateful to Samson Abramsky, Guido Bacciagaluppi, Andrei Khrennikov, Jan-Åke Larsson, and Patrick Suppes for helpful discussions. Proofs of statements {#appendix:proofs} ==================== In this appendix, we describe how the analytic results of the main text were obtained for each of the expressions , , , and of the main text. EPR-Bell: Contextuality-by-Default ---------------------------------- Following the computations of Dzhafarov and Kujala [@dzhafarov_all-possible-couplings_2013 Text S3], or the more general formulation in Ref. [@dzhafarov_generalizing_2014], it can be shown that the observable distributions with probabilities given by the matrices $$\begin{tabular}{c|cc|ccc|cc|ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc} \cline{2-3} \cline{7-8} & \ensuremath{\mathbf{B}_{1,1}=+1} & \ensuremath{\mathbf{B}_{1,1}=-1} & & \quad{} & & \ensuremath{\mathbf{B}_{1,2}=+1} & \ensuremath{\mathbf{B}_{1,2}=-1} & \tabularnewline\cline{1-4} \cline{6-9} \multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{1,1}=+1}} & \ensuremath{p_{1,1}} & \ensuremath{a_{1}-p_{1,1}} & \multicolumn{1}{c|}{\ensuremath{a_{1}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{A}_{1,2}=+1} & \ensuremath{p_{1,2}} & \ensuremath{a_{1}-p_{1,2}} & \multicolumn{1}{c|}{\ensuremath{a_{1}}}\tabularnewline\multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{1,1}=-1}} & \ensuremath{b_{1}-p_{1,1}} & \ensuremath{1-a_{1}-b_{1}+p_{1,1}} & \multicolumn{1}{c|}{\ensuremath{1-a_{1}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{A}_{1,2}=-1} & \ensuremath{b_{2}-p_{1,2}} & \ensuremath{1-a_{1}-b_{2}+p_{1,2}} & \multicolumn{1}{c|}{\ensuremath{1-a_{1}}}\tabularnewline\cline{1-4} \cline{6-9} & \ensuremath{b_{1}} & \ensuremath{1-b_{1}} & & & & \ensuremath{b_{2}} & \ensuremath{1-b_{2}} & \tabularnewline\cline{2-3} \cline{7-8} \multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & & & \multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & \tabularnewline\cline{2-3} \cline{7-8} & \ensuremath{\mathbf{B}_{2,1}=+1} & \ensuremath{\mathbf{B}_{2,1}=-1} & & & & \ensuremath{\mathbf{B}_{2,2}=+1} & \ensuremath{\mathbf{B}_{2,2}=-1} & \tabularnewline\cline{1-4} \cline{6-9} \multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{2,1}=+1}} & \ensuremath{p_{2,1}} & \ensuremath{a_{2}-p_{2,1}} & \multicolumn{1}{c|}{\ensuremath{a_{2}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{A}_{2,2}=+1} & \ensuremath{p_{2,2}} & \ensuremath{a_{2}-p_{2,2}} & \multicolumn{1}{c|}{\ensuremath{a_{2}}}\tabularnewline\multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{2,1}=-1}} & \ensuremath{b_{1}-p_{2,1}} & \ensuremath{1-a_{2}-b_{1}+p_{2,1}} & \multicolumn{1}{c|}{\ensuremath{1-a_{2}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{A}_{2,2}=-1} & \ensuremath{b_{2}-p_{2,2}} & \ensuremath{1-a_{2}-b_{2}+p_{2,2}} & \multicolumn{1}{c|}{\ensuremath{1-a_{2}}}\tabularnewline\cline{1-4} \cline{6-9} & \ensuremath{b_{1}} & \ensuremath{1-b_{1}} & & & & \ensuremath{b_{2}} & \ensuremath{1-b_{2}} & \tabularnewline\cline{2-3} \cline{7-8} \end{tabular}\label{eq:obs}$$ are compatible with the connections $$\begin{tabular}{c|cc|ccc|cc|cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc} \cline{2-3} \cline{7-8} & \ensuremath{\mathbf{A}_{1,2}=+1} & \ensuremath{\mathbf{A}{}_{1,2}=-1} & & \quad{} & & \ensuremath{\mathbf{B}_{2,1}=+1} & \ensuremath{\mathbf{B}_{2,1}=-1} & & \quad{}\quad{}\tabularnewline\cline{1-4} \cline{6-9} \multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{1,1}=+1}} & \ensuremath{a_{1}-\alpha_{1}} & \ensuremath{\alpha_{1}} & \multicolumn{1}{c|}{\ensuremath{a_{1}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{B}_{1,1}=+1} & \ensuremath{b_{1}-\beta_{1}} & \ensuremath{\beta_{1}} & \multicolumn{1}{c|}{\ensuremath{b_{1}}} & \tabularnewline\multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{1,1}=-1}} & \ensuremath{\alpha_{1}} & \ensuremath{1-a_{1}-\alpha_{1}} & \multicolumn{1}{c|}{\ensuremath{1-a_{1}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{B}_{1,1}=-1} & \ensuremath{\beta_{1}} & \ensuremath{1-b_{1}-\beta_{1}} & \multicolumn{1}{c|}{\ensuremath{1-b_{1}}} & \tabularnewline\cline{1-4} \cline{6-9} & \ensuremath{a_{1}} & \ensuremath{1-a_{1}} & & & & \ensuremath{b_{1}} & \ensuremath{1-b_{1}} & & \tabularnewline\cline{2-3} \cline{7-8} \multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & & & \multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & & \tabularnewline\cline{2-3} \cline{7-8} & \ensuremath{\mathbf{A}_{2,2}=+1} & \ensuremath{\mathbf{A}{}_{2,2}=-1} & & & & \ensuremath{\mathbf{B}_{2,2}=+1} & \ensuremath{\mathbf{B}_{2,2}=-1} & & \tabularnewline\cline{1-4} \cline{6-9} \multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{2,1}=+1}} & \ensuremath{a_{2}-\alpha_{2}} & \ensuremath{\alpha_{2}} & \multicolumn{1}{c|}{\ensuremath{a_{2}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{B}_{1,2}=+1} & \ensuremath{b_{2}-\beta_{2}} & \ensuremath{\beta_{2}} & \multicolumn{1}{c|}{\ensuremath{b_{2}}} & \tabularnewline\multicolumn{1}{|c|}{\ensuremath{\mathbf{A}_{2,1}=-1}} & \ensuremath{\alpha_{2}} & \ensuremath{1-a_{2}-\alpha_{2}} & \multicolumn{1}{c|}{\ensuremath{1-a_{2}}} & \multicolumn{1}{c|}{} & \ensuremath{\mathbf{B}_{1,2}=-1} & \ensuremath{\beta_{2}} & \ensuremath{1-b_{2}-\beta_{2}} & \multicolumn{1}{c|}{\ensuremath{1-b_{2}}} & \tabularnewline\cline{1-4} \cline{6-9} & \ensuremath{a_{2}} & \ensuremath{1-a_{2}} & & & & \ensuremath{b_{2}} & \ensuremath{1-b_{2}} & & \tabularnewline\cline{2-3} \cline{7-8} \end{tabular}\label{eq:conn}$$ if and only if $$\begin{array}{l} s_{0}\!\left(\left\langle \mathbf{A}_{1,1}\mathbf{B}_{1,1}\right\rangle \!,\left\langle \mathbf{A}_{1,2}\mathbf{B}_{1,2}\right\rangle \!,\left\langle \mathbf{A}_{2,1}\mathbf{B}_{2,1}\right\rangle \!,\left\langle \mathbf{A}_{2,2}\mathbf{B}_{2,2}\right\rangle \right)\\ \le6-s_{1}\!\left(\left\langle \mathbf{A}_{1,1}\mathbf{A}_{1,2}\right\rangle \!,\left\langle \mathbf{A}_{2,1}\mathbf{A}_{2,2}\right\rangle \!,\left\langle \mathbf{B}_{1,1}\mathbf{B}_{2,1}\right\rangle \!,\left\langle \mathbf{B}_{1,2}\mathbf{B}_{2,2}\right\rangle \right), \end{array}\label{eq:CbD-s0}$$ $$\begin{array}{l} s_{1}\!\left(\left\langle \mathbf{A}_{1,1}\mathbf{B}_{1,1}\right\rangle \!,\left\langle \mathbf{A}_{1,2}\mathbf{B}_{1,2}\right\rangle \!,\left\langle \mathbf{A}_{2,1}\mathbf{B}_{2,1}\right\rangle \!,\left\langle \mathbf{A}_{2,2}\mathbf{B}_{2,2}\right\rangle \right)\\ \le6-s_{0}\!\left(\left\langle \mathbf{A}_{1,1}\mathbf{A}_{1,2}\right\rangle \!,\left\langle \mathbf{A}_{2,1}\mathbf{A}_{2,2}\right\rangle \!,\left\langle \mathbf{B}_{1,1}\mathbf{B}_{2,1}\right\rangle \!,\left\langle \mathbf{B}_{1,2}\mathbf{B}_{2,2}\right\rangle \right). \end{array}\label{eq:CbD-s1}$$ where $$\begin{aligned} s_{0}(x_{1},\dots,x_{n}) & =\max\{\pm x_{1}\pm\dots\pm x_{n}:\text{even \# of \ensuremath{-}'s}\},\\ s_{1}(x_{1},\dots,x_{n}) & =\max\{\pm x_{1}\pm\dots\pm x_{n}:\text{odd \# of \ensuremath{-}'s}\},\end{aligned}$$ and where we use the parameterization by the 12 expectation variables defined as $$\left\langle \mathbf{A}_{i,j}\mathbf{B}_{i,j}\right\rangle =\left(4p_{ij}-1\right)-\left(2a_{i}-1\right)-\left(2b_{j}-1\right),\label{eq:ABcorr}$$ $$\left\langle \mathbf{A}_{i,1}\mathbf{A}_{i,2}\right\rangle =1-4\alpha_{i}=1-2\Pr\left[\mathbf{A}_{i,1}\ne\mathbf{A}_{i,2}\right],\label{eq:Acorr}$$ $$\left\langle \mathbf{B}_{1,j}\mathbf{B}_{2,j}\right\rangle =1-4\beta_{j}=1-2\Pr\left[\mathbf{B}_{1,j}\ne\mathbf{B}_{2,j}\right],\label{eq:Bcorr}$$ $$\left\langle \mathbf{A}_{i}\right\rangle =2a_{i}-1,\label{eq:Amarginal}$$ $$\left\langle \mathbf{B}_{j}\right\rangle =2b_{j}-1,\label{eq:Bmarginal}$$ for $i,j\in\text{\{1,2\}}.$ Writing the inequalities and in terms of these expectations rather than in terms of probabilities is the most economic way of presenting the 128 non-trivial inequalities of the system, as the marginal probabilities $a_{1},a_{2},b_{1},b_{2}$ (or expectations $\left\langle \mathbf{A}_{1}\right\rangle ,\left\langle \mathbf{A}_{2}\right\rangle ,\left\langle \mathbf{B}_{1}\right\rangle ,\left\langle \mathbf{B}_{2}\right\rangle $) vanish in this form. However, it should be noted that in addition to these 128 inequalities, the form of the observed distributions and connections itself imposes further 28 trivial constraints on the 12 expectation variables of the system: the probabilities within each $2\times2$ matrix in and should be nonnegative and sum to one. 16 of these trivial constraints pertain to the observed distributions and 12 to the connections. In terms of the expectations, these trivial constraints correspond to $$-1+|\left\langle \mathbf{A}\right\rangle +\left\langle \mathbf{B}\right\rangle |\le\left\langle \mathbf{A}\mathbf{B}\right\rangle \le1-|\left\langle \mathbf{A}\right\rangle -\left\langle \mathbf{B}\right\rangle |,\label{eq:trivial}$$ for given marginals for each pair $(\mathbf{A},\mathbf{B})$ of random variables in –. This expands to four inequalities for each of the observed distributions and to three inequalities for each of the connections (the two upper bounds in coincide when $\left\langle \mathbf{A}\right\rangle =\left\langle \mathbf{B}\right\rangle $). Although these trivial constraints can usually be assumed implicitly, it is important to keep them explicitly in the system for the next step. Adding the equation $$\begin{aligned} \Delta= & \Pr\left[\mathbf{A}_{1,1}\ne\mathbf{A}_{1,2}\right]+\Pr\left[\mathbf{A}_{2,1}\ne\mathbf{A}_{2,2}\right]\\ & +\Pr\left[\mathbf{B}_{1,1}\ne\mathbf{B}_{2,1}\right]+\Pr\left[\mathbf{B}_{1,2}\ne\mathbf{B}_{2,2}\right]\\ = & 2-\frac{1}{2}\left(\left\langle \mathbf{A}_{1,1}\mathbf{A}_{1,2}\right\rangle +\left\langle \mathbf{A}_{2,1}\mathbf{A}_{2,2}\right\rangle \right.\\ & +\left.\left\langle \mathbf{B}_{1,1}\mathbf{B}_{2,1}\right\rangle +\left\langle \mathbf{B}_{1,2}\mathbf{B}_{2,2}\right\rangle \right)\end{aligned}$$ to the system and then eliminating the connection correlations $\left\langle \mathbf{A}_{1,1}\mathbf{A}_{1,2}\right\rangle $, $\left\langle \mathbf{A}_{2,1}\mathbf{A}_{2,2}\right\rangle $, $\left\langle \mathbf{B}_{1,1}\mathbf{B}_{2,1}\right\rangle $, $\left\langle \mathbf{B}_{1,2}\mathbf{B}_{2,2}\right\rangle $ from the system using the Fourier–Motzkin elimination algorithm, we obtain the system $$\begin{aligned} & -1+\frac{1}{2}S_{CHSH}\le\Delta\le4-\left[-1+\frac{1}{2}S_{CHSH}\right],\label{eq:delta1}\\ & 0\le\Delta\le4-\left(\left|\left\langle \mathbf{A}_{1}\right\rangle \right|+\left|\left\langle \mathbf{A}_{2}\right\rangle \right|+\left|\left\langle \mathbf{B}_{1}\right\rangle \right|+\left|\left\langle \mathbf{B}_{2}\right\rangle \right|\right),\label{eq:delta2}\end{aligned}$$ where we denote $$\begin{array}{l} S_{CHSH}=s_{1}\big(\left\langle \mathbf{A}_{1,1}\mathbf{B}_{1,1}\right\rangle ,\left\langle \mathbf{A}_{1,2}\mathbf{B}_{1,2}\right\rangle ,\\ \phantom{S_{CHSH}=s_{1}\big(}\left\langle \mathbf{A}_{2,1}\mathbf{B}_{2,1}\right\rangle ,\left\langle \mathbf{A}_{2,2}\mathbf{B}_{2,2}\right\rangle \big) \end{array}$$ as in the main text. This means that $\Delta$ is compatible with the given observed probabilities if and only if the above inequalities are satisfied. Since the set of possible values of $\Delta$ constrained by (\[eq:delta1\]) and (\[eq:delta2\]) is known to be nonempty, it follows that the minimum value of $\Delta$ is always given by $$\Delta_{\min}=\max\left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} .$$ EPR-Bell: Negative probabilities -------------------------------- The analogous result for the negative probabilities approach is that the observable distributions are obtained as the marginals of some negative probability joint of $\mathbf{A}_{1}=\mathbf{A}_{1,1}=\mathbf{A}_{1,2},$ $\mathbf{A}_{2}=\mathbf{A}_{2,1}=\mathbf{A}_{2,2}$, $\mathbf{B}_{1}=\mathbf{B}_{1,1}=\mathbf{B}_{2,1},$ and $\mathbf{B}_{2}=\mathbf{B}_{1,2}=\mathbf{B}_{2,2}$ given by $$\begin{aligned} & \Pr\left[\mathbf{A}_{1}=a'_{1},\mathbf{\, A}_{2}=a'_{2},\,\mathbf{B}_{1}=b'_{1},\,\mathbf{B}_{2}=b'_{2}\right]\\ & \,=p^{+}(a'_{1},a'_{2},b'_{1},b'_{2})-p^{-}(a'_{1},a'_{2},b'_{1},b'_{2}),\end{aligned}$$ $a'_{1},a'_{2},b'_{1},b'_{2}\in\{1,-1\}$, for some nonnegative functions $p^{+}$ and $p^{-}$ having a total probability mass value $$M=\sum_{a'_{1},a'_{2},b'_{1},b'_{2}}p^{+}(a'_{1},a'_{2},b'_{1},b'_{2})+p^{-}(a'_{1},a'_{2},b'_{1},b'_{2})$$ if and only if $M\ge1+\Gamma_{\min}$, where $$\Gamma_{\min}=\max\left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} .$$ The computations are similar to those of the CbD approach, but there are two general differences. First, in the CbD approach, the convex range of the possible observed and connection expectations – over the convex polytope of all possible joints is obtained by looking at these expectations at the $2^{8}$ vertices defining the polytope of all joints and then applying a computer algorithm to find the set of inequalities delineating the extreme values of the expectations at these vertices. However, in the negative probabilities approach, the joint is represented by the $2^{4}$ differences of the positive and negative components of the distribution and so, although these $2\cdot2^{4}$ components are nonnegative as in the CbD approach, they do not need to sum to one. Hence, the joint is represented by a convex cone rather than a bounded polytope. Still, a convex cone is a special case of a general polytope and can be handled by the same algorithms that we have used in the CbD approach. Second, we do not need to apply the Fourier–Motzkin elimination algorithm here as we have defined $M$ directly by the representation of the joint so there are no extra variables we would need to eliminate. This difference, however, is not really a difference between the two approaches, as we could have done the same in the CbD approach as well: we could have defined $\text{\ensuremath{\Delta}}$ directly based on the joint of all eight variables without explicitly defining the connection correlations –, and then we would have obtained the result directly from the half-space representation, as we do in the negative probabilities approach. Leggett–Garg: Contextuality-by-Default -------------------------------------- The results for Leggett–Garg $\mathbf{Q}_{1},\mathbf{Q}_{2},\mathbf{Q}_{3}$ can be obtained in the same way as for the EPR-Bell systems. In the CbD approach, the observed correlations $\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle $, $\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\rangle $, $\left\langle \mathbf{Q}_{2,3}\mathbf{Q}_{3,2}\right\rangle $, $\left\langle \mathbf{Q}_{1}\right\rangle =\left\langle \mathbf{Q}_{1,2}\right\rangle =\left\langle \mathbf{Q}_{1,3}\right\rangle $, $\left\langle \mathbf{Q}_{2}\right\rangle =\left\langle \mathbf{Q}_{2,1}\right\rangle =\left\langle \mathbf{Q}_{3,2}\right\rangle $, $\left\langle \mathbf{Q}_{3}\right\rangle =\left\langle \mathbf{Q}_{3,1}\right\rangle =\left\langle \mathbf{Q}_{3,2}\right\rangle $ are consistent with the connection correlations $\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{1,3}\right\rangle $, $\left\langle \mathbf{Q}_{2,1}\mathbf{Q}_{2,3}\right\rangle $, $\left\langle \mathbf{Q}_{3,1}\mathbf{Q}_{3,2}\right\rangle $ if and only if these connection correlations are realizable with the given marginals (i.e., each correlation $\left\langle \mathbf{AB}\right\rangle $ has to satisfy $-1+|\left\langle \mathbf{A}\right\rangle +\left\langle \mathbf{B}\right\rangle |\le\left\langle \mathbf{A}\mathbf{B}\right\rangle \le1-|\left\langle \mathbf{A}\right\rangle -\left\langle \mathbf{B}\right\rangle |$ as discussed in the EPR-Bell case above) and satisfy $$\begin{array}{l} s_{0}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle ,\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\rangle ,\left\langle \mathbf{Q}_{2,3}\mathbf{Q}_{3,2}\right\rangle \right)\\ +s_{1}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{1,3}\right\rangle ,\left\langle \mathbf{Q}_{2,1}\mathbf{Q}_{2,3}\right\rangle ,\left\langle \mathbf{Q}_{3,1}\mathbf{Q}_{3,2}\right\rangle \right)\le4, \end{array}$$ $$\begin{array}{l} s_{1}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle ,\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\rangle ,\left\langle \mathbf{Q}_{2,3}\mathbf{Q}_{3,2}\right\rangle \right)\\ +s_{0}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{1,3}\right\rangle ,\left\langle \mathbf{Q}_{2,1}\mathbf{Q}_{2,3}\right\rangle ,\left\langle \mathbf{Q}_{3,1}\mathbf{Q}_{3,2}\right\rangle \right)\le4 \end{array}$$ These two inequalities expand to 32 linear inequalities and there are 21 trivial constraints. Denoting $$\begin{aligned} \Delta & =\Pr\left[\mathbf{Q}_{1,2}\!\ne\!\mathbf{Q}_{1,3}\right]+\Pr\left[\mathbf{Q}_{2,1}\!\ne\!\mathbf{Q}_{2,3}\right]+\Pr\left[\mathbf{Q}_{3,1}\!\ne\!\mathbf{Q}_{3,2}\right]\\ & =\frac{3}{2}-\frac{1}{2}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{1,3}\right\rangle +\left\langle \mathbf{Q}_{2,1}\mathbf{Q}_{2,3}\right\rangle +\left\langle \mathbf{Q}_{3,1}\mathbf{Q}_{3,2}\right\rangle \right)\end{aligned}$$ and eliminating the connection correlations from the system using the Fourier–Motzkin algorithm, we obtain the system $$\begin{aligned} -\frac{1}{2}+\frac{1}{2}S_{LG}\le\Delta & \le3-\left[-\frac{1}{2}+\frac{1}{2}S_{LG}^{0}\right],\\ 0\le\Delta & \le3-\left|\left\langle \mathbf{Q}_{1}\right\rangle \right|-\left|\left\langle \mathbf{Q}_{2}\right\rangle \right|-\left|\left\langle \mathbf{Q}_{3}\right\rangle \right|,\end{aligned}$$ where we denote $$\begin{aligned} S_{LG} & =s_{1}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle ,\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\rangle ,\left\langle \mathbf{Q}_{2,3}\mathbf{Q}_{3,2}\right\rangle \right),\\ S_{LG}^{0} & =s_{0}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle ,\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\rangle ,\left\langle \mathbf{Q}_{2,3}\mathbf{Q}_{3,2}\right\rangle \right).\end{aligned}$$ That is, $\Delta$ is consistent with the observed probabilities if and only if the above inequalities are satisfied. It follows that the minimum value of $\Delta$ is given by $$\Delta_{\min}=\max\left\{ 0,\frac{1}{2}S_{LG}-\frac{1}{2}\right\} .$$ Leggett–Garg: Negative probabilities ------------------------------------ With the same additional comments as in the negative probability calculations for the EPR-Bell case, our calculations show that the observable probabilities $\Pr\left[\mathbf{Q}_{12}=q_{1},\mathbf{Q}_{21}=q_{2}\right]$, $\Pr\left[\mathbf{Q}_{13}=q_{1},\mathbf{Q}_{31}=q_{3}\right]$, $\Pr\left[\mathbf{Q}_{23}=q_{2},\mathbf{Q}_{32}=q_{3}\right]$, $q_{1},q_{2},q_{3}\in\{0,1\},$ can be obtained as the marginals of a negative probability jpd of $\mathbf{Q}_{1}=\mathbf{Q}_{1,2}=\mathbf{Q}_{1,3}$, $\mathbf{Q}_{2}=\mathbf{Q}_{2,1}=\mathbf{Q}_{3,2}$, and $\mathbf{Q}_{3}=\mathbf{Q}_{3,1}=\mathbf{Q}_{3,2}$ with the total probability mass of $M$ if and only if $M\ge1+\Gamma_{\min}$, where $$\Gamma_{\min}=\max\left\{ 0,\frac{1}{2}S_{LG}-\frac{1}{2}\right\} .$$ [10]{} Abramsky, S., Brandenburger, A.: The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13, 113036 (2011). Abramsky, S., Brandenburger, A.: An operational interpretation of negative probabilities and no-signalling models. In Horizons of the Mind. A Tribute to Prakash Panangaden, Lecture Notes in Computer Science No. 8464, edited by F. van Breugel, E. Kashefi, C. Palamidessi, and J. Rutten (Springer Int. Pub., 2014) pp. 5975. Al-Safi, S., Short, A.: Simulating all nonsignaling correlations via classical or quantum theory with negative probabilities. Phys. Rev. Lett. 111, 170403 (2013). Bacciagaluppi, G.: Leggett-Garg inequalities, pilot waves and contextuality. Int. J. Quant. Found., 1(1), 1-17 (2015). Chaves, R., Fritz, T.: Entropic approach to local realism and noncontextuality. Phys. Rev. A 85, 032113 (2012). de Barros, J.A.: Decision making for inconsistent expert judgments using negative probabilities. Lecture Notes in Computer Science 8369, 257 (2014). de Barros, J.A., Suppes, P.: Inequalities for dealing with detector inefficiencies in Greenberger-Horne-Zeilinger type experiments. Phys. Rev. Lett. 84, 793 (2000). de Barros, J.A., Suppes, P.: Probabilistic results for six detectors in a three-particle GHZ experiment. In Chance in Physics, Lecture Notes in Physics, Vol. 574, edited by J. Bricmont, D. Durr, M. C. Galavotti, G. Ghirardi, F. Petruccione, and N. Zanghi (2001) p. 213. de Barros, J. A., Oas, G., Suppes, P.: Negative probabilities and counterfactual reasoning on the double-slit experiment. In J.-Y. Beziau, D. Krause, & J. B. Arenhart (Eds.), Conceptual Clarification: Tributes to Patrick Suppes (1992-2014). London: College Publications (2015). de Barros, J.A., Oas, G.: Negative probabilities and counter-factual reasoning in quantum cognition. Phys. Scripta T163, 014008 (2014). Dzhafarov, E., Kujala, J.: All-Possible-Couplings approach to measuring probabilistic context. PLoS ONE 8, e61712 (2013). Dzhafarov, E., Kujala, J.: No-Forcing and No-Matching theorems for classical probability applied to quantum mechanics. Found. Phys., 44, 248-265 (2014). Dzhafarov, E., Kujala, J.: Embedding quantum into classical: Contextualization vs conditionalization. PLoS One 9, e92818 (2014). Dzhafarov, E., Kujala, J.: contextuality is about identity of random variables. Phys. Scripta T163, 014009 (2014). Dzhafarov, E., Kujala, J.: random variables recorded under mutually exclusive conditions: Contextuality-by-Default. In H. Liljenström (Ed.) Advances in Cognitive Neurodynamics IV (pp. 405-410) (2015). Dzhafarov, E., Kujala, J.: A qualified kolmogorovian account of probabilistic contextuality. Lecture Notes in Computer Science 8369, 201 (2014). Dzhafarov, E., Kujala, J.: probabilistic contextuality in EPR/Bohm-type systems with signaling allowed. arXiv:1406.0243 [\[]{}quant-ph, q-bio[\]]{} (2014). Dzhafarov, E., Kujala, J.: generalizing Bell-type and Leggett-Garg-type inequalities to systems with signaling. arXiv:1407.2886 [\[]{}quant-ph[\]]{} (2014). Dzhafarov, E., Kujala, J., Larsson, J.-Å.: contextuality in three types of quantum-mechanical systems. Found. Phys. DOI 10.1007/s10701-015-9882-9 (2014). Dzhafarov, E., Kujala, J., Larsson, J.-Å.: Contextuality-by-Default: A brief overview of ideas, concepts, and terminology. arXiv:1504.00530 [\[]{}quant-ph[\]]{} (2015). Fine, A.: Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett. 48, 291 (1982). Grudka, A., Horodecki, K., Horodecki, M., Horodecki, P., Horodecki, R., Joshi, P., Kłobus, W., Wójcik, A.: Quantifying contextuality. Phys. Rev. Lett. 112, 120401 (2014). Halmos, P.: Measure Theory. Springer-Verlag, New York, NY (1974). Kleinmann, M., Gühne, O., Portillo, J., Larsson, J.-Å., Cabello, A.: Memory cost of quantum contextuality. New J. Phys. 13, 113011 (2011). Khrennikov, A. (2005). The principle of supplementarity: A contextual probabilistic viewpoint to complementarity, the interference of probabilities, and the incompatibility of variables in quantum mechanics. Found. Phys. 35, 1655-1693. Khrennikov, A. (2008). Bell-Boole inequality: Nonlocality or probabilistic incompatibility of random variables? Entropy 10, 19-32. Khrennikov, A. (2009). Contextual approach to quantum formalism. Springer: Berlin. Kurzyński, P., Ramanathan, R., Kaszlikowski, D.: entropic test of quantum contextuality. Phys. Rev. Lett. 109, 020404 (2012). Kujala, J., Dzhafarov, E., J.-Å. Larsson, J.-Å.: Necessary and Sufficient Conditions for Maximal Noncontextuality in a Broad Class of Quantum Mechanical Systems. arXiv:1412.4724 [\[]{}quant-ph[\]]{} (2014). Kujala, J., Dzhafarov, E. Proof of a conjecture on contextuality in cyclic systems with binary variables. arXiv:1503.02181 [\[]{}quant-ph[\]]{} (2015). Larsson, J.-Å.: A Kochen-Specker inequality. EPL (Europhysics Letters) 58, 799 (2002). Leggett, A., Garg, A.: Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett. 54, 857 (1985). Oas, G., de Barros, J.A. Carvalhaes, C.: Exploring non-signalling polytopes with negative probability. Phys. Scripta T163, 014034 (2014). Simon, C., Brukner, Č., Zeilinger, A.: Hidden-variable theorems for real experiments. Phys. Rev. Lett., 86(20):44274430 (2001). Specker, E.P.: The logic of propositions which are not simultaneously decidable. In The Logico-Algebraic Approach to Quantum Mechanics, The University of Western Ontario Series in Philosophy of Science No. 5a, edited by C. A. Hooker (Springer Netherlands, 1975) pp. 135140. Spekkens, R.W.: Negativity and contextuality are equivalent notions of nonclassicality. Phys. Rev. Lett. 101, 020401 (2008). Suppes, P., Zanotti, M.: When are probabilistic explanations possible? Synthese 48, 191 (1981). Svozil, K.: How much contextuality? Natural Computing 11, 261 (2012). Winter, A.: What does an experimental test of quantum contextuality prove or disprove? J. Phys. A: Math. Theor., 47, 424031 (2014). [^1]: This example has the same structure as the Leggett-Garg system [@leggett_quantum_1985]. A special version of it was examined by Suppes and Zanotti [@suppes_when_1981], and also by Specker [@specker_logic_1975]. [^2]: No-signaling condition is a fundamental limitation of any approach with noncontextually labeled random variables, including NP. [^3]: This formulation is predicated on no-signaling, which we assume throughout this paper. CbD has been generalized to situations when this condition is violated [@dzhafarov_probabilistic_2014; @bacciagaluppi_leggett-garg_2014; @dzhafarov_generalizing_2014; @KDL2014; @DKL2014; @KDconjecture; @DKL_overview].

--- abstract: 'The Isolated Horizons (IH) formalism, together with a simple phenomenological model for colored black holes has been used to predict non-trivial formulae that relate the ADM mass of the solitons and hairy Black Holes of Gravity-Matter system on the one hand, and several horizon properties of the black holes in the other. In this article, the IH formalism is tested numerically for spherically symmetric solutions to an Einstein-Higgs system where hairy black holes were recently found to exist. It is shown that the mass formulae still hold and that, by appropriately extending the current model, one can account for the behavior of the horizon properties of these new solutions. An empirical formula that approximates the ADM mass of hairy solutions is put forward, and some of its properties are analyzed.' author: - Alejandro Corichi - Ulises Nucamendi - Marcelo Salgado date: 'November 8, 2005' title: | Scalar hairy black holes and scalarons\ in the isolated horizons formalism --- Introduction ============ In recent years, the introduction of the Isolated Horizon (IH) formalism [@PRL; @Ashtekar:2000yj; @AFK] has proved to be useful to gain insight into the static sector of theories admitting “hair” [@CS; @ACS]. Firstly, it has been found that the Horizon Mass of the black hole (BH), a notion constructed out of purely quasi-local quantities, is related in a simple way to the ADM masses of both the colored black hole and the solitons of the theory [@CS]. Second, a simple model for colored black holes as bound states of regular black holes and solitons has allowed to provide heuristic explanations for the behavior of horizon quantities of those black holes [@ACS]. Third, the formalism is appropriate for the formulation of [*uniqueness conjectures*]{} for the existence of unique stationary solutions in terms of horizon “charges" [@CS]. Finally, the combination of the Mass formula, together with the fact that in theories such as Einstein-Yang-Mills-Higgs (EYMH) different “branches” of static solutions merge, has allowed to have a formula for the difference of soliton masses in terms of black hole quantities [@Kleihaus:2000kv; @ACS; @CNS]. Many of these predictions have been confirmed in more general situations and for other matter couplings [@horizon; @horizon2]. For a recent review on IH (including hair) see [@AK2], and for a review of hairy black holes see [@volkov]. In this article, we explore further the consequences of the IH formalism in the static sector of the theory. In particular, we explore the behavior of a recently found family of hairy static spherically symmetric (SSS) solutions to the Einstein-Higgs system [@SN], where the scalar potential is allowed to be negative and therefore, the existing no-hair theorems [@hair] do not apply. In the standard treatment of stationary black holes with killing horizons, one is always restoring to several concepts that use asymptotic information very strongly [@HeuslerB]. On the other hand, the IH formalism only uses quasi-local information defined on the horizon, allowing it to prove very general results involving only these quasi-local quantities. The IH formalism has proved to be generalizable, in the scalar sector, even to the non-minimal coupling regime, where the energy conditions required for the consistency of the formalism are much weaker [@ACS2]. In the present paper we shall restrict our attention to the minimally coupled case, and for a particular form of the scalar potential for which static solutions are known to exist [@SN]. We will study the one parameter family of solutions (that could be labelled by its geometric radius $r_\Delta$) and compare its properties with those of hairy black holes in other theories, such as EYM, where the phenomenological predictions of the IH formalism have been shown to work very well [@ACS; @horizon]. As we will show, we find that the mass formulae relating BH and soliton ADM masses works also well, but the model of a hairy black hole as a bound state of a soliton and a bare black holes exhibits some new unexpected features. As we shall see, one needs to slightly modify the model from its original formulation in Ref.[@ACS]. Once this modification is made the model can again explain all the qualitative behavior of the hairy BH solutions. The structure of the paper is as follows: In Sec. \[sec:2\] we review the consequences of the IH formalism for hairy solitons and BH solutions. In Sec. \[sec:3\] we review the SSS found recently in the Einstein-Higgs sector. Section \[sec:4\] is the main section of the paper. In it, we show the numerical evidence for the mass formulae and the phenomenological predictions of the model. Unlike the EYM case, there are some unexpected features, such as the binding energy becoming positive. We then propose a modification of the formalism to deal with such situations. We show that with these modifications, the model can still account for the geometrical phenomena found in several theories. In Sec. \[sec:5\] we explore the situation of the collapse of a hairy black hole and use the model to put bound on the total possible energy to be radiated. These results should be of some relevance to full dynamical numerical evolutions of such black holes. In Sec. \[sec:6\] we propose an empirical formula for the horizon and ADM masses of scalar hairy black holes that can also be applied to the EYM case. Finally, we end with a discussion in Sec. \[sec:7\]. Consequences of the Isolated horizons formalism {#sec:2} =============================================== In recent years, a new framework tailored to consider situations in which the black hole is in equilibrium (“nothing falls in”), but which allows for the exterior region to be dynamical, has been developed. This [*Isolated Horizons*]{} (IH) formalism is now in the position of serving as starting point for several applications. Notably, for the extraction of physical quantities in numerical relativity and also for quantum entropy calculations [@PRL; @AK2]. The basic idea is to consider space-times with an interior boundary (to represent the horizon), satisfying quasi-local boundary conditions ensuring that the horizon remains ‘isolated’. Although the boundary conditions are motivated by geometric considerations, they lead to a well defined action principle and Hamiltonian framework. Furthermore, the boundary conditions imply that certain ‘quasi-local charges’, defined at the horizon, remain constant ‘in time’, and can thus be regarded as the analogous of the global charges defined at infinity in the asymptotically flat context. The isolated horizons Hamiltonian framework allows to define the notion of [*Horizon Mass*]{} $M_\Delta$, as a function of the ‘horizon charges’ (hereafter, the subscript “$\Delta$" stands for a quantity at the horizon). In the Einstein-Maxwell and Einstein-Maxwell-Dilaton systems considered originally [@Ashtekar:2000yj], the horizon mass satisfies a Smarr-type formula and a generalized first law in terms of quantities defined exclusively at the horizon (i.e. without any reference to infinity). The introduction of non-linear matter fields like the Yang-Mills field has brought unexpected subtleties to the formalism [@CS]. However, one still is in the position of defining a Horizon Mass, and furthermore, this Horizon Mass satisfies a first law. An isolated horizon is a non-expanding null surface generated by a (null) vector field $l^a$. The IH boundary conditions imply that the acceleration $\kappa$ of $l^a$ ($l^a\nabla_al^b=\kappa l^b$) is constant on the horizon $\Delta$. However, the precise value it takes on each point of phase space (PS) is not determined a-priori. On the other hand, it is known that for each vector field $t^a_o$ on space-time, the induced vector field $X_{t_{o}}$ on phase space is Hamiltonian if and only if there exists a function $E_{t_{o}}$ such that $\delta E_{t_{o}}=\Omega (\delta,X_{t_{o}})$, [*for any vector field $\delta$ on PS*]{}. This condition can be re-written as, $ \delta E_{t_{o}}=\frac{\kappa_{t_{o}}}{8\pi G}\,\delta a_{\Delta} + {\rm work\;\; terms}$. Thus, the first law arises as a necessary and sufficient condition for the consistency of the Hamiltonian formulation. Thus, the allowed vector fields $t^a$ will be those for which the first law holds. Note that there are as many ‘first laws’ as allowed vector fields $l^a\widehat{=} \,t^a$ on the horizon. However, one would like to have a [*Physical First Law*]{}, where the Hamiltonian $E_{t_{o}}$ be identified with the ‘physical mass’ $M_{\Delta}$ of the horizon. This amounts to finding the ‘right $\kappa$’. This ‘normalization problem’ can be easily overcome in the EM system [@Ashtekar:2000yj]. In this case, one chooses the function $\kappa=\kappa(a_\Delta, Q_\Delta)$ as the corresponding function for the [*static*]{} solution with charges $(a_\Delta, Q_\Delta)$. However, for the EYM system, this procedure is not as straightforward. A consistent viewpoint is to abandon the notion of a globally defined horizon mass on Phase Space, and to define, for each value of $n=n_o$ (which labels different branches of the solutions), a canonical normalization $t^a_{n_o}$ that yields the Horizon Mass $M^{(n_o)}_{\Delta}$ for the $n_o$ branch [@CS; @AFK]. The horizon mass takes the form (from now on we shall omit the $n_0$ label), $$M_{\Delta}(r_\Delta)=\frac{1}{2G_0}\int_0^{r_\Delta} \beta(r) \, \d r\, ,$$ with $r_\Delta$ the horizon radius. Here $\beta(r_{\Delta})$ is related to the surface gravity as follows $\beta(r_{\Delta})=2r_{\Delta}\kappa(r_{\Delta})$. Furthermore, one can relate the horizon mass $M_{\Delta}$ to the ADM mass of static black holes. Recall first that general Hamiltonian considerations imply that the total Hamiltonian, consisting of a term at infinity, the ADM mass, and a term at the horizon, the Horizon Mass, is constant on every connected component of static solutions (provided the evolution vector field $t_0^a$ agrees with the static Killing field everywhere on this connected component) [@Ashtekar:2000yj; @AFK]. In the Einstein-Yang-Mills case, since the Hamiltonian is constant on any branch, we can evaluate it at the solution with zero horizon area. This is just the soliton, for which the horizon area $a_\Delta$, and the horizon mass $M_{\Delta}$ vanish. Hence we have that $H^{(t_0)} = M_{\rm sol}$. Thus, we conclude [@CS]: $$\label{ymmass} M_{\rm sol} = M_{\rm ADM} - M_{\Delta}\, ,$$ Thus, the ADM mass contains two contributions, one attributed to the black hole horizon and the other to the outside ‘hair’, captured by the ‘solitonic residue’. The formula (\[ymmass\]), together with some energetic considerations [@ACS], lead to the model of a colored black hole as a bound state of an ordinary, ‘bare’, black hole and a ‘solitonic residue’, where the ADM mass of the colored black hole of radius $r_{\Delta}$ is given by the ADM mass of the soliton plus the horizon mass of the ‘bare’ black hole plus the binding energy: $$\label{ymbinding} M_{\rm ADM} = M_{\rm sol} + M_{\Delta}= M^0_\Delta + M_{\rm sol} + E_{\rm bind}\,\,\,,$$ with $E_{\rm bind}= M_{\Delta} - M^0_\Delta$. Simple considerations about the behavior of the ADM masses of the colored black holes and the solitons, together with some expectations of this model (such as demanding for a non-positive binding energy) give raise to several predictions about the behavior of the horizon parameters [@ACS]. Among the predictions, we have: i\) The absolute value of the binding energy decreases as $r_\Delta$ increases. ii\) $\beta(r_{\Delta})$, as a function of $r_\Delta$, is a positive function, bounded above by $\beta_{(0)}(r_{\Delta})=1$. iii\) The curve $\beta(r)$, as functions of $r$ intersect the $r=0$ axis at distinct points between $0$ and $1$, and never intersect. Finally, iv\) The curve for $\beta$, for large value of its argument, becomes asymptotically tangential to the curve $\beta_{(0)}(r_\Delta)=1$. One of the features of these solutions in, say, EYM is that there is no limit for the size of the black hole. That is, if we plot the ADM mass of the BH as function of the radius $r_\Delta$ we get an infinite number of curves, each of the intersecting the $r_{\Delta}=0$ line at the value of the soliton mass, and never intersecting each other. The purpose of this paper is to test the mass formula (\[ymmass\]), for the scalar hairy solutions found in Ref. [@SN] and also to confront the predictions i)-iv) (obtained for the colored EYM BH model [@ACS]), with the corresponding properties for the scalar hairy BH. In the next section we will review the scalar hairy solutions, and afterwards we shall study their horizon properties. Scalar solitons and black holes in Einstein-Higgs theory {#sec:3} ======================================================== Let us consider the theory of a scalar field minimally coupled to gravity described by the total action: $$S_{\rm tot}[g_{\mu\nu},\phi] = \int \sqrt{-g}\left\{ \frac{R}{16\pi} - \left[\frac{1}{2} (\nabla^{\beta}\phi)(\nabla_{\beta} \phi) + V(\phi) \right]\right\} {\rm d}^4 x \label{acttot}$$ (units where $G_0=c=1$ are employed). The field equations following from the variation of the action (\[acttot\]) are, $$\label{eqsmov} G_{\mu\nu} = 8\pi \left\{(\nabla_\mu\phi)\nabla_\nu\phi - g_{\mu\nu}\left[\frac{1}{2} (\nabla^{\beta}\phi) (\nabla_{\beta} \phi) + V(\phi)\right] \right\} \ ,$$ and, $$\begin{aligned} \Box \phi = {\frac{\partial V(\phi)}{\partial \phi}} \ . \label{eqssca}\end{aligned}$$ It is well known that asymptotically flat static spherically symmetric solutions representing black holes solutions to the Einstein-Higgs equations do not exist if the scalar matter satisfies the weak energy condition (WEC) due to the existence of the so called scalar no-hair theorems [@hair]. Recently, numerical evidence for the existence of asymptotically flat and static spherically symmetric solutions representing scalar hairy black holes (SHBH) and scalar solitons ([*scalarons*]{}; hereafter SS) have been found in theories represented by the action (\[acttot\]) and with a scalar potential non-positive-semidefinite [@SN] given by the asymmetric potential, $$\begin{aligned} \label{potential} V(\phi) &=& \frac{\lambda}{4}\left[ (\phi-a)^2 - \frac{4(\eta_1 + \eta_2)}{3} (\phi-a) + 2\eta_1\eta_2 \right] (\phi-a)^2 \,\,\,,\end{aligned}$$ where $\lambda$, $\eta_i$ and $a$ are constants. For this class of potential one can see that, for $\eta_1>2\eta_2>0$, $\phi=a$ corresponds to the local minimum, $\phi=a+\eta_1$ is a global minimum and $\phi=a+\eta_2$ is a local maximum (see Fig. 1). The key point in the shape of the potential, $V(\phi)$, for the asymptotically flat solutions to exist, is that the local minimum $V^{\rm local}_{\rm min}=V(a)$ is also a zero of $V(\phi)$ (to see [@SN] for an detailed analysis for the existence of these solutions). Moreover, $V(\phi)$ is not positive definite (we assume $\lambda>0$), which leads to a violation of the WEC and therefore the scalar no-hair theorems [@hair] can not be applied to this case. Qualitative shape of the scalar-field potential $V(\phi)$ as given by Eq. (\[potential\]) used to construct the asymptotically flat black hole and soliton solutions. \[fig0\] In order to describe the asymptotically flat SHBH and SS, we use a standard parametrization for the metric and the scalar field describing spherically symmetric and static spacetimes $$\begin{aligned} ds^2 &=& - \left(1-\frac{2m(r)}{r}\right) e^{2\delta(r)} dt^2 + \left(1-\frac{2m(r)}{r}\right)^{-1} dr^2 + r^2 d\Omega^2 \,\,\,\,\,, \label{metric} \\ \phi &=& \phi(r) \,\,\,\,\,, \label{scalar}\end{aligned}$$ For SHBH we demand regularity on the event horizon $r_{\Delta}$ which implies the conditions, $$m_{\Delta} = \frac{r_{\Delta}}{2}, \,\,\,\,\,\, \delta(r_{\Delta}) = \delta_{\Delta}, \,\,\,\,\,\, \phi(r_{\Delta}) = \phi_{\Delta}, \label{EHconditions1}$$ $$(\partial_{r}\phi)_{\Delta} = \frac{r_{\Delta}(\partial_{\phi}V)_{\Delta}}{(1 - 8\pi r_{\Delta}^2 V_{\Delta})} \,\,\,, \,\,\,\,\,\, (\partial_{r}m)_{\Delta} = 4\pi r_{\Delta}^2 V_{\Delta} \,. \label{EHconditions2}$$ For SS we impose regularity at the origin of coordinates $r=0$, $$m(0) = 0, \,\,\,\,\,\, \delta(0) = \delta_{0}, \,\,\,\,\,\, \phi(0) = \phi_{0},\,\,\,\,\,\, (\partial_{r}\phi)_{0} = 0 \,. \label{origencond}$$ where $\delta_{0}$ and $\phi_{0}$ are to be found such as to obtain the desired asymptotic conditions. In addition to the regularity conditions, we impose asymptotically flat conditions on the spacetime for SHBH and SS: $$m(\infty) = M_{\rm ADM}, \,\,\,\,\,\, \delta(\infty) = 0, \,\,\,\,\,\, \phi(\infty) = \phi_{\infty} \,\,\,\,\,\, . \label{asympcond}$$ Above, the value $\phi_{\infty}$ corresponds to the local minimum of $V(\phi)$. $M_{\rm ADM}$ is the ADM mass associated with a SHBH or SS configuration. For a given theory, the family of SHBH configurations is parametrized by the free parameter $A_{\Delta}$ which specifies the area of the black hole horizon. Therefore for SHBH, $M_{\rm ADM}=M_{\rm ADM}(A_{\Delta})$, or equivalently $M_{\rm ADM}=M_{\rm ADM}(r_{\Delta})$ since in our coordinates the horizon area $A_{\Delta}=4\pi r_{\Delta}^2$. The value $\phi_{\Delta}=\phi_{\Delta}(r_{\Delta})$ is a shooting parameter rather than an arbitrary boundary value which is determined so that the asymptotic flat conditions are satisfied. On the other hand, for SS the value $\phi_0$ is the shooting parameter, and the corresponding configuration is characterized by a unique $M_{\rm ADM}^{\rm sol}$. Finally, the surface gravity of a spherically symmetric static black hole can be calculated from the general expression of spacetimes admitting a Killing horizon [@HeuslerB]: $$\kappa = \left[ -\frac{1}{4}\nabla^2 \alpha \right]^{1/2}_{r=r_{\Delta}} \,\,\,\,\,\,, \label{defsg0}$$ where $\nabla^2$ stands for the Laplacian operator associated with the stationary metric and $\alpha= (\partial_t,\partial_t)$ is the norm of the time-like (static) Killing field which is null at the horizon. For the present case, $\alpha= g_{tt}= - \left(1-\frac{2m(r)}{r}\right) e^{2\delta(r)}$. From the above formula, one can obtain the following useful expression $$\kappa = \lim_{r \rightarrow r_{\Delta}} \left\{\frac{1}{2} \frac{\partial_r g_{tt}}{\sqrt{g_{tt}g_{rr}}} \right\} \,\,\,\,\,\, . \label{defsg}$$ For the election of the parametrization of the metric (\[metric\]) we have $$\kappa (r_{\Delta}) = \frac{1}{2r_{\Delta}} e^{\delta(r_{\Delta})} \left[ 1 - 2(\partial_{r}m)_{\Delta} \right] \,\,\,\,\,\, . \label{sg}$$ Introducing (\[EHconditions2\]) in (\[sg\]) we obtain the final expression for the surface gravity of the SHBH $$\kappa (r_{\Delta}) = \frac{1}{2r_{\Delta}} e^{\delta(r_{\Delta})} \left[ 1 - 8\pi r_{\Delta}^2 V_{\Delta} \right] \,\,\,\,\,\, . \label{sgSHBH}$$ In the next section, we shall analyze these solutions from the perspective of the IH formalism. Mass formulae {#sec:4} ============= The ADM mass (solid lines), the horizon BH mass (dash-dotted lines), and the mass of the Schwarzschild solution (dashed-lines) plotted as functions of $r_\Delta$ (first panel). The second panel depicts similar quantities using logarithmic scales to appreciate better their behavior for small $r_\Delta$. The soliton mass $M_{\rm sol}\approx 3.827$. \[masses\] $\beta(r_\Delta)=2\kappa\,r_\Delta$ is plotted as function of $r_\Delta$. Note that it approaches asymptotically the value $\approx 1.24$. Here $\beta(0)\approx 0.324$. \[beta\] Let us now turn to the straightforward application of the IH formalism mentioned in the Sec. II to the case of SHBH and SS in the Einstein-Higgs theory with action given by (\[acttot\]) and $V(\phi)$ by Eq. (\[potential\]) [@SN]. As in Ref. [@SN], we shall take the specific values $\eta_1=0.5$, $\eta_2=0.1$ and $a=0$; all the quantities (e.g., $M_{\rm ADM}$ and $r_\Delta$) have been rescaled as appropriate using $1/\sqrt{\lambda}$ as a length-unit. The first consequence coming from the IH formalism is that the horizon mass associated to the SHBH takes the form[^1], $$\label{MH} M_{\Delta}(r_\Delta)=\frac{1}{2}\int_0^{r_\Delta}\beta(r) \, \d r\, ,$$ where $\beta(r_{\Delta})=2r_{\Delta}\kappa(r_{\Delta})$ \[the value of $\kappa (r_{\Delta})$ is given by (\[sgSHBH\])\]. We have dropped the subindex $(n)$ in the expression (\[MH\]) because in the Einstein-Higgs system considered here there is only one branch of static spherically symmetric SHBH labelled by its horizon radius $r_\Delta$ (the corresponding scalar configurations do not have nodes). Additionally, there is another branch of static spherically symmetric BH given by the family of Schwarzschild BH’s labelled by its corresponding horizon radius $r_\Delta$ and with horizon mass $$\label{MHS} M_{\Delta}^{\rm Schwarz}(r_\Delta)= r_\Delta/2 \,\,\,\,.$$ The second consequence coming from the IH formalism is that on the entire branch of SHBH we can expect the following identity to be true $$\label{EHmass} M_{\rm ADM}= M_{\rm sol} + M_{\Delta} \,\,\,,$$ where $M_{\rm sol}$ is the ADM mass of the SS obtained taking the limit $r_\Delta \rightarrow 0$ of the branch of SHBH and $M_{\rm ADM}$ is the ADM mass corresponding to the SHBH with horizon radius $r_\Delta$, and the horizon mass is given by (\[MH\]). Thus, in a similar way to the EYM theory, the total ADM Mass of the solution contains two contributions, one attributed to the horizon of the SHBH and the other to the outside ‘hair’, captured by the SS. We have performed numerical explorations for SHBH for a large range of values of the horizon radius (in normalized units) and have checked the identity (\[EHmass\]). We have found complete agreement within the numerical uncertainties. This can be seen in Fig. \[masses\], where the identity was checked up to $r_{\Delta}=250$. Figure \[beta\] depicts the behavior of $\beta(r_{\Delta})$. Unlike the EYM model, where $\beta\approx 1$ for large $r_{\Delta}$, in this model $\beta\approx 1.24$ asymptotically. Figure \[BHconf\] shows an example of a BH solution with large $r_{\Delta}$ (for a small BH see fig.2 of Ref. [@SN]). Large-black-hole configuration constructed with $V(\phi)$ as given by Eq. (\[potential\]) with parameters $\eta_1= 0.5$, $\eta_2= 0.1$, $a= 0$, and $r_{\Delta}=150/\sqrt{\lambda}$, $\phi_{\Delta}\sim 0.26111$. The upper panels depict the scalar field and the mass function respectively. The latter converges to $M_{\rm ADM}\sim 93.096 /\sqrt{\lambda}$. The lower panels depicts the metric potentials (the first is a zoom of the second): $\sqrt{-g_{tt}}$ (solid line), $\sqrt{g_{rr}}$ (dashed line), $e^{\delta}$ (dash-dotted line) and $\delta$ (dotted line). \[BHconf\] A physical model of SHBH ------------------------ The extrapolation of the model of a hairy black hole as a bound state of an ordinary, ‘bare’, black hole and a ‘solitonic residue’ (first applied successfully to the colored BH’s in the EYM theory) [@ACS] does not apply directly to the SHBH because the straightforward generalization of the formula (\[ymbinding\]) as $$\label{PM} M_{\rm ADM} = M_{\rm sol} + M_{\Delta}= M^{\rm schwarz}_\Delta + M_{\rm sol} + E_{\rm bind}\,\,\,\,,$$ where $E_{\rm bind}= M_{\Delta} - M^{\rm schwarz}_\Delta$, to our case, has the problem that the binding energy changes sign, becoming positive for BH larger that $r_{\Delta}\approx 30$, and then increasing in absolute value as $r_\Delta$ gets larger. That is, $$E_{\rm bind} \sim B r_\Delta \,\,\,\,,$$ (where $B$ is a constant whose value depends on the specific model) for $r_\Delta \gg 1$, which is contrary to the expected feature of a negative binding energy as in the EYM case (i.e. the prediction (i) mentioned above will not be satisfied). In order to appreciate the origin of the failure of this feature, let us recall that we can write $M_{\rm ADM}$ in terms of the Schwarzschild mass and the mass of the ‘hair’ as, $$\label{FM} M_{\rm ADM}(r_\Delta) = M^{\rm schwarz}_\Delta(r_\Delta) + M_{\rm hair}(r_\Delta) \,\,\,,$$ where $$M_{\rm hair}(r_\Delta) = - \int^{\infty}_{r_\Delta} r^2 {T}^{t}_{t} \d r \,\,\,\,, \label{mhairint0}$$ By rescaling the $r-$coordinate in terms of $r_\Delta$, we can rewrite the mass of the hair. It becomes then, $$M_{\rm hair}(r_\Delta) = - r_\Delta \int^{\infty}_{1} x^2 \tilde{T}^{t}_{t} \d x \,\,\,\,, \label{mhairint}$$ with $$\begin{aligned} x &=& \frac{r}{r_\Delta}\,\,\,,\\ \tilde{T}^{t}_{t} &=& r_\Delta^2 T^{t}_{t} \,\,\,,\\ T^{t}_{t} &=& - \left[ \left(1-\frac{2m(r)}{r}\right) \frac{(\partial_r \phi)^2}{2} + V(\phi) \right]\,\,\,.\end{aligned}$$ Now, for $x\gg 1$, the integral in Eq.(\[mhairint\]) becomes almost independent of $r_\Delta$, and in fact the numerical analysis provides the following value $$M_{\rm hair}(r_\Delta\gg 1) \sim B\, r_{\Delta} \,\,\,.$$ where $B\approx 0.12$. Now, since $M^{\rm schwarz}_\Delta(r_\Delta)= r_\Delta/2$ we have then $$M_{\rm ADM}(r_\Delta\gg 1) \sim C r_\Delta \,\,\,.$$ where now $C\approx 0.62$. Therefore we conjecture that $C$ is a constant that depends of the matter-theory involved. For the EYM case, one can easily show that the scaling properties of the hair contribution of the energy makes the equivalent of the integral of Eq.(\[mhairint\]) to behave like $1/r_\Delta$ rather than $r_\Delta$. Therefore, for the EYM case, $C=1/2$. As we now show, this subtle difference in both theories makes that the binding energy expression used in the EYM cannot be used straightforwardly for the Einstein-Higgs theory analyzed here. From (\[PM\]) and (\[FM\]) we find that $$E_{\rm bind}(r_\Delta\gg 1) \sim M_{\rm hair}(r_\Delta\gg 1) - M_{\rm sol} \sim B\, r_\Delta - M_{\rm sol},$$ then $E_{\rm bind}$ scales as $r_\Delta$ when $r_\Delta \gg 1$ (remember that $M_{\rm sol}$ is a constant) [@comment]. It is clear that the sign of the binding energy as such defined changes sign and grows with the size of the black hole. To deal with this situation we now proceed to propose a modification of the model of a hairy black hole as a bound state in order to adapt it to the more general case. Our proposal consists in redefining the binding energy as, $$E_{\rm bind}^{\rm new}(r_\Delta) = E_{\rm bind}(r_\Delta) - B\,r_\Delta = M_{\Delta}(r_\Delta) - M^{\rm schwarz}_\Delta(r_\Delta) - B\,r_{\Delta}\, ,$$ By this procedure we have ‘renormalized’ the binding energy by subtracting the divergent term. That is, the new expression for the binding energy is $$\label{ebindnew} E_{\rm bind}^{\rm new}(r_\Delta) =M_{\Delta}(r_\Delta) - C\,r_{\Delta}\, .$$ This definition shares now exactly the same properties as the original expression for the EYM. Thus, it vanishes at $r_\Delta=0$, and decreases monotonically to the negative value $-M_{\rm sol}$. It is interesting to note that this new definition reduces to the old one for the EYM case, since as we previously remarked $C_{\rm EYM}=1/2$. Both binding energies are plotted. The ‘old’ binding energy $E_{\rm bind}(r_\Delta)$ is shown to become positive and approaches asymptotically a straight line. The ‘new’ binding energy $E_{\rm bind}^{\rm new}(r_\Delta)$ is also plotted (continuous line), showing the expected behavior. \[bindE\] The formula (\[PM\]) can now be reformulated as, \[PMnew\] M\_[ADM]{}(r\_) &=& M\_[sol]{} + M\_(r\_)\ &=& M\^[schwarz]{}\_(r\_)+ M\_[sol]{} + E\_[bind]{}\^[new]{}(r\_) + B r\_\ & = & Cr\_ + M\_[sol]{} + E\_[bin]{}\^[new]{}(r\_), Thus, we would get the same structure for the ADM mass of the hairy black hole where now the mass of the ‘bare black hole’ would be equal to $\tilde{M}^0_\Delta= (1/2+B) r_\Delta= C r_\Delta$. However it is not clear what the origin of this extra term ($Br_\Delta$) is, and we could very well have assigned it to the soliton mass to form a new ‘solitonic residue´ with mass $M_{\rm sol}+B r_\Delta$ (whose interpretation however seems somewhat obscure). We must explore more in order to decide which interpretation is best suited for our model. As we stressed, the constant $C$ depends on the theory considered; for the EYM and EYMH theories, $C=1/2$. It would be interesting to explore (in addition to the current analysis, see below) whether other theories admitting hair posses values of $C$ different from $1/2$. To end this section, let us rewrite the form of the hairy mass in terms of the old binding energy, in order to understand the behavior of the scalar system. First, let us note using Eqs. (\[PM\]) and (\[FM\]) that $$\label{gg} M_{\rm hair}(r_\Delta)= M_{\rm ADM}(r_\Delta)- M^{\rm schwarz}_\Delta(r_\Delta)= M_{\rm sol}+E_{\rm bind} \,\,\,,$$ so the binding energy is $$\label{be} E_{\rm bind}=M_{\rm hair}(r_\Delta)-M_{\rm sol}= -\left[ \int^{\infty}_{r_\Delta} r^2 {T}^{t}_{t} \d r -\int^{\infty}_{0} r^2 {{T_o}^{t}}_{t}\, \d r \right] \,\,\,,$$ where ${{T_o}^{t}}_{t}$ is the stress-energy tensor of the solitonic regular solution. This equation can be rewritten as, $$\label{be2} E_{\rm bind}= - \left[ \int^{\infty}_{r_\Delta} r^2 ({T}^{t}_{t}- {{T_o}^{t}}_{t})\, \d r -\int^{r_\Delta}_{0} r^2 {{T_o}^{t}}_{t} \d r \right] \,\,\,.$$ Here we can identify the first term as the difference between the hair of the BH and the “hair" of the soliton. Of course we are comparing the quantities (the integrands) that live on different manifolds, but the total integral is well defined. What happens in the EYM case is that both stress tensors behave very much alike, for the exterior region ($r>r_\Delta$) and for large values of $r_\Delta$, and thus the only term that contributes is the second one, that gives the ADM mass of the soliton (recall that $r_\Delta \gg 1$, that is for BH’s much larger than the characteristic size of the soliton (of order one in this dimension-less units), so the integral captures most of the soliton mass). In the scalar field case, the fact that the binding energy is proportional to $r_\Delta$, for large black holes, is captured by the fact that the BH contribution to the first term in (\[be2\]) is dominating. It would be interesting to explore this issue in other gravity-matter systems. Instability and final state {#sec:5} =========================== The next question we want to consider has to do with the following situation. Consider the case where a hairy black hole of geometrical radius $r_\Delta$ is slightly perturbed and therefore it decays. The final state will be, one expects, a black hole that in its near horizon geometry resembles the Schwarzschild solution, with the scalar field taking the value where the potential has a local minima and vanishes. This means that in this process the “scalar charge" at the horizon, namely the value $\phi_\Delta$ must change. One can make an argument similar to the one in Ref. [@ACS] to conclude that, in that situation, the horizon must grow in the process and therefore, the available energy to be radiated can not all be radiated to infinity; part of it must [*fall into the black hole*]{}. Let us now recall the estimate for the upper bound of the total energy to be radiated. ![[]{data-label="fig:4"}](figure6.eps) This figure illustrates a physical process where an initial configuration with an isolated horizon $\Delta_{\rm in}$ is perturbed and the final state contains another isolated horizon $\Delta_{\rm fin}$. The first step is to assume that the process illustrated in Fig. \[fig:4\] takes place. Then, we assume that in the initial surface there was an isolated horizon $\Delta_{\rm in}$ and after the initial unstable configuration has decayed, with part of the energy falling through the horizon and the rest radiating away to infinity, we are left with a horizon $\Delta_{\rm fin}$ of a hairless black hole (with $r_\Delta^{\rm fin}>r_\Delta^{\rm in}$). If we denote by $E_{{\cal I}^+}$ the energy radiated to future null infinity ${\cal I}^+$, and given that the ADM energy does not change in the process, we have $$M_{\rm ADM}=M_\Delta(r_\Delta^{\rm in})+M_{\rm sol}=M_\Delta^{\rm schwarz}(r_\Delta^{\rm fin})+E_{{\cal I}^+}\,,$$ which can be rewritten as, $$M_{\rm sol}+E^{\rm in}_{\rm bind}= M_\Delta^{\rm schwarz}(r_\Delta^{\rm fin})- M_\Delta^{\rm schwarz}(r_\Delta^{\rm in}) + E_{{\cal I}^+}\, .$$ On the right-hand-side note that the first two terms can be identified with $\Delta M^{0}_{\Delta}$, namely the change in (bare) horizon mass, while the second term corresponds to the radiated energy. Thus, it is natural to identify the quantity on the left as the [*available energy*]{} $E_{\rm avail}$ on the system. We can then write, $$E_{\rm avail}= M_{\rm ADM}- M_\Delta^{\rm schwarz}(r_\Delta^{\rm in})= M_{\rm sol}+E_{\rm bind} (r^{\rm in}_{\Delta} )=M_{\rm sol}+E^{\rm new}_{\rm bind}(r^{\rm in}_{\Delta})+B\,r_{\Delta}^{\rm in}\label{eavai}$$ There are several comments regarding this quantity. First, we note that there is a qualitative change in the behavior of $E_{\rm avail}(r^{\rm in}_\Delta)$ as function of the initial horizon radius, as in the EYM case. Its functional dependence is very similar to the binding energy since they differ only by the soliton mass. In the EYM case the available energy was equal to the soliton mass when there was no initial black hole (there is no energy used in binding the BH), and decreases as the radius increases. For very large black holes, the available energy goes to zero. For the scalar case under consideration here, we have a different behavior. The available energy decreases for small black holes but starts to increase and grows linearly with $r_\Delta$. The fact that in the EYM case the available energy went to zero for large BH´s was interpreted as meaning that those black holes were ‘less unstable’. This expectation was confirmed by the fact that the frequencies of the linear perturbations was decreasing with the radius of the initial BH [@Bizon-Chmaj; @ACS]. It is natural then to ask the same question for the scalar black holes. We have computed the frequencies of the (single) unstable mode $\psi(t,r)= \chi(r) e^{\imath \sigma t}$ present (where $\sigma^2$ turns to be always negative), as a function of the horizon radius and plotted it in Fig. \[Efrec\] (where $\psi(t,r)$ represents a linear perturbation of $\phi(r)$; see Ref.[@SN] for the details). The unstable-mode frequency $\omega=\sqrt{-\sigma^2}$ of the perturbed BH and soliton ($r_\Delta=0$) is plotted as a function of the horizon radius $r_\Delta$. Note that the frequency decreases as the horizon radius increases. \[Efrec\] As can be seen from the figure \[Efrec\], the frequencies still decrease as the black holes become larger, which is the same behavior observed in the EYM case. It is convenient then to reconsider the meaning of ‘less unstable’. In the scalar field case considered here, numerical investigations of the dynamical evolution of the soliton as initial state show that the system is unstable [@evolution]. The dynamical evolution of the system depends on the sign of the initial perturbation on the extrinsic curvature. For one sign of the perturbation, the system collapses and forms a black hole with a final isolated horizon, while for the other sign the system expands as a domain wall and gets therefore radiated to infinity (for details see [@evolution]). One should then expect that the dynamical evolution of slightly perturbed hairy black holes will show a similar qualitative behavior. In that case, for one sign of the perturbation one might expect the situation considered before, namely that the scalar field collapses and the BH grows. For the other sign, one can imagine that there could be, in some situations, an expanding wall that radiates away while leaving a “naked black hole". The pressing question in whether, in that case, this residue would be a Schwarzschild like or an AdS like black hole. This question arises since, for the soliton collapse in the case of the expanding wall, the region around the origin resembles an AdS spacetime with an effective negative cosmological constant generated by the (non-positive) potential. One might need in that case a new interpretation of the formalism. It is our belief that one needs to clarify what the criteria should be for regarding the system as slightly unstable or very unstable, other than the frequency of its perturbations. This and a full clarification of the nature of the resulting bare black hole could be achieved whenever full numerical simulations of dynamical evolution staring from scalar hairy black holes become available. Let us return our discussion to Eq. (\[eavai\]). The first thing to note is that due to the characteristic behavior of these solutions, for horizons larger than $r_\Delta\approx 30$, the horizon mass of the hairy black hole becomes larger than the Schwarzschild horizon mass of the same radius. This is also the point at which the binding energy becomes positive. One can thus speculate that the black holes of this radius and larger will have more violent collapses with a larger fraction of the available energy radiated away. Finally, from Eq. (\[eavai\]) one could interpret that again, the term $(B r_\Delta )$ could be associated to the soliton mass to form a solitonic residue that, together with the new binding energy allows us to have the same qualitative features of the heuristic model of [@ACS]. Again, a more detailed analysis will have to wait for the numerical investigations of the fully dynamical process. An empirical formula {#sec:6} ==================== The non-linear behavior of the Einstein-Matter equations and the non-trivial relations between the masses and the horizon radius posses a challenge to obtain an analytical formula for $M_{\rm ADM}(r_\Delta)$. One should expect that there is in general no closed analytical formula for the masses of hairy BH. We have discovered that the following empirical formula reproduces the qualitative and quantitative features of the numerical analysis $$\begin{aligned} M_{\rm ADM}^{\rm emp} (r_\Delta) &=& \sqrt{ \left(M_{\rm sol} + \frac{D}{2C} \right)^2 + C^2 r_\Delta^2 + Dr_\Delta } - \frac{D}{2C} \,\,\,\nonumber \\ &=& \overline{M_{\rm sol}}\left\{ \sqrt{ \left[1 + \frac{Cr_\Delta}{\overline{M_{\rm sol}}}\right]^2 + F} - 1\right\} \,\,\,. \label{memp}\end{aligned}$$ where $$D= \frac{2C\beta_0M_{\rm sol}}{2C-\beta_0}\,\,\,,\,\,\, \overline{M_{\rm sol}} = \frac{\beta_0 M_{\rm sol}}{2C-\beta_0} \,\,\,,\,\,\,F= \left(\frac{2C}{\beta_0}\right)^2 -1$$ $\beta_0$ being the value of $\beta$ at $r_\Delta=0$. The formula (\[memp\]) has the following nice properties: 1. $M_{\rm ADM}^{\rm emp}(0)= M_{\rm sol}$. 2. $M_{\rm ADM}^{\rm emp} (r_\Delta)$ is a monotonically increasing function of $r_\Delta$. 3. For large $r_\Delta$, $M_{\rm ADM}^{\rm emp}\rightarrow C r_\Delta$. 4. The relative error between $M_{\rm ADM}^{\rm emp}$ and the numerical one is less that $10\%$. These errors become very small for small and large $r_\Delta$. 5. One can define an empirical binding energy by using $E_{\rm bin}^{\rm emp}(r_\Delta)= M_{\rm ADM}^{\rm emp}- M_{\rm sol} - Cr_\Delta$ \[where the first two terms provide the empirical horizon mass; here we are using Eq. (\[ebindnew\]) \]. This formula reproduces very well the numerical results. 6. One can then obtain a fit for $\beta$ as follows $$\beta^{\rm em}(r_\Delta) = 2\frac{dM_{\rm ADM}^{\rm emp}}{dr_\Delta} = \frac{2r_\Delta C^2 + D}{\sqrt{ \left(M_{\rm sol} + \frac{D}{2C} \right)^2 + C^2 r_\Delta^2 + Dr_\Delta } } = \frac{2C\left(1 + \frac{Cr_\Delta}{\overline{M_{\rm sol}}}\right)} {\sqrt{ \left[1 + \frac{Cr_\Delta}{\overline{M_{\rm sol}}}\right]^2 + F}} \,\,\,.$$ This formula reproduces the qualitative shape of the numerical $\beta$, such as its exact value at the origin, its monotonically increasing behavior and its asymptotic value $2C$ for large $r_\Delta$. 7. For the Schwarzschild case ($C=1/2$, $M_{\rm sol}=0$), one obtains the expected results: $M_{\rm ADM}^{\rm emp}(r_\Delta)= r_\Delta/2$, $\beta^{\rm em}(r_\Delta) \equiv 1$, $E_{\rm bind}^{\rm emp}(r_\Delta) \equiv 0$. Clearly by adding terms of the form $r_\Delta^\theta$ ($1<\theta<2$) inside the square root of Eq.(\[memp\]) one could improve the fit between the numerical results and the analytical formula. In Figure \[fits\] we compare between the empirical formula and the numerical values of the hairy scalar black holes, for the ADM mass, screened surface gravity $\beta$ and the binding energy. The empirical formula can be used also for the EYM case with $C=1/2$ and the corresponding values of $M_{\rm sol}$ and $D$. Figure \[fitseym\] compares the numerical values of the EYM $n=1$-branch with those obtained from the empirical formula. We conjecture that the empirical formula can work also for different $n$, by using their corresponding values $M_{\rm sol}^{n}$, and $\beta_0^{n}$. Moreover, we also speculate that such a formula can hold for other theories admitting hair, such as in the Einstein-Skyrme and Einstein-sphaleron models. It remains to be investigated what are the values of $C$, $M_{\rm sol}$ and $\beta_0$ for such cases. Now, we can further use the first law of thermodynamics $\delta M= \kappa \delta A_\Delta/(8\pi)$ for $M_{\rm ADM}^{\rm emp}$, and obtain the following prediction $$\label{pred} \kappa\left(M_{\rm ADM}^{\rm emp} + \frac{D}{2C}\right)= C^2 + \frac{D}{2r_\Delta} \,\,\,.$$ Since the properties 1)$-$7) show that the analytical results obtained from Eq. (\[memp\]) work particularly well for large and small $r_\Delta$, the most reliable consequence of (\[pred\]) is a remarkable simple relation between the surface gravity and the ADM mass for sufficiently large hairy black holes: $$\label{pred2} \kappa M_{\rm ADM}\approx C^2 \,\,\,.$$ Note that Eq. (\[pred2\]) is consistent for the Schwarzschild case ($C=1/2$, $D=0$, $M_{\rm ADM}=r_\Delta/2$), where the identity $\kappa M_{\rm ADM}\equiv 1/4$, holds exactly. The three panels depict the ADM-mass, $\beta$, and binding energy $E_{\rm bind}^{\rm new}$ , respectively, as a function of the horizon radius. The solid lines correspond to the values obtained from a numerical analysis and the dashed lines were obtained from the empirical formulae described in the main text. Note the good qualitative behavior of the empirical formulae. Remarkably good fits to the more precise numerical values are obtained for small and large $r_\Delta$. The values for the empirical formula are $C\approx 0.62$, $M_{\rm sol}\approx 3.827$ and $\beta_0\approx 0.324$. \[fits\] Same as Fig.\[fits\], for the Einstein-Yang-Mills theory ($n=1$ colored black holes). Here the values for the empirical formulae are $C=1/2$, $M_{\rm sol}\approx 0.828$ and $\beta_0\approx 0.126$. \[fitseym\] We have performed a non-exhaustive analysis of the solutions with respect to variations of some of the parameters of the scalar potential Eq. (\[potential\]). Notably, we have computed the effect of the variation of $\eta_2$ on the global quantities. It is to note that changing $\eta_2$ modifies the potential barrier between the global and the local minimum. In fact, the closer the value $\eta_2$ to $\eta_1/2$, the less negative is $V(a+\eta_1)= \lambda \eta_1^3\left[2\eta_2 -\eta_1\right]/12$, and therefore the potential approaches the conditions where the no-hair theorems apply. The details of the solutions then depend in a non-trivial fashion between the interplay of the negative global minimum (in order to avoid the applicability of the non-hair theorems) and the height of the potential barrier. Figure  \[multgraf\] depicts different global quantities as a function of the horizon radius for five different values of $\eta_2$. The soliton mass ($r_\Delta=0$) as well as the ADM and horizon masses (for large $r_\Delta$) tend to increase with $\eta_2$. Remarkably, the empirical formulae continue to provide reasonable good results by changing the corresponding values of their parameters $\vec{P}_{\eta_2}:= \left(M_{\rm sol}, C,\beta_0\right)_{\eta_2}$. The quality of the fit to the numerical values can be appreciated by the dashed curves of Fig.  \[multgraf\] which were computed with the empirical formulae. Panels 1-4 depict the ADM-mass, horizon mass, $\beta$ and the binding energy respectively as a function of $r_\Delta$. The solid lines were obtained from the numerical analysis while the dashed lines were computed using the empirical formulae. The lines are associated with the five different values used for $\eta_2= 0.1,0.11,0.12,0.13,0.14$ with $\eta_1=0.5$ fixed. As seen from bottom to top (for large $r_\Delta$) the plots of panels 1-3 correspond to $\eta_2$ in increasing order (in panel 4 the order is reversed). The values of the parameters $\vec{P}_{\eta_2}= \left(M_{\rm sol}, C,\beta_0\right)_{\eta_2}$ used in the empirical formulae are $\vec{P}_{0.1}\approx \left(3.82,0.62,0.32\right)$, $\vec{P}_{0.11}\approx \left(5.22,0.66,0.24\right)$, $\vec{P}_{0.12}\approx \left(7.47,0.72,0.17\right)$ $\vec{P}_{0.13}\approx \left(11.78,0.82,0.11\right)$ $\vec{P}_{0.14}\approx \left(23.51,1.01,0.06\right)$. \[multgraf\] Discussion {#sec:7} ========== Let us first summarize our results. By solving numerically Einstein’s equations for static solutions of a self-gravitating scalar field, we have analyzed the behavior of several spacetime quantities as functions of the black hole horizon radius. We have found that the ADM mass of the spacetimes exhibits two types of behavior: it is similar to other ‘hairy’ theories for small black holes, but its behavior changes dramatically for large black holes. In particular the ADM mass of large BH scales not as $r_\Delta/2$ as in other theories (EYM, EYMH, etc), but the proportionality constant (with respect to horizon radius) takes a different value depending on the form of the potential ($C\approx 0.63$ for $\eta_2=0.1$). In this article we have analyzed the consequences of this fact for a model based on the isolated horizons formalism. In such a model, a hairy black hole is viewed as a bound state of a soliton (which we have) and a ‘bare black hole’. The binding energy is found to be negative in EYM and EYMH, but in our case, for large BH, the binding energy becomes positive and grows linearly with $r_\Delta$ [^2]. This fact leads to several possibilities. We have seen that it is possible to modify the original model by ‘renormalizing’ the binding energy in such a way that the newly defined energy has the same qualitative behavior as in the EYM system. The price one has to pay is the need to reinterpret either a new ‘solitonic residue’, or a new bare black hole. As a first attempt towards giving a definite answer to this question, we analyzed the frequency of the unstable mode of the linearized perturbation, and found that the behavior is the same as in EYM. This suggests that the proper physical interpretation is still unclear and that further numerical dynamical investigations are needed to fully settle the question. In particular, the two different regimes of the theory might have some consequences in the dynamical evolution of slightly perturbed BHs, where one could conjecture a different qualitative behavior for small and large black holes, regarding the endpoint of evolution and the nature of the [*bare*]{} black hole to which the solution settles. We have also conjectured that the constant that fixed the proportionality between ADM mass and horizon radius for large BH’s is a theory-dependent constant, which would in particular imply that axi-symmetric non-spherical BH solutions to the gravity-scalar field system would have the same asymptotic behavior, for each given potential. It would be worth studying other gravity-matter systems, such as non-minimally coupled scalars, to see whether they posses a different proportionality constant (work is in progress in these directions). We have shown also that a very simple heuristic analytic formula captures the essential qualitative behavior of the ADM mass of the hairy scalar BH’s, specially for small and large values of the horizon radius. We have conjectured that such formula can also be useful for EYM and more general hairy black holes. It remains a theoretical challenge to fully understand the origin of such simple formula. Perhaps the most important conclusion from the present work is the lesson that hairy black holes for different matter systems exhibit new, and sometimes, unexpected behavior. This also point out to the need of a proper and deeper understanding of the reason why the heuristic hairy black hole model works so well for the system that it does, and whether the phenomenological modifications we have proposed here stand the test of full numerical investigations. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank A. Ashtekar and D. Sudarsky for discussions. This work was in part supported by grants DGAPA-UNAM IN122002, and IN119005. U.N. acknowledges partial support from SNI, and Grants No. 4.8 CIC-UMSNH, No. PROMEP PTC-61 and No. CONACYT 42949-F. [99]{} A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krishnan, J. Lewandowski and J. Wisniewski, “Isolated horizons and their applications,” Phys. Rev. Lett. [**85**]{}, 3564 (2000), [gr-qc/0006006]{}. A. Ashtekar, C. Beetle and S. Fairhurst, “Mechanics of Isolated Horizons,” Class. Quant. Grav.  [**17**]{}, 253 (2000), [gr-qc/9907068]{}. A. Ashtekar and A. Corichi, “Laws governing isolated horizons: Inclusion of dilaton couplings,” Class. Quant. Grav.  [**17**]{}, 1317 (2000), [gr-qc/9910068]{}. A. Ashtekar, S. Fairhurst and B. Krishnan, “Isolated horizons: Hamiltonian evolution and the first law,” Phys. Rev. D [**62**]{}, 104025 (2000), [gr-qc/0005083]{}. A. Corichi and D. Sudarsky, “Mass of colored black holes,” Phys. Rev. D [**61**]{}, 101501(R) (2000), [gr-qc/9912032]{}; A. Corichi, U. Nucamendi and D. Sudarsky, “Einstein-Yang-Mills isolated horizons: Phase space, mechanics, hair and conjectures,” Phys. Rev. D [**62**]{}, 044046 (2000), [gr-qc/0002078]{}. A. Ashtekar, A. Corichi and D. Sudarsky, “Hairy black holes, horizon mass and solitons,” Class. Quant. Grav. [**18**]{}, 919 (2001), [gr-qc/0011081]{}. B. Kleihaus and J. Kunz, “Non-Abelian black holes with magnetic dipole hair,” Phys. Lett. B [**494**]{}, 130 (2000), [hep-th/0008034]{}. A. Corichi, U. Nucamendi and D. Sudarsky, “A mass formula for EYM solitons,” Phys. Rev. D [**64**]{}, 107501 (2001), [arXiv:gr-qc/0106084]{}. B. Kleihaus, J. Kunz, A. Sood and M. Wirschins, “Horizon properties of Einstein-Yang-Mills black hole,” Phys. Rev. D [**65**]{}, 061502(R) (2002), [arXiv:gr-qc/0110084]{}. R. Ibadov, B. Kleihaus, J. Kunz and M. Wirschins, “New black hole solutions with axial symmetry in Einstein-Yang-Mills theory,” Phys. Lett. B [**627**]{}, 180 (2005) [arXiv:gr-qc/0507110]{}. A. Ashtekar and B. Krishnan, “Isolated and dynamical horizons and their applications,” Living Rev. Rel.  [**7**]{}, 10 (2004), [arXiv:gr-qc/0407042]{}. M. S. Volkov and D. V. Gal’tsov, “Gravitating non-Abelian solitons and black holes with Yang-Mills fields,” Phys. Rept.  [**319**]{}, 1 (1999), [arXiv:hep-th/9810070]{}. U. Nucamendi and M. Salgado, “Scalar hairy black holes and solitons in asymptotically flat spacetimes,” Phys. Rev. D [**68**]{}, 044026 (2003), [arXiv:gr-qc/0301062]{}. M. Heusler, J. Math. Phys. [**33**]{}, 3497 (1992); D. Sudarsky, Class. Quantum Grav. [**12**]{}, 579 (1995); J.D. Bekenstein, Phys. Rev. D[**51**]{}, R6608 (1995). M. Heusler, [*Black Hole Uniqueness Theorems*]{}, Cambridge Univ. Press, Cambridge (1996) A. Ashtekar, A. Corichi and D. Sudarsky, “Non-minimally coupled scalar fields and isolated horizons,” Class. Quant. Grav.  [**20**]{}, 3413 (2003), [arXiv:gr-qc/0305044]{}. Another example of hairy BH’s with the property that the masses $M_{\rm hair}$ scale also linearly with $r_\Delta$ are the hairy BH’s of the Einstein-Skyrme system. To see this we take the equations of motion for the Einstein-Skyrme model as described in the subsection (7.4) from the reference [@volkov]; specifically, we take the equation (7.37): $$\begin{aligned} \frac{dm(r)}{dr}&=& N \left[ \frac{r^2}{2} + \sin^2 \chi \right] \left(\frac{d\chi}{dr}\right)^2 + \left[ r^2 + \frac{\sin^2 \chi}{2} \right] \frac{\sin^2 \chi}{r^2} \label{7.37}\end{aligned}$$ where the metric for static and spherically symmetric configurations is: $$\begin{aligned} ds^2 &=& -\sigma^2(r) N(r) dt^2 + \frac{1}{N(r)} dr^2 + r^2 d\Omega^2\end{aligned}$$ with $$\begin{aligned} N(r) = 1 - 2m(r)/r,\end{aligned}$$ and the Skyrmion field $\chi(r)$ depending on the dimensionless coordinate $r$; to continuation we define a new coordinate as $x=r/r_\Delta$ and after integration rewrite (\[7.37\]) as\ $$\begin{aligned} M_{\rm ADM}(r_\Delta) = \frac{r_\Delta}{2} &+& M_{\rm hair}(r_\Delta) \nonumber \\ =\frac{r_\Delta}{2} &+& r_\Delta \int^\infty_1 \left[\frac{N x^2}{2} \left(\frac{d\chi}{dx}\right)^2 + \sin^2 \chi \right] dx \nonumber \\ &+& \frac {1}{r_\Delta} \int^\infty_1 \left[N \left(\frac{d\chi}{dx}\right)^2 + \frac{\sin^2 \chi}{2x^2} \right] (\sin^2 \chi) dx \label{4}\end{aligned}$$ The behavior for $\chi(x)$ can see from left panel in the fig. 14. of such reference. From equation (7.41) we see the asymptotic behavior for $\chi(x)$ as $x \rightarrow \infty$ (with $a$ a constant): $\chi \approx ax^{-2}$. Altough this analysis shows that $M_{\rm ADM}(r_\Delta)$ grows linearly with $r_\Delta$ for $r_\Delta\gg 1$, we can not apply the limit $r_\Delta \rightarrow \infty$ to (\[4\]) because the existence of Skyrme’s black holes is limited to configurations with horizon radius $r_\Delta \lesssim r_\Delta^{max}(\kappa)$ (here $\kappa\equiv 4\pi G_0 f^2$ is the coupling constant of the theory and $r_\Delta^{max}(\kappa)$ is a maximal value depending of $\kappa$) and $\kappa \lesssim \kappa_{bh}^{max}$. The value of $\kappa_{bh}^{max}$ is of the order of $0.0315$. At the other hand, the existence of solitons is permitted for $\kappa \lesssim 0.0437$. In the Eisntein-Higgs at hand, $r_\Delta$ seems to be limited by the precision of the shooting method. For large $r_\Delta$, the shooting parameter $\phi_\Delta$ approaches a below limit ($\approx 0.26$ for the model $\vec{P}_{0.1}$ of fig. \[multgraf\]). It is unknown if this limit for the SHBH configurations is a fundamental limit or not. It turns then interesting to construct a similar model based in the isolated-horizon formalism for the Einstein-Skyrme system to compare similarities and differences with the model presented here [@Nielsen]. P. Bizoń and T. Chmaj, “Remark on the formation of colored black holes via fine-tuning,” Phys. Rev. D [**61**]{}, 067501 (2000) M. Alcubierre, J. A. Gonzalez and M. Salgado, “Dynamical evolution of unstable self-gravitating scalar solitons,” Phys. Rev. D [**70**]{}, 064016 (2004), [arXiv:gr-qc/0403035]{}. Y. Brihaye and B. Hartmann, “Deformed black strings in 5-dimensional Einstein-Yang-Mills theory,” [arXiv:gr-qc/0503102]{}. A. Nielsen, “Skyrme Black Holes in the Isolated Horizon Formalism”, in preparation [^1]: We remind the reader our choice of units $G_0=c=1$. [^2]: Recently, another system in 5 dimensions was shown to posses a positive binding energy as well [@hart].

--- abstract: | The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_1$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P, Q \in A_1$ then $A_1 = K \langle P, Q \rangle$. The Weyl algebra $A_1$ is a $\Z$-graded algebra. We prove that the Dixmier Conjecture holds if the elements $P$ and $Q$ are sums of no more than two homogeneous elements of $A$ (there is no restriction on the total degrees of $P$ and $Q$).\ [*Key Words: the Weyl algebra, the Dixmier Conjecture, automorphism, endomorphism, a $\Z$-graded algebra.*]{} [*Mathematics subject classification 2010: 16S50, 16W20, 16S32, 16W50.*]{} address: - | Department of Pure Mathematics\ University of Sheffield\ Hicks Building\ Sheffield S3 7RH, UK - | Lehrstuhl D für Mathematik\ RWTH Aachen University\ 52062 Aachen, Germany author: - 'V. V. Bavula' - 'V. Levandovskyy' title: A remark on the Dixmier Conjecture --- Introduction ============ In the paper, $K$ is a field of characteristic zero and $K^*:=K \setminus\{0\}$. The algebra $A_1 := K \langle X, Y \mid [Y, X]=1 \rangle$ is called the [*first Weyl algebra*]{} where $[Y, X]= YX - XY$. The $n$’th tensor power of $A_1$, $A_n := A_1^{\otimes n} = \underbrace{A_1 \otimes \cdots \otimes A_1}_{n \text{ times}}$, is called the [*$n$’th Weyl algebra*]{}. The algebra $A_n$ is a simple Noetherian domain of Gel’fand-Kirillov dimension $\GK(A_n)=2n$, it is canonically isomorphic to the algebra of polynomial differential operators $K \langle X_1, \ldots, X_n, \partial_1, \ldots, \partial_n \rangle$ (where $\partial_i = \tfrac{\partial}{\partial x_i}$) via $X_i \mapsto X_i$, $Y_i \mapsto \partial_i$ for $i=1,\ldots,n$. In his seminal paper [@Dix], Dixmier (1968) found explicit generators for the group $G = \Aut_K(A_1)$ of $K$-automorphisms of the Weyl algebra $A_1$. Namely, the group $G$ is generated by the obvious automorphisms: $$(X, Y) \mapsto (X, Y + \l X^n), \quad (X, Y) \mapsto (X + \l Y^n, Y), \quad (X, Y) \mapsto (\mu X, \mu^{-1} Y)$$ where $\l \in K$, $\mu \in K^*$ and $n \in \Nat_{+}:=\{1, 2, \ldots\}$. In [@Dix], Dixmier posed six problems: The first problem of Dixmier (in the list) asks if [*every endomorphism of the Weyl algebra $A_1$ is an automorphism*]{}, i.e., given elements $P, Q$ of $A$ such that $[P, Q]=1$, do they generate the algebra $A_1$? A similar problem but for the $n$’th Weyl algebra is called the [*Dixmier Conjecture*]{}. Problems 3 and 6 have been solved by Joseph [@josclA1] (1975), Problem 5 and Problem 4 (in the case of homogeneous elements) have been solved by Bavula [@Bav-DixPr5] (2005). The Dixmier Conjecture implies the [*Jacobian Conjecture*]{} (see [@BCW]) and the inverse implication is also true (see [@Tsuchimoto-2005] and [@Belov-Konts-2007]); a short proof is given in [@Bav-DCJC-2005]; see also [@Adjam-vdEs-2007]). In [@Bav-RenQn], it is shown that for each $K$-endomorphism $\phi: A_n \to A_n$ its image is very large, i.e., the left $A_{2n}$-module ${}^{\phi}{A_n}^{\phi}$ is a holonomic $A_{2n}$-module (where for all $a, b \in A_n$ and $c \in {}^{\phi}{A_n}^{\phi}$, $a \cdot c \cdot b := \phi(a) c \phi(b)$). In particular, it has finite length with simple holonomic factors over $A_{2n}$ (see [@Bav-RenQn] for details). To prove that the Dixmier Conjecture holds for the Weyl algebra $A_n$ it remains to show that the length is 1. Note, that the Gel’fand-Kirillov dimension of a simple $A_{2n}$-module can be $2n, 2n+1, \ldots, 4n-1$, and the last case is the generic case. In [@Bav-T1DC], it is shown that every algebra endomorphism of the algebra $\CI_1 = K\langle x, \partial, \int \rangle$ of polynomial integro-differential operators is an automorphism and it is conjectured that the same result holds for $\CI_n := \CI_1^{\otimes n}=K\langle x_1, \ldots , x_n, \der_1, \ldots , \der_n , \int_1, \ldots , \int_n\rangle$. The Weyl algebra $A = \oplus_{i\in\Z} A_{1,i}$ is a $\Z$-graded algebra ($A_{1,i} A_{1,j} \subseteq A_{1,i+j}$ for all $i, j\in \Z$) where $A_{1,0}=K[H]$, $H=YX$ and, for $i\geq 1$, $A_{1,i} = K[H]X^i$ and $A_{1,-i} = K[H]Y^i$. For a nonzero element $a$ of $A_1$, the number of [*nonzero homogeneous*]{} components is called the [*mass*]{} of $a$, denoted by $m(a)$. For example, $m(\alpha X^i)=1$ for all $\alpha \in K[H]\setminus\{0\}$ and $i\geq 1$. The aim of this paper is to prove the following theorem. \[20Sep16\] Let $P, Q$ be elements of the first Weyl algebra $A_1$ with $m(P)\leq 2$ and $m(Q)\leq 2$. If $[P,Q]=1$ then $P = \tau(Y)$ and $Q=\tau(X)$ for some automorphism $\tau\in\Aut_K (A_1)$. Proof of Theorem \[20Sep16\] ============================ [**The Weyl algebra is a generalized Weyl algebra**]{}. Let $D$ be a ring with an automorphism $\s $ and a central element $a$. The [**generalized Weyl algebra**]{} $A=D(\sigma, a)$ of degree 1, is the ring generated by $D$ and two indeterminates $X$ an $Y$ subject to the relations [@Bav-FA-1991]: $$X\alpha=\sigma(\alpha)X \ {\rm and}\ Y\alpha=\sigma^{-1}(\alpha)Y,\; {\rm for \; all }\; \alpha \in D, \ YX=a \ {\rm and}\ XY=\sigma(a).$$ The algebra $A={\oplus}_{n\in \mathbb{Z}}\, A_n$ is a $\mathbb{Z}$-graded algebra where $A_n=Dv_n$, $v_n=X^n\,\, (n>0), \,\,v_n=Y^{-n}\,\, (n<0), \,\,v_0=1.$ It follows from the defining relations that $$v_nv_m=(n,m)v_{n+m}=v_{n+m}<n,m>$$ for some elements $(n,m)=\s^{-n-m}(<n,m>)\in D$. If $n>0$ and $m>0$ then $$\begin{aligned} n\geq m & : & (n,-m)=\sigma^n(a)\cdots \sigma^{n-m+1}(a), \,\, (-n,m)=\sigma^{-n+1}(a)\cdots \sigma^{-n+m}(a), \\ n\leq m & : & (n,-m)=\sigma^{n}(a)\cdots \sigma(a),\,\,\, (-n,m)=\sigma^{-n+1}(a)\cdots a,\end{aligned}$$ in other cases $(n,m)=1$. Let $K[H]$ be a polynomial ring in a variable $H$ over the field $K$, $\s :H\ra H-1$ be the $K$-automorphism of the algebra $K[H]$ and $a=H$. The first Weyl algebra $A_1=K\langle X, Y \mid YX-XY=1\rangle$ is isomorphic to the generalized Weyl algebra $$A_1\simeq K[H](\s , H),\; X \mapsto X,\; Y \mapsto Y,\; YX \mapsto H.$$ We identify both these algebras via this isomorphism, that is $A_1=K[H](\s , H)$ and $H=YX$. If $n>0$ and $m>0$ then $$\begin{aligned} n\geq m & : & (n,-m)= (H-n)\cdots (H-n+m-1),\; (-n,m)=(H+n-1)\cdots (H+n-m), \\ n\leq m & : & (n,-m)=(H-n)\cdots (H-1), \; (-n,m)=(H+n-1)\cdots H,\end{aligned}$$ in other cases $(n,m)=1$. The localization $B=S^{-1}A_1$ of the Weyl algebra $A_1$ at the Ore subset $S=K[H]\backslash \{ 0\}$ of $A_1$ is the [*skew Laurent polynomial ring*]{} $B=K(H)[X, X^{-1}; \s ]$ with coefficients from the field $K(H)=S^{-1}K[H]$ of rational functions where $\s \in \Aut_K \, K(H)$ and $\s (H)=H-1$. The map $A_1\ra B$, $a \mapsto a/1$ is an algebra monomorphism. We identify the algebra $A_1$ with its image in the algebra $B$ via $A_1 \ra B, \;\; X \mapsto X, \;\; Y \mapsto HX^{-1}$. The algebra $B=\oplus_{i\in \mathbb{Z}}\, B_i$ is a $\mathbb{Z}$-graded algebra where $B_i=K(H)X^i$. The algebra $A_1$ is a $\mathbb{Z}$-graded subalgebra of $B$. A polynomial $f(H)=\l_nH^n+\l_{n-1}H^{n-1}+\cdots +\l_0\in K[H]$ of degree $n$ is called a [*monic*]{} polynomial if the [*leading coefficient*]{} $\l_n$ of $f(H)$ is $1$. A rational function $h\in K(H)$ is called a [*monic*]{} rational function if $h=f/g$ for some monic polynomials $f,g$. A homogeneous element $u=\alpha x^n$ of $B$ is called [*monic*]{} if $\alpha $ is a monic rational function. We can extend the concept of degree of polynomial to the field of rational functions by the rule $\deg \, h = \deg\, f - \deg\, g$ where $h=f/g\in K[H]$. If $h_1, h_2 \in K(H)$ then $\deg\, h_1h_2=\deg\,h_1 +\deg\, h_2$ and $\deg (h_1+h_2)\leq \max \{ \deg\,h_1 , \deg\, h_2\}$. We denote by ${\rm sign} (n)$ and by $|n|$ the [*sign*]{} and the [*absolute value*]{} of $n\in \mathbb{Z}$, respectively. Let $A$ be an algebra and $a\in A$. The subalgebra of $A$, $C_A(a)=\{ b \in A \mid ab=ba \}$, is called the [*centralizer*]{} of the element $a$ in $A$. \[cheab\] [*(Centralizer of a Homogeneous Element of the Algebra $B$)*]{} 1. Let $u=\alpha X^n$ be a monic element of $B_n$ with $n\neq 0$. Then the centralizer $C_B(u)=K[v,v^{-1}]$ is a Laurent polynomial ring for a unique element $v=\beta X^{{\rm sign} (n) s}$ where $s$ is the least positive divisor of $n$ for which there exists an element $\beta =\beta_s \in K(H)$, necessarily monic and uniquely defined, such that $$\label{bap} \beta \,\s^s (\beta )\, \s^{2s}(\beta ) \cdots \s^{(n/s -1)s}(\beta )=\alpha, \;\; {\rm if}\;\; n>0,$$ $$\label{bam} \beta \, \s^{-s} (\beta ) \,\s^{-2s}(\beta ) \cdots \s^{-(|n|/s -1)s}(\beta )=\alpha, \;\; {\rm if}\;\; n<0.$$ 2. Let $u\in K(H)\backslash K$. Then $C_B(u)=K(H)$. Let $A_{1,+} := K[H][X; \sigma]$ and $A_{1,-} := K[H][Y; \sigma^{-1}]$. The algebras $A_{1,+}$ and $A_{1,-}$ are (skew polynomial) subalgebras of $A_1$. \[a20Sep16\] If $u \in A_{1,\pm}\setminus\{0\}$ then $C_A(u) \subseteq A_{1,\pm}$. The $K$-automorphism of the Weyl algebra $A_1$, $$\label{xiaut} \xi: A_1 \to A_1, \; X \mapsto Y, \; Y \mapsto -X,$$ reverses the $\Z$-grading of the Weyl algebra $A_1$, that is $$\label{xiaut1} \xi(A_{1,i}) = A_{1,-i} \text{ for all } z\in\Z.$$ By the [*degree*]{} of an element of $A_1$ we mean its [*total degree*]{} with respect to the canonical generators $X$ and $Y$ of $A_1$. Let $A_{1, \leq i} := \{ p\in A \mid \deg(p)\leq i\}$ for $i\in\Nat$. Then $\{A_{1, \leq i}\}_{i\in\Nat}$ is the standard filtration of the algebra $A_1$ associated with the generators $X$ and $Y$. For all $i\in\Z\setminus\{0\}$ and $f\in K[H]\setminus K$, $$\label{degsf} \deg \s^i(f) = \deg f \text{ and } \deg (1 - \s^i)(f) = \deg f - 1.$$ [**Proof of Theorem \[20Sep16\]**]{}: [*(i)*]{} [*If $P, Q\in A_{1,\leq 1}$ then $P = \tau(Y)$ and $Q=\tau(X)$ for some $\tau\in\Aut_K (A_1)$*]{}: Clearly, $P = aY + bX + \l$ and $Q = cY +dX + \mu$ for some $a, b, c, d, \l, \mu \in K$. Then $1 = [P,Q]=ad-bc$. So, the automorphism $\tau$ can be chosen of the form $\tau(Y) = aY + bX + \l$ and $\tau(X) = cY +dX + \mu$. So, till the end of the proof we assume that at least one of the polynomials $P$ or $Q$ does not belong to the space $A_{1, \leq 1}$. In view of the relation $1 = [P,Q] = [-Q,P]$, we can assume that $P \notin A_{1, \leq 1}$. In view of Equation (\[xiaut1\]), we can assume that the highest homogeneous part of $P$, say $P_p \in A_{1,p}$, satisfies the condition that $p \geq 2$. Since $m(P)\leq 2$, either $P = P_p$ (if $m(P)=1$) or otherwise $P = P_r + P_p$ for some nonzero $P_r \in A_{1,r}$ where $r<p$. [*(ii)*]{} $(m(P), m(Q)) \neq (1,1)$: Suppose that $m(P)= m(Q)=1$, we seek a contradiction. Then $P =\alpha X^p$ and $Q=\beta Y^p$ for some nonzero polynomials $\alpha, \beta\in K[H]$. Then $$1 = [P,Q] = \alpha \s^p(\beta) (p,-p) - \beta \s^{-p}(\alpha) (-p,p)$$ $$= \alpha \s^p(\beta) (p,-p) - \beta \s^{-p}(\alpha) \s^{-p}((p,-p)) = (1-\s^{-p})(\alpha \s^p(\beta) (p,-p)).$$ Since $p \geq 2$ (or $P\notin A_{1, \leq 1}$), $0 = \deg 1 = \deg\, (1-\s^{-p})(\alpha \s^p(\beta) (p,-p)) = \deg \alpha + \deg \beta +\deg\, (p,-p) -1$ (by Equation (\[degsf\])) $\geq 0+0+p-1 \geq 2-1=1$, a contradiction. [*(iii)*]{} $(m(P), m(Q)) \neq (1,2)$: Suppose that $m(P)=1$ and $m(Q)=2$. Then $P = \alpha X^p$ for some $p\geq 2$ and $Q=Q_s + Q_q$ where $Q_s \in A_{1,s}$, $Q_q \in A_{1,q}$ and $s<q$. By Lemma \[a20Sep16\], the equality $[P,Q]=1$ implies that $[P, Q_s]=1$ and $[P, Q_q]=0$. By the case (ii), this is not possible. [*(iv)*]{} Suppose that $m(P)=2$ and $m(Q)=1$. Then $P = P_r + P_p$ and $Q=Q_q$. By Lemma \[a20Sep16\] the equality $[P,Q]=1$ implies that $[P_p, Q_q]=0$ and $[P_r, Q_q]=1$. Then, $q\geq 0$, by Lemma \[a20Sep16\]. The case $q=0$ is not possible since then both $P_r, Q_q\in K[H]$ and this would contradict the equality $[P_r, Q_q]=1$. Therefore, $q>0$. Then $P_r = \beta Y^q$ and $Q_q = \alpha X^q$ for some nonzero elements $\beta, \alpha \in K[H]$. Then $$-1 = [Q_q, P_r] = (1-\s^{-q}) (\alpha \s^p(\beta) (q,-q) )$$ implies that $0 = \deg (-1) = \deg\, (1-\s^{-q}) (\alpha \s^p(\beta) (q,-q) ) = \deg \alpha + \deg \beta +q -1$, by Equation (\[degsf\]). Hence, $q=1$, $\alpha, \beta \in K^*$ and $\beta = -\alpha^{-1}$. Then $P, Q \in A_{1,\leq 1}$, and, by the statement (i), the pair $(P,Q)$ is obtained from the pair $(Y,X)$ by applying an automorphism of $A_1$. [*(v)*]{} $(m(P), m(Q)) \neq (2,2)$: Since $m(P)=m(Q)=2$, we can write $P = P_r + P_p$ and $Q = Q_s + Q_q$ as sums of homogeneous elements where $r < p$, $P_r \in A_{1,r}$, $P_p \in A_{1,p}$ and $s<q$, $Q_s \in A_{1,s}$, $Q_q \in A_{1,q}$. The equality $[P,Q]=1$ implies that $[P_r, Q_s]=0$ and $[P_p, Q_q]=0$ (see Lemma \[a20Sep16\]). By Lemma \[a20Sep16\], the elements $r$ and $s$ have the same sign (i.e., either $r<0, s<0$ or $r=s=0$ or $r>0, s>0$) and also the elements $p$ and $q$ have the same sign. Since $p\geq 2$, we must have $q>0$. Suppose that $r\geq 0$, we seek a contradiction. Then $s\geq 0$ and so the elements $P$ and $Q$ are elements of the subring $A_{1,+}= \oplus_{i \geq 0} K[H] X^i$. Now, $$K[H] \ni 1 = [P, Q] \in [A_{1,+}, A_{1,+}] \subseteq \oplus_{i \geq 1} K[H] X^i,$$ a contradiction. Therefore, $r<0$ and $s<0$. The equality $1=[P,Q]=[P_r, Q_q] + [P_p, Q_s]$ and Lemma \[a20Sep16\] imply that $r+q=0$ and $p+s=0$, that is $r=-q$ and $s=-p$. So, $$P = P_{-q} + P_p \text{ and } Q = Q_{-p} + Q_q.$$ The elements $P_p$ and $P_{-q}$ are homogeneous elements of the Weyl algebra $A_1$. The Weyl algebra $A_1$ is a homogeneous subalgebra of the algebra $K(H)[X, X^{-1}; \s] = K(H)[Y, Y^{-1}; \s^{-1}]$ where $K(H)$ is the field of rational functions in the variable $H$ and the automorphism $\s$ of $K(H)$ is given by the rule $\s(H) = H-1$. By [@Bav-DixPr5 Proposition 2.1(1)], the centralizer $C_B(P_p)$ of the element $P_p$ in $B$ is a Laurent polynomial algebra $K[\alpha X^n, (\alpha X^n)^{-1}]$ for some nonzero element $\alpha\in K(H)$ and $n \geq 1$. In general, $\alpha \notin K[H]$. Similarly, $C_B(P_{-q}) = K[\beta Y^m, (\beta Y^m)^{-1}]$ for some nonzero element $\beta \in K(H)$ and $m \geq 1$. Since $[P_p, Q_q]=0$, $Q_q \in C_B(P_p)$ and $$P_p = \lambda(P_p) (\alpha X^n)^i = \lambda(P_p) \alpha \s^n(\alpha) \cdots \s^{n(i-1)}(\alpha) X^{ni} = \alpha_{n,i} X^p,$$ $$Q_q = \lambda(Q_q) (\alpha X^n)^j = \lambda(Q_q) \alpha \s^n(\alpha) \cdots \s^{n(j-1)}(\alpha) X^{nj} = \alpha'_{n,j} X^q,$$ for some nonzero scalars $\lambda(P_p), \lambda(Q_q) \in K^*$ and some $i \geq 1$ and $j\geq 1$ where $$\alpha_{n,i} = \lambda(P_p) \alpha \s^n(\alpha) \cdots \s^{n(i-1)}(\alpha) \in K[H], \; p=ni,$$ $$\alpha'_{n,j} = \lambda(Q_q) \alpha \s^n(\alpha) \cdots \s^{n(j-1)}(\alpha) \in K[H], \; q=nj.$$ Since $[P_{-p}, Q_{-p}]=0$, $Q_{-p} \in C_B(P_{-q})$ and $$P_{-q} = \lambda(P_{-q}) (\beta Y^m)^s = \lambda(P_{-q}) \beta \s^{-m}(\beta)\cdots \s^{-m(s-1)}(\beta) Y^{ms} = \beta_{m,s} Y^p,$$ $$Q_{-p} = \lambda(Q_{-p}) (\beta Y^m)^t = \lambda(Q_{-p}) \beta\s^{-m}(\beta)\cdots \s^{-m(t-1)}(\beta) Y^{mt} = \beta'_{m,t} Y^q,$$ for some nonzero scalars $\lambda(P_{-q}), \lambda(Q_{-p}) \in K^*$ and some $s \geq 1$ and $t\geq 1$ where $$\beta_{m,s} = \lambda(P_{-q}) \beta \s^{-m}(\beta) \cdots \s^{-m(s-1)}(\beta) \in K[H],\; p=ms,$$ $$\beta'_{m,t} = \lambda(Q_{-p})\beta \s^{-m}(\beta) \cdots \s^{-m(t-1)}(\beta) \in K[H], \; q=mt.$$ Now, $$1 = [P,Q] = [P_p, Q_{-p}] + [P_{-q}, Q_q] = [\alpha_{n,i} X^p, \beta_{m,t}' Y^p ] + [\beta_{m,s} Y^q , \alpha_{n,j}' X^q]$$ $$= \alpha_{n,i} \s^p (\beta'_{m,t}) (p,-p) - \beta'_{m,t} \s^{-p}(\alpha_{n,i}) (-p,p)$$ $$+ \beta_{m,s} \s^{-q}(\alpha'_{n,j}) (-q,q) - \alpha'_{n,j} \s^{q}(\beta_{m,s}) (q,-q).$$ Using the equalities $(-p,p)=\s^{-p}((p,-p))$ and $(-q,q)=\s^{-q}((q,-q))$, the last equality above can be rewritten as follows $$\label{1=ab} 1 = (1-\s^{-p})(a) + (1-\s^{-q})(b)$$ where $a = \alpha_{n,i} \s^{p}(\beta'_{m,t}) (p,-p) \in K[H]$ and $b = \alpha'_{n,j} \s^{q}(\beta_{m,s}) (q,-q) \in K[H]$. Recall that $P = P_{-q} + P_p$, $Q = Q_{-p} + Q_q$, $$\label{2=ab} p = mt=ni \geq 2 \text{ and } q=ms=nj \geq 1.$$ Suppose that $p=q$, and so $P = P_{-p} + P_p$, $Q = Q_{-p} + Q_p$. Then $Q=\lambda P_p$ for some $\lambda\in K^*$. Since $1=[P,Q]=[P, Q-\lambda P]$, $m(P)=2$ and $m(Q-\lambda P)=1$. By the case (iv), the pair $(P, Q-\lambda P)$ is obtained from the pair $(Y, X)$ by applying an automorphism of the Weyl algebra $A_1$. So, either $p<q$ or $p>q$. In view of $(P,Q)$-symmetry ($1=[P,Q]=[-Q,P]$), it suffices to consider, say, the first case only. Since $p<q$, the equalities (\[2=ab\]) imply that $i<j$ and $t<s$. Then, using Equation (\[degsf\]) and the fact that $\deg (p,-p) = p$ for all $p \geq 1$, we see that $$\deg a = \deg \alpha_{n,i} + \deg \beta'_{m,t} + p -1,$$ $$\deg b = \deg \alpha'_{n,j} + \deg \beta_{m,s} + q -1.$$ Since $i<j$ and $t<s$, $\deg \alpha_{n,i} < \deg \alpha'_{n,j}$ and $\deg \beta'_{m,t} < \deg \beta_{m,s}$. In particular, $\deg a < \deg b$. This equality contradicts Equation (\[1=ab\]) since, by Equation (\[degsf\]), $$0=\deg 1 = \deg a - 1 - \deg b +1 = \deg a - \deg b >0.$$ This means that the cases $p<q$ and $p>q$ are impossible. The proof of the theorem is complete. $\Box$ \[24Sep16\] Let $P, Q$ be elements of the first Weyl algebra $A_1$ with $m(P)=1$ or $m(Q)=1$. If $[P,Q]=1$ then $P = \tau(Y)$ and $Q=\tau(X)$ for some automorphism $\tau\in\Aut_K (A_1)$. [**Proof:**]{} Without loss of generality we may assume $m(Q)=1$ and $m(P)\geq 3$. That is $Q = Q_q$ and $P = \sum_{i\in I} P_i$, where $I \subset \Z$ is a finite set, $q\in \Z\setminus\{0\}$ and the elements $Q_q$ and $P_i$ are homogeneous in $A_1$. By Equation (\[xiaut1\]), we may assume that $q>0$. Then $1 = [P,Q]=\sum_i [P_i, Q_q]$ implies that $-q\in I$, $[P_{-q}, Q_q]=1$ and $[P_j, Q_q]=0$ for all $j\in I$ such that $j\neq -q$. By Theorem \[20Sep16\], $$q=1, Q_1 = \l X \text{ and } P_{-1} = \l^{-1}Y \text{ for some }\l\in K^*.$$ By Lemma \[a20Sep16\], $C := P - P_{-1} \in C_A(X)=K[X]$. Then $P = \tau(Y)$ and $Q=\tau(X)$ where $\tau: A_1 \to A_1$, $X \mapsto \l X$, $Y \mapsto \l^{-1} Y + C$, is an automorphism. Acknowledgements {#acknowledgements .unnumbered} ================ This paper has been written during the visit of V. V. Bavula to Aachen in 2016, which was supported by the Graduiertenkolleg “Experimentelle und konstruktive Algebra" of the German Research Foundation (DFG). The second author has been supported by Project II.6 of SFB-TRR 195 “Symbolic Tools in Mathematics and their Applications” of the DFG. [99]{} K. Adjamagbo and A. R. P. van den Essen, A proof of the equivalence of the Dixmier, Jacobian and Poisson Conjectures, [*Acta Mathematica Vietnamica*]{} [**32**]{} (2007), no. 3, 15–23. H. Bass, E. H. Connel and D. Wright, The Jacobian Conjecture: reduction of degree and formal expansion of the inverse, [*Bull. Amer. Math. Soc. (New Series)*]{}, [**7**]{} (1982), 287–330. V. V. Bavula, Finite-dimensionality of $Ext^n$ and $Tor_n$ of simple modules over a class of algebras, [*Funct. Anal. Appl.*]{} [**25**]{} (1991), no. 3, 229–230. V. V. Bavula, Dixmier’s Problem 5 for the Weyl Algebra, [*J. Algebra*]{} [**283**]{} (2005), no. 2, 604–621. V. V. Bavula, A Question of Rentschler and the Problem of Dixmier, [*Ann. of Math.*]{} [**154**]{} (2001), no. 3, 683–702. V. V. Bavula, The Jacobian Conjecture implies the Dixmier Problem, (2005). arxiv:math/0512250 (3 pages). V. V. Bavula, An Analogue of the Conjecture of Dixmier is true for the ring of polynomial integro-differential operators, [*J. Algebra*]{} [**372**]{} (2012), 237–250. A. Belov-Kanel and M. Kontsevich, The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture, [*Moscow Math. J.*]{} [**7**]{} (2007), no. 2, 209–218. J. Dixmier, Sur les algèbres de Weyl, [*Bull. Soc. Math. France*]{} [**96**]{} (1968), 209–242. A. Joseph, The Weyl algebra—semisimple and nilpotent elements, [*Amer. J. Math.*]{} [**97**]{} (1975), no. 3, 597–615. Y. Tsuchimoto, Endomorphisms of Weyl algebra and $p$-curvatures, [*Osaka J. Math.*]{} [**42**]{} (2005), no. 2, 435-452.

--- abstract: 'Parallel tempering simulates at many quark masses simultaneously, by changing the mass during the simulation while remaining in equilibrium. The algorithm is faster than pure HMC if more than one mass is needed, and works better the smaller the smallest mass is.' address: | Centre for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan.\ email:boyd rccp.tsukuba.ac.jp author: - 'G. BOYD[^1]' title: | \ Tempered Fermions in the Hybrid Monte Carlo Algorithm --- INTRODUCTION ============ The standard algorithms used for full QCD are painfully slow. Physically large objects, like instantons, pions etc., may well decorrelate much more slowly than, say, the plaquette [@us]. Tempered algorithms [@us; @ST; @st_biel] promote a parameter of the theory to a dynamical variable that changes during the simulation, which has tremendous potential for speeding up slow simulations. They have been successfully implemented in $\beta$ for spin glasses, U(1) [*etc*]{}. For QCD at zero temperature promoting $\beta$ does not help (although at high temperature, around the chiral phase transition it probably will), but promoting the quark mass, and allowing it to change during the simulation, does speed things up. The mass is only changed if the configuration is simultaneously in the equilibrium distribution of both masses. So tempering is always in equilibrium and requires no re-weighting etc. afterwards. The minimum gain comes from running at heavier (faster) masses between independent configurations. The maximum gain comes if the relevant auto-correlations are smaller for larger masses. TEMPERING ========= The quark mass, $a{\ensuremath{m_{\text{q}}}}$ for staggered or $\kappa$ for Wilson fermions, becomes a dynamic variable, and may take a different value for each trajectory of, say, the hybrid Monte Carlo algorithm [@hmca]. (Regard $m$ below as the corresponding $\kappa$ for Wilson fermions.) The masses used belong to an ordered set with $N_{m}$ elements, $[m_{\text{min}}, ... ,m_{\text{max}}]$. The only requirement is that the action histograms of neighbouring masses overlap. There are two types of tempering, simulated and parallel tempering. The idea behind simulated tempering in QCD, investigated in [@st_biel], is very simple, and as it is the basis for the parallel tempering investigated here it will be described first. In simulated tempering you add to the probability distribution a constant $g_{i}$ for each mass $m_{i}$, which indicates roughly where the half-way point between the action histograms of mass $m_{i}$ and $m_{i+1}$ is. The original QCD probability distribution $P(U,\phi)$ now becomes $$P(U,\phi,i) \propto \exp[-S(U,\phi,\beta,m_{i}) + g_{i}].$$ This distribution is simulated using your favourite algorithm for fixed quark mass (eg., HMC here), combined with Metropolis steps to change from $m_i$ to $m_{i\pm 1}$. The constants $g_{i}$ are only to enable the masses to change both up and down, and do not affect the physics. The hybrid Monte Carlo algorithm insures that the correct Gibbs distribution is generated at each value of the mass, and the tempering Metropolis step insures that the mass only changes if the configuration is part of the equilibrium distribution of both masses. The constants $g_{i}$ can be chosen freely, depending on what seems best for the simulation. They do not affect the physics, only the frequency with which each mass is visited. The $g_{i}$ can be fixed by choosing, for example, to visit each mass with equal probability, $P(i)=1/N_{m}$. Then $g_{i} = -\ln Z_{i}$, ie. the original free energy at fixed mass $m_{i}$. The choice is arbitrary, though, and can be optimized for speed. The simulation only needs $g_{i+1}-g_{i}$, and a good starting point is to take the first two terms below: $$\Delta g = - \langle{\bar{\psi}}\psi\rangle V\delta m - \langle\chi\rangle V(\delta m)^{2} + O((\delta m)^{3}) \label{eq:delg}$$ where $\langle{\bar{\psi}}\psi\rangle$ and $\langle \chi\rangle$ are the chiral condensate and susceptibility. The requirement of overlapping histograms implies that $\delta m$ satisfies $$\delta m \sim 1 /\sqrt{\langle\chi\rangle V} \sim m_{\sigma}/\sqrt{V}.$$ The overhead depends on the step size $\delta m$. As the susceptibility is related to the scalar meson mass, $\chi = A_{\sigma}/m_{\sigma}^{2}$, the step size is large in the chiral limit. Hence simulated tempering becomes more effective for very small quark masses, with the gain in speed more than compensating for the $N_{m}^{2}$ cost of having additional masses. The volume dependence above is in units of some relevant correlation length, so taking the continuum limit in fixed physical volume does not cause the method to break down. A further improvement comes if you temper in a parallel way. If $N_{m}$ different masses are needed, you can happily do $N_{m}$ different simulated tempering runs simultaneously on different computers. You can even go one better, and put them together on one computer in a way that removes the need for the constants $g_{i}$ and improves the performance. This is called parallel tempering; results are presented in the next section. In parallel tempering you first generate one (thermalised) configuration $C_{i}$ at each of the masses. Then use a Metropolis step to decide whether the configuration $C_{i}$ at mass $m_{i}$, and configuration $C_{i+1}$ at mass $m_{i+1}$ should be swapped for the next trajectory. If this is done, the next trajectory will run with $C_{i+1}$ at mass $m_{i}$, and $C_{i}$ at mass $m_{i+1}$. After each trajectory one starts trying to swap masses 1 and 2, moving up through the list, ending by trying to swap the configurations at masses $N_{m}-1$ and $N_{m}$. This method doesn’t need any constants $g_{i}$, changes the mass at two rather than one configuration, and has the further advantage of generating a configuration at each mass every trajectory. Also, trajectories way out in the tail of the distribution stand a chance of moving more than one step in mass. This doesn’t happen very often though! RESULTS ======= ------------- ----- ---- --------- --------- \(T) 0.020 475 15 4.9(14) 3.6(10) \(T) 0.024 350 21 3.3(8) 2.8(6) \(T) 0.028 283 24 4.2(15) 3.3(10) \(T) 0.032 231 21 3.5(9) 2.7(10) \(T) 0.036 193 – 3.6(12) 2.8(6) (HMC) 0.020 475 – 9.0(21) 8.4(17) (HMC) 0.040 166 – 8.5(20) 6.9(15) (HMC) 0.060 84 – 7.3(13) 5.8(13) ------------- ----- ---- --------- --------- : Parameters and results from the tempered (T) and a standard HMC run. The trajectories have unit length. The last three columns give the acceptance rate for the Metropolis mass changes, and the integrated autocorrelation times of the plaquette and $2\times 2$ Wilson loop.[]{data-label="tab:tauint"} Two sets of runs are in progress with four staggered fermions on an $8^{3}\times 12$ lattice. The tempered one has five masses, $\{0.020, 0.024, 0.028, 0.032, 0.036 \}$, called set ‘T’. For comparison there are three standard HMC runs at masses 0.020, 0.040 and 0.060. The run parameters are given in table \[tab:tauint\]. So far about 2000 units in $\tau$ have been run for T, and about 3500 for the HMC run. Which of the five configurations of the tempered run used $m=0.020$ at which time can be seen in the time history of figure \[fig:mass\]. It is clear that all five configurations have run at each mass about equally often, as required. The acceptance rate for transitions between masses is also shown in table \[tab:tauint\], and lies around 20%. Another run with closer masses, $\{0.020, 0.023, 0.026, 0.029, 0.032 \}$, yielded rates about ten percentage points higher. An acceptance rate of around 20% to 30% seems optimal. A measure of the speed of an algorithm is $\tau_{\text{int}}^{O}$, the integrated auto-correlation [@tauint] for an observable $O$. With insufficient data, $N_{\text{data}}<1000\tau_{\text{int}}$, an accurate value cannot be obtained. The largest value for $\tau_{\text{int}}^{O}$ of all observables $O$ defines the number of independent configurations. For full QCD the global topological charge $Q$ seems to be the slowest observable[@us] . However, on this size lattice the topology (field theoretic definition) turned out to depend on the action used for cooling, and is probably not well defined[^2]. The plaquette and Wilson loops up to $2\times 2$ seem to have the most well defined auto-correlation, and have been used here. In figure \[fig:cvect\] the auto-correlation function for the $2\times 2$ Wilson loop at $m=0.020$ from the tempering and a standard HMC run is plotted. The integrated autocorrelations obtained for both the plaquette and the $2\times 2$ Wilson loop are given in table \[tab:tauint\]. These turn out to be about 20% smaller than the slope parameter needed to fit the central part of the correlator in figure \[fig:cvect\] to an exponential. For observables from the tempered run, $\tau_{\text{int}}$ is about three times smaller than from the standard HMC run. The computer time needed per tempered trajectory, $T_{\text{T}}$, compared with the time for a single HMC run at the smallest mass yields $T_{\text{T}}= 3.22 T^{m=0.02}_{\text{HMC}}$. CONCLUSIONS =========== Tempering yields an integrated auto-correlation that is about a factor of three smaller than the HMC run, although much better data is needed to make this reliable! Tempering costs about three times more than the single HMC run at the smallest mass, so it is clearly faster if more than one mass is required, as is usually the case! For realistic simulations, using improved actions on large, smooth lattices at very small quark masses, tempered methods are very likely to be of benefit. This is especially true for Wilson fermions, where many $\kappa$ values are needed in order to extract meaningful physics. Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by HCM-Fellowship contract ERBCHBGCT940665, and currently the Japanese Society for the Promotion of Science. The author is grateful to both the Pisa and Tsukuba lattice groups for many useful discussions related to this work. [10]{} G. Boyd [*et al.*]{}, Proceedings, Lattice 96, IFUP-TH 47/96, hep-lat/96008123; B. Allés, [*et al.*]{}, Phys. Lett [**B 389**]{} (1996) 107; K. Bitar [*et al.*]{}, Phys. Rev. [**D42**]{} (1990) 3794. E. Marinari and G. Parisi, Europhys. Lett. [**19**]{} (1992) 451; A. P. Lyubartsev [*et al.*]{}, J. Chem. Phys. [**96**]{} (1992) 1776; L. Fernández [*et al.*]{}, J. Phys. I France [**5**]{} (1995) 1247; E. Marinari, proceedings, Bielefeld workshop on Multi-scale Phenomena, October 1996. and cond-mat/9612010. G. Boyd, proceedings, Bielefeld workshop on Multi-scale Phenomena, October 1996; hep-lat/9701009. S. Gottlieb [*et al.*]{}, Phys. Rev. [**D 35**]{} (1987) 2531; A. D. Kennedy, Nucl. Phys. [**B**]{} (Proc. Suppl.) [**30**]{} (1993) 96. See, for example, A. D. Sokal, in [*Quantum Fields on the Computer*]{}, Ed. M. Creutz, World Scientific 1992. [^1]: Combined proceedings for Lattice 97, Edinburgh and the International Workshop ’Lattice QCD on Parallel Computers’, University of Tsukuba, Japan. [^2]: The cooling actions tested used $1\times 1$ and $1\times 2$ loops with various values of the coefficients. On a given configuration different choices for the action lead to completely different global topological charges.

--- abstract: 'We show that, by studying the arrival times of radio pulses from highly-magnetized transient beamed sources, it may be possible to detect light pseudo-scalar particles, such as axions and axion-like particles, whose existence could have considerable implications for the strong-CP problem of QCD as well as the dark matter problem in cosmology. Specifically, such light bosons may be detected with a much greater sensitivity, over a broad particle mass range, than is currently achievable by terrestrial experiments, and using indirect astrophysical considerations. The observable effect was discussed in Chelouche & Guendelman (2009), and is akin to the Stern-Gerlach experiment: the splitting of a photon beam naturally arises when finite coupling exists between the electro-magnetic field and the axion field. The splitting angle of the light beams linearly depends on the photon wavelength, the size of the magnetized region, and the magnetic field gradient in the transverse direction to the propagation direction of the photons. If radio emission in radio-loud magnetars is beamed and originates in regions with strong magnetic field gradients, then splitting of individual pulses may be detectable. We quantify the effect for a simplified model for magnetars, and search for radio beam splitting in the 2GHz radio light curves of the radio loud magnetar XTEJ1810-197.' address: - 'Physics Department, Ben-Gurion University, Beer-Sheva 84105, Israel; guendel@bgu.ac.il' - 'Department of Physics, University of Haifa, Haifa 31905, Israel ; doron@sci.haifa.ac.il' author: - 'Eduardo I. Guendelman' - Doron Chelouche title: 'Radio-loud Magnetars as Detectors for Axions and Axion-like Particles' --- Introduction ============ The Peccei-Quinn mechanism (Peccei & Quinn 1977) was devised to elegantly solve to the strong-CP problem of quantum chromodynamics (QCD). This was accomplished by postulating a new quantum field and a new class of particles associated with it. The particles are pseudo-scalars that couple very weakly to the electromagnetic (EM) field. It later became apparent that such particles, termed axions, could also provide a solution to the dark matter problem (Khlopov 1999 and references therein). Besides QCD axions there are also the putative axion-like particles (ALPs), which may be related to the quintessence field, and whose existence is predicted by many versions of string theory. To date, however, there is no evidence for the existence of such particles and it is not clear that the Peccei-Quinn solution actually works. There is a longstanding interest in determining the physical properties of axions/ALPs. At present, laboratory experiments and astrophysical bounds imply that their coupling constant to the electromagnetic field, $g<10^{-10}\,{\rm GeV}^{-1}$ (see Chelouche et al. 2009 for summary). Mass limits are less stringent: if QCD axions are concerned, then their mass is probably $>10^{-6}$eV since otherwise the Universe would over-close, in contrast to observations. These limits, however, do not apply for ALPs. Here we follow the formalism given in Chelouche & Guendelman (2009; see also Guendelman 2008a,b,c) who outlined a new effect that arises from the coupling between the electromagnetic field and the axion field. The effect has the advantage of having unique observational signatures, which can be visible down to very small values of the (unknown) coupling constant compared to those accessible by other methods. Below, we outline the effect of beam splitting and look for it in the radio light curve of a radio-loud magnetar. Splitting in in-homogenous magnetic fields ========================================== The interaction term in the Lagrangian for the electromagnetic and the axion field is of the form $$\displaystyle L_{\rm int} = \frac{1}{4} g \tilde{F}^{\mu \nu}F_{\mu \nu}a = g{\bf E}\cdot {\bf B} a$$ where $E$ is the electric field (associated with the photon), $B$ the magnetic field, and $a$ the axion field. $g$ is the unknown coupling of particles to the EM field ($F_{\mu \nu}$, and its dual, $\tilde{F}^{\mu \nu}$). The full Lagrangian for the system can be written as $$L= -\frac{1}{4}F^{\mu \nu}F_{\mu \nu} -\frac{1}{2}m_\gamma^2A^2 + \frac{1}{2} \partial_\mu a \partial ^\mu a -\frac{1}{2} m_a^2 a^2 +L_{\rm int}$$ which is comprised of the free EM Lagrangian (including an effective photon mass, $m_\gamma$, term which takes into account potential refractive index in the medium) and the Klein-Gordon equations for free particles having a rest-mass $m_a$. In the absence of $L_{\rm int}$, photons and particles (e.g., axions/ALPs) are well defined energy states of the system. However, for finite coupling, the equation of motion for the photon-particle system takes the form (Raffelt & Stodolsky 1988)[^1], $$\left [ {\bf k}^2 -\omega^2+\left \vert \begin{array}{cc} m_\gamma^2 & -gB_\| \omega \\ -gB_\| \omega & m_a^2 \end{array} \right \vert \right ] \left ( \begin{array}{l} \gamma \\ a \end{array} \right )=0, \label{mat}$$ where $\omega$ is the photon energy and $B_\|$ the magnetic field in the direction of the photon polarization (the photon’s $E$ field). Clearly, neither pure photon nor pure axion/ALP states are eigenstates of the system but rather some combination of them. Let us now focus on the limit $$\vert m_a^2-m_\gamma(\omega)^2 \vert \ll gB_\| \omega. \label{cond2}$$ This condition is met either near resonance where $m_\gamma^2 \simeq m_a^2$ or when both masses are individually smaller than $\sqrt{gB_\| \omega}$ (which limit is actually met is immaterial). The eigenstates of equation \[mat\] are then given by $$\left \vert \psi \right >_- = \left [ \left \vert \gamma \right > + \left \vert a \right > \right ]/\sqrt{2},~~\left \vert \psi \right >_+ = \left [ \left \vert \gamma \right > - \left \vert a \right > \right ]/\sqrt{2} \label{psi}$$ where $\left \vert a \right >$ is the axion state and $\left \vert \gamma \right >$ is the photon state. The eigenvalues are $m_\pm^2= \pm gB_\| \omega$. By analogy with optics, these masses are related to effective refractive indices: $n_\pm=1+\delta n_\pm\simeq 1-{m_\pm^2}/{2\omega^2}$ (for $\vert \delta n_\pm\vert \ll1$) meaning that different paths through a refractive medium would be taken by the rays. We note that there is no dependence on the particle or photon mass so long as equation \[cond2\] is satisfied. In terms of the refractive index, and in complete analogy to mechanics, the equation of motion for a ray may be found by minimizing the action $\int d{\bf s} n({\bf s})$. It is straightforward to show (Chelouche & Guendelman 2009) that the momentum imparted on each state is $$\delta p_y^\pm = \mp (g/2) \int dz \left ( \partial B_x/\partial y \right ), \label{dpy}$$ where $B_x=B_x(y,z)$, and is taken to be parallel to the photon polarization (Fig. 1). Clearly, each of the beams will be affected in a similar way while gaining opposite momenta so that the total momentum is zero and the classical wave packet travels in a straight line (along the $z$-axis). This effect is analogous to the Stern-Gerlach experiment (Fig. 1). In the limit $n_\pm \simeq 1$, the separation angle between the beams is $$\delta \phi \simeq \frac{2\vert \delta p_y \vert}{p}, \label{theta}$$ where $p$ is the beam momentum along the propagation direction, i.e., the $z$-axis. This expression holds for small splitting angles and assumes relativistic particles. The magnitude of splitting depends on the relative angle between the photon polarization and the magnetic field, as well as on the geometry (and strength) of the magnetic field, which are poorly understood in magnetars. To gain a qualitative understanding of the magnitude of the effect, we assume $B\sim \int dz \left ( \partial B_x/\partial y \right ) $, where $B$ is the magnetic field in region through which the photon propagates. Taking current limits on the value of the coupling constant of $g\sim 10^{-10}\,{\rm GeV}^{-1}$, a photon frequency of 2GHz, and a typical magnetar magnetic field, $B\sim 10^{15}$G of we find that typical splitting phase between the photon beams is $$\delta \phi \sim 1\left ( \frac{g}{10^{-10}\,{\rm GeV}^{-1}} \right ) \left ( \frac{B}{10^{15}\,{\rm G}} \right ) \left ( \frac{\omega}{2\,{\rm GHz}} \right )^{-1}\,{\rm rad}$$ (here $\omega$ is the photon frequency in GHz, and $B$ is the magnetic field in Gauss). Provided that the intrinsic pulse emitted by the magnetar is narrower than the splitting angle, a double pulse is expected to appear due to the effect of splitting. In fact, in cases where the pulses are highly beamed, hence narrow in phase (see below), considerably smaller values of the coupling constant, $g$, may be probed. Furthermore, by choosing to work at lower frequencies, smaller coupling constants may be probed. This allows one to search for ALPs in a previously unexplored phase space using the radio light curves of radio-loud magnetars. Figure 2 shows an example for what a narrow radio pulse, typical of radio-loud magnetars (see below) would look like when split due to photon-particle coupling in a highly magnetized object ($B=10^{16}$G) observed at a radio frequency of 300MHz. Clearly, splitting may be discernible down to very low values of the coupling constant. We note that, depending on the strength of the magnetic field (and to some degree also on the poorly constrained plasma density), and the contribution of the vacuum birefringence term (Adler 1971) to the effective photon mass, two split pulses or one shifted pulse (with the shifting angle being $\delta \phi /2$) will be observed. In the latter case, light curves in two or more bands are required to detect the effect. The full treatment of such issues is beyond the scope of this contribution, and is discussed at some length in Chelouche & Guendelman (2009). Below we consider only the effect of splitting when studying the (monochromatic) light curve of a radio-loud magnetar. The case of XTEJ1810-197 ======================== We adopt a pragmatic approach when searching for the effect of photon-axion coupling-induced splitting in magnetars. Given the large uncertainties in the physics of magnetars and their radio emission mechanisms, we do not know whether the effect of splitting or shifting should be observed. In addition, it is certainly possible that not all radio pulses originate from the same region in the magnetosphere, and while in some cases splitting will be observed, in other cases beam shifting will be the relevant effect. Similarly, if different pulses are emitted from different regions having different polarizations with respect to a complicated, tangled geometry of the field, then many different splitting or shifting angles are predicted. For these reasons, we aim to study the statistics of phase differences between pulses. Should all radio pulses be emitted from the same region, the data will reflect on the typical phase difference between pulses. This phase difference, which is induced by photon-particle coupling, may be discernible from other phase difference scales, which relate to the physics of the radio-emitting regions in the magnetar. We consider the 2GHz radio observations of the radio-loud magnetar XTEJ1810-197 whose data were published by Camilo et al. (2006). A total of 40 object rotations were recorded, with the light curve of a few individual rotations shown in figure \[mag1\]. As discussed in Camilo et al. (2006), the light curves are characterized by narrow transient radio pulses and, as such, are very different than those typically observed toward pulsars. When averaging the light curves of individual rotations, evidence for quasi-periodicity appears, whereby the bulk of the radio emission is confined to certain orbital phases, akin to the better studied pulsar phenomenon. Aiming to statistically study the difference in arrival phases of radio pulses, we first need to positively identify the numerous, potentially weak, narrow transient features in the light curves of XTEJ1810-197. To this end, we devised the following “peak-finder” algorithm: for each light curve (rotation), we define the (initial) standard deviation of the light curve, $\sigma$. Only those peaks that satisfy $(f(t)-\left < f \right >)/\sigma>3$ \[$f(t)$ is the time-series and $\left < f \right >$ is its mean\], are identified as peaks, and are then removed from the observed light curve. A new standard deviation, $\sigma$, is calculated for the reduced light curve, and the peak identification algorithm is executed leading to new significant peaks being identified. The scheme iterates until $\sigma$ between successive iterations converges to better than 0.01%. Figure \[mag1\] shows the results of the peak identification algorithm for a light curve corresponding to a single stellar revolution. Clearly, all peaks lie well above the fluctuating background. The phase stamps of individual peaks are identified with their maxima. In cases were a multi-maxima ridge exists, the time stamps for individual maxima is recorded. We analyze the data from individual revolutions to be less sensitive to non-stationary effects in the light curve (e.g., a varying noise level between stellar revolutions). All peaks from all stellar revolutions were identified and their phase stamps, relative to the first revolution, logged. We then evaluate the phase difference distribution taking into account all peak pairs. The results are shown in figure \[mag2\]. A clear peak is observed, by definition, at around the stellar orbital period ($\delta \phi/2\pi=1$). Two small peaks at $\delta \phi/2\pi \sim 0.3,~0.7$ are due to the secondary pulse at $\phi/2\pi \sim 0.87$ (see the inset of Fig. \[mag2\]). A second significant time-scale is apparent at a phase difference of $\delta \phi/2\pi \lesssim 0.2$. This scale roughly corresponds to the phase width of the mean main pulse (a second mean pulse exists at $\phi/2\pi \sim 0.87$ and is not shown here). Interestingly, we cannot positively identify any particular phase difference scale for $\delta \phi/2\pi <0.2$, as might be expected due to the effect of splitting. In fact, the distribution is qualitatively consistent with the predictions from a purely random origin for the radio pulses (see dotted line in Fig. \[mag2\]). Further analysis is underway. Based on our preliminary analysis, we cannot find supporting evidence for beam splitting in the 2GHz light curve of XTEJ1810-197. There remains the open possibility that, for this object and this particular waveband, we are in the regime of beam shifting, and light curve comparison with [*simultaneous*]{} observations in other wavebands may be able to detect it. Given our limited understanding of magnetars and their radio emission processes, we do not claim to interpret our null result as a limit on the photon-ALP coupling constant, $g$, or on the existence of light bosons. Conclusions =========== We show that the effect of beam splitting due to finite coupling between the axion field and the electromagnetic field (Chelouche & Guendelman 2009) may be observable in the radio-light curves of radio-loud magnetars for a plausible range of values corresponding to the properties of magnetars and photon-to-axion coupling strength. The phase between the split pulses depends linearly on the magnetic field, the photon-particle coupling constant, and on the photon wavelength. As such, this effect can be used to detect axions and ALPs with much greater sensitivity than photon-axion/ALP oscillations. Our predictions indicate that, for narrow (beamed) radio pulses, the phase between the split pulses is likely to be $\lesssim 1$rad at 2GHz. Such a timescale will contribute to the statistics of phase differences between pulses, whose underlying form is determined by radio emission processes in the magnetar itself. Searching for discernible phase difference scales, which can be related to the beam-splitting effects, in the 2GHz light curve of XTEJ1810-197, shows no clear characteristic phase scale in the range $0.1-1$rad. Interestingly, a preliminary analysis shows that the data is qualitatively consistent with narrow pulsed emission being drawn from a random process. While this could be used to shed light on the radio emission mechanism in magnetars, we cannot draw any conclusions at this stage concerning the existence of light bosons or their coupling to the electromagnetic field. We thank Scott Ransom for providing us the 2GHz data for XTEJ1810-197 in electronic form. Adler, S. L. 1971, Annals of Physics, 67, 599 Camilo, F., Ransom, S. M., Halpern, J. P., Reynolds, J., Helfand, D. J., Zimmerman, N., & Sarkissian, J. 2006, Nature, 442, 892 Chelouche, D., Rabad[á]{}n, R., Pavlov, S. S., & Castej[ó]{}n, F. 2009, ApJS, 180, 1 Chelouche, D., & Guendelman, E. I. 2009, ApJL, 699, L5 Guendelman, E. I. 2008, Modern Physics Letters A, 23, 191 Guendelman, E. I. 2008, Physics Letters B, 662, 227 Guendelman, E. I. 2008, Physics Letters B, 662, 445 Khlopov, M. Y. 1999, Cosmoparticle Physics, World Scientific Peccei, R. D., & Quinn, H. R. 1977, Physical Review Letters, 38, 1440 Raffelt, G., & Stodolsky, L. 1988, Phys. Rev. D., 37, 1237 [^1]: Unless otherwise stated, we work in natural units so that $\hbar=c=1$.

--- abstract: 'In this paper, we study the entanglement properties of a spin-1 model the exact ground state of which is given by a Matrix Product state. The model exhibits a critical point transition at a parameter value $a=0$. The longitudinal and transverse correlation lengths are known to diverge as $a\rightarrow0$. We use three different entanglement measures $S(i)$ (the one-site von Neumann entropy), $S(i,j)$ (the two-body entanglement) and $G(2,n)$ (the generalized global entanglement) to determine the entanglement content of the MP ground state as the parameter $a$ is varied. The entanglement length, associated with $S(i,j)$, is found to diverge in the vicinity of the quantum critical point $a=0$. The first derivative of the entanglement measure $E$ $(=S(i),\: S(i,j))$ w.r.t. the parameter $a$ also diverges. The first derivative of $G(2,n)$ w.r.t. $a$ does not diverge as $a\rightarrow0$ but attains a maximum value at $a=0$. At the QCP itself all the three entanglement measures become zero. We further show that multipartite correlations are involved in the QPT at $a=0$.' author: - Amit Tribedi and Indrani Bose title: Quantum Critical Point and Entanglement in a Matrix Product Ground State --- Department of Physics Bose Institute 93/1, Acharya Prafulla Chandra Road Kolkata - 700 009, India I. INTRODUCTION {#i.-introduction .unnumbered} =============== The entanglement characteristics of the ground states of many body Hamiltonians describing condensed matter systems constitute an important area of study in quantum information theory. Entanglement is an essential resource in quatum computation and communication protocols. Condensed matter, specially, spin systems have been proposed as candidate systems for the realization of some of the protocols. Entanglement provides a measure of non-local quantum correlations in the system and it is of significant interest to determine how the correlations associated with the ground state of the system change as one or more than one parameter of the system is changed. The focus on ground state characteristics arises from the possibility of quantum phase transitions (QPTs) which occur at temperature $T=0$ (when the system is in its ground state) and are driven solely by quantum fluctuations [@key-1]. A QPT is brought about by tuning a parameter, either external or intrinsic to the Hamiltonian, to a special value termed the transition point. In thermodynamic critical phenomena, the thermal correlation length diverges and the thermodynamic quantities become singular as the critical point is approached. In the quantum case, the correlation length diverges in the vicinity of the QCP and the ground state properties develop non-analytic features. An issue of considerable interest is whether the quantum correlations, like the usual correlation functions, become long-ranged near the QCP. In a wider perspective, the major goal is to acquire a clear understanding of the variation in entanglement characteristics as a tuning parameter is changed. QPTs have been extensively studied in spin systems both theoretically and experimentally. In recent years, several theoretical studies have been undertaken to elucidate the relationship between QPTs and entanglement in spin systems [@key-2; @key-3; @key-4; @key-5; @key-6; @key-7]. In particular, a number of entanglement measures have been identified which develop special features close to the transition point. One such measure is concurrence which quantifies the entanglement between two spins $(S=\frac{1}{2})$. At a QCP, as illustrated by a class of exactly-solvable spin models $(S=\frac{1}{2})$, the derivative of the ground state concurrence has a logarithmic singularity though the concurrence itself is non-vanishing upto only next-nearest-neighbour-distances between two spins [@key-2; @key-3]. Discontinuities in the ground state concurrence have been shown to characterize first order QPTs [@key-8; @key-9; @key-10]. Later, Wu et al. [@key-5] showed that under some general assumptions a first order QPT, associated with a discontinuity in the first derivative of the ground state energy, gives rise to a discontinuity in a bipartite entanglement measure like concurrence and negativity. Similarly, a discontinuity or a divergence in the first derivative of the same entanglement measure is the signature of a second order phase transition with a discontinuity or a divergence in the second derivative of the ground state energy. Another measure of entanglement, studied in the context of QPTs, is the entropy of entanglement between a block of $L$ adjacent spins in a chain with the rest of the system [@key-4]. At the QCP, the entropy of entanglement diverges logarithmically with the length of the block. There is, however, no direct relation with the long range correlations in the system. A number of entanglement measures have recently been proposed which are characterized by a diverging length scale, the entanglement length, close to a QCP. The localizable entanglement (LE) between two spins is defined as the maximum average entanglement that can be localized between them by performing local measurements on the rest of the spins [@key-11]. The entanglement length sets the scale over which the LE decays. The two-body entanglement $S(i,j)$ is a measure of the entanglement between two separated spins, at sites $i$ and $j$, and the rest of the spins [@key-7]. Let $\rho(i,j)$ be the reduced density matrix for the two spins, obtained from the full density matrix by tracing out the spins other than the ones at sites $i$ and $j$. The two body entanglement $S(i,j)$ is given by the von Neumann entropy $$S(i,j)=-Tr\,\rho(i,j)\, log_{2}\,\rho(i,j)\label{1}$$ In a translationally invariant system, $S$ depends only on the distance $n=\mid j-i\mid$. As pointed out in [@key-7], the spins that are entangled with one or both the spins at sites $i$ and $j$ contribute to $S$. The following results have been obtained in the case of the $S=\frac{1}{2}$ exactly solvable anisotropic $XY$ model in a transverse magnetic field. The model, away from the isotropic limit, belongs to the universality class of the transverse Ising model. The two-body entanglement $S(i,j)$ has a simple dependence on the spin correlation functions in the large $n$ limit. Away from the critical point, $S(i,j)$ is found to saturate over a length scale $\xi_{E}$ as $n$ increases. Near the QCP, one obtains $$S(i,j)-S(\infty)\sim n^{-1}\, e^{-\frac{n}{\xi_{E}}}\label{2}$$ The entanglement length (EL), $\xi_{E}$, has an interpretation similar to that in the case of LE. The EL diverges with the same critical exponent as the correlation length at the QCP. $S(i,j)$ thus captures the long range correlations associated with a QPT. At the critical point itself, $S(i,j)-S(\infty)$ has a power-law decay, i.e., $S(i,j)-S(\infty)\sim n^{-\frac{1}{2}}$. In the limit of large $n$, the first derivative of $S(i,j)$ w.r.t. a Hamiltonian parameter develops a $\lambda-$like cusp at the critical point. The universality and a finite-size scaling of the entanglement have also been demonstrated. The one-site von Neumann entropy $$S(i)=-Tr\,\rho(i)\, log_{2}\,\rho(i)\label{3}$$ is also known to be a good indicator of a QPT [@key-3]. It provides a measure of how a single spin at the site $i$ is entangled with the rest of the system. The reduced density matrix $\rho(i)$ is obtained from the full density matrix by tracing out all the spins except the one at the site $i$. Oliveira et al. [@key-6] have proposed a generalized global entanglement (GGE) measure $G(2,n)$ which quantifies multipartite entanglement (ME). $G(2,n)$ for a translationally symmetric system is given by $$G(2,n)=\frac{d}{d-1}[1-\sum_{l,m=1}^{d^{2}}\mid[\rho(j,j+n)]_{lm}\mid^{2}]\label{4}$$ where $\rho(j,j+n)$ is the reduced density matrix of dimension $d$. The factor 2 in $G(2,n)$ indicates that the reduced density matrix is that for a pair of particles. Wu et al. [@key-5] considered QPTs characterized by non-analyticities in the derivatives of the ground state energy. These arise from the non-analyticities in one or more of the elements of the reduced density matrix. In terms of the GGE, a discontinuity in $G(2,n)$ signals a first order QPT, brought about by a discontinuity in one or more of the elements, $[\rho_{j,j+n}]_{lm}$ of the reduced density matrix [@key-6]. A discontinuity or divergence in the first derivative of $G(2,n)$ w.r.t. the tuning parameter occurs due to a discontinuity or divergence in the first derivetives of one or more of the elements of the reduced density matrix. The associated QPT is of second order. Non-analyticities in $G(2,n)$ and its derivatives thus serve as indicators of QPTs. In the case of the $XY$ $S=\frac{1}{2}$ spin chain, the GGE measure shows a diverging EL as the QCP is approached. The EL $\xi_{E}=\frac{\xi_{C}}{2}$ where $\xi_{C}$ is the usual correlation length. Thus, both the length scales diverge with the same critical exponent near the QCP. The relationship between entanglement and QPTs has mostly been explored for spin-$\frac{1}{2}$ systems. The entanglement properties of the ground states of certain spin$-1$ Hamiltonians have been studied using different measures [@key-11; @key-12; @key-13]. Numerical studies show that the LE has the maximal value for the ground state of the spin-1 Heisenberg antiferromagnet with open boundary conditions (OBC) [@key-13]. In the case of the spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) model [@key-14], the result can be proved exactly. A class of spin-1 models, the $\phi$-deformed AKLT models, is characterized by an exponentially decaying LE with a finite EL $\xi_{E}$. The length $\xi_{E}$ diverges at the point $\phi=0$ though the conventional correlation length remains finite [@key-13]. A recent study [@key-15] shows that in the case of spin-1 systems, the use of LE for the detection of QPTs is not feasible. An example is given by the $S=1$ XXZ Heisenberg antiferromagnet with single-ion anisotropy. The model has a rich phase diagram with six different phases. The LE is found to be always 1 in the entire parameter region and hence is insensitive to QPTs. The ground states of certain spin-1 models have an exact representation in terms of matrix product states (MPS) [@key-16; @key-17; @key-18]. The ground state of the spin-1 AKLT model, termed a valence bond solid (VBS) state, is an example of an MPS. The ground state is characterized by short-ranged spin-spin correlations and a hidden topological order known as the string order. The excitation spectrum of the model is further gapped. In the MPS formalism, ground state expectation values like the correlation functions are easy to calculate. This has made it particularly convenient to study phase transitions in spin models with MP states as exact ground states [@key-17]. The transitions identified so far include both first and second order transitions and are brought about by the tuning of the Hamiltonian parameters. The second order transition in the class of finitely correlated MP states, however, differs from the conventional QPT in one important respect. The spin correlation function is always of the form $A_{C}\, e^{-\frac{n}{\xi_{C}}}$ for large $n$. The correlation length $\xi_{C}$ diverges as the transition point is approached. The pre-factor $A_{C}$, however, vanishes at the transition point [@key-17]. This is in contrast to the power-law decay of the correlation function at a conventional QCP. Some distinct features of QPTs in MP states have recently been identified [@key-19]. One of these relates to the analyticity of the ground state energy density for all values of the tuning parameter. In a conventional QPT, the energy density becomes non-analytic at the QCP. The MP states appear to provide an ideal playground for exploring novel types of QPTs. In this paper, we consider a spin-1 model, the exact ground state of which is given by an MP state [@key-20]. The model has a rich phase diagram with a number of first order phase transitions and a critical point transition. We study the entanglement properties of the ground state with a view to pinpoint the special features which appear close to the critical point. This is done by using three different entanglement measures, namely, the single-site, two-body and generalized global entanglement defined earlier. II. REDUCED DENSITY MATRIX OF MP {#ii.-reduced-density-matrix-of-mp .unnumbered} ================================= GROUND STATE {#ground-state .unnumbered} ============ We consider a spin-1 chain Hamiltonian proposed by Klümper et al. [@key-20] which describes a large class of antiferromagnetic (AFM) spin-1 chains with MP states as exact ground states. The Hamiltonian satisfies the symmetries : (i) rotational invariance in the $x-y$ plane, (ii) invariance under $S^{z}\rightarrow-S^{z}$ and (iii) translation and parity invariance. The Hamiltonian has the general form $$H=\sum_{j=1}^{L}h_{j,\, j+1}$$ $$h_{j,\, j+1}=\alpha_{0}A_{j}^{2}+\alpha_{1}(A_{j}B_{j}+B_{j}A_{j})+\alpha_{2}B_{j}^{2}+\alpha_{3}A_{j}+\alpha_{4}B_{j}(1+B_{j})+$$ $$+\alpha_{5}((S_{j}^{z})^{2}+(S_{j+1}^{z})^{2}+C\label{5}$$ where $L$ is the number of sites in the chain and periodic boundary conditions (PBC) hold true. The parameters $\alpha_{j}$ are real and $C$ is a constant. The nearest-neighbour (n.n.) interactions are $$\begin{array}{c} A_{j}=S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}\\ B_{j}=S_{j}^{z}S_{j+1}^{z}\end{array}\label{6}$$ The constant $C$ in Eq. (5) may be adjusted so that the ground state eigenvalue of $h_{j,\, j+1}=0$. Hence $$h_{j,\, j+1}\geq0\quad\Rightarrow H\geq0\label{7}$$ i.e., $H$ has only non-negative eigenvalues. In the AFM case, the $z$-component of the total spin of the ground state $S_{tot}^{z}=0$. Klümper et al. showed that in a certain subspace of the $\alpha_{j}-$parameter space , the AFM ground state has the MP form. Let $\left|0\right\rangle $ and $\left|\pm\right\rangle $ be the eigenstates of $S^{z}$ with eigenvalues $0$, $+1$ and $-1$ respectively. Define a $2\times2$ matrix at each site $j$ by $$g_{j}=\left(\begin{array}{cc} \left|0\right\rangle & -\sqrt{a}\left|+\right\rangle \\ \sqrt{a}\left|-\right\rangle & -\sigma\left|0\right\rangle \end{array}\right)\label{8}$$ with non-vanishing parameters $a,\,\sigma\neq0$. The global AFM state is written as $$\left|\psi_{0}\,(a,\,\sigma)\right\rangle =Tr\,(g_{1}\otimes g_{2}\otimes......\otimes g_{L})\label{9}$$ where ‘$\otimes$’ denotes a tensor product. One can easily check that $S_{tot}^{z}\left|\psi_{0}\right\rangle =0$, i.e., the state is AFM. One now demands that the state $\left|\psi_{0}\,(a,\,\sigma)\right\rangle $ is the exact ground state of the Hamiltonian $H$ with eigenvalues $0$. For this, it is sufficient to show that $$h_{j,\, j+1}\:(g_{j}\otimes g_{j+1})=0\label{10}$$ Eq. (3) and (10) are satisfied provided the following equalities $$\begin{array}{cc} 1)\,\sigma=sign(\alpha_{3}), & 2)\, a\,\alpha_{0}=\alpha_{3}-\alpha_{1},\\ 3)\,\alpha_{5}=\mid\alpha_{3}\mid+\alpha_{0}(1-a^{2}), & 4)\,\alpha_{2}=\alpha_{0}a^{2}-2\mid\alpha\mid\end{array}\label{11}$$ and inequalities $$a\neq0,\:\alpha_{3}\neq0,\:\alpha_{4}>0,\:\alpha_{0}>0\label{12}$$ hold true. The state $\left|\psi_{0}\,(a,\,\sigma)\right\rangle $ is the ground state of the Hamiltonian $(5)$ with ground state energy zero provided the equalities in $(8)$ are satisfied. The inequalities constrain the other eigenvalues of $h_{j,j+1}$ to be positive. If the inequalities are satisfied, the ground state can be shown to be unique for any chain length $L$. Also, in the thermodynamic limit $L\rightarrow\infty$, the excitation spectrum has a gap $\Delta$. With equality signs in the inequalities $(12)$, the state $\left|\psi_{0}\,(a,\,\sigma)\right\rangle $ is still the ground state but is no longer unique. The spin$-1$ model has the typical feature of a Haldane-gap (HG) antiferromagnet. In fact, the AKLT model is recovered as a special case with $a=2,\:\sigma=1,\:\alpha_{3}=3\alpha_{0}>0,\:\alpha_{2}=-2\alpha_{0}$ and $\alpha_{4}=3\alpha_{0}$. The state $(9)$ now represents the VBS state. Using the transfer matrix method [@key-16], the ground state correlation functions can be calculated in a straightforward manner. The results are $(L\rightarrow\infty,\, r\geq2)$ : Longitudinal correlation function $$\left\langle S_{1}^{z}\, S_{r}^{z}\right\rangle =-\frac{a^{2}}{(1-|a|)^{2}}\left(\frac{1-|a|}{1+|a|}\right)^{r}\label{13}$$ Transverse correlation function$$\left\langle S_{1}^{x}\, S_{r}^{x}\right\rangle =-|a|\,[\sigma+sign\, a]\left(\frac{-\sigma}{1+|a|}\right)^{r}\label{14}$$ The correlations $(13)$ and $(14)$ decay exponentially with the longitudinal and transverse correlation lengths given by$$\xi_{l}^{-1}=ln\left|\frac{1+|a|}{1-|a|}\right|,\;\;\xi_{t}^{-1}=ln(1+|a|)\label{15}$$ Furthermore, the string order parameter has a non-zero expectation value in the ground state. One finds that the correlation lengths diverge as $a\rightarrow0$. At the point $a=0$, the correlation functions given by Eq. $(13)$ and $(14)$ are zero. At a conventional QCP, the correlation functions have a power-law decay. We will, however, refer to the point as a QCP since the correlation lengths diverge as the point is approached. A consequence of the diverging correlation length is that the excitation spectrum of the spin-1 model, which is gapped (the Haldane phase) for $a>0$, becomes gapless at the critical point $a=0$ [@key-20]. The presence or absence of a gap in the excitation spectrum of a system is reflected in the low temperature thermodynamic properties of the system. Furthermore, the string order parameter has a non-zero expectation value in the ground state for $a>0$ and becomes zero at $a=0$ indicating the appearance of a new phase. Refs. [@key-16; @key-17] provide several other examples of spin-1 models with finitely correlated MP states as exact ground states. All these models exhibit critical point transitions with features similar to those in the case of the spin-1 model described by the Hamiltonian in Eq. $(5)$. We now focus on the entanglement properties of the MP ground state (Eq. $(9)$). We consider $a$ to be $\geq0$ and $\sigma=+1$ in Eq. $(8)$. The one-site reduced density matrix $\rho(i)$ (Eq. $(3)$) obtained by tracing out all the spins except the $i$-th spin from the ground state density matrix $\rho=\left|\psi_{0}\right\rangle $$\left\langle \psi_{0}\right|$, can be calculated using the transfer matrix method [@key-16]. The density matrix, from Eq. $(9)$, is$$\rho=\left|\psi_{0}\right\rangle \left\langle \psi_{0}\right|=\sum_{\{ n_{\alpha},m_{\alpha}\}}g_{n_{1}n_{2}}\, g_{n_{2}n_{3}}........g_{n_{L}n_{1}}\: g_{m_{1}m_{2}}^{\dagger}\, g_{m_{2}m_{3}}^{\dagger}......g_{m_{L}m_{1}}^{\dagger}\label{16}$$ The summation is over all the indices, $n_{i}$, $m_{i}$, $i=1,2,.....L$. We define a $4\times4$ matrix $f$ (the elements of which are operators) at any lattice site as $$f_{\mu_{1}\mu_{2}}\Rightarrow f_{(n_{1},m_{1})(n_{2},m_{2})}\equiv g_{n_{1}n_{2}}\, g_{m_{1}m_{2}}^{\dagger}\label{17}$$ The convention of the ordering of the multi-indices is $\mu=1,2,3,4\:\leftrightarrow(11),(12),(21),(22)$. Thus, $f$ can be written as $$f=\left(\begin{array}{cccc} \left|0\right\rangle \left\langle 0\right| & -\sqrt{a}\left|0\right\rangle \left\langle 1\right| & -\sqrt{a}\left|1\right\rangle \left\langle 0\right| & a\left|1\right\rangle \left\langle 1\right|\\ \sqrt{2}\left|0\right\rangle \left\langle -1\right| & -\left|0\right\rangle \left\langle 0\right| & -a\left|1\right\rangle \left\langle -1\right| & \sqrt{a}\left|1\right\rangle \left\langle 0\right|\\ \sqrt{a}\left|-1\right\rangle \left\langle 0\right| & -a\left|-1\right\rangle \left\langle 1\right| & -\left|0\right\rangle \left\langle 0\right| & \sqrt{a}\left|0\right\rangle \left\langle 1\right|\\ a\left|-1\right\rangle \left\langle -1\right| & -\sqrt{a}\left|-1\right\rangle \left\langle 0\right| & -\sqrt{a}\left|0\right\rangle \left\langle -1\right| & \left|0\right\rangle \left\langle 0\right|\end{array}\right)\label{18}$$ Also,$$\rho(i)=Tr_{1,..L}^{i}\left|\psi_{0}\right\rangle \left\langle \psi_{0}\right|\label{19}$$ where the trace is over all the spins except the $i$-th one. The transfer matrix $F$ at a site $m$ is obtained by taking the trace over $f$ at the same site, i.e., $$F_{m}=\sum_{k}\left\langle k\right|f_{m}\left|k\right\rangle \label{20}$$ where the states $\left|k\right\rangle $ are the states $\left|0\right\rangle $, $\left|\pm1\right\rangle $. The transfer matrix $F$ is obtained as$$F=\left(\begin{array}{cccc} 1 & 0 & 0 & a\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ a & 0 & 0 & 1\end{array}\right)\label{21}$$ The eigenvalues are $$\varepsilon_{1}=1+a,\:\varepsilon_{2}=1-a,\:\varepsilon_{3}=-1,\:\varepsilon_{4}=-1\label{22}$$ The corresponding eigenvectors are$$\begin{array}{cc} \left|e_{1}\right\rangle =\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1\\ 0\\ 0\\ 1\end{array}\right),\quad & \left|e_{2}\right\rangle =\frac{1}{\sqrt{2}}\left(\begin{array}{c} -1\\ 0\\ 0\\ 1\end{array}\right)\\ \left|e_{3}\right\rangle =\left(\begin{array}{c} 0\\ 1\\ 0\\ 0\end{array}\right),\quad & \left|e_{4}\right\rangle =\left(\begin{array}{c} 0\\ 0\\ 1\\ 0\end{array}\right)\end{array}\label{23}$$ From Eq. $(20)$,$$\rho(i)=\frac{\sum_{\alpha=1}^{4}\left\langle e_{\alpha}\right|F^{L-1}f\left|e_{\alpha}\right\rangle }{\sum_{\alpha=1}^{4}\left\langle e_{\alpha}\right|F^{L}\left|e_{\alpha}\right\rangle }\label{24}$$ The factor in the denominator takes care of the condition $Tr\,\rho=1$. On taking the thermodynamic limit $L\rightarrow\infty$, we get $$\rho(i)=\varepsilon_{1}^{-1}\,\left\langle e_{1}\right|f\left|e_{1}\right\rangle \label{25}$$ In the $\left|0,\pm1\right\rangle $ basis, the reduced density matrix becomes$$\rho(i)=\left(\begin{array}{ccc} \frac{1}{1+a} & 0 & 0\\ 0 & \frac{a}{2(1+a)} & 0\\ 0 & 0 & \frac{a}{2(1+a)}\end{array}\right)\label{26}$$ The calculation of the two-site reduced density matrix $\rho(i,j)$ follows in the same manner. $\rho(i,j)$ is given by$$\rho(i,j)=Tr_{1,..L}^{i,j}\left|\psi_{0}\right\rangle \left\langle \psi_{0}\right|\label{27}$$ where the trace is taken over all the spins except the $i$-th and $j$-th ones. $$\rho(i,j)=\frac{\sum_{\alpha=1}^{4}\left\langle e_{\alpha}\right|F^{i-1}f\, F^{j-i-1}f\, F^{L-j}\left|e_{\alpha}\right\rangle }{\sum_{\alpha=1}^{4}\left\langle e_{\alpha}\right|F^{L}\left|e_{\alpha}\right\rangle }\label{28}$$ In the thermodynamic limit $L\rightarrow\infty$, $\rho(i,j)$ reduces to $$\rho(i,j)=\sum_{\alpha=1}^{4}\varepsilon_{\alpha}^{-2}\,\left(\frac{\varepsilon_{\alpha}}{\varepsilon_{1}}\right)^{n+1}\,\left\langle e_{1}\right|f\left|e_{\alpha}\right\rangle \left\langle e_{\alpha}\right|f\left|e_{1}\right\rangle \label{29}$$ where $n=|j-i|$. The matrix $\rho(i,j)$ is a $9\times9$ matrix and defined in the two-spin basis states $\left|lm\right\rangle $ with the ordering $$\left|lm\right\rangle \equiv\left|11\right\rangle ,\left|10\right\rangle ,\left|01\right\rangle ,\left|1-1\right\rangle ,\left|-11\right\rangle ,\left|00\right\rangle ,\left|0-1\right\rangle ,\left|-10\right\rangle ,\left|-1-1\right\rangle \label{30}$$ The non-zero matrix elements, $b_{pq}$ $(p=1,...,9,\, q=1,...,9)$, of $\rho(i,j)$ are : $$\begin{array}{c} b_{11}=b_{99}=\frac{a^{2}}{4(1+a)^{2}}-\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}\\ b_{22}=b_{33}=b_{77}=b_{88}=\frac{a}{2(1+a)^{2}}\\ b_{44}=b_{55}=\frac{a^{2}}{4(1+a)^{2}}+\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}\\ b_{23}=b_{32}=b_{46}=b_{64}=b_{56}=b_{65}=b_{78}=b_{87}=\frac{a}{2(1+a)}\left(-\frac{1}{1+a}\right)^{n}\\ b_{66}=\frac{1}{(1+a)^{2}}\end{array}\label{31}$$ It is easy to check that $\rho(i,j)$ has a block-diagonal form. III. ENTANGLEMENT MEASURES $\; S(i)$, $S(i,j)$, $G(2,n)$ {#iii.-entanglement-measures-si-sij-g2n .unnumbered} ======================================================== We now determine the entanglement content of the ground state $\left|\psi_{0}\right\rangle $ (Eq. $(9)$) of the Hamiltonian (Eq. $(5)$) using the entanglement measures $S(i)$, $S(i,j)$, and $G(2,n)$. The calculations are carried out for different values of the parameter $a$ in Eq. $(8)$. The ultimate aim is to probe the special features, if any, of entanglement in the vicinity of the QCP at $a=0$. From Eq. $(3)$ and $(26)$, the one-site entanglement $$S(i)=\frac{1}{1+a}[(1+a)log_{2}\,(1+a)-a\, log_{2}\, a+a]\label{32}$$ Figure $1$ (top) shows the variation of $S(i)$ w.r.t. $a$. The one-site entanglement has the maximum possible value $log_{2}\,3$. This is attained at the AKLT point $a=2$. The VBS state is in this case the exact ground state. In the VBS state, each spin-1 at a specific lattice site can be considered as a symmetric combination of two spin-$\frac{1}{2}$’s [@key-14]. In the VBS state, each spin-$\frac{1}{2}$ at a particular lattice site forms a spin singlet with a spin-$\frac{1}{2}$ at a neighbouring lattice site. $S(i)$ has the value zero at the QCP $a=0$. Figure $1$ (bottom) shows the variation of $\frac{\partial S(i)}{\partial a}$ with the parameter $a$. The derivative diverges as the QCP is approached. This is the expected behaviour at the QCP of a conventional QPT. In the latter case, however, $S(i)$ has the maximum value at the QCP [@key-3]. From Eq. $(1)$ and $(31)$, the two-body entanglement $S(i,j)$ is $$S(i,j)=-\sum_{i=1}^{9}\lambda_{i}\, log_{2}\lambda_{i}\label{33}$$ Where $\lambda_{i}$’s are the eigenvalues of the reduced density matrix $\rho(i,j)$. These are given by$$\begin{array}{c} \lambda_{1}=\lambda_{2}=\frac{a}{2(1+a)^{2}}-\frac{a}{2(1+a)}\left(-\frac{1}{1+a}\right)^{n}\\ \lambda_{3}=\lambda_{4}=\frac{a}{2(1+a)^{2}}+\frac{a}{2(1+a)}\left(-\frac{1}{1+a}\right)^{n}\\ \lambda_{5}=\frac{a^{2}}{4(1+a)^{2}}+\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}\\ \lambda_{6}=\lambda_{7}=\frac{a^{2}}{4(1+a)^{2}}-\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}\\ \lambda_{8}=\frac{1}{2}\left(\frac{a^{2}}{4(1+a)^{2}}+\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}+\frac{1}{1+a^{2}}\right)-\frac{1}{2(1+a)}\\ \left(\frac{(a^{2}-4)^{2}}{16(1+a)^{2}}+2a^{2}\left(-\frac{1}{1+a}\right)^{2n}+\frac{a^{4}}{16(1-a)^{2}}\left(\frac{1-a}{1+a}\right)^{2n}+\frac{a^{2}(a^{2}-4)}{8(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{2n}\right)^{\frac{1}{2}}\\ \lambda_{9}=\frac{1}{2}\left(\frac{a^{2}}{4(1+a)^{2}}+\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}+\frac{1}{1+a^{2}}\right)+\frac{1}{2(1+a)}\\ \left(\frac{(a^{2}-4)^{2}}{16(1+a)^{2}}+2a^{2}\left(-\frac{1}{1+a}\right)^{2n}+\frac{a^{4}}{16(1-a)^{2}}\left(\frac{1-a}{1+a}\right)^{2n}+\frac{a^{2}(a^{2}-4)}{8(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{2n}\right)^{\frac{1}{2}}\end{array}\label{34}$$ Knowing the reduced density matrix $\rho(i,j)$, the correlation functions $\left\langle S_{i}^{\alpha}\, S_{j}^{\alpha}\right\rangle \;$ ($\alpha=x,y,z$) can be calculated in the usual manner. One then recovers the expressions in Eq. $(13)$ and $(14)$ ($r=n+1$, where $n=|j-i|$). Figure $2$ (top) shows the variation of $S(i,j)$ as a function of $a$ for $n=1000$. Figure $2$ (bottom) shows the variation of the derivative $\frac{\partial S(i,j)}{\partial a}$ w.r.t. $a$ for the same value of $n$. The maximum of $S(i,j)$ is at the AKLT point $a=2$ and has the value zero at $a=0$. For large $n$, the derivative $\frac{\partial S(i,j)}{\partial a}$ diverges near the QCP at $a=0$. The last fearure is characteristic of a conventional QPT [@key-7]. We next calculate the GGE $G(2,n)$ (Eq. $(4)$). This is easily done as the matrix elements of the reduced density matrix (Eq. $(3)$) are known. Figure $3$ (top) shows the variation of $G(2,n)$ versus $a$ for $n=1000$. Figure $3$ (bottom) shows the plot of $\frac{\partial G(2,n)}{\partial a}$ against $a$. Again $G(2,n)$ has the maximum value at the AKLT point and is zero at $a=0$. The derivative $\frac{\partial G(2,n)}{\partial a}$ does not diverge as $a\rightarrow0$ in contrast to the case of a conventional QPT [@key-6]. The derivative, however, attains the maximum value at the QCP $a=0$. Figure $4$ (top) shows the plots of $S(i)$, $S(i,j)$, and $G(2,n)$ versus $a$ for $n=1000$. Figure $4$ (bottom) shows the plots of the first derivatives of the same quantities w.r.t. $a$ for $n=1000$. The plots are shown for comparing the different entanglement measures. We next determine the EL $\xi_{E}$ and its variation w.r.t. the parameter $a$. We consider the entanglement measure $S(i,j)$ for this purpose. Close to the QCP $a=0$ and in the limit of large $n$, one can write$$S(n=|j-i|)-S(\infty)\sim A_{e}\, e^{-\frac{n}{\xi_{E}}}\label{35}$$ The longitudinal and transverse correlation functions, $p_{n}^{z}=\left\langle S_{1}^{z}S_{n+1}^{z}\right\rangle $ and $p_{n}^{x}=\left\langle S_{1}^{x}S_{n+1}^{x}\right\rangle $ are given by Eq. $(13)$ and $(14)$ with $r=n+1$. For $a<1$, $p_{n}^{z}$ decays faster than $p_{n}^{x}$ with $n$. The eigenvalues $\lambda_{i}$’s, $i=1,...9$, can be expressed in terms of the correlation functions $p_{n}^{z}$ and $p_{n}^{x}$. For large $n$, the contributions from $p_{n}^{z}$ can be ignored. The eigenvalues now reduce to the expressions $$\begin{array}{c} \lambda_{1}=\lambda_{2}=\frac{a}{2(1+a)^{2}}-4\, p_{n}^{x}\\ \lambda_{3}=\lambda_{4}=\frac{a}{2(1+a)^{2}}+4\, p_{n}^{x}\\ \lambda_{5}=\frac{a^{2}}{4(1+a)^{2}}\\ \lambda_{6}=\lambda_{7}=\frac{a^{2}}{4(1+a)^{2}}\\ \lambda_{8}=\frac{1}{2}\left(\frac{a^{2}}{4(1+a)^{2}}+\frac{1}{1+a^{2}}\right)-\frac{1}{2}\left(\frac{(a^{2}-4)^{2}}{16(1+a)^{2}}+2\,\left(p_{n}^{x}\right)^{2}\right)^{\frac{1}{2}}\\ \lambda_{9}=\frac{1}{2}\left(\frac{a^{2}}{4(1+a)^{2}}+\frac{1}{1+a^{2}}\right)+\frac{1}{2}\left(\frac{(a^{2}-4)^{2}}{16(1+a)^{2}}+2\,\left(p_{n}^{x}\right)^{2}\right)^{\frac{1}{2}}\end{array}\label{36}$$ From Eq. $(1)$ and $(36)$, one ultimately arrives at the expressions$$S(n=|j-i|)-S(\infty)\sim A_{e}^{'}\,\left(p_{n}^{x}\right)^{2}\sim A_{e}\, e^{-\frac{n}{\xi_{E}}}\label{37}$$ The pre-factor $A_{e}=0$ at the QCP $\: a=0$ . The EL $\:\xi_{E}$ is given by $$\xi_{E}=\frac{\xi_{t}}{2}=\frac{1}{2\, ln(1+a)}\label{38}$$ where $\xi_{t}$ is the transverse correlation length (Eq. $(15)$). In the case of the $S=\frac{1}{2}$ anisotropic XY model in a transverse field, an expression similar to that in Eq. $(37)$ is obtained close to the QCP in the limit of large $n$ [@key-7]. The pre-factor in this case, however, does not vanish at the QCP but has a power-law dependence on $n$. Figure $5$ shows the variation of $\xi_{E}$ w.r.t. $a$ based on the entanglement measure $S(i,j)$. The total correlations, with both classical and quantum components, between two sites $i$ and $j$ are quantified in terms of the quantum mutual information [@key-21; @key-22]$$I_{ij}=S(i)+S(j)-S(i,j)\label{39}$$ As explained in [@key-21], a comparison of the singular behaviour of $S(i)$ with that of $I_{ij}$ allows one to determine whether two-point ($Q2$) or multipartite ($QS$) quantum correlations are important in a QPT. Figure $6$ (top) shows a plot of $I_{ij}$ versus $a$ for $n=1000$. Figure $6$ (bottom) shows the variation of $\frac{\partial I_{ij}}{\partial a}$ versus $a$ for $n=1000$. The derivative does not diverge as $a\rightarrow0$, a behaviour distinct from that of $\frac{\partial S(i)}{\partial a}$ close to $a=0$. The difference in the singular behaviour of quantities associated with $S(i)$ and $I_{ij}$ indicates that multipartite quantum correlations are involved in the QPT.  **FIG. 1:** Plot of $S(i)$ (top) and $\frac{\partial S(i)}{\partial a}$ (bottom) vs. $a$ .  **FIG.** **2:** Plot of $S(i,j)$ (top) and $\frac{\partial S(i,j)}{\partial a}$ (bottom) as a function of $a$.  **FIG. 3:** Plot of $G(2,n)$ (top) and $\frac{\partial G(2,n)}{\partial a}$ (bottom) as a function of $a$.  **FIG. 4:** Plots of $S(i)$ ($q$), $S(i,j)$ ($r$), and $G(2,n)$ ($p$) (top) and the corresponding first derivatives w.r.t. $a$ (bottom) as a function of $a$. $E$ represents the entanglement measure.  **FIG.** **5:** Plot of EL as a function of $a$.  **FIG.** **6:** Plots of $I_{ij}$ (top) and $\frac{\partial I_{ij}}{\partial a}$ (bottom) as a function of $a$. IV. DISCUSSIONS {#iv.-discussions .unnumbered} =============== The MP states provide exact representations of the ground states of several spin models in low dimensions [@key-17]. The remarkable features of such states arises from the fact that complicated many body states have a simple factorized form. The simplicity in structure makes the calculation of the ground state expectation values particularly easy to perform. The spin-1 AKLT model is a well-known example of a spin model in 1d the exact ground state of which (a VBS state) has an MP representation. The AKLT model and the spin-1 Heisenberg AFM chain belong to the same universality class [@key-23]. The insight gained from the study of models in the MP formalism is expected to be of relevance in understanding the properties of more physical systems. The MP states also serve as good trial wave functions for standard spin models. The MP representation lies at the heart of the powerful density matrix renormalization group (DMRG) method and provides the basis for several interesting developments in quantum information [@key-24]. Studies of the entanglement characteristics of the MP states have begun only recently. The QPTs which occur in such states have characteristics different from those of conventional QPTs. It is thus of considerable interest to determine whether the entanglement content of MP states develops special features close to a QCP. In this paper, we consider a spin-$1$ model the exact ground state of which is of the MP form over a wide range of parameter values. The model exhibits a novel QPT in that the longitudinal and transverse correlation lengths diverge as the QCP is approached but the correlation functions vanish at the QCP. In a conventional QPT, the correlation functions have a power-law decay at the QCP. In the spin-1 model, the divergence of correlation lengths is accompanied by the excitation gap going to zero. The string order parameter, which has a non-zero expectation value in the MP state for $a>0$, vanishes at the QCP $a=0$. The distinctive signatures indicate the appearance of a new phase. We study the entanglement properties of the MP state for different values of the parameter $a$. The measures used are $S(i)$ (one-site von Neumann entropy), $S(i,j)$ (two-body entanglement) and $G(2,n)$ (GGE). All the entanglement measures have zero value at the QCP so that the ground state is disentangled at that point. As seen from the different plots, figures $(1)$, $(2)$, $(3)$, and $(4)$, the entanglement content, as measured by $E=$$S(i)$, $S(i,j)$ and $G(2,n)$, has a slow variation w.r.t. $a$ for $a>2$. At the AKLT point $a=2$, $E$ reaches its maximum value and as $a$ is reduced further, the magnitude of $E$ falls rapidly to approach zero value at $a=0$. The study of conventional QPTs shows that $E$ is maximum at a QCP [@key-3; @key-6; @key-7]. Also, $\frac{\partial E}{\partial a}$ diverges as the QCP is approached. The EL, $\xi_{E}$, as calculated from $S(i,j)$ and $G(2,n)$ for large $n$, also diverges with $\xi_{E}=\frac{\xi_{C}}{2}$ where $\xi_{C}$ is the usual correlation length. In the case of the spin-1 model under consideration, $\frac{\partial E}{\partial a}$ diverges as $a\rightarrow0$ when $E=S(i)$ and $S(i,j)$. The EL, $\xi_{E}$, as calculated from $S(i,j)$ in the large $n$ limit, also diverges with $\xi_{E}=\frac{\xi_{C}}{2}$. One now has the interesting situation that the entanglement content of the MP state decreases as $a\rightarrow0$ but the entanglement is spread over larger distances. The derivative $\frac{\partial G(2,n)}{\partial a}$, however, does not diverge as $a\rightarrow0$ but attains a maximum value at the QCP. The results can be understood by noting that in all the three cases, $E=S(i),\: S(i,j)$ and $G(2,n)$, the reduced density matrices $\rho(i)$ and $\rho(i,j)$ smoothly approach the forms associated with pure states as the parameter $a$ tends to zero. The matrix elements of the reduced density matrices do not develop non-analyticities in the parameter region of interest. Thus, the energy density, calculated from the reduced density matrix $\rho(i,j)$, does not develop a non-analyticity at the QCP. The derivative $\frac{\partial G(2,n)}{\partial a}$ depends upon the first derivatives of the matrix elements of $\rho(i,j)$. Since the latter is analytic for $a\geq0$, $\frac{\partial G(2,n)}{\partial a}$ does not diverge in the whole parameter regime including the point $a=0$. In the cases of the entanglement measures $E=S(i)$ and $S(i,j)$, the derivative $\frac{\partial E}{\partial a}$ diverges as $a\rightarrow0$ due to the divergence of $log_{2}\, a$ in the same limit. The divergence is thus due to the special form of the von Neumann entropy and occurs for $n\geq1$. A recent work [@key-25] provides another example of such a singularity. Though $\frac{\partial G(2,n)}{\partial a}$ does not diverge or become discontinuous at the QCP $a=0$, it attains its maximum value at the point. The first derivative of the string order parameter w.r.t. $a$ also attains its maximum value at $a=0$ though the order parameter itself vanishes at the point. Figure $(4)$ shows that amongst the three entanglement measures $E=$$S(i)$, $S(i,j)$ and $G(2,n)$, used in this study to obtain a quantitative estimate of the entanglement content of the MP ground state, the measure $S(i,j)$ yields the largest value of the entanglement at different values of $a$. The difference in the singular behaviour of the measures $S(i)$ and the mutual information entropy $I_{ij}$ as $a\rightarrow0$ indicates that multipartite quantum correlations are involved in the QPT. In summary, the present study identifies the entanglement characteristics of the MP ground state of a spin-1 model close to the critical point $a=0$. The features are distinct from those associated with conventional QPTs. Several spin models are known for which the MP states are the exact ground states [@key-17]. Some of these models have interesting phase diagrams exhibiting both first and second order phase transitions. It will be of interest to extend the present study to other spin models (both $S=\frac{1}{2}$ and $1$) in order to identify the universal characteristics of QPTs in MP states. ACKNOWLEDGMENT {#acknowledgment .unnumbered} ============== A. T. is supported by the Council of Scientific and Industrial Research, India, under Grant No. 9/15 (306)/ 2004-EMR-I. [10]{} S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 2000); S. L. Sondhi et al., Rev. Mod. Phys. 69, 315 (1997). A. Osterloh, L. Amico, G. Falci and R. Fazio, Nature 416, 608 (2002). T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002). G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, Phys. Rev. Lett. 90, 0279011 (2004). L. -A. Wu, M. S. Sarandy and D. A. Lidar, Phys. Rev. Lett 93, 250404 (2004). T. R. de Oliveira, G. Rigolin, M. C. de Oliveira and E. Miranda, Phys. Rev. Lett. 97, 170401 (2006). H.- D. Chen, cond-mat/0606126. I. Bose and E. Chattopadhyay, Phys. Rev. A 66, 062320 (2002). F. C. Alcaraz, A. Saguia and M. S. Sarandy, Phys. Rev. A 70, 032333 (2004). J. Vidal, R. Mosseri and J. Dukelsky, Phys. Rev. A 69, 054101 (2004). F. Verstraete, M. Popp and J. I. Cirac, Phys. Rev. Lett. 92, 027901 (2004). H. Fan, V. Korepin and V. Roychoudhury, Phys. Rev. Lett. 93, 227203 (2004). F. Verstraete, M. A. Martin-Delgado and J. I. Cirac, Phys. Rev. Lett. 92, 087201 (2004). I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki, Phys. Rev. Lett. 59, 779 (1987); Commun. Math. Phys. 115, 447 (1988). L. Campos Venuti and M. Roncaglia, Phys. Rev. Lett. 94, 207207 (2005). A. Klümper, A. Schadschneider and J. Zittartz, Z. Phys. B 87, 281-287 (1992). A. K. Kolezhuk and H. J. Mikeska, Int. J. Mod. Phys. B 12, 2325 (1998) and references therein. M. Fannes, B. Nachtergaele and R. F. Werner, Europhys. Lett. 10, 633 (1989); Commun. Math. Phys. 144, 443 (1992). M. M. Wolf, G. Ortiz, F. Verstraete and J. I. Cirac, Phys. Rev. Lett. 97, 110403 (2006). A. Klümper, A. Schadschneider and J. Zittartz, Europhys. Lett. 24, 293 (1993). A. Anfossi, P. Giorda, A. Montorsi and F. Traversa, Phys. Rev. Lett. 95, 056402 (2005). A. Anfossi, P. Giorda, A. Montorsi and F. Traversa, cond-mat/0611091. S. Brehmer, H. -J. Mikeska and U. Neugebauer, J. Phys.: Condens. Matter 8, 7161 (1996). D. Pérez-García, F. Verstraete, M. M. Wolf and J. I. Cirac, quant-ph/0608197. M. Cozzini, R. Ionicioiu and P. Zanardi, Cond-mat/0611727.

**Inequalities Becker-Stark at extreme points** Ling Zhu$^{1}$ and Cristinel Mortici$^{2}$ $^{1}$Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China, zhuling0571@163.com $^{2}$Valahia University of Târgovişte, Bd. Unirii 18, 130082, Târgovişte, Romania, cristinel.mortici@hotmail.com $$$$ **Abstract:** *The aim of this work is to extend Becker-Stark inequalities near the origin and* $\pi /2.$ **Keywords:** Inequalities; approximations; monotonicity; convexity; Becker-Stark inequalities **MSC:** 26D20 Introduction and Motivation =========================== In 1978, M. Becker and L. E. Stark presented the inequalities$$\frac{8}{\pi ^{2}-4x^{2}}<\frac{\tan x}{x}<\frac{\pi ^{2}}{\pi ^{2}-4x^{2}},\ \ \ 0<x<\frac{\pi }{2}. \label{a}$$These inequalities were proven to be of great interest through the researchers, since they were extended in different forms in the recent past. We refer to [@bk]-[@zzz] and all references therein. It is true that inequalities (1) and some of recent improvements are nice through its symmetric form, but let us remember the practical importance of an inequality which is to provide some bounds for a given expression. Observe that near $\pi /2,$ the right-hand side inequality (1) becomes weak, as soon as$$\lim_{x\rightarrow \left( \pi /2\right) _{-}}\left( \frac{\pi ^{2}}{\pi ^{2}-4x^{2}}-\frac{\tan x}{x}\right) =\infty .$$The left-hand side inequality (1) is somehow motivated by the fact that$$\lim_{x\rightarrow \left( \pi /2\right) _{-}}\left( \pi ^{2}-4x^{2}\right) \frac{\tan x}{x}=8.$$Having in mind these remarks, we assume that good estimates of $\left( \tan x\right) /x$ near $\pi /2$ are obtained using expressions of the form$$\frac{\tan x}{x}\approx \frac{8+\omega \left( x\right) }{\pi ^{2}-4x^{2}}\ ,\ \ \ x<\frac{\pi }{2},$$with $\omega \left( x\right) $ tending to zero, as $x$ approaches $\pi /2,$ with $x<\pi /2.$ More precisely, we propose the following double inequality of Becker-Stark type, on a neighborhood of $\pi /2.$ **Theorem 1.** *For every* $x\in \left( 0.373,\pi/2\right) $ *in the left-hand side and for every* $x\in \left( 0.301,\pi/2\right) $ *in the right-hand side*$,$ *the following inequalities hold true:*$$\frac{8+a\left( x\right) }{\pi ^{2}-4x^{2}}<\frac{\tan x}{x}<\frac{8+b\left( x\right) }{\pi ^{2}-4x^{2}}, \label{t}$$*where*$$a\left( x\right) =\frac{8}{\pi }\left( \frac{\pi }{2}-x\right) +\left( \frac{16}{\pi ^{2}}-\frac{8}{3}\right) \left( \frac{\pi }{2}-x\right) ^{2}$$*and*$$b\left( x\right) =a\left( x\right) +\left( \frac{32}{\pi ^{3}}-\frac{8}{3\pi }\right) \left( \frac{\pi }{2}-x\right) ^{3}.$$ We think to similar comments on the behavior of left-hand inequality (1) near the origin. In connection with this problem, we propose the following improvement. **Theorem 2.** *For every real number* $x\in \left( 0,1.371\right) ,$ *the following inequality holds true:*$$\frac{\tan x}{x}<\frac{\pi ^{2}-\left( 4-\frac{1}{3}\pi ^{2}\right) x^{2}-\left( \frac{4}{3}-\frac{2}{15}\pi ^{2}\right) x^{4}}{\pi ^{2}-4x^{2}}. \label{r}$$ As a fact to remember, if we wish to obtain accurate approximations of $\left( \tan x\right) /x,$ using Becker-Stark inequalities, then the best constant at the numerator near $0$ is $\pi ^{2},$ while the best constant at the numerator near $\pi /2$ is $8.$ The Proofs ========== *Proof of Theorem 1.* Inequalities (\[t\]) are equivalent to $f<0$ on $\left( 0.373,\pi/2 \right) $ and $g>0$ on $\left( 0.301,\pi/2 \right) ,$ where$$f\left( x\right) =\arctan \left( x\cdot \frac{8+a\left( x\right) }{\pi ^{2}-4x^{2}}\right) -x\text{ \ \ and \ \ \ }g\left( x\right) =\arctan \left( x\cdot \frac{8+b\left( x\right) }{\pi ^{2}-4x^{2}}\right) -x.$$If $p_{k},$ $q_{k}$ are polynomial functions, we use the following derivation formula$$\left( \arctan \frac{p_{k}\left( x\right) }{q_{k}\left( x\right) }-x\right) ^{\prime }=\frac{p_{k}{}^{\prime }\left( x\right) q_{k}\left( x\right) -p_{k}\left( x\right) q_{k}{}^{\prime }\left( x\right) -p_{k}^{2}\left( x\right) -q_{k}^{2}\left( x\right) }{p_{k}^{2}\left( x\right) +q_{k}^{2}\left( x\right) }. \label{d}$$With $p_{1}\left( x\right) =x\left( 8+a\left( x\right) \right) $ and $q_{1}\left( x\right) =\pi ^{2}-4x^{2},$ we get$$f^{\prime }\left( x\right) =\frac{\left( \pi -2x\right) ^{3}u\left( x\right) }{9\pi ^{4}\left( p_{1}^{2}\left( x\right) +q_{1}^{2}\left( x\right) \right) }, \label{f}$$where$$\begin{aligned} u\left( x\right) &=&\allowbreak 144\pi ^{3}-15\pi ^{5}+x\left( 432\pi ^{2}-42\pi ^{4}\right) \\ &&+\allowbreak x^{2}\left( 96\pi ^{3}-432\pi -4\pi ^{5}\right) +x^{3}\left( 8\pi ^{4}-96\pi ^{2}+288\right) .\end{aligned}$$Now we present the following steps in our proof: - $u^{\prime \prime }$ is a first degree polynomial funcion, strictly increasing, with $u^{\prime \prime }\left( 0.373\right) =\allowbreak 1058.\,\allowbreak 803...>0,$ so $u^{\prime \prime }>0$ on $\left( 0.373,\pi/2\right) .$ - $u^{\prime }$ is strictly increasing, with $u^{\prime }\left( 0.373\right) =\allowbreak 517.\,\allowbreak 421...>0,$ so $u^{\prime }>0$ on $\left( 0.373,\pi/2 \right) .$ - $u$ is strictly increasing, with $u\left( 0.373\right) =\allowbreak 0.168...,$ so $u>0$ on $\left( 0.373,\pi/2\right) .$ By (\[f\]), $f$ is strictly increasing, with limit $f\left(\pi/2-0\right) =0,$ so $f<0$ and the first inequality (\[t\]) is proved. With $p_{2}\left( x\right) =x\left( 8+b\left( x\right) \right) $ and $q_{2}\left( x\right) =\pi ^{2}-4x^{2}$ in (\[d\])$,$ we get$$g^{\prime }\left( x\right) =-\frac{\left( \pi -2x\right) ^{4}v\left( x\right) }{9\pi ^{6}\left( p_{2}^{2}\left( x\right) +q_{2}^{2}\left( x\right) \right) }, \label{g}$$where$$\begin{aligned} v\left( x\right) &=&\allowbreak 18\pi ^{6}-180\pi ^{4}+x\left( 60\pi ^{5}-576\pi ^{3}\right) +x^{4}\left( 4\pi ^{4}-96\pi ^{2}+576\right) \\ &&+\allowbreak x^{3}\left( 240\pi ^{3}-1152\pi -12\pi ^{5}\right) +x^{2}\left( 864\pi ^{2}-168\pi ^{4}+9\pi ^{6}\right) \allowbreak .\end{aligned}$$Now we present the following steps in our proof: - $v^{\prime \prime }$ is a second degree polynomial with minimum at $x_{0}=-2.\,\allowbreak 067....$ As $v^{\prime \prime }\left( 0.301\right) =1921.\,\allowbreak 145...>0,$ it results that $v^{\prime \prime }>0$ on $\left( 0.301,\pi/2\right) .$ - $v^{\prime }$ is strictly increasing, with $v^{\prime }\left( 0.301\right) =\allowbreak 1035.\,\allowbreak 057...>0,$ so $v^{\prime }>0$ on $\left( 0.301,\pi/2 \right) .$ - $v$ is strictly increasing, with $v\left( 0.301\right) =\allowbreak 0.434\,386\,67>0,$ so $v>0$ on $\left( 0.301,\pi/2 \right) .$ By (\[g\]), $g$ is strictly decreasing, with limit $g\left( \pi/2-0\right) =0,$ so $g>0$ on $\left( 0.301,\pi/2 \right) $ and Theorem 1 is proved.$\square $ *Proof of Theorem 2.* We rewrite (\[r\]) in the form $h>0$ on $\left( 0,1.371\right) ,$ where$$h\left( x\right) =\arctan \left( x\cdot \frac{\pi ^{2}-\left( 4-\frac{1}{3}\pi ^{2}\right) x^{2}-\left( \frac{4}{3}-\frac{2}{15}\pi ^{2}\right) x^{4}}{\pi ^{2}-4x^{2}}\right) -x.$$With $p_{3}\left( x\right) =x\left( \pi ^{2}-\left( 4-\frac{1}{3}\pi ^{2}\right) x^{2}-\left( \frac{4}{3}-\frac{2}{15}\pi ^{2}\right) x^{4}\right) $ and $q_{3}\left( x\right) =\pi ^{2}-4x^{2}$ in (\[d\]), we get $$h^{\prime }\left( x\right) =-\frac{x^{6}w\left( x^{2}\right) }{225\left( p_{3}^{2}\left( x\right) +q_{3}^{2}\left( x\right) \right) }, \label{az}$$ where $$w\left( t\right) =85\pi ^{4}-840\pi ^{2}+t^{2}\left( 4\pi ^{4}-80\pi ^{2}+400\right) +t\left( 20\pi ^{4}-440\pi ^{2}+2400\right) .$$This function $w$ is a second degree polynomial function with minimum at $t_{0}=-40.\,\allowbreak 844...$ . As $w\left( 1.881\right) =\allowbreak -0.0037...<0,$ it follows that $w\left( t\right) <0$ for every $t\in \left( 0,1.881\right) .$ That is $w\left( x^{2}\right) <0,$ for every $x\in \left( 0,1.371\right) $ (Notice that $\sqrt{1.881}=\allowbreak 1.\,\allowbreak 371\,495...$). By (\[az\]), $h$ is strictly increasing on $\left( 0,1.371\right) ,$ with $h\left( 0\right) =0,$ so $h\left( x\right) >0,$ for every $x\in \left( 0,1.371\right) .$$$$$ The work of the second author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI project number PN-II-ID-PCE-2011-3-0087. [9]{} M. Becker and E. L. Stark, On hierarchy of polynomial inequalities for $\tan (x)$, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 602–633 (1978), p. 133–138. C.-P. Chen and W.-S. Cheung, Sharp Cusa and Becker-Stark Inequalities, J. Inequal. Appl., (2011). H.-F. Ge, New Sharp Bounds for the Bernoulli Numbers and Refinement of Becker-Stark Inequalities, J. Appl. Math., Article ID 137507, (2012). Zh.-J. Sun and L.Zhu, Simple proofs of the Cusa–Huygens–type and Becker–Stark–type inequalities, J. Math. Inequal., 7 (2013), no. 4, 563-567. L. Zhu and J.-K. Hua, Sharpening the Becker-Stark inequalities, J. Inequal. Appl., Article ID 931275 (2010). L. Zhu, Sharp Becker-Stark-type inequalities for Bessel functions, J. Inequal. Appl., Article ID 838740 (2010). L. Zhu, A Refinement of the Becker–Stark Inequalities, Mat. Zametki, 93:3 (2013), 401–406. J.-L. Zhao, Q.-M. Luo, B.-N. Guo, F. Qi, Remarks on inequalities for the tangent function, Hacettepe J. Math. Stat., 41 (2012), no. 4, 499-506.

--- abstract: 'Coulomb exchange interactions of electrons in the $\nu=3$ quantum Hall state are determined from two inter-Landau level spin-flip excitations measured by resonant inelastic light scattering. The two coupled collective excitations are linked to inter-Landau level spin-flip transitions arising from the N=0 and N=1 Landau levels. The strong repulsion between the two spin-flip modes in the long-wave limit is clearly manifested in spectra displaying Coulomb exchange contributions that are comparable to the exchange energy for the quantum Hall state at $\nu=1$ . Theoretical calculations within the Hartree-Fock approximation are in a good agreement with measured energies of spin-flip collective excitations.' author: - 'A. B. Van’kov' - 'T. D. Rhone' - 'A. Pinczuk' - 'I. V. Kukushkin' - 'Loren N. Pfeiffer' - 'Ken W. West' title: 'Observation of exchange Coulomb interactions in the quantum Hall state at $\nu=3$' --- The exchange Coulomb interaction energy of electrons on Landau levels (LL) plays key roles in quantum Hall systems, particularly at odd values of the filling factor, $\nu=nhc/eB$ (where $n$ is the areal density), when the 2D electron system evolves into a quantum Hall ferromagnet. One way to probe the exchange interaction is by measurements of energies of collective spin-flip excitations. The simplest one is the spin-wave, in which Landau orbital quantization does not change. At odd filling factors the spin wave energy in the short wavelength limit is predicted to have a large exchange contribution, resulting in an enhanced spin gap [@by81; @kallin84]. However, the actual energy values measured in activated transport experiments turned out to be significantly below theoretical estimates. These discrepancies occur in both the integer and fractional quantum Hall regimes. Possible reasons for the discrepancies lie in impact of spin-textures (skyrmions) [@schmeller; @dethlef; @grosh], and of weak residual disorder potential  [@usher; @dolgo; @khrapai]. Experimental venues to access Coulomb exchange interactions also emanate from determinations of collective excitation modes in the spin degree of freedom. At odd integer values of filling factor the long-wavelength spin-wave is a minimum energy collective excitation. The long-wavelength spin-wave mode approaches the unshifted Zeeman energy [@by81; @kallin84; @Larmor] and carries marginal information about the electron-electron interaction. In contrast, inelastic light scattering methods enable the direct determination of exchange Coulomb interactions from measurements of spin-flip collective excitations across cyclotron gaps  [@nu1pinczuk; @nu1vankov; @nu1temp]. In these spin-flip (SF) excitations there is simultaneous change in Landau quantization and in orientation of spin. The long wavelength SF excitations represent probes that are nearly insensitive to perturbations on length scales exceeding the characteristic size of the quasiparticle-quasihole pair magnetoexciton that is of the order of the magnetic length $l_o=(\hbar c/eB)^{1/2}$, where $B$ is the perpendicular component of magnetic field. At $\nu=1$ the electron-electron interaction affects the energy of the long-wave cyclotron SF mode, which involves the change of the Landau quantization number by +1. This mode is shifted upwards from the cyclotron energy by about half the full exchange energy in the large momentum (small wave length) spin wave. Studies of the cyclotron SF mode at $\nu=1$ have shown that the Coulomb exchange contributions to its energy scale as $\sqrt{B}$ and its value is softened by the spread of the electron wave-function in the direction normal to the 2D-plane. Theoretical predictions are in good agreement with measured mode energies determined as function of magnetic field and quantum well width  [@nu1pinczuk; @nu1vankov]. We report inelastic light scattering measurements of collective inter-Landau level excitations in the quantum Hall state at $\nu=3$. All collective excitations are identified in spectra of inelastic light scattering and their energies are compared with theoretical calculations. We identified two coupled cyclotron SF modes arising from the N=0 and N=1 Landau levels and interpreted the results in terms of Coulomb exchange interactions. We determined that these coupled cyclotron SF modes at $\nu=3$ are subject to large Coulomb exchange interactions that are comparable to the exchange energy in the quantum Hall state at $\nu=1$. There is great current interest in the roles of the spin degree of freedom in the remarkable quantum Hall phases that emerge in the N=1 Landau level  [@csathy; @dean; @trevor]. The finding reported here is that exchange Coulomb interactions in the N=1 Landau level are comparable to those in the N=0 level. This comparatively simple result suggest that the exotic collective states that emerge in the partially populated N=1 level are linked to the differences in correlation effects between the two levels. Figure1a shows the schematic representation of five lowest energy collective excitations in case of filling factor $\nu$=3. They are shown as magnetoexcitons consisting of an electron promoted from a filled Landau level and bound to an effective hole left in the “initial” LL. This representation is exact in the limit of strong magnetic field where the parameter $r_c=E_c/\hbar \omega_c$ is small enough [@by81; @by83; @kallin84]. Here $E_c$ is the characteristic Coulomb energy scale and $\hbar \omega_c$ is the cyclotron energy. The set of dispersion curves of the collective modes can be described in the following way [@kallin84]: $$E_{m, \delta S_z}(k)=m \hbar \omega_c + g \mu_B B \delta S_z + \Delta E_{m, \delta S_z}(k),$$ where $m$ is the change in the LL index, $g \mu_B B \delta S_z$ is the bare Zeeman energy associated with spin-flip. The last term $\Delta E_{m, \delta S_z}(k)$ is alone responsible for the dispersion and comprises all contributions from the many-body Coulomb interaction and exchange energies in the initial and the excited states. In the present discussion we focus on the excitation spectrum with $m=0$ and $m=1$. At $\nu=3$ the four inter-LL transitions with $m=1$ shown in Fig.1a are not independent. They couple via the Coulomb interaction to yield two pairs of excitations. For the two inter-LL excitations with changes in the charge degree of freedom with $\delta S_z=0$ we have the in-phase magnetoplasmon (MP) mode and the antiphase plasmon (AP) mode. For the the two excitations with changes in the spin degree of freedom with $\delta S_z=-1$ the two coupled modes are cyclotron spin-flip excitations SF1 and SF2. In first-order perturbation theory the dispersion curves of the coupled modes are expressed as follows: $$\begin{aligned} E_{1,2}(k) & = & \frac{{\cal E}_1(k)+{\cal E}_2(k)}{2} \pm {}\nonumber\\ & \pm & \sqrt{\left( \frac{{\cal E}_1(k)-{\cal E}_2(k)}{2} \right)^2 + \Delta_{12}(k)^2},\end{aligned}$$ where ${\cal E}_{1,2}(k)$ are the energies of single transitions either with or without spin-flip, $\Delta_{12}(k)$ – is the term, responsible for coupling. For MP and AP excitations this theory yields a vanishing Coulomb term $\Delta E(k)$ in the long-wavelength limit. Unlike MP, for which the Kohn’s theorem [@Kohn] is valid, the experimental values of the energy of AP mode are red-shifted relative to the cyclotron energy at integer filling factors $\nu \geq 2$. The experimental results were reported in Refs.\[\] and the explanation was given in the framework of the second-order perturbation theory [@nu2dickm; @DrozdovKulik]. We calculated the wave vector dispersions of SF1 and SF2 at $\nu=3$ in terms of matrix elements $\tilde{V}^{(1)}_{\alpha \beta \gamma \delta}(k)$ introduced in Ref.\[\]: $$\begin{aligned} {\cal E}_1(k) & = & \hbar \omega_c + \left | g \mu_B B \right | + \Sigma_{0\,\uparrow,1\,\downarrow}- \tilde{V}^{(1)}_{1001}(k)\\ {\cal E}_2(k) & = & \hbar \omega_c + \left | g \mu_B B \right | + \Sigma_{1\,\uparrow,2\,\downarrow}- \tilde{V}^{(1)}_{2112}(k)\nonumber\\ \Delta_{12}(k) & = & \tilde{V}^{(1)}_{1102}(k)\nonumber\end{aligned}$$ where $\Sigma_{0\,\uparrow,1\,\downarrow}=\tilde{V}^{(1)}_{0000}(0)+\tilde{V}^{(1)}_{0101}(0)-\tilde{V}^{(1)}_{1010}(0)$ and $\Sigma_{1\,\uparrow,2\,\downarrow}=\tilde{V}^{(1)}_{1010}(0)+\tilde{V}^{(1)}_{1111}(0)-\tilde{V}^{(1)}_{2020}(0)$ are the differences of exchange-self energies in the excited and ground states for two single spin-flip transitions between adjacent LLs depicted on Fig.1a. The calculated dispersion curves for all four inter-Landau level excitations at $B_{\perp}=$5.3T are plotted on Fig.1b by solid lines. For comparison with the experiment, performed on the 24nm quantum well, it was essential to take into account the finite thickness of the 2D-electron system. For this the Fourier component of the effective [*e-e*]{} interaction potential $\vartheta(q)=2\pi e^2/\varepsilon q$ was multiplied by the geometric form-factor $F(q)$ calculated via the self-consistent solution of the Poisson’s and Schrödinger’s equations [@lu93]. ![\[fig1\] (a): Schematic representation of the formation of collective modes at $\nu=3$ from single-electron transitions. Spin-wave (SW) is described as a single spin-flip transition within half-filled LL$_1$. MP and AP are formed as inphase and antiphase combinations of two inter-LL transitions with $\delta S_z=0$ (painted in green). Cyclotron spin-flip modes SF1 and SF2 arise from analogous combinations of inter-LL transitions with $\delta S_z=-1$ (painted in red). (b): Dispersion curves of inter-LL excitations calculated at $B_{\perp}=5.3$T within the first-order Hartree-Fock approximation are shown. Here the finite thickness of the 2D electron system is taken into account via the geometric form-factor. Dashed line represents the dispersion of cyclotron spin-flip mode at $\nu=1$ and the same magnetic field. (c): The zoomed-in image of the long-wavelength region of Fig.1b, painted in light grey. The dashed vertical line indicates the experimental in-plane momentum ${\rm k}^*=5.3 \times 10^4\,{\rm cm}^{-1} $. Open circles represent the experimental data.](Fig1.eps){width=".48\textwidth"} Both cyclotron spin-flip modes at $\nu=3$ are significantly blue-shifted from the cyclotron energy and are nearly dispersionless in the long-wavelength limit (Fig.1b,c). Furthermore, they strongly repulse each other especially at small momenta. As a result, the Coulomb energy of SF2 in the long wavelength limit is even larger than that of analogous inter-LL spin-flip mode in a fully spin-polarized quantum Hall state $\nu=1$ [@nu1pinczuk; @nu1vankov]. The Coulomb energy of the long-wavelength mode SF2 is just 15% smaller than that of spin wave at $k \to \infty$ (shown on the inset to Fig.2) being the exchange energy of electrons on the LL$_1$. On the contrary, the energy of SF1 proves out to be pushed down. One of the intriguing results of this calculation is that the highest energy spin-flip excitation SF2 corresponds to the antiphased combination of two single electron transitions and SF1 corresponds to the inphase combination. In this aspect the situation is opposite to the case of $\delta S_z=0$ modes MP (inphase) and AP (antiphase). As was shown in case of AP [@nu2chg; @nu2dickm] the first-order perturbation theory gives somewhat overestimated energy in the long-wavelength limit. Although second-order corrections are exactly computed only for AP at $k=0$, they are likely to be of the same order also for SF1 and SF2. ![\[fig2\] Inelastic light scattering spectra of intra-LL SW mode at $\nu = 3$ and $B=5.3\ {\rm T}$ taken at different laser photon energies (shown on the left). The inset shows the SW dispersion curve calculated within the Hartree-Fock approximation [@kallin84], for a 24nm- wide quantum well. At the experimental in-plane momentum the energy of SW is indistinguishable from ${\rm E_Z}$, shown by a dashed arrow.](Fig3.eps){width=".48\textwidth"} The inelastic light scattering measurements were performed on a high quality ${\rm GaAs}/{\rm Al}_{0.3}{\rm Ga}_{0.7}{\rm As}$ single quantum well of width 24nm. The electron density is $n=3.85 \times 10^{11}$cm$^{-2}$ and low temperature mobility above $17 \times 10^6$cm$^2$/V$\cdot$sec. The sample was mounted on the cold finger of a ${}^3{\rm He}/{}^4{\rm He}$ dilution refrigerator that is inserted in the cold bore of a superconducting magnet. The refrigerator is equipped with windows for optical access. Cold finger temperature was held mostly at $T=40$mK or 1.7K for one part of the experiment. The backscattering geometry was used at an angle $\theta=20^\circ$ with the normal of the sample surface. The perpendicular component of the magnetic field is $B=B_T$cos$\theta$ and $B_T$ is the total magnetic field. Resonant inelastic light scattering spectra were obtained by tuning the incident photon energy of a Ti:sapphire laser close to the fundamental optical gap of GaAs to enhance the light scattering cross section. The power density was kept below $10^{-4}$W/cm$^2$ for the measurements at temperatures around 40mK. The in-plane momentum, transferred to the excitations at the employed experimental geometry was about $5.3 \times 10^4$cm$^{-1}$. The scattered signal was dispersed by a triple grating spectrometer T-64000 working in additive and subtractive modes and analyzed by a charge-coupled device camera. The combined resolution of the system was about 0.02meV. In order to distinguish between inelastic light scattering and luminescence lines in spectra, the special test was employed – when varying the incident photon energy, inelastic light scattering lines traced the laser path, while luminescence lines did not change their spectral position. The resonant enhancement of the intensities of light scattering spectra of the long wavelength spin-wave (SW) mode at $\nu=3$ is displayed in Fig.2. The SW is at the bare Zeeman energy with $\left | g \right |=0.37$ and corresponds to the leftmost part of the mode dispersion shown in the inset to Fig.2. Very small changes in the laser photon energy (by $\sim0.5$meV) dramatically affect the line intensity, indicating the importance of resonant excitation in these experiments. The strong SW seen in Fig.2 is consistent with the ferromagnetic character of the quantum Hall state at $\nu=3$. Similar resonant excitation conditions prevail in the observation of the inter-Landau level excitations reported below. To capture light scattering spectra of inter-LL excitations, the incident photon energies were chosen in such a way as to excite electrons from the valence band to the second or third Landau levels in the conduction band. Typical spectra are measured at two laser positions (1538.2meV and 1550.0meV) shown on Fig.3. The magnetoplasmon and antiphased plasmon are seen shifted from the cyclotron energy $\hbar \omega_c$=8.65meV (depicted by an arrow on Fig.3). The blue shift of the MP results from the 2D-plasma energy at the non-zero in-plane momentum used in the experiment. In fact, the MP is the only dispersive mode in the range of experimentally accessible momenta (see Fig.1c). The energy of AP is below the CR by 0.19meV. This energy shift is somewhat smaller than that measured for 18nm quantum well in Ref.\[\], which is a consequence of the strong dependence of the effective Coulomb interaction strength on the quantum well width. Developed in Ref.\[\] theory gives $\Delta E_{\rm{AP}}(0) \approx-0.25$meV for this magnetic field and quantum well width. ![\[fig2\] Inelastic light scattering spectra of inter-LL excitations at $\nu = 3$ and $B_{\perp}=5.3\ {\rm T}$. Upper spectrum is taken at the incident photon energy $\hbar \omega_{exc}=1538.2\ {\rm meV}$, the lower one – at 1550.0meV. The position of $\hbar \omega_{c}$=8.65meV is shown by the arrow. Grey vertical columns mark inelastic light scattering lines. The rest of the spectrum is composed of the luminescence bands.](Fig2_0810.eps){width=".48\textwidth"} We focus here on the two cyclotron spin-flip modes SF1 and SF2 which are blue-shifted from $\hbar \omega_c$ by 1.13meV and 4.3meV respectively. We have compared these experimental values to those calculated theoretically within the first-order Hartree-Fock approximation taking into account the actual width of the quantum well (see Fig.1c). The discrepancy is of the order of the negative second order corrections such as for AP plasmon. The Coulomb energy of SF2 is close to the estimated full exchange energy of electrons on LL$_1$. The latter is represented by the energy limit of shortwave SW at $\nu=3$. This asymptotical value is about three fourth of analogous quantity at fully spin polarized state $\nu=1$. We also find a marked dependence on magnetic field in which lines SF1 and SF2 are observed only in the narrow interval $\Delta B \simeq 0.3$T around $\nu=3$. Outside this range of fields and filling factors the lines disappear from the spectrum. From this fact we conclude that these excitations are inherent to filling factor $\nu=3$. To summarize, by means of inelastic light scattering we have observed and identified four inter-Landau level collective excitations and intra-LL spin-wave at $\nu=3$. Among these excitations there are two cyclotron spin-flip modes, which interact repulsively in the long-wave limit. As a result, the upper of them (SF2) acquires a huge exchange contribution to the energy, comparable with the theoretically estimated exchange energy of electrons on the first Landau level. The experimentally measured energies of all excitations are in a good agreement with the Hartree-Fock calculations taking into account the finite thickness of the 2D-electron system. The authors acknowledge support from the U.S. Civilian Research and Development Foundation and the Russian Foundation for Basic Research. T.D.R. and A. P. are supported by the National Science Foundation under Grant No. DMR-0352738 and DMR-0803445, and by the Nanoscale Science and Engineering Initiative of the National Science Foundation under Award Number CHE-0117752 and CHE-0641523. T.D.R. and A.P. were also supported by the Department of Energy under Grant No. DE-AIO2-04ER46133, and by a research grant from the W. M. Keck Foundation. [99]{} , JETP Lett. [**33**]{}, 143 (1981). C. Kallin, and B. I. Halperin, Phys. Rev. B [**30**]{}, 5655 (1984). , Sov. Phys. JETP [**58**]{}, 1062 (1983). A. Schmeller, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. [**75**]{}, 4290 (1995). A. F. Dethlefsen, R. J. Haug, K. Vyborny, O. Certik, and A. Wojs, Phys. Rev. B  [**74**]{}, 195324 (2006). J. G. Groshaus, I. Dujovne, Y. Gallais, C. F. Hirjibehedin, A. Pinczuk, Y.-W. Tan, H. Stormer, B. S. Dennis, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett.  [**100**]{}, 046804 (2008). A. Usher, R. J. Nicholas, J. J. Harris, and C.T. Foxon, Phys. Rev. B [**41**]{}, 1129 (1990). V. T. Dolgopolov, A. A. Shashkin, V. Aristov, D. Schmerek, W. Hansen, J. P. Kotthaus, and M. Holland, Phys. Rev. Lett. [**79**]{}, 729 (1997). V. S. Khrapai, A. A. Shashkin, E. L. Shangina, V. Pellegrini, F. Beltram, G. Biasiol, L. Sorba, Phys. Rev. B [**72**]{}, 035344 (2005). M. Dobers, K. von Klitzing, G. Weimann, Phys. Rev. B [**38**]{}, 5453 (1988). A. Pinczuk, B.S. Dennis, D. Heiman, C. Kallin, L. Brey, C. Tejedor, S. Schmitt-Rink, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. [**68**]{}, 3623 (1992). A. B. Van’kov, L. V. Kulik, I. V. Kukushkin, V. E. Kirpichev, S. Dickmann, V. M. Zhilin, J. H. Smet, K. v. Klitzing, and W. Wegscheider, Phys. Rev. Lett. [**97**]{}, 246801 (2006). A. S. Zhuravlev, A. B. Van’kov, L. V. Kulik [*et al*]{}., Phys. Rev. B [**77**]{}, 155404 (2008). G. A. Csathy, J. S. Xia, C. L. Vicente, E. D. Adams, N. S. Sullivan, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. [**94**]{}, 146801 (2005). C. R. Dean, B. A. Piot, P. Hayden, S. Das Sarma, G. Gervais, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. [**101**]{}, 186806 (2008). T. D. Rhone, J. Yan, Y. Gallais, A. Pinczuk, L. Pfeiffer, and K. West, http://meetings.aps.org/link/BAPS.2009.MAR.J23.1. W. Kohn, Phys. Rev. [**123**]{}, 1242 (1961). M. A. Eriksson, A. Pinczuk, B. S. Dennis, S. H. Simon, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. [**82**]{}, 2163 (1999). L. V. Kulik, I. V. Kukushkin, S. Dickmann et al., Phys. Rev. B [**72**]{}, 073304 (2005). S. Dickmann and I. V. Kukushkin,Phys. Rev. B [**71**]{}, 241310(R) (2005). M. S-C. Luo, Sh.L. Chuang, S. Schmitt-Rink, and A. Pinczuk, Phys. Rev. B [**48**]{}, 11086 (1993). L. V. Kulik, S. Dickmann, I. K. Drozdov, A. S. Zhuravlev, V. E. Kirpichev, I. V. Kukushkin, S. Schmult, and W. Dietsche, Phys. Rev. B [**79**]{}, 121310(R) (2009).

--- abstract: 'We study the Knapp-Stein $R$-groups of the inner forms of $\SL(N)$ over a non-archimedean local field of characteristic zero, by using restriction from the inner forms of $\GL(N)$. As conjectured by Arthur, these $R$-groups are then shown to be naturally isomorphic to their dual avatars defined in terms of L-parameters. The $2$-cocycles attached to $R$-groups can be described as well. The proofs are based on the results of K. Hiraga and H. Saito. We also construct examples to illustrate some new phenomena which do not occur in the case of $\SL(N)$ or classical groups.' author: - Kuok Fai Chao - 'Wen-Wei Li' bibliography: - 'Restriction.bib' title: 'Dual $R$-groups of the inner forms of $\SL(N)$' --- Introduction ============ Let $G$ be a connected reductive group over a local field $F$ and $G(F)$ be the locally compact group of the $F$-points of $G$. The study of the tempered representations of $G(F)$ is a crucial ingredient of the monumental work of Harish-Chandra on his Plancherel formula. Denote by $\Pi_{\text{temp}}(G)$ the set of isomorphism classes of irreducible tempered representations, and by $\Pi_{2,\text{temp}}(G)$ its subset of representations which are square-integrable modulo the center. Roughly speaking, elements in $\Pi_{\text{temp}}(G)$ can be obtained as subrepresentations of $I^G_P(\sigma)$, where $P=MU$ is a parabolic subgroup, $\sigma \in \Pi_{2,\text{temp}}(M)$ and $I^G_P(\sigma)$ is the normalized parabolic induction. Assuming the knowledge of square-integrable representations, the study of $\Pi_{\text{temp}}(G)$ then boils down to that of the decomposition of $I^G_P(\sigma)$, for $P$ and $\sigma$ as above. Knapp, Stein and Silberger (for the non-archimedean case) described the decomposition of $I^G_P(\sigma)$ in terms of the *Knapp-Stein $R$-group* $R_\sigma$. More precisely, we have a central extension of groups $$1 \to \C^\times \to \tilde{R}_\sigma \to R_\sigma \to 1$$ defined using the normalized intertwining operators $R_P(w, \sigma)$. It is the set $\Pi_-(\tilde{R}_\sigma)$ of the irreducible representations of $\tilde{R}_\sigma$ by which $\C^\times$ acts by $z \mapsto z \cdot\identity$ which governs the decomposition of $I^G_P(\sigma)$. Equivalently, we are given a cohomology class $\mathbf{c}_\sigma \in H^2(R_\sigma, \C^\times)$ attached to this central extension. The group $R_\sigma$ itself suffices to determine whether $I^G_P(\sigma)$ is reducible or not. To extract further information, such as the description of elliptic tempered representations, some knowledge about $\tilde{R}_\sigma$ is also needed. We refer the reader to [@Ar93 §2] for details. On the other hand, the tempered part of the local Langlands correspondence predicts a map $\phi \mapsto \Pi_\phi$ which assigns a finite subset $\Pi_\phi$ of $\Pi_{\text{temp}}(G)$ to every bounded L-parameter $\phi \in \Phi_{\text{bdd}}(G)$, taken up to equivalence, such that $$\Pi_{\text{temp}}(G) = \bigsqcup_{\phi \in \Phi_{\text{bdd}}(G)} \Pi_\phi.$$ The internal structure of the tempered L-packets $\Pi_\phi$ is conjectured to be controlled by the $S$-group $S_\phi := Z_{\hat{G}}(\Im(\phi))$. More precisely, following [@Ar06], one has to introduce a central extension $$1 \to \tilde{Z}_\phi \to \tilde{\mathscr{S}}_\phi \to \mathscr{S}_\phi \to 1$$ of finite groups defined in terms of $S_\phi$. The L-packet $\Pi_\phi$ should be in bijection with a set $\Pi(\tilde{\mathscr{S}}_\phi, \chi_G)$ of representations of $\tilde{\mathscr{S}}_\phi$ where $\chi_G$ is a character of $\tilde{Z}_\phi$ coming from Galois cohomology. The relevant definitions will be reviewed later in this article. The tempered local Langlands correspondence is expected to behave well under normalized parabolic induction, namely for $P=MU$ as above and $\phi_M \in \Phi_{\text{bdd}}(M)$, we deduce $\phi \in \Phi_{\text{bdd}}(G)$ by composing $\phi_M$ with the inclusion $\Lgrp{M} \to \Lgrp{G}$ of L-groups, which is well-defined up to conjugacy. Then $\Pi_\phi$ should be the union of the irreducible constituents of $I^G_P(\sigma)$, where $\sigma$ ranges over the elements of $\Pi_{\phi_M}$. A natural question arises: Is it possible to describe $R_\sigma$, or even $\tilde{R}_\sigma$, in terms of the $S$-groups? For archimedean $F$ this has been answered by Shelstad [@Sh82]; in that case, the extension $\tilde{R}_\sigma \to R_\sigma$ splits and $R_\sigma$ is abelian of exponent two. For general $F$ of characteristic zero, Arthur proposed a generalization in [@Ar89-unip §7] as follows. For every $\phi \in \Phi_{\text{bdd}}(G)$ coming from $\phi_M \in \Phi_{2,\text{bdd}}(M)$ (i.e. a parameter for $M$ which is square-integrable modulo the center), he introduced the *dual $R$-group* (also known as the *endoscopic $R$-group*) $R_\phi \simeq \mathscr{S}_{\phi}/\mathscr{S}_{\phi_M}$ and a subgroup $R_{\phi, \sigma} \subset R_\phi$ for every $\sigma \in \Pi_{\phi_M}$. Arthur conjectures a natural isomorphism $$R_{\phi, \sigma} \simeq R_\sigma.$$ This has been verified for quasisplit classical groups and unitary groups by Arthur [@ArEndo] and Mok [@Mok1], respectively. In their construction of L-packets, the dual $R$-groups play a pivotal role through the *local intertwining relations* (see [@ArEndo Chapter 2]). It turns out that in these cases, we have $R_\phi = R_{\phi,\sigma}$ and $\tilde{R}_\sigma \to R_\sigma$ splits (see [@ArEndo §6.5]). Similar results are obtained independently by Ban, Goldberg and Zhang [@BZ05; @Go11; @BG12] for non-archimedean $F$. For quaternionic unitary groups, see [@Ha04]. We shall assume hereafter that $F$ is a non-archimedean local field of characteristic zero. Another good test ground for Arthur’s conjectures is the group $\SL(N)$ and its inner forms. Indeed, the case $N=2$ is the genesis of endoscopy [@LL79]; for general $N$, the local Langlands correspondence for the inner forms $G^\sharp$ of $\SL(N)$ is established in [@HS12], at least in the tempered case. This is based on the local Langlands correspondence for the inner forms $G$ of $\GL(N)$, which satisfies the following nice properties: - the L-packets $\Pi_\phi$ for $G$ are all singletons; - for any parabolic subgroup $P=MU$ and $\sigma \in \Pi_{\text{temp}}(M)$, the induced representation $I^G_P(\sigma)$ is irreducible. In fact, the latter property holds for all unitary $\sigma$, known as Tadić’s property (U0) [@Sec09]. The (tempered) local Langlands correspondence for $G^\sharp$ can be obtained by restriction from $G(F)$ to $G^\sharp(F)$; the procedure is somehow dual to the natural projection of L-groups $$\mathbf{pr}: \Lgrp{G} \twoheadrightarrow \Lgrp{G^\sharp}.$$ The same recipe can be applied to any Levi subgroup $M$, with respect to $M^\sharp := M \cap G^\sharp$. The method of restriction provides a convenient device, but we still have to study the internal structure of L-packets for $G^\sharp$ and their behaviour under normalized parabolic induction. For the quasisplit case $G^\sharp = \SL(N)$, such issues can be addressed by the multiplicity-one property of Whittaker models. In that case, the Knapp-Stein $R$-groups are studied in depth in [@GK81; @GK82; @Sh83; @Ta92; @Go94]. Roughly speaking, let $\sigma^\sharp \in \Pi_{2,\text{temp}}(M^\sharp)$ which lies in the L-packet $\Pi_{\phi^\sharp}$. We may choose $\sigma \in \Pi_{2,\text{temp}}(M)$ such that $\sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$. Set $\pi := I^G_P(\sigma)$, which is irreducible, then $R_{\sigma^\sharp}$ is described in terms of $$X^G(\pi) = \left\{ \eta \in (G(F)/G^\sharp(F))^D : \eta \otimes \pi \simeq \pi \right\}$$ and its analogue $X^M(\sigma)$ for the Levi subgroup $M$ with respect to $M^\sharp$, where $(G(F)/G^\sharp(F))^D$ means the group of continuous characters of $G(F)/G^\sharp(F)$. It is then easy to relate $R_{\sigma^\sharp}$ with $R_{\phi^\sharp}$, and we deduce a canonical isomorphism $R_{\phi^\sharp} = R_{\phi^\sharp, \sigma^\sharp} \simeq R_{\sigma^\sharp}$ as well as a splitting for $\tilde{R}_{\sigma^\sharp} \twoheadrightarrow R_{\sigma^\sharp}$. Note that we used the notations $\phi^\sharp$, $\sigma^\sharp$, etc. to denote the objects attached to $G^\sharp$ and its Levi subgroups. Whittaker models are no longer available for the non-quasisplit inner forms $G^\sharp$ of $\SL(N)$. What saves the day is the work of Hiraga and Saito [@HS12]. They defined a central extension $$1 \to \C^\times \to S^G(\pi) \to X^G(\pi) \to 1$$ and related it to the central extension of $\mathscr{S}$-groups alluded above. This allows us to study the internal structure of the L-packets obtained by restriction. In the main Theorem \[prop:main\], we will prove, among others, that there are 1. a canonical isomorphism $R_{\phi^\sharp, \sigma^\sharp} \simeq R_{\sigma^\sharp}$ as conjectured by Arthur; 2. a “concrete” description of the dual $R$-groups for $G^\sharp$, namely $$\begin{aligned} R_{\phi^\sharp} & \simeq X^G(\pi)/X^M(\sigma), \\ R_{\phi^\sharp, \sigma^\sharp} & \simeq Z^M(\sigma)^\perp/X^M(\sigma); \end{aligned}$$ 3. a description of the class $\mathbf{c}_{\sigma^\sharp} \in H^2(R_{\sigma^\sharp}, \C^\times)$ attached to $\tilde{R}_{\sigma^\sharp} \twoheadrightarrow R_{\sigma^\sharp}$, in terms of the obstruction for extending the representation $\rho$ of $\tilde{\mathscr{S}}_{\phi^\sharp_M}$ to the preimage of $R_{\phi, \sigma^\sharp}$ in $\tilde{\mathscr{S}}_{\phi^\sharp}$. We refer the reader to §\[sec:res-2\] for unexplained notations. Note that the description of $\mathbf{c}_{\sigma^\sharp}$ is also conjectured by Arthur; see [@Ar96 p.537], [@Ar08 §3] or [@ArEndo §2.4] for further discussions. Arthur’s conjecture on $R$-groups for the inner forms of $\SL(N)$ is thus verified. The examples are probably more interesting, however. In §\[sec:examples\] we will give conceptual constructions of $\phi^\sharp$ and $\sigma^\sharp$ as above such that 1. $\tilde{R}_{\sigma^\sharp} \twoheadrightarrow R_{\sigma^\sharp}$ is not split, or 2. $R_{\phi^\sharp, \sigma^\sharp} \subsetneq R_{\phi^\sharp}$. Such phenomena do not occur to the quasisplit classical groups, the quaternionic unitary groups, or $\SL(N)$. The first example is perhaps more surprising, since $\tilde{R}_{\sigma^\sharp} \twoheadrightarrow R_{\sigma^\sharp}$ always splits for generic inducing data. Keys [@Ke87 §6] constructed a Knapp-Stein $R$-group with non-split cocycle in the non-connected setting; our example seems to be the only known case for connected reductive groups. In both examples, the relation between $R_{\sigma^\sharp}$ and the $S$-groups is crucial. In view of the possible applications to automorphic representations, one should also consider certain nontempered unitary representations, namely those appearing in the *A-packets*; see Remark \[rem:nontempered\] for a short discussion. Shortly after the release of the first version of our preprint, we were informed of the independent work [@CG12] by Choiy and Goldberg that treats the same problems except that of cocycles. Despite some overlaps, their work has a completely different technical core, namely the transfer of Plancherel measures between inner forms, which should have wide-ranging applications. #### Organization of this article In §\[sec:preliminaries\], we recapitulate the formalism of normalized intertwining operators and Knapp-Stein $R$-groups. We follow Arthur’s notations in [@Ar89-IOR1; @Ar93] closely. In particular, the $R$-group $R_\sigma$ is regarded as a quotient of the isotropy group $W_\sigma$, instead of a subgroup. In §\[sec:res\], we set up a general formalism of restriction of representations. These results are scattered in [@Sh83; @Ke87; @Ta92; @HS12], just to mention a few. In view of the possible sequels of this work, the behaviour under restriction of normalized intertwining operators is treated in generality. A special assumption is made in §\[sec:res-2\] (Hypothesis \[hyp:irred\]), namely the parabolically induced representations in question should be irreducible. We are then able to deduce finer information on $R$-groups and their cocycles in this setting. The arguments are not too difficult, but require some careful manipulations. In §\[sec:SL\], we will specialize to the inner forms of $\SL(N)$ and reformulate the results of Hiraga and Saito [@HS12] on the local Langlands correspondence and the $S$-groups. In order to study parabolic induction , we also have to generalize these results to the Levi subgroups. In §\[sec:dual-R\], we recapitulate Arthur’s definition of dual $R$-groups via the omnipresent commutative diagram in Proposition \[prop:diagram\]. The results obtained earlier can then be easily assembled, and Arthur’s conjecture on $R$-groups for the inner forms of $\SL(N)$ (Theorem \[prop:main\]) follows. #### Acknowledgements The authors would like to thank Kwangho Choiy and David Goldberg for communicating their results to us, as well as the referee for meticulous reading. The second-named author is grateful to Dipendra Prasad for his interest in this work and helpful comments. Preliminaries {#sec:preliminaries} ============= Conventions ----------- #### Local fields Throughout this article, $F$ always denotes a non-archimedean local field of characteristic zero. We set - $\Gamma_F$: the absolute Galois group of $F$, defined with respect to a chosen algebraic closure $\bar{F}$; - $W_F$: the Weil group of $F$; - $\WD_F := W_F \times \SU(2)$: the Weil-Deligne group of $F$; - $|\cdot| = |\cdot|_F$: the normalized absolute value of $F$; - $q_F$: the cardinality of the residue field of $F$. When discussing the canonical family of normalizing factors for $\GL(N)$ and its inner forms, we will also fix a non-trivial additive character $\psi_F: F \to \C^\times$. The usual Galois cohomology over $F$ is denoted by $H^\bullet(F,\cdot)$. Continuous cohomology of $W_F$ is denoted by $H^\bullet_\text{cont}(W_F, \cdots)$; the groups of continuous cocycles are denoted by $Z^\bullet_\text{cont}(W_F, \cdots)$. #### Groups and representations For an $F$-group scheme $G$, the group of its $F$-points is denoted by $G(F)$; subgroups of $G$ mean the closed subgroup schemes. The identity connected component of $G$ is denoted by $G^0$. The center of $G$ is denoted by $Z_G$. Centralizers (resp. normalizers) in $G$ are denoted by $Z_G(\cdot)$ (resp. $N_G(\cdot)$). The algebraic groups over $\C$ are identified with their $\C$-points. The derived group of $G$ is denoted by $G_\text{der}$. Now assume $G$ to be connected reductive. A simply connected cover of $G_\text{der}$, which is unique up to isomorphism, is denoted by $G_\text{SC} \to G_\text{der}$. We denote the adjoint group of $G$ by $G_\text{AD} := G/Z_G$. For every subgroup $H$ of $G$, we denote by $H_\text{sc}$ (resp. $H_\text{ad}$) the preimage of $H$ in $G_\text{SC}$ (resp. image in $G_\text{AD}$). The same formalism pertains to connected reductive $\C$-groups as well. The definitions of the dual group $\Lgrp{G} = \hat{G} \rtimes W_F$ and the L-parameters will be reviewed in §\[sec:res-L\]. The symbol $\Ad(\cdots)$ denotes the adjoint action of an abstract group on itself, namely $\Ad(x): g \mapsto gxg^{-1}$. For any division algebra $D$ over $F$ and $n \in \Z_{\geq 1}$, we denote by $\GL_D(n)$ the group of invertible elements in $\End_D(D^n)$, where $D^n$ is viewed as a right $D$-module. It is also regarded as a connected reductive $F$-group. The representations considered in this article are all over $\C$-vector spaces. For a connected reductive $F$-group $G$, we define - $\Pi(G)$: the set of equivalence classes of irreducible smooth representations of $G(F)$; - $\Pi_{\text{unit}}(G)$: the subset consisting of unitary (i.e. unitarizable) representations; - $\Pi_{\text{temp}}(G)$: the subset consisting of tempered representations; - $\Pi_{2,\text{temp}}(G)$: the subset consisting of unitary representations which are square-integrable modulo the center. For an abstract group $S$, we will also denote by $\Pi(S)$ the set of its irreducible representations up to equivalence. The central character of $\pi \in \Pi(G)$ is denoted by $\omega_\pi$. The group of morphisms (resp. the set of isomorphisms) in the category of representations of $G(F)$ is denoted by $\Hom_G(\cdots)$ (resp. $\Isom_G(\cdots)$). For any topological group $H$, we set $$H^D := \{\chi: H \to \C^\times, \text{ continuous character} \}.$$ For any representation $\pi$ of $G(F)$ and any $\eta \in G(F)^D$, we write $\eta\pi := \eta \otimes \pi$ for abbreviation. Also note that $\pi$ and $\eta\pi$ have the same underlying $\C$-vector spaces. If $M$ is a subgroup of $G$ and $\pi$ is a smooth representation of $G(F)$, we shall denote the restriction of $\pi$ to $M(F)$ by $\pi|_M$. #### Combinatorics Let $G$ be a connected reductive $F$-group. We employ the following notations in this article. Let $M$ be a Levi subgroup, we write - $\mathcal{P}(M)$: the set of parabolic subgroups of $G$ with Levi component $M$; - $\mathcal{L}(M)$: the set of Levi subgroups of $G$ containing $M$; - $\mathcal{F}(M)$: the set of parabolic subgroups of $G$ containing $M$; - $W(M) := N_G(M)(F)/M(F)$: the Weyl group (in a generalized sense) relative to $M$; The Levi decompositions are written as $P=MU$, where $U$ denotes the unipotent radical of $P$. For $M$ chosen, the opposite parabolic of $P=MU$ is denoted by $\bar{P} = M\bar{U}$. When we have to emphasize the role of $G$, the notations $\mathcal{P}^G(M)$, $\mathcal{L}^G(M)$, $\mathcal{F}^G(M)$ and $W^G(M)$ will be used. Let $w \in W(M)$ with a representative $\tilde{w} \in G(F)$. For $\sigma \in \Pi(M)$, we define $\tilde{w}\sigma$ to be the representation on the same underlying vector space, with the new action $$(\tilde{w}\sigma)(m) := \sigma(\tilde{w}^{-1} m \tilde{w}), \quad m \in M(F).$$ The equivalence class of $\tilde{w}\sigma$ depends only on $w \in W(M)$, and we will write $w\sigma$ instead, if there is no confusion. Define $\mathcal{X}(G) := \Hom_{F-\text{grp}}(G, \Gm)$ and $\mathfrak{a}_G := \Hom(\mathcal{X}(G), \R)$. For every Levi subgroup $M$, there is a canonically split short exact sequence of finite-dimensional $\R$-vector spaces $$0 \to \mathfrak{a}_G \to \mathfrak{a}_M \leftrightarrows \mathfrak{a}^G_M \to 0.$$ The linear duals of these spaces are denoted by $\mathfrak{a}^*_G$, etc. We also write $\mathfrak{a}_{G,\C} := \mathfrak{a}_G \otimes_\R \C$, etc. Sometime we also write $\mathfrak{a}_P$ instead of $\mathfrak{a}_M$ if $P=MU$. The Harish-Chandra map $H_G: G(F) \to \mathfrak{a}_G$ is the homomorphism characterized by $$\angles{\chi, H_G(x)} = \log|\chi(x)|_F, \quad \chi \in \mathcal{X}(G).$$ For $\lambda \in \mathfrak{a}^*_{G,\C}$ and $\pi \in \Pi(G)$, we define $\pi_\lambda \in \Pi(G)$ by $$\begin{gathered} \label{eqn:pi-twist} \pi_\lambda := e^{\angles{\lambda, H_G(\cdot)}} \otimes \pi.\end{gathered}$$ Fix a minimal parabolic subgroup $P_0 = M_0 U_0$ of $G$. We define $\Delta_0$, $\Delta_0^\vee$ to be the set of simple roots and coroots, which form bases of $(\mathfrak{a}^G_{M_0})^*$ and $\mathfrak{a}^G_{M_0}$, respectively. The set of positive roots is denoted by $\Sigma_0$, and its subset of reduced roots by $\Sigma^{\text{red}}_0$. They form a bona fide root system. For every $P = MU \supset P_0$, we define $\Delta_P \subset \Sigma_P \subset \Sigma^{\text{red}}_P$ by taking the set of nonzero restrictions to $(\mathfrak{a}^G_M)^*$ of elements in $\Delta_0 \subset \Sigma_0 \subset \Sigma^\text{red}_0$. To each $\alpha \in \Sigma_P$ we may associate the coroot $\alpha^\vee \in \mathfrak{a}^G_M$: it is defined as the restriction of the coroot in $\Delta^\vee_0$. For a given $P$, the objects above are independent of the choice of $P_0$. We can emphasize the role of $G$ by using the notations $\Delta^G_P$, etc. whenever needed. #### Induction We always consider a parabolic subgroup $P=MU$ of $G$. The modulus character of $P(F)$ is denoted by $\delta_P$, i.e. $$(\text{left Haar measure}) = \delta_P \cdot (\text{right Haar measure}).$$ The usual smooth induction functor is denoted by $\Ind(\cdots)$. The normalized parabolic induction functor from $P$ to $G$ is denoted by $I^G_P(\cdot) := \Ind^G_P(\delta^{\frac{1}{2}}_P \otimes \cdot)$. Recall that for $\sigma \in \Pi(M)$ with underlying vector space $V_\sigma$, we use the usual model to realize $I^G_P(\sigma)$ as the space of functions $\varphi: G(F) \to V_\sigma$ such that $\varphi$ is invariant under right translation by an open compact subgroup of $G(F)$, and that $\varphi(umx) = \delta_P(m)^{\frac{1}{2}}\sigma(m)(\varphi(x))$ for all $m \in M(F)$, $u \in U(F)$. The group $G(F)$ acts on this function space by the right regular representation. For $\sigma, \sigma' \in \Pi(M)$ and $f \in \Hom_M(\sigma, \sigma')$, the induced morphism is denoted by $I^G_P(f)$; it sends $\varphi$ to $f(\varphi)$. Normalized intertwining operators --------------------------------- Our basic reference for normalized intertwining operators is [@Ar93]. Consider the following data - $G$: a connected reductive $F$-group. - $M$: a Levi subgroup of $G$, - $P,Q \in \mathcal{P}(M)$, - $\sigma: M(F) \to \Aut_\C(V_\sigma)$: a smooth representation of $M(F)$ of finite length, - $\lambda \in \mathfrak{a}^*_{M,\C}$. For every $\alpha \in \Delta_P$, we denote by $r_\alpha$ the smallest positive rational number such that $r_\alpha \cdot \alpha^\vee$ lies in the lattice $H_M(M(F))$. We define $$\begin{gathered} \label{eqn:alpha-check} \check{\alpha} := r_\alpha \alpha^\vee .\end{gathered}$$ By recalling , we form the normalized parabolic induction $I^G_P(\sigma_\lambda)$, $I^G_Q(\sigma_\lambda)$. Their underlying spaces are denoted by $I^G_P(V_{\sigma_\lambda})$, $I^G_Q(V_{\sigma_\lambda})$. The standard intertwining operator $$J_{Q|P}(\sigma_\lambda): I^G_P(\sigma_\lambda) \longrightarrow I^G_Q(\sigma_\lambda)$$ is defined by the absolutely convergent integral $$\begin{gathered} \label{eqn:J-int} (J_{Q|P}(\sigma_\lambda)\varphi)(x) = \int_{U_P(F) \cap U_Q(F) \backslash U_Q(F)} \varphi(u x) \dd u, \quad x \in G(F),\end{gathered}$$ when $\angles{\Re(\lambda), \alpha^\vee } \gg 0$ for all $\alpha \in \Sigma^\text{red}_P \cap \Sigma^\text{red}_{\bar{Q}}$; see [@Wa03 IV.1] for the precise meaning of absolute convergence. Recall that upon choosing a special maximal compact open subgroup $K \subset G(F)$ in good position relative to $M$, these induced representations can be realized on a vector space that is independent of $\lambda$. It is known that $J_{Q|P}(\sigma_\lambda)$ is a rational function in the variables $$\left\{ q_F^{-\angles{\lambda, \check{\alpha}}} : \alpha \in \Delta_P \right\}.$$ In particular, as a function in $\lambda$, $J_{Q|P}(\sigma_\lambda)$ admits a meromorphic continuation to $\mathfrak{a}^*_{M,\C}$. When $\sigma \in \Pi_{\text{temp}}(M)$, it is known that is absolutely convergent for $\angles{\Re(\lambda), \alpha^\vee} > 0$ for all $\alpha \in \Sigma^\text{red}_P \cap \Sigma^\text{red}_{\bar{Q}}$. Moreover, as a meromorphic family of operators, it satisfies $\text{ord}_{\lambda=0} (J_{Q|P}(\sigma_\lambda)) \geq -1$. Henceforth we assume $\sigma$ irreducible, i.e. $\sigma \in \Pi(M)$. Take any $P \in \mathcal{P}(M)$, define the $j$-functions as $$\begin{gathered} \label{eqn:j-function} j(\sigma_\lambda) := J_{P|\bar{P}}(\sigma_\lambda) J_{\bar{P}|P}(\sigma_\lambda).\end{gathered}$$ It is known that $\lambda \mapsto j(\sigma_\lambda)$ a scalar-valued meromorphic function, which is not identically zero. Moreover, $j(\sigma_\lambda)$ is independent of $P$ and admits a product decomposition $$j(\sigma_\lambda) = \prod_{\alpha \in \Sigma^\text{red}_P} j_\alpha(\sigma_\lambda)$$ where $j_\alpha$ denotes the $j$-function defined relative to the Levi subgroup $M_\alpha \in \mathcal{L}(M)$ such that $\Sigma^{M_\alpha,\text{red}}_M = \{\pm \alpha\}$. Now assume $\sigma \in \Pi_{2,\text{temp}}(M)$. In this paper, we define Harish-Chandra’s $\mu$-function as the meromorphic function $$\mu(\sigma_\lambda) := j(\sigma_\lambda)^{-1}.$$ Accordingly, $\mu$ also admits a product decomposition $\mu = \prod_\alpha \mu_\alpha$. It is analytic and non-negative for $\lambda \in i\mathfrak{a}^*_M$. Note that our definitions of $j$-functions and $\mu$-functions depend on the choice of Haar measures on unipotent radicals. In particular, our $\mu$-function differs from that in [@Wa03 V.2] by some harmless constant. \[def:normalizing\] In this article, a family of normalizing factors is a family of meromorphic functions on the $\mathfrak{a}^*_{M,\C}$-orbits in $\Pi(M)$, for all Levi subgroup $M$ of $G$, written as $$r_{Q|P}(\sigma_\lambda), \quad P,Q \in \mathcal{P}(M), \sigma_\lambda \in \mathfrak{a}^*_{M,\C}$$ satisfying the following conditions. First of all, we define the corresponding normalized intertwining operators as $$R_{Q|P}(\sigma_\lambda) := r_{Q|P}(\sigma_\lambda)^{-1} J_{Q|P}(\sigma_\lambda)$$ which is a meromorphic family (in $\lambda$) of intertwining operators $I^G_P(\sigma_\lambda) \to I^G_Q(\sigma_\lambda)$. We shall also assume that a family of normalizing factors is chosen for every proper Levi subgroup. 1. For all $P, P', P'' \in \mathcal{P}(M)$, we have $R_{P''|P}(\sigma_\lambda) = R_{P''|P'}(\sigma_\lambda) R_{P'|P}(\sigma_\lambda)$. 2. If $\sigma \in \Pi_\text{unit}(M)$, then $$R_{Q|P}(\sigma_\lambda) = R_{P|Q}(\sigma_{-\bar{\lambda}})^*, \quad \lambda \in \mathfrak{a}_{M,\C}^* .$$ In particular, $R_{Q|P}(\sigma)$ is a well-defined unitary operator. 3. This family is compatible with conjugacy, namely $$R_{gQg^{-1}|gPg^{-1}}(g\sigma_\lambda) = \ell(g) R_{Q|P}(\sigma_\lambda) \ell(g)^{-1}$$ for all $g \in G(F)$, where $\ell(g)$ is the map $\varphi(\cdot) \mapsto \varphi(g^{-1} \cdot)$ 4. We have $$r_{Q|P}(\sigma_\lambda) = \prod_{\alpha \in \Sigma_P^\text{red} \cap \Sigma_{\bar{Q}}^\text{red}} r^{\tilde{M}_\alpha}_{\overline{P_\alpha}|P_\alpha}(\sigma_\lambda),$$ where $P_\alpha := P \cap M_\alpha$, and $r^{M_\alpha}_{\overline{P_\alpha}|P_\alpha}$ comes from the family of normalizing factors for $M_\alpha$. 5. Let $S=LU \in \mathcal{F}(M)$ containing both $P$ and $Q$, then $R_{Q|P}(\sigma_\lambda)$ is the operator deduced from $R^L_{P \cap L|Q \cap L}(\sigma_\lambda)$ by the functor $I^G_S(\cdot)$. 6. The function $\lambda \mapsto r_{Q|P}(\sigma_\lambda)$ is rational in the variables $\left\{ q_F^{-\angles{\lambda, \check{\alpha}}} : \alpha \in \Delta_P \right\}$. 7. If $\sigma \in \Pi_\text{temp}(M)$, then the meromorphic function $\lambda \mapsto r_{Q|P}(\sigma_\lambda)$ is invertible whenever $\Re\angles{\lambda, \alpha^\vee} > 0$ for all $\alpha \in \Delta_P$. Observe that $\mathbf{R}_2$ is equivalent to that $r_{Q|P}(\sigma_\lambda) = \overline{r_{P|Q}(\sigma_{-\bar{\lambda}})}$ for $\sigma \in \Pi_\text{unit}(M)$, as the unnormalized operators $J_{Q|P}(\sigma_\lambda)$ satisfy a similar condition. Similarly, $\mathbf{R}_3$ is equivalent to that $r_{gQg^{-1}|gPg^{-1}}(g\sigma_\lambda) = r_{Q|P}(\sigma_\lambda)$. The fundamental result about the normalizing factors is that they exist [@Ar89-IOR1 Theorem 2.1]. \[rem:normalization\] According to Langlands [@Lan76 Appendix 2], there is a conjectural canonical family of normalizing factors $r_{Q|P}(\sigma_\lambda)$ in terms of local factors, namely $$r_{Q|P}(\sigma_\lambda) = \varepsilon(0, \rho_{Q|P}^\vee \circ \phi_\lambda, \psi_F)^{-1} L(0, \phi_\lambda, \rho_{Q|P}^\vee) L(1, \phi_\lambda, \rho_{Q|P}^\vee)^{-1},$$ where - $\phi_\lambda$ is the Langlands parameter for $\sigma_\lambda$; - let $\hat{\mathfrak{u}}_Q$ (resp. $\hat{\mathfrak{u}}_P$) denote the Lie algebra of the unipotent radical of the dual parabolic subgroup $\hat{Q}$ (resp. $\hat{P}$) in $\hat{G}$; - $\rho_{Q|P}$ is the adjoint representation of $\Lgrp{M}$ on $\hat{\mathfrak{u}}_Q/(\hat{\mathfrak{u}}_Q \cap \hat{\mathfrak{u}}_P)$, and $\rho_{Q|P}^\vee$ denotes its contragredient; - $\psi_F: F \to \C^\times$ is a chosen non-trivial additive character. We will invoke this description only in the case $G=\GL_F(n)$. In that case, the local factors in sight are essentially those associated with pairs $(\phi_1, \phi_2)$ where $\phi_1, \phi_2$ are among the L-parameters parametrizing the components of $\sigma$. Such Artin local factors are known to agree with their representation-theoretic avatars, say those defined by Rankin-Selberg convolution or by the Langlands-Shahidi method. \[rem:construction-normalization\] The construction of normalizing factors can be reduced to the case that $M$ is a maximal proper Levi subgroup of $G$ and $\sigma \in \Pi_{2,\text{temp}}(M)$, as illustrated in [@Ar89-IOR1]. Let us give a quick sketch of this reduction. 1. In view of $\mathbf{R}_4$, we are led to the case $M$ maximal proper. Moreover, it suffices to verify $\mathbf{R}_3$ for the representatives in $G(F)$ of the elements in $W(M) := N_{G(F)}(M)/M(F)$, which has at most two elements. 2. Assume that $\sigma \in \Pi_\text{temp}(M)$. By the classification of tempered representations, there exist a parabolic subgroup $R=M_R U_R$ of $M$ and $\tau \in \Pi_\text{temp}(M_R)$ such that $\sigma \hookrightarrow I^M_R(\tau)$. The pair $(M,\tau)$ is unique up to conjugacy. There is a unique element $P(R)$ in $\mathcal{P}(M_R)$, characterized by the properties - $P(R) \subset P$, - $P(R) \cap M = R$. Consequently, parabolic induction in stages gives $I^G_P I^M_R(\tau) = I^G_{P(R)}(\tau)$. The same construction works when $P$ is replaced by $Q$. Set $$r_{Q|P}(\sigma_\lambda) := r_{Q(R)|P(R)}(\tau_\lambda).$$ In view of $\mathbf{R}_5$ together with parabolic induction in stages, we see that $R_{Q|P}(\sigma_\lambda)$ is the restriction of $R_{Q(R)|P(R)}(\sigma_\lambda)$ to $I^G_P(\sigma_\lambda)$. The required conditions can be readily verified. 3. For general $\sigma$, we may realize it as the Langlands quotient $I^M_R(\tau_\mu) \twoheadrightarrow \sigma$, where $R = M_R U_R$ is a parabolic subgroup of $M$, $\tau \in \Pi_\text{temp}(M_R)$ and $\mu \in \mathfrak{a}^*_{M_R}$ satisfies $\Re\angles{\mu, \beta^\vee} > 0$ for all $\beta \in \Delta^M_R$. The triplet $(M,\tau,\mu)$ is again unique up to conjugacy. Let $P,Q \in \mathcal{P}(M)$, define $P(R), Q(R) \in \mathcal{P}(M_R)$ as before and set $$r_{Q|P}(\sigma_\lambda) := r_{Q(R)|P(R)}(\tau_{\lambda+\mu}).$$ Recall that $$\Ker[I^M_R(\tau_\mu) \twoheadrightarrow \sigma] = \Ker(J^M_{\bar{R}|R}(\tau_\mu)).$$ The condition $\mathbf{R}_7$ applied to $\tau_\mu$ tells us that $\Ker(J^M_{\bar{R}|R}(\tau_\mu)) = \Ker(R^M_{\bar{R}|R}(\tau_\mu))$. Idem for $\mu$ replaced by $\mu+\lambda$. Using $\mathbf{R}_5$, one sees that $R_{Q(R)|P(R)}(\tau_{\lambda+\mu})$ factors into $R_{Q|P}(\sigma_\lambda)$ on $I^G_P(\sigma_\lambda)$. All conditions except $\mathbf{R}_2$ follow from this. The proof of $\mathbf{R}_2$ requires somehow more efforts to deal with the unitarizability of Langlands quotients; the reader can consult [@Ar89-IOR1 p.30] for details. 4. Reverting to our original assumption that $M$ is maximal proper and $\sigma \in \Pi_{2,\text{temp}}(M)$, it is clear that it remains to verify conditions $\mathbf{R}_1$, $\mathbf{R}_2$, $\mathbf{R}_3$, $\mathbf{R}_6$, $\mathbf{R}_7$. Furthermore, one can reduce $\mathbf{R}_3$ to the assertion that $r_{w\bar{P}w^{-1}|wPw^{-1}}(w(\sigma_\lambda)) = r_{\bar{P}|P}(\sigma_\lambda)$, for $w \in W(G)$ being the non-trivial element in $W(M)$ if it exists. Knapp-Stein $R$-groups ---------------------- Fix a family of normalizing factors for $G$. Assume henceforth that $M$ is a Levi subgroup of $G$ and $\sigma \in \Pi_{2,\text{temp}}(M)$. Define the isotropy group $$W_\sigma := \{w \in W(M) : w\sigma \simeq \sigma \}.$$ Fix $P \in \mathcal{P}(M)$. For $w \in W(M)$ with a representative $\tilde{w} \in G(F)$, we define the operator $r_P(\tilde{w}, \sigma) \in \Isom_G(I^G_P(\sigma), I^G_P(\tilde{w}\sigma))$ by $$\begin{gathered} \label{eqn:r_P} r_P(\tilde{w}, \sigma): I^G_P(\sigma) \xrightarrow{R_{w^{-1}Pw|P}(\sigma)} I^G_{w^{-1}Pw}(\sigma) \xrightarrow{[\ell(\tilde{w}): \phi \mapsto \phi(\tilde{w}^{-1} \cdot)]} I^G_P(\tilde{w}\sigma).\end{gathered}$$ We notice the property that for any $w, w' \in W(M)$ with representatives $\tilde{w}, \tilde{w}' \in G(F)$, we have $$\begin{gathered} \label{eqn:r_P-property} r_P(\tilde{w}\tilde{w}', \sigma) = r_P(\tilde{w}, \tilde{w}'\sigma) \circ r_P(\tilde{w}', \sigma).\end{gathered}$$ Assume now $w \in W_\sigma$. Choose a representative $\tilde{w}$ of $w$ and $\sigma(\tilde{w}) \in \Isom_M(\tilde{w}\sigma, \sigma)$ to define the operator $$R_P(\tilde{w}, \sigma) := I^G_P(\sigma(\tilde{w})) \circ r_P(\tilde{w}, \sigma).$$ The class $R_P(\tilde{w}, \sigma) \text{ mod } \C^\times$ is independent of the choices of $\sigma(\tilde{w})$ and the representative $\tilde{w}$. We also have $$\begin{gathered} R_P(\tilde{w}, \sigma) \in \Aut_G(I^G_P(\sigma)), \\ R_P(\tilde{w}, \sigma) R_P(\tilde{w}', \sigma) = R_P(\tilde{w} \tilde{w}', \sigma) \mod \C^\times, \quad w, w' \in W_\sigma.\end{gathered}$$ Now we can define the Knapp-Stein $R$-group as follows. $$\begin{aligned} W^0_\sigma & := \{w \in W_\sigma : R_P(\tilde{w},\sigma) \in \C^\times \identity \}, \\ R_\sigma & := W_\sigma/W^0_\sigma.\end{aligned}$$ We will also make use of the following alternative description of $R_\sigma$. The subgroup $W^0_\sigma$ is the Weyl group of the root system on $\mathfrak{a}_M$ composed of the multiples of the roots in $\{ \alpha \in \Sigma^\mathrm{red}_P : \mu_\alpha(\sigma)=0 \}$. Given any Weyl chamber $\mathfrak{a}^+_\sigma \subset \mathfrak{a}_M$ for the aforementioned root system, there is then a unique section $R_\sigma \hookrightarrow W_\sigma$ that sends $r \in R_\sigma$ to the representative $w \in W_\sigma$ such that $w\mathfrak{a}^+_\sigma = \mathfrak{a}^+_\sigma$. Consequently, we can write $W_\sigma = W^0_\sigma \rtimes R_\sigma$. In the literature, $R_\sigma$ is sometimes viewed as a subgroup of $W_\sigma$ in this manner, eg. [@Go06]. Write $V_\sigma$ (resp. $I^G_P(V_\sigma)$) for the underlying vector space of the representation $\sigma$ (resp. $I^G_P(\sigma)$). It follows that $w \mapsto R_P(\tilde{w}, \sigma)$ induces a projective representation of $R_\sigma$ on $I^G_P(V_\sigma)$, where $\tilde{w} \in G(F)$ is any representative of $w \in W_\sigma$. We denote this projective representation provisionally by $r \mapsto R_P(r, \sigma)$, for $r \in R_\sigma$. There is a standard way to lift $R_P(\cdot,\sigma)$ to an authentic representation of some group $\tilde{R}_\sigma$ which sits in a central extension $$1 \to \C^\times \to \tilde{R}_\sigma \to R_\sigma \to 1,$$ such that $\C^\times$ acts by $z \mapsto z \cdot \identity$. Namely, we can set $\tilde{R}_\sigma$ to be the group of elements $(r, M[r]) \in R_\sigma \times \Aut_\C(I^G_P(V_\sigma))$ such that $M[r] \text{ mod } \C^\times$ gives $R_P(r, \sigma)$. The lifted representation, denoted by $\tilde{r} \mapsto R_P(\tilde{r},\sigma)$, is then $\tilde{r} = (r, M[r]) \mapsto M[r]$. Such a central extension by $\C^\times$ that lifts $R_P(\cdot, \sigma)$ is unique up to isomorphism. Note that the central extension above can also be described by the class $\mathbf{c}_\sigma \in H^2(R_\sigma, \C^\times)$ coming from the $\C^\times$-valued $2$-cocycle $c_\sigma$ defined by $$\begin{gathered} \label{eqn:R-cocycle} R_P(\widetilde{r_1 r_2}, \sigma) = c_\sigma(r_1, r_2) R_P(\tilde{r}_1, \sigma) R_P(\tilde{r}_2, \sigma), \quad r_1, r_2 \in R_\sigma\end{gathered}$$ where we choose a preimage $\tilde{r} \in \tilde{R}_\sigma$ for every $r \in R_\sigma$. Fix a preimage $\tilde{r} \in \tilde{R}_\sigma$ for every $r \in R_\sigma$, then the operators $\{ R_P(\tilde{r}, \sigma) : r \in R_\sigma \}$ form a basis of $\End_G(I^G_P(\sigma))$. Following Arthur, we reformulate this fundamental result as follows. Let $$\begin{aligned} \Pi_\sigma(G) & := \{ \text{irreducible constituents of } I^G_P(\sigma) \}/\simeq , \\ \Pi_-(\tilde{R}_\sigma) & := \{ \rho \in \Pi(\tilde{R}_\sigma) : \forall z \in \C^\times, \; \rho(z) = z\cdot\identity \}.\end{aligned}$$ Note that $\Pi_\sigma(G)$, $\Pi_-(\tilde{R}_\sigma)$ are both finite sets, and each $\rho \in \Pi_-(\tilde{R}_\sigma)$ is finite-dimensional. Let $\mathcal{R}$ be the representation of $\tilde{R}_\sigma \times G(F)$ on $I^G_P(V_\sigma)$ defined by $$\mathcal{R}(\tilde{r}, x) = R_P(\tilde{r}, \sigma) I^G_P(\sigma,x), \quad \tilde{r} \in \tilde{R}_\sigma, \; x \in G(F).$$ Then there is a decomposition $$\begin{gathered} \label{eqn:R-decomp} \mathcal{R} \simeq \bigoplus_{\rho \in \Pi_-(\tilde{R}_\sigma)} \rho \boxtimes \pi_\rho, \end{gathered}$$ where $\rho \mapsto \pi_\rho$ is a bijection from $\Pi_-(\tilde{R}_\sigma)$ to $\Pi_\sigma(G)$, characterized by . Consequently, $I^G_P(\sigma)$ is irreducible if and only if $R_\sigma=\{1\}$. When $G$ is quasisplit and $\sigma$ is generic with respect to a given Whittaker datum for $M$, the work of Shahidi [@Sh90] furnishes 1. a canonical family of normalizing factors $r_{Q|P}(\sigma)$; 2. a canonically defined homomorphism $w \mapsto R_P(w, \sigma)$; 3. a canonical splitting of the central extension $1 \to \C^\times \to \tilde{R}_\sigma \to R_\sigma \to 1$. These properties are not expected in general. Indeed, we shall see in Example \[ex:nonsplit\] that (iii) may fail. The formalism above depends not only on $(M,\sigma)$, but also on the choice of $P \in \mathcal{P}(M)$. One can easily pass to another choice $Q \in \mathcal{P}(M)$ by transport of structure using $R_{Q|P}(\sigma)$. For example, one has $$\begin{aligned} r_Q(\tilde{w}, \sigma) & = R_{P|Q}(\tilde{w}\sigma)^{-1} r_P(\tilde{w}, \sigma) R_{P|Q}(\sigma), \\ R_Q(\tilde{w}, \sigma) & = R_{P|Q}(\sigma)^{-1} R_P(\tilde{w}, \sigma) R_{P|Q}(\sigma) \end{aligned}$$ for all $w \in W(M)$ with a representative $\tilde{w} \in G(F)$ and some chosen $\sigma(\tilde{w})$. Restriction {#sec:res} =========== Let $G$, $G^\sharp$ be connected reductive $F$-groups such that $$G_\text{der} \subset G^\sharp \subset G.$$ Restriction of representations {#sec:res-rep} ------------------------------ In this subsection, we will review the basic results in [@Ta92 §2] and [@HS12 Chapter 2] concerning the restriction of a smooth representation from $G(F)$ to $G^\sharp(F)$. The objects associated to $G^\sharp$ are endowed with the superscript $\sharp$, eg. $\pi^\sharp \in \Pi(G^\sharp)$. \[prop:lifting\] Let $\pi \in \Pi(G)$, then $\pi|_{G^\sharp}$ decomposes into a finite direct sum of smooth irreducible representations. Each irreducible constituent of $\pi|_{G^\sharp}$ has the same multiplicity. Conversely, every $\pi^\sharp \in \Pi(G^\sharp)$ embeds into $\pi|_{G^\sharp}$ for some $\pi \in \Pi(G)$. If the central character $\omega_{\pi^\sharp}$ is unitary, one can choose $\pi$ so that $\omega_\pi$ is also unitary. \[prop:res-disjoint\] Let $\pi_1, \pi_2 \in \Pi(G)$. The following are equivalent: 1. $\Hom_{G^\sharp}(\pi_1, \pi_2) \neq \{0\}$; 2. $\pi_1|_{G^\sharp} \simeq \pi_2|_{G^\sharp}$; 3. there exists $\eta \in (G(F)/G^\sharp(F))^D$ such that $\eta \pi_1 \simeq \pi_2$. For $\pi \in \Pi(G)$, we define a finite “packet” of smooth irreducible representations of $G^\sharp(F)$ as $$\Pi_\pi := \{\text{irreducible constituents of } \pi|_{G^\sharp} \}/\simeq .$$ Consequently, Proposition \[prop:res-disjoint\] implies that $\Pi(G^\sharp) = \bigsqcup_\pi \Pi_\pi$, when $\pi$ is taken over the $(G(F)/G^\sharp(F))^D$-orbits in $\Pi(G)$. \[prop:heredity\] Let $\pi \in \Pi(G)$ and assume that $\omega_\pi$ is unitary. Let $\mathbf{P}$ be one of the following properties of smooth irreducible representations of $G(F)$ or $G^\sharp(F)$: 1. unitary, 2. tempered, 3. square-integrable modulo the center, 4. cuspidal. Then we have equivalences of the form $$[ \pi \text{ satisfies } \mathbf{P} ] \Leftrightarrow [\exists \pi^\sharp \in \Pi_\pi, \; \pi^\sharp \text{ satisfies } \mathbf{P} ] \Leftrightarrow [\forall \pi^\sharp \in \Pi_\pi, \; \pi^\sharp \text{ satisfies } \mathbf{P} ].$$ Now comes the decomposition of $\pi|_{G^\sharp}$. Let $\pi \in \Pi(G)$ with the underlying $\C$-vector space $V_\pi$. Note that $V_{\eta\pi} = V_\pi$ for all $\eta \in (G(F)/G^\sharp(F))^D$. Introduce the following groups $$\begin{aligned} X^G(\pi) & := \{\eta \in (G(F)/G^\sharp(F))^D : \eta\pi \simeq \pi \}, \\ S^G(\pi) & := \langle I^G_\eta \in \Isom_G(\eta\pi, \pi) : \eta \in X^G(\pi) \rangle \; \subset \Aut_{G^\sharp}(\pi). \end{aligned}$$ Observe that $\Isom_G(\eta\pi, \pi)$ is a $\C^\times$-torsor by Schur’s lemma, and an element $I^G_\eta \in S^G(\pi)$ uniquely determines $\eta$. The group law is given by composition in $\Aut_\C(V_\pi)$, namely by $$\Isom_G(\eta\pi, \pi) \times \Isom_G(\eta'\pi, \pi) = \Isom_G(\eta'\eta\pi, \eta'\pi) \times \Isom_G(\eta'\pi, \pi) \to \Isom_G(\eta'\eta\pi, \pi)$$ for all $\eta, \eta' \in X^G(\pi)$. Thus we obtain a central extension of locally compact groups $$\begin{gathered} \label{eqn:res-S} 1 \to \C^\times \to S^G(\pi) \to X^G(\pi) \to 1, \end{gathered}$$ where the first arrow is $z \mapsto z \cdot \identity$ and the second one is $I^G_\eta \mapsto \eta$. Also note that $X^G(\pi)=X^G(\xi\pi)$, $S^G(\pi)=S^G(\xi\pi)$ for any $\xi \in (G(F)/G^\sharp(F))^D$. *Attention*: implicit in the notations above is the reference to $G^\sharp$, which is usually clear from the context. Indications to $G^\sharp$ will be given when necessary. It is easy to see that $X^G(\pi)$ is finite abelian. As in the setting of $R$-groups, we define the finite set $$\Pi_-(S^G(\pi)) := \left\{ \rho \in \Pi(S^G(\pi)) : \forall z \in \C^\times, \; \rho(z) = z\cdot\identity \right\}.$$ \[prop:S-decomp\] Let $\mathfrak{S} = \mathfrak{S}(\pi)$ be the representation of $S^G(\pi) \times G^\sharp(F)$ on $V_\pi$ defined by $$\mathfrak{S}(I,x) = I \circ \pi(x), \quad I \in S^G(\pi), x \in G^\sharp(F).$$ Then there is a decomposition $$\begin{gathered} \label{eqn:S-decomp} \mathfrak{S} \simeq \bigoplus_{\rho \in \Pi_-(S^G(\pi))} \rho \boxtimes \pi^\sharp_\rho, \end{gathered}$$ where $\rho \mapsto \pi^\sharp_\rho$ is a bijection from $\Pi_-(S^G(\pi))$ to $\Pi_\pi$, characterized by . Relation to parabolic induction {#sec:res-ind} ------------------------------- Let $P$ be a parabolic subgroup of $G$ with a Levi decomposition $P=MU$. In this article, we denote systematically $$\begin{aligned} P^\sharp & := P \cap G^\sharp, \\ M^\sharp & := M \cap M^\sharp.\end{aligned}$$ Then $P^\sharp$ is a parabolic subgroup of $G^\sharp$ with Levi decomposition $P^\sharp = M^\sharp U$, since every unipotent subgroup of $G$ is contained in $G_\text{der}$. The map $P \mapsto P^\sharp$ (resp. $M \mapsto M^\sharp$) induces a bijection between the parabolic subgroups (resp. Levi subgroups) of $G$ and $G^\sharp$, which leaves the unipotent radicals intact. We also have a canonical identification $W(M^\sharp) = W(M)$. In what follows, we will fix Haar measures on the unipotent radicals of parabolic subgroups of $G$ and $G^\sharp$, which are compatible with the identifications above. Obviously, the modulus functions satisfy $\delta_P(m) = \delta_{P^\sharp}(m)$ for all $m \in M^\sharp(F)$. \[prop:res-induction\] Let $\sigma \in \Pi(M)$. Then we have the following isomorphism between smooth representations of $G^\sharp(F)$ $$\begin{aligned} I^G_P(\sigma)|_{G^\sharp} & \longrightarrow I^{G^\sharp}_{P^\sharp}(\sigma|_{M^\sharp}) \\ \varphi & \longmapsto \varphi|_{G^\sharp(F)} \end{aligned}$$ which is functorial in $\sigma$. Upon recalling the definitions of $I^G_P(\sigma)$ and $I^{G^\sharp}_{P^\sharp}(\sigma|_{M^\sharp})$ as function spaces, the assertion follows from the canonical isomorphisms $$P^\sharp(F) \backslash G^\sharp(F) = (P^\sharp \backslash G^\sharp)(F) \rightiso (P \backslash G)(F) = P(F) \backslash G(F)$$ and the fact that $\delta_P|_{M^\sharp} = \delta_{P^\sharp}$. The next result will not be used in this article; we include it only for the sake of completeness Recall that the normalized Jacquet functor $r^G_P$ is the left adjoint of $I^G_P$. Idem for $r^{G^\sharp}_{P^\sharp}$. \[prop:res-Jacquet\] Let $\pi \in \Pi(G)$. The restriction of representations induces an isomorphism $$r^G_P(\pi)|_{M^\sharp} \rightiso r^{G^\sharp}_{P^\sharp}(\pi|_{G^\sharp})$$ between smooth representations of $M^\sharp(F)$, which is functorial in $\pi$. Evident. Let $M$ be a Levi subgroup of $G$. Then the inclusion map $M \hookrightarrow G$ induces an isomorphism between locally compact abelian groups $$M(F)/M^\sharp(F) \rightiso G(F)/G^\sharp(F).$$ The inclusion map induces an isomorphism $M/M^\sharp \hookrightarrow G/G^\sharp$ as $F$-tori. Hence the short exact sequence $1 \to M^\sharp \to M \to M/M^\sharp \to 1$ and its avatar for $G$ provide a commutative diagram of pointed sets with exact rows $$\xymatrix{ 1 \ar[r] & G^\sharp(F) \ar[r] & G(F) \ar[r] & (G/G^\sharp)(F) \ar[r] & H^1(F, G^\sharp) \\ 1 \ar[r] & M^\sharp(F) \ar[u] \ar[r] & M(F) \ar[u] \ar[r] & (M/M^\sharp)(F) \ar@{=}[u] \ar[r] & H^1(F, M^\sharp) \ar[u] }.$$ Fix a parabolic subgroup $P$ of $G$ with a Levi decomposition $P=MU$. The rightmost vertical arrow factorizes as $$H^1(F, M^\sharp) \to H^1(F,P^\sharp) \to H^1(F, G^\sharp).$$ The first map is an isomorphism whose inverse is induced by $P^\sharp \twoheadrightarrow P^\sharp/U = M^\sharp$. It is well known that the second map is injective, hence so is the composition. A simple diagram chasing shows that $M(F)/M^\sharp(F) \rightiso G(F)/G^\sharp(F)$, as asserted. \[prop:char-G-M\] The restriction map induces an isomorphism $$(G(F)/G^\sharp(F))^D \rightiso (M(F)/M^\sharp(F))^D.$$ Here is a trivial but important consequence: any $\eta \in (M(F)/M^\sharp(F))^D$ is invariant under the $W(M)$-action. Relation to intertwining operators ---------------------------------- Let $M$ be a Levi subgroup of $G$. First of all, observe that there is a natural decomposition $$\mathfrak{a}^*_M = \mathfrak{a}^*_{M^\sharp} \oplus \mathfrak{b}^*$$ where $$\mathfrak{b}^* := \mathcal{X}(M/M^\sharp) \otimes_\Z \R \hookrightarrow \mathfrak{a}^*_G .$$ Henceforth, we shall identify $\mathfrak{a}^*_{M^\sharp}$ as a vector subspace of $\mathfrak{a}^*_M$. Idem for their complexifications. This is compatible with restrictions in the following sense $$(\sigma_\lambda)|_{M^\sharp} = (\sigma|_{M^\sharp})_\lambda, \quad \sigma \in \Pi(M), \; \lambda \in \mathfrak{a}^*_{M^\sharp, \C}.$$ \[prop:compatible-intertwinner\] Let $\sigma \in \Pi(M)$, $P, Q \in \mathcal{P}(M)$. For $\lambda \in \mathfrak{a}^*_{M^\sharp, \C}$ in general position, the following diagram is commutative $$\xymatrix{ I^G_P(\sigma_\lambda)|_{G^\sharp} \ar[d]_{\simeq} \ar[rrr]^{J_{Q|P}(\sigma_\lambda)} & & & I^G_Q(\sigma_\lambda)|_{G^\sharp} \ar[d]^{\simeq} \\ I^{G^\sharp}_{P^\sharp}(\sigma_\lambda|_{M^\sharp}) \ar[rrr]_{J_{Q^\sharp|P^\sharp}(\sigma_\lambda|_{M^\sharp})} & & & I^{G^\sharp}_{Q^\sharp}(\sigma_\lambda|_{M^\sharp}) }$$ where the vertical isomorphisms are those defined in Lemma \[prop:res-induction\]. It suffices to check this for $\Re(\lambda)$ in the cone of absolute convergence of the integrals defining $J_{Q|P}$ and $J_{Q^\sharp|P^\sharp}$. The commutativity then follows from and the definition of the isomorphism in Lemma \[prop:res-induction\]. \[prop:mu-compatibility\] Let $\sigma \in \Pi_{2,\mathrm{temp}}(M)$, $\sigma^\sharp \in \Pi_{2,\mathrm{temp}}(M^\sharp)$ such that $\sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$. Then for all $\lambda \in \mathfrak{a}^*_{M^\sharp,\C}$, we have $\mu(\sigma_\lambda)=\mu(\sigma^\sharp_\lambda)$; more precisely, $$\mu_\alpha(\sigma_\lambda) = \mu_\alpha(\sigma^\sharp_\lambda), \quad \forall \alpha \in \Sigma^{\mathrm{red}}_P = \Sigma^{\mathrm{red}}_{P^\sharp}, \; P \in \mathcal{P}(M).$$ In view of our choice of measures on unipotent radicals, the identities of $\mu$-functions follow from and Lemma \[prop:compatible-intertwinner\]. \[prop:mu-invariance\] Let $\sigma \in \Pi(M)$ and $\eta \in (G(F)/G^\sharp(F))^D$. Then we have $$j(\sigma) = j(\eta\sigma).$$ In particular, for $\sigma \in \Pi_{2,\text{temp}}(M)$, we have $\mu(\sigma)=\mu(\eta\sigma)$. In view of the definition of $j$-function , it suffices to observe that for all $P,Q \in \mathcal{P}(M)$ and $\lambda \in \mathfrak{a}^*_{M,\C}$ in general position, the following diagram commutes $$\xymatrix{ \eta I^G_P(\sigma_\lambda) \ar[rr]^{J_{Q|P}(\sigma_\lambda)} \ar[d]_{\simeq} & & \eta I^G_Q(\sigma_\lambda) \ar[d]^{\simeq} \\ I^G_P(\eta\sigma_\lambda) \ar[rr]_{J_{Q|P}(\eta\sigma_\lambda)} & & I^G_Q(\eta\sigma_\lambda) }$$ where the vertical arrows are given by $\varphi(\cdot) \mapsto \eta(\cdot)\varphi(\cdot)$; note that we used the natural identification $\Hom_G(\pi_1, \pi_2) = \Hom_G(\eta\pi_1, \eta\pi_2)$ for all $\pi_1, \pi_2 \in \Pi(G)$. Indeed, the commutativity can be seen from the definition of $J_{Q|P}(\cdot)$ when $\Re(\lambda)$ lies in the cone of absolute convergence. \[prop:normalizing-invariance\] One can choose a family of normalizing factors for $G$ such that $$r_{Q|P}(\sigma_\lambda) = r_{Q|P}(\eta\sigma_\lambda)$$ for all $(M, \sigma)$, $P,Q \in \mathcal{P}(M)$ and $\eta \in (G(F)/G^\sharp(F))^D$ which is unitary. Given such a family of normalizing factors, one can define normalizing factors $r_{Q^\sharp|P^\sharp}(\sigma^\sharp_\lambda)$ for those $\sigma^\sharp$ such that $\omega_{\sigma^\sharp}$ is unitary, by setting $$\begin{gathered} \label{eqn:r-equality} r_{Q^\sharp|P^\sharp}(\sigma^\sharp_\lambda) := r_{Q|P}(\sigma_\lambda), \quad \lambda \in \mathfrak{a}^*_{M^\sharp,\C} \end{gathered}$$ where $\sigma \in \Pi(M)$ is as in Proposition \[prop:lifting\] with $\omega_\sigma$ unitary. Moreover, let $\sigma$, $\sigma^\sharp$ be as above and $\iota: \sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$ be an embedding. Let $P,Q \in \mathcal{P}(M)$, $w \in W(M)$ with a representative $\tilde{w} \in G^\sharp(F)$. The following diagrams of $G^\sharp(F)$-representations are commutative $$\xymatrix{ I^G_P(\sigma_\lambda) \ar[rr]^{R_{Q|P}(\sigma_\lambda)} & & I^G_Q(\sigma_\lambda) \\ I^{G^\sharp}_{P^\sharp}(\sigma^\sharp_\lambda) \ar@{^{(}->}[u] \ar[rr]_{R_{Q^\sharp|P^\sharp}(\sigma^\sharp_\lambda)} & & I^{G^\sharp}_{Q^\sharp}(\sigma^\sharp_\lambda) \ar@{^{(}->}[u] } \qquad \xymatrix{ I^G_P(\sigma_\lambda) \ar[rr]^{r_P(\tilde{w}, \sigma_\lambda)} & & I^G_P(\tilde{w}(\sigma_\lambda)) \\ I^{G^\sharp}_{P^\sharp}(\sigma^\sharp_\lambda) \ar[rr]_{r_{P^\sharp}(\tilde{w}, \sigma^\sharp_\lambda)} \ar@{^{(}->}[u] & & I^{G^\sharp}_{P^\sharp}(\tilde{w}(\sigma^\sharp_\lambda)) \ar@{^{(}->}[u] }$$ for $\lambda \in \mathfrak{a}^*_{M,\C}$ in general position, where the vertical arrows are given by $$I^{G^\sharp}_{P^\sharp}(\sigma^\sharp_\lambda) \xrightarrow{I^{G^\sharp}_{P^\sharp}(\iota)} I^{G^\sharp}_{P^\sharp}(\sigma_\lambda|_{M^\sharp}) \xrightarrow{\sim} I^G_P(\sigma_\lambda)|_{G^\sharp}.$$ Observe that the asserted invariance under $\eta$-twist is satisfied by Langlands’ conjectural family of normalizing factors in Remark \[rem:normalization\], since they are defined in terms of local factors through the adjoint representation of $\Lgrp{M}$ on the Lie algebra of $\hat{G}$. We will show that $r_{Q|P}(\sigma_\lambda) = r_{Q|P}(\eta\sigma_\lambda)$ by reviewing the construction in Remark \[rem:construction-normalization\]. More precisely, we will start from the square-integrable case and show that the equality is preserved throughout the inductive construction. To begin with, suppose that $M$ is maximal proper and $\sigma \in \Pi_{2,\text{temp}}(M)$. Suppose that $r_{Q|P}(\sigma_\lambda)$ is chosen so that $\mathbf{R}_1$, $\mathbf{R}_2$, $\mathbf{R}_3$, $\mathbf{R}_6$, $\mathbf{R}_7$ are satisfied. Put $r_{Q|P}(\eta\sigma_\lambda) := r_{Q|P}(\sigma_\lambda)$, then all the conditions above except $\mathbf{R}_1$ are trivially satisfied for $\eta\sigma$. As for $\mathbf{R}_1$, all what we need to check is that when $Q=\bar{P}$, $$r_{P|Q}(\sigma_\lambda) r_{Q|P}(\sigma_\lambda) = j(\eta\sigma_\lambda), \quad \lambda \in \mathfrak{a}^*_{M,\C}.$$ By Proposition \[prop:mu-invariance\], the right hand side is equal to $j(\sigma_\lambda)$, hence the equality holds. This completes the case of $\sigma \in \Pi_{2,\text{temp}}(M)$. Suppose now $\sigma \in \Pi_{\text{temp}}(M)$. By the classification of tempered representations, we may write $\sigma \hookrightarrow I^M_R(\tau)$ for some parabolic subgroup $R=M_R U_R$ of $M$ and $\tau \in \Pi_{2,\text{temp}}(M_R)$. Twisting everything by $\eta$, we obtain $\eta\sigma \hookrightarrow I^M_R(\eta\tau)$ in the classification of tempered representations. We have $r_{Q|P}(\sigma) = r_{Q(R)|P(R)}(\tau) = r_{Q(R)|P(R)}(\eta\tau)$ by the previous case; on the other hand, the inductive construction of normalizing factors says that $r_{Q|P}(\eta\sigma)=r_{Q(R)|P(R)}(\eta\tau)$, hence $r_{Q|P}(\eta\sigma)=r_{Q|P}(\sigma)$. The case of general $\sigma$ is similar. We may write $\sigma$ as the Langlands quotient $I^M_R(\tau_\mu) \twoheadrightarrow \sigma$ where $R=M_R U_R$ is as before, $\tau \in \Pi_\text{temp}(M_R)$ and $\Re\angles{\mu,\alpha^\vee} > 0$ for all $\alpha \in \Delta^M_R$. Twisting everything by $\eta$, we have $I^M_R(\eta\tau_\mu) \twoheadrightarrow \eta\sigma$, which is still a Langlands quotient. The inductive construction of normalizing factors says that $r_{Q|P}(\sigma) = r_{Q(R)|P(R)}(\tau_{\mu+\lambda})$. Repeating the arguments for the previous case, it follows that $r_{Q|P}(\sigma) = r_{Q|P}(\eta\sigma)$. Now we can check that $$r_{Q^\sharp|P^\sharp}(\sigma^\sharp_\lambda) := r_{Q|P}(\sigma_\lambda)$$ is well-defined. Recall that $\omega_{\sigma^\sharp}$ and $\omega_\sigma$ are assumed to be unitary. If $\sigma'$ is another choice such that $\sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$ and $\omega_{\sigma'}$ is unitary, then there exists $\eta$ such that $\sigma \simeq \eta\sigma'$. This would imply that $\eta|_{Z_G(F)}$ is unitary, hence so is $\eta$ itself. Therefore $r_{Q|P}(\eta\sigma_\lambda)=r_{Q|P}(\sigma_\lambda)$. Finally, the commutativity of the diagram results from Lemma \[prop:compatible-intertwinner\] and . Relation to $R$-groups {#sec:rel-R} ---------------------- In this subsection, we will fix - a parabolic subgroup $P=MU$ of $G$; - and the corresponding parabolic subgroup $P^\sharp := P \cap G^\sharp = M^\sharp U$ of $G^\sharp$; - $\sigma^\sharp \in \Pi_{2,\text{temp}}(M^\sharp)$; - $\sigma \in \Pi_{2,\text{temp}}(M)$ such that $\sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$. Given $\sigma^\sharp$, the existence of such a $\sigma$ is guaranteed by Propositions \[prop:lifting\] and \[prop:heredity\]. \[prop:W\^0-equality\] Under the identification $W(M)=W(M^\sharp)$, we have $W^0_\sigma = W^0_{\sigma^\sharp}$. Since both sides are generated by root reflections, it suffices to fix $\alpha \in \Sigma^{\text{red}}_P = \Sigma^{\text{red}}_{P^\sharp}$ and show that $s_\alpha \in W^0_\sigma$ if and only if $s_\alpha \in W^0_{\sigma^\sharp}$, where $s_\alpha$ denotes the root reflection with respect to $\alpha$. By [@Wa03 Proposition IV.2.2], $\mu_\alpha(\sigma)=0$ implies $s_\alpha \sigma \simeq \sigma$. Idem for $\sigma^\sharp$ instead of $\sigma$. According to the description of $W^0_\sigma$ (resp. $W^0_{\sigma^\sharp}$) in terms of $\mu$-functions, we obtain $$\mu_\alpha(\sigma)=0 \Leftrightarrow s_\alpha \in W^0_\sigma \quad (\text{resp. } \mu_\alpha(\sigma^\sharp)=0 \Leftrightarrow s_\alpha \in W^0_{\sigma^\sharp}).$$ On the other hand, Proposition \[prop:mu-compatibility\] implies $\mu_\alpha(\sigma)=\mu_\alpha(\sigma^\sharp)$. The assertion follows immediately. \[def:L\] Set $$\begin{aligned} L(\sigma) & := \left\{ \eta \in (M(F)/M^\sharp(F))^D : \exists w \in W(M), \; w\sigma \simeq \eta\sigma \right\}, \\ L(\sigma^\sharp) & := \left\{ \eta \in (M(F)/M^\sharp(F))^D : \exists w \in W(M), \; w\sigma \simeq \eta\sigma, \; w\sigma^\sharp \simeq \sigma^\sharp \right\}. \end{aligned}$$ They are subgroups of $(M(F)/M^\sharp(F))^D$. Indeed, let $\eta, \eta' \in L(\sigma)$ and $w, w' \in W(M)$ such that $\eta\sigma \simeq w\sigma$, $\eta'\sigma \simeq w'\sigma$. Then one has $$\begin{gathered} \label{eqn:group-law-L} \eta'\eta\sigma \simeq \eta' w\sigma = w \eta'\sigma \simeq ww'\sigma. \end{gathered}$$ Hence $\eta\eta' \in L(\sigma)$. The case of $L(\sigma^\sharp)$ is similar. Note that $X^M(\sigma) \subset L(\sigma^\sharp) \subset L(\sigma)$. There is an obvious counterpart for the Weyl group, namely $$\bar{W}_\sigma := \left\{ w \in W(M) : \exists \eta \in (M(F)/M^\sharp(F))^D, \; w\sigma \simeq \eta\sigma \right\}.$$ It is clear that $\bar{W}_\sigma \supset W_\sigma$. On the other hand, Proposition \[prop:res-disjoint\] implies that $\bar{W}_\sigma \supset W_{\sigma^\sharp}$. The following result is clear in view of the preceding definitions. \[prop:barGamma\] There is a homomorphism given by $$\begin{aligned} \bar{\Gamma}: \bar{W}_\sigma & \longrightarrow L(\sigma)/X^M(\sigma) \\ w & \longmapsto \text{ the } \left[ \eta \text{ mod } X^M(\sigma) \right] \text{ such that } w\sigma \simeq \eta\sigma \end{aligned}$$ which satisfies 1. $\bar{\Gamma}$ is surjective; 2. $\Ker(\bar{\Gamma}) = W_\sigma$; 3. the preimage of $L(\sigma^\sharp)/X^M(\sigma)$ is $W_{\sigma^\sharp} W_\sigma$. Let $$\Gamma: \bar{W}_\sigma/W_\sigma \rightiso L(\sigma)/X^M(\sigma)$$ be the isomorphism obtained from $\bar{\Gamma}$ in the previous Lemma. \[prop:Goldberg\] Set $$R_\sigma[\sigma^\sharp] := (W_\sigma \cap W_{\sigma^\sharp})/W^0_{\sigma^\sharp}$$ which is legitimate by Lemma \[prop:W\^0-equality\]. It is a subgroup of $R_{\sigma^\sharp}$. 1. The homomorphism $\bar{\Gamma}$ induces an isomorphism $$\Gamma: R_{\sigma^\sharp}/R_\sigma[\sigma^\sharp] \rightiso L(\sigma^\sharp)/X^M(\sigma).$$ 2. If $R_\sigma = \{1\}$, or equivalently if $I^G_P(\sigma)$ is irreducible, then $\Gamma$ induces an isomorphism $R_{\sigma^\sharp} \rightiso L(\sigma^\sharp)/X^M(\sigma)$. Consequently, $R_{\sigma^\sharp}$ is abelian in this case. Lemma \[prop:barGamma\] gives an isomorphism $$W_{\sigma^\sharp}/(W_\sigma \cap W_{\sigma^\sharp}) \rightiso L(\sigma^\sharp)/X^M(\sigma)$$ that can be viewed as a restriction of $\Gamma$. By Lemma \[prop:W\^0-equality\], we can take the quotients by $W^0_\sigma=W^0_{\sigma^\sharp}$ on the left hand side. The first assertion follows immediately. For the second assertion, it suffices to note that $R_\sigma[\sigma^\sharp]$ embeds into $R_\sigma$ as well, since $W^0_\sigma=W^0_{\sigma^\sharp}$. L-parameters {#sec:res-L} ------------ Let $G$ be a connected reductive $F$-group equipped with a quasisplit inner twist $$\psi: G \times_F \bar{F} \to G^* \times_F \bar{F}.$$ We identify $\hat{G}$ with $\widehat{G^*}$, thus $\Lgrp{G}=\Lgrp{G^*}$. The reader should recall that the definition of the complex reductive group $\widehat{G^*}$ and the $\Gamma_F$-action thereof depend on the choice of a $\Gamma_F$-stable splitting $(B^*, T^*, (E_\alpha)_{\alpha \in \Delta(B^*,T^*)})$ (also known as an $F$-splitting) of $G^*(\bar{F})$ (see [@Ko84 §1]). These choices permit to define a correspondence $M^* \leftrightarrow \Lgrp{M^*}$ between the conjugacy classes of Levi subgroups of $G^*$ and their dual avatars inside $\Lgrp{G^*}$. Using the inner twist $\psi$, it also makes sense to say if a Levi subgroup $M^*$ of $G^*$ comes from $G$; this notion only depends on the conjugacy classes of Levi subgroups. For any Levi subgroup $M$ of $G$, there is a canonical bijection between $W^G(M)$ and $W^{\hat{G}}(\hat{M})$ coming from the bijection between roots and coroots. An L-parameter for $G^*$ is a homomorphism $$\phi: \WD_F \to \Lgrp{G^*} = \Lgrp{G}$$ such that - $\phi$ is an L-homomorphism, i.e. the composition of $\phi$ with the projection $\Lgrp{G} \to W_F$ equals $\WD_F \to W_F$; - $\phi$ is continuous; - the projection of $\Im(\phi)$ to $\hat{G}$ is formed of semisimple elements. Two L-parameters $\phi_1$, $\phi_2$ are called equivalent, denoted by $\phi_1 \sim \phi_2$, if they are conjugate by $\hat{G}$. We say that $\phi$ is bounded if the projection of $\Im(\phi)$ to $\hat{G}$ is bounded (i.e. relatively compact); this property depends only on the equivalence class of $\phi$. Given an L-parameter $\phi$, we define $$S_\phi := Z_{\hat{G}}(\Im(\phi)).$$ The connected component $S^0_\phi$ is a connected reductive subgroup of $\hat{G}$. We record the following basic properties. 1. the Levi subgroups $\Lgrp{M^*} \subset \Lgrp{M}$ which contain $\Im(\phi)$ minimally are conjugate by $S^0_\phi$. 2. Let $\Lgrp{M^*}$ be a Levi subgroup containing $\Im(\phi)$ minimally, then $Z_{\widehat{M^*}}^{\Gamma_F, 0}$ is a maximal torus of $S^0_\phi$. Indeed, these assertions follow from [@Bo79 Proposition 3.6] and its proof, applied to the subgroup $\Im(\phi)$ of $\Lgrp{G}$. So far, everything depends only on the quasisplit inner form $G^*$. We say that $\phi$ is $G$-relevant if $M^*_\phi$ corresponds to a Levi subgroup of $G$; in this case, we write $M^*_\phi = M_\phi$. Put $$\begin{aligned} \Phi(G) & := \{\phi : \WD_F \to \Lgrp{G}, \; \phi \text{ is a $G$-relevant L-parameter}\}/\sim, \\ \Phi_\text{bdd}(G) & := \{\phi \in \Phi(G) : \phi \text{ is bounded} \}, \\ \Phi_{2,\text{bdd}}(G) & := \{\phi \in \Phi_\text{bdd}(G) : M_\phi = G \}.\end{aligned}$$ Since the relevance condition is vacuous if $G=G^*$, we have $\Phi(G) \subset \Phi(G^*)$, etc. Now let $G^\sharp$ be a subgroup of $G$ such that $G_\text{der} \subset G^\sharp \subset G$. We will study the lifting of L-parameters from $G^\sharp$ to $G$, which is in some sense dual to the restriction of representations. In what follows, the L-groups of $G$ and $G^\sharp$ will be defined using compatible choices of quasisplit inner twists and $F$-splittings. There is a natural, $\Gamma_F$-equivariant central extension $$1 \to \hat{Z}^\sharp \to \hat{G} \xrightarrow{\mathbf{pr}} \widehat{G^\sharp} \to 1$$ which is dual to $G^\sharp \to G$; here $\hat{Z}^\sharp$ is the $\C$-torus dual to $G/G^\sharp$. For $\phi \in \Phi(G)$, we shall set $\phi^\sharp := \mathbf{pr} \circ \phi \in \Phi(G^\sharp)$. When this equality holds, $\phi$ is called a lifting of $\phi^\sharp$. \[prop:lifting-parameter\] For any $\phi^\sharp \in \Phi(G^\sharp)$, there exists a lifting $\phi \in \Phi(G)$ of $\phi^\sharp$ which is unique up to twists by $H^1_\mathrm{cont}(W_F, \hat{Z}^\sharp)$. If $\phi^\sharp \in \Phi_\mathrm{bdd}(G^\sharp)$ (resp. $\phi^\sharp \in \Phi_{2,\mathrm{bdd}}(G^\sharp)$), then $\phi$ can be chosen so that $\phi \in \Phi_\mathrm{bdd}(G)$ (resp. $\phi \in \Phi_{2,\mathrm{bdd}}(G)$). Note that by local class field theory, $H^1_\mathrm{cont}(W_F, \hat{Z}^\sharp)$ parametrizes the continuous characters of $(G(F)/G^\sharp(F))^D$. The existential part is just [@Lab85 Théorème 8.1] and the uniqueness follows easily. Assume that $\phi^\sharp \in \Phi_\mathrm{bdd}(G^\sharp)$ (resp. $\phi^\sharp \in \Phi_{2,\mathrm{bdd}}(G^\sharp)$) and let $\phi$ be any lifting of $\phi^\sharp$; we have to show that there exists a continuous $1$-cocycle $a: W_F \to \hat{Z}^\sharp$ such that the twisted L-parameter $a\phi$ is bounded (resp. bounded and satisfying $M_{a\phi}=M_\phi=G$). Note that there exists a central isogeny of connected reductive groups $$G^\sharp \times C \to G$$ given by multiplication, where $C$ is some subtorus of $Z^0_G$. Hence $C \to G/G^\sharp$ is also an isogeny. By duality, we obtain a $\Gamma_F$-equivariant central isogeny of connected reductive complex groups $$\hat{G} \to \widehat{G^\sharp} \times \hat{C}.$$ Let $\phi'$ be the composition of $\phi$ (projected to the $\hat{G}$ component) with the aforementioned central isogeny. Let us show that $\Im(\phi')$ is bounded upon twisting $\phi$. The first component of $\phi'$ is automatically bounded since $\phi^\sharp$ is. On the other hand, $\hat{C}$ is isogeneous to $\hat{Z}^\sharp$, therefore upon replacing $\phi$ by $a\phi$ for some suitable $1$-cocycle $a: W_F \to \hat{Z}^\sharp$, the second component can be made bounded. Hence $a\phi$ is a bounded L-parameter. To finish the proof, it remains to observe that assuming $\phi^\sharp = \mathbf{pr} \circ \phi$, the preimage of $M^\sharp_{\phi^\sharp}$ in $\hat{G}$ is equal to $M_\phi$. Here we record a construction related to inner forms which will be required later. Recall that the inner forms of $G^*$ are parametrized by $H^1(F, G^*_\text{AD})$. Kottwitz [@Ko84 §6] defined the “abelianization” map $\text{ab}^1: H^1(F, G^*_\text{AD}) \to (Z^{\Gamma_F}_{\hat{G}_\text{SC}})^D$ between pointed sets. Hence we can associate to $G$ a character $\chi_G$ of $Z^{\Gamma_F}_{\hat{G}_\text{SC}}$, namely by $$\label{eqn:chi-G}\begin{split} \{ \text{inner forms of } G^* \} = H^1(F, G^*_\text{AD}) \xrightarrow{\text{ab}^1} (Z^{\Gamma_F}_{\hat{G}_\text{SC}})^D, \\ G \longmapsto \left[ \chi_G : Z^{\Gamma_F}_{\hat{G}_\text{SC}} \to \C^\times \right]. \end{split}$$ Restriction, continued {#sec:res-2} ====================== This section is devoted to the study of restriction under parabolic induction. As before, we fix connected reductive $F$-groups $G$, $G^\sharp$ such that $G_\text{der} \subset G^\sharp \subset G$. We also fix a Levi subgroup $M$ of $G$ and $P \in \mathcal{P}(M)$. The bijection between Levi subgroups (resp. parabolic subgroups) $M \mapsto M^\sharp$ (resp. $P \mapsto P^\sharp$) is defined in §\[sec:res-ind\]. The normalizing factors for $G$, $G^\sharp$ are chosen as in Theorem \[prop:normalizing-invariance\] for the representations with unitary central character. Let $\sigma \in \Pi(M)$. We shall make the following (rather restrictive) hypothesis on $\sigma$ throughout this section. \[hyp:irred\] We assume that $\pi := I^G_P(\sigma)$ is irreducible. Embedding of central extensions {#sec:embedding} ------------------------------- \[prop:S-embedding\] Let $\sigma \in \Pi(M)$ and $\pi := I^G_P(\sigma) \in \Pi(G)$. 1. Under the identification of Corollary \[prop:char-G-M\], we have $X^M(\sigma) \hookrightarrow X^G(\pi)$. 2. Let $\omega \in X^M(\sigma)$, $I^M_\omega \in \Isom_M(\omega\sigma, \sigma)$, define the operator $I^G_\omega$ as the composition of $$\begin{aligned} A_\omega: \omega I^G_P(\sigma) & \longrightarrow I^G_P(\omega\sigma) \\ \varphi & \longmapsto \omega(\cdot)\varphi(\cdot) \end{aligned}$$ with $I^G_P(I^M_\omega): I^G_P(\omega\sigma) \rightiso I^G_P(\sigma)$. Then $I^G_\omega \in \Isom_G(\omega\pi, \pi)$. 3. We have the following commutative diagram of groups with exact rows $$\xymatrix{ 1 \ar[r] & \C^\times \ar[r] & S^G(\pi) \ar[r] & X^G(\pi) \ar[r] & 1 \\ 1 \ar[r] & \C^\times \ar@{=}[u] \ar[r] & S^M(\sigma) \ar@{^{(}->}[u] \ar[r] & X^M(\sigma) \ar@{^{(}->}[u] \ar[r] & 1 }$$ where the arrow $S^M(\sigma) \to S^G(\pi)$ is the map $I^M_\omega \mapsto I^G_\omega$ defined above. It follows from the definition that $I^G_\omega \in S^G(\pi)$ for all $\omega \in X^M(\sigma)$, hence $X^M(\sigma) \subset X^G(\pi)$. On the other hand, $I^G_\omega$ is simply the map $\varphi \mapsto \omega(\cdot) I^M_\omega(\varphi(\cdot))$. It is clear that $I^M_\omega \mapsto I^G_\omega$ is a group homomorphism. The commutativity of the diagram is then clear. We denote by $\mathbf{K}_0(\Pi_-(S^M(\sigma)))$ the space of virtual characters of $S^M(\sigma)$ generated by the elements of $\Pi_-(S^M(\sigma))$. Similarly, $\mathbf{K}_0(\Pi_\sigma)$ denotes the space of virtual characters of $M^\sharp(F)$ generated by the elements of $\Pi_\sigma$. The bijection $\rho \mapsto \pi^\sharp_\rho$ in Theorem \[prop:S-decomp\] extends to an isomorphism $\mathbf{K}_0(\Pi_-(S^M(\sigma))) \rightiso \mathbf{K}_0(\Pi_\sigma)$. Assuming $\pi = I^G_P(\sigma)$ irreducible, we have the analogous isomorphism $\mathbf{K}_0(\Pi_-(S^G(\pi))) \rightiso \mathbf{K}_0(\Pi_\pi)$, as well as the linear maps $$\begin{aligned} \text{Ind}^{S^G(\pi)}_{S^M(\sigma)}: & \mathbf{K}_0(\Pi_-(S^M(\sigma))) \longrightarrow \mathbf{K}_0(\Pi_-(S^G(\pi))), \\ I^{G^\sharp}_{P^\sharp}: & \mathbf{K}_0(\Pi_\sigma) \longrightarrow \mathbf{K}_0(\Pi_\pi),\end{aligned}$$ given by the usual induction and normalized parabolic induction, respectively. Note that Lemma \[prop:res-induction\] is invoked here. \[prop:K\_0-diagram\] The following diagram commutes. $$\xymatrix{ \mathbf{K}_0(\Pi_-(S^G(\pi))) \ar[rr]^{\simeq} & & \mathbf{K}_0(\Pi_\pi) \\ \mathbf{K}_0(\Pi_-(S^M(\sigma))) \ar[rr]_{\simeq} \ar[u]^{\mathrm{Ind}^{S^G(\pi)}_{S^M(\sigma)}} & & \mathbf{K}_0(\Pi_\sigma) \ar[u]_{I^{G^\sharp}_{P^\sharp}} }$$ To prove this, some harmonic analysis on the groups $S^M(\sigma)$, $S^G(\pi)$ is needed. These groups are infinite; nonetheless, the usual theory carries over as we are only concerned about the representations in $\Pi_-(S^G(\pi))$, $\Pi_-(S^M(\sigma))$ or their contragredients. The worried reader may reduce $S^G(\pi) \to X^G(\pi)$ (resp. $S^M(\sigma) \to X^M(\sigma)$) to a central extension by $\mu_m := \{z \in \C^\times : z^m = 1 \}$ for some $m \in \Z$, which is always possible. Let $\sigma^\sharp \in \Pi_\sigma$ and $\rho \in \Pi_-(S^M(\sigma))$ be the corresponding element. Define $$\begin{aligned} \tau & := \text{Ind}^{S^G(\pi)}_{S^M(\sigma)}(\rho) \in \mathbf{K}_0(\Pi_-(S^G(\pi))), \\ \pi^\sharp & := I^{G^\sharp}_{P^\sharp}(\sigma^\sharp) \in \mathbf{K}_0(\Pi_\pi). \end{aligned}$$ We have to show that $\tau$ corresponds to $\pi^\sharp$. To begin with, set $$\sigma[I^M_\omega] := \sigma(\cdot) \circ I^M_\omega : M(F) \to \Aut_\C(V_\sigma), \quad I^M_\omega \in S^M(\sigma)$$ where $V_\sigma$ is the underlying vector space of $\sigma$. Then $(\sigma[I^M_\omega],\sigma)$ is a smooth $\omega$-representation of $M(F)$, that is, $$\sigma[I^M_\omega](xy) = \omega(y) \sigma[I^M_\omega](x)\sigma(y), \quad x,y \in M(F).$$ This notion appears in the study of automorphic induction, and more generally it fits into the formalism of twisted endoscopy. Cf. [@L10 §0.4]. It is easy to see that $\Theta_\sigma[I^M_\sigma] := \Tr\sigma[I^M_\sigma]$ is well-defined as a distribution on $M(F)$. We may restrict $\sigma[I^M_\omega]$ to $M^\sharp(F)$; by abuse of notations, the corresponding distribution, which is also well-defined by Proposition \[prop:lifting\], is again denoted by $\Theta_\sigma[I^M_\sigma]$. The same definition applies to $\pi$. Theorem \[prop:S-decomp\] implies the following identity of distributions on $M^\sharp(F)$ $$\begin{gathered} \label{eqn:sigma-sharp-inversion} \Theta_{\sigma^\sharp} = \frac{1}{|X^M(\sigma)|} \sum_{\omega \in X^M(\sigma)} \Tr(\rho^{\vee})\left( I^M_\omega \right) \cdot \Theta_\sigma[I^M_\omega] \end{gathered}$$ where $\rho^\vee$ is the contragredient of $\rho$ and $I^M_\omega \in S^M(\sigma)$ is any preimage $\omega$; the summand does not depend on the choice of $I^M_\omega$. Define $Z^M(\sigma)$ to be the subgroup of elements $\omega \in X^M(\sigma)$ such that every preimage of $\omega$ in $S^M(\sigma)$ is central. Define $Z^G(\pi)$ similarly. The sum in can be taken over $Z^M(\sigma)$, since $\rho|_{\C^\times} = \identity$ implies that $\Tr(\rho^{\vee})$ is zero outside the center. Let $\pi^\sharp_1 \in \mathbf{K}_0(\Pi_\pi)$ be the character corresponding to $\tau$. By the same reasoning, there is an identity of distributions on $G^\sharp(F)$ $$\begin{gathered} \label{eqn:pi-sharp_1-inversion} \Theta_{\pi^\sharp_1} = \frac{1}{|X^G(\pi)|} \sum_{\eta \in Z^G(\pi)} \Tr(\tau^{\vee})\left( I^G_\eta \right) \cdot \Theta_\pi[I^G_\eta] \end{gathered}$$ where $I^G_\eta \in S^G(\pi)$ is any preimage of $\eta$, as before. It remains to show $\Theta_{\pi^\sharp_1}(f^\sharp) = \Theta_{\pi^\sharp}(f^\sharp)$ for every $f^\sharp \in C^\infty_c(G^\sharp(F))$. Choose a special maximal compact open subgroup $K \subset G(F)$ in good position relative to $M$, and set $K^\sharp := K \cap G^\sharp(F)$. Equip $K$ and $K^\sharp$ with appropriate Haar measures that are compatible with the Iwasawa decomposition (see [@Wa03 I.1]). The parabolic descent of characters implies $$\begin{gathered} \label{eqn:pi-sharp-inversion} \Theta_{\pi^\sharp}(f^\sharp) = \Theta_{\sigma^\sharp}(f^\sharp_{P^\sharp}) = \frac{1}{|X^M(\sigma)|} \sum_{\omega \in Z^M(\sigma)} \Tr(\rho^{\vee})\left( I^M_\omega \right) \cdot \Theta_\sigma[I^M_\sigma](f^\sharp_{P^\sharp}), \end{gathered}$$ where $$f^\sharp_{P^\sharp}(m) = \delta^{\frac{1}{2}}_P(m) \iint_{U(F) \times K^\sharp} f^\sharp(k^{-1}muk) \dd u \dd k, \quad m \in M^\sharp(F).$$ Since $\tau = \text{Ind}^{S^G(\pi)}_{S^M(\sigma)}(\rho)$, we have $$\Tr(\tau^\vee)\left( I^G_\eta \right) = \begin{cases} \dfrac{1}{|X^M(\sigma)|} \sum_{\xi \in X^G(\pi)} \Tr(\rho^\vee) \left((I^G_\xi)^{-1} I^M_\eta I^G_\xi \right), & \text{if } \eta \in X^M(\sigma), \; I^M_\eta \mapsto I^G_\eta, \\ 0, & \text{otherwise}, \end{cases}$$ where $I^G_\xi \in S^G(\pi)$ is any preimage of $\xi$. Cf. [@Se67 Proposition 20]. This may be rewritten as $$\Tr(\tau^\vee)\left( I^G_\eta \right) = \begin{cases} \dfrac{|X^G(\pi)|}{|X^M(\sigma)|} \Tr(\rho^\vee) \left(I^M_\eta \right), & \text{if } \eta \in X^M(\sigma) \cap Z^G(\pi), \; I^M_\eta \mapsto I^G_\eta, \\ 0, & \text{otherwise}. \end{cases}$$ We claim that $\Theta_\pi[I^G_\omega](f^\sharp) = \Theta_\sigma[I^M_\omega](f^\sharp_{P^\sharp})$ if $I^M_\sigma \mapsto I^G_\sigma \in S^G(\pi)$. First of all, note that $\Theta_\pi[I^G_\omega]$ is the normalized parabolic induction of $\Theta_\sigma[I^M_\omega]$ in the setting of $\omega$-representations [@L10 §1.7 and §3.8]. Hence we have the parabolic descent of $\omega$-characters [@L10 Théorème 3.8.2], namely $$\Theta_\pi[I^G_\omega](f) = \Theta_\sigma[I^M_\omega](f_{P,\omega}), \quad f \in C^\infty_c(G(F))$$ where $$f_{P,\omega}(m) = \delta^{\frac{1}{2}}_P(m) \iint_{U(F) \times K} \omega(k) f(k^{-1}muk) \dd u \dd k, \quad m \in M(F).$$ To prove the claim, let us sketch how to “restrict” the $\omega$-character relation above to $G^\sharp(F)$. There exists a compact open subgroup $C \subset Z_G(F)$, verifying 1. $C \cap G^\sharp(F) = \{1\}$; 2. $C \subset K$; 3. $\omega$ and $\omega_\sigma$ are trivial on $C$; 4. the multiplication maps $C \times G^\sharp(F) \hookrightarrow G(F)$ and $C \times M^\sharp(F) \hookrightarrow M(F)$ are submersive. Define $\mathbbm{1}_C$ to be the constant function $1$ on $C$. Choose the unique Haar measure on $C$ such that the submersions above preserve measures locally. Given $f^\sharp \in C^\infty_c(G^\sharp(F))$, we set $f = \text{vol}(C)^{-1} \mathbbm{1}_C \otimes f^\sharp$ on $C \times G^\sharp(F)$, and zero elsewhere. For such $f$, by inspecting the proof in [@L10 Proposition 1.8.1], we may redefine $f_{P,\omega}$ by taking the double integral of $f(k^{-1}muk)$ over $U(F) \times K^\sharp$, so that $f_{P,\omega} = \text{vol}(C)^{-1} \mathbbm{1}_C \otimes f^\sharp_{P^\sharp}$ on $C \times M^\sharp(F)$ and zero elsewhere. Therefore $$\begin{aligned} \Theta_\pi[I^G_\omega](f) & = \Theta_\pi[I^G_\omega](f^\sharp), \\ \Theta_\sigma[I^M_\sigma](f_{P,\omega}) & = \Theta_\sigma[I^M_\sigma](f^\sharp_{P^\sharp}). \end{aligned}$$ Hence our claim follows. All in all, becomes $$\Theta_{\pi^\sharp_1}(f^\sharp) = \frac{1}{|X^M(\sigma)|} \sum_{\omega \in X^M(\sigma) \cap Z^G(\pi)} \Tr(\rho^\vee)(I^M_\omega) \cdot \Theta_\sigma[I^M_\omega](f^\sharp_{P^\sharp})$$ where $I^M_\sigma \in S^M(\sigma)$ is any preimage of $\omega$ and $I^M_\sigma \mapsto I^G_\sigma \in S^G(\pi)$. In comparison with , it suffices to show that $\Theta_\sigma[I^M_\omega](f^\sharp_{P^\sharp}) = 0$ if $\omega \in Z^M(\sigma)$ but $\omega \notin Z^G(\pi)$. Indeed, for $I \in S^G(\pi)$, we have $$\begin{aligned} \Theta_\sigma[I^M_\omega](f^\sharp_{P^\sharp}) & = \Theta_\pi[I^G_\omega](f^\sharp) = \Tr\mathfrak{S}(I^G_\omega, f^\sharp) \\ & = \Tr\mathfrak{S}(I^{-1} I^G_\omega I, f^\sharp) \end{aligned}$$ since $\mathfrak{S}$ is a representation of $S^G(\pi) \times G^\sharp(F)$. Since $I$ is arbitrary and $\mathfrak{S}|_{\C^\times \times \{1\}} = \identity$, we conclude that $\Theta_\sigma[I^M_\omega](f^\sharp_{P^\sharp}) \neq 0$ only if $\omega \in Z^G(\pi)$. Description of $R$-groups {#sec:desc-R} ------------------------- Let $w \in W(M)$ with a chosen representative $\tilde{w} \in G^\sharp(F)$. Recall the operator $r_P(\tilde{w}, \sigma): I^G_P(\sigma) \to I^G_P(\tilde{w}\sigma)$ defined in , which is the composition of $R_{w^{-1}Pw|P}(\sigma): I^G_P(\sigma) \to I^G_{w^{-1}Pw|P}(\sigma)$ with the isomorphism $$\begin{aligned} \ell(\tilde{w}): I^G_{w^{-1}Pw}(\sigma) & \longrightarrow I^G_P(\tilde{w}\sigma) \\ \varphi(\cdot) & \longmapsto \varphi(\tilde{w}^{-1} \cdot).\end{aligned}$$ For $\eta \in (G(F)/G^\sharp(F))^D$, recall the isomorphism $A_\eta$ defined as $$\begin{aligned} \eta I^G_P(\sigma) & \longrightarrow I^G_P(\eta\sigma) \\ \varphi(\cdot) & \longmapsto \eta(\cdot)\varphi(\cdot).\end{aligned}$$ Note that the representations $\sigma$, $\eta\sigma$, $\tilde{w}\sigma$ and $\eta\tilde{w}\sigma$ share the same underlying vector space $V_\sigma$. As usual, we will compose the operators above after appropriate twists by $\eta$ or $\tilde{w}$. For example, given $\eta, \eta'$, we may define $A_\eta A_{\eta'}$ which is equal to $A_{\eta\eta'}: \eta\eta' I^G_P(\sigma) \to I^G_P(\eta\eta' \sigma)$. \[prop:L-X\^G\] Let $L(\sigma) \subset (M(F)/M^\sharp(F))^D$ be the subgroup defined in Definition \[def:L\]. Upon identifying $M(F)/M^\sharp(F)$ and $G(F)/G^\sharp(F)$, we have $$L(\sigma) \subset X^G(\pi).$$ If $\sigma \in \Pi_{2,\mathrm{temp}}(M)$, then equality holds. Let $\eta \in L(\sigma)$. By definition, there exists $w \in W(M)$ with a representative $\tilde{w} \in G^\sharp(F)$ such that $\eta\sigma \simeq \tilde{w}\sigma$. Hence $\eta\pi \simeq I^G_P(\eta\sigma) \simeq I^G_P(\tilde{w}\sigma)$. There is also an isomorphism $r_P(\tilde{w}^{-1}, \tilde{w}\sigma): I^G_P(\tilde{w}\sigma) \rightiso I^G_P(\sigma)$. Hence $\eta\pi \simeq \pi$. Assume $\sigma \in \Pi_{2,\mathrm{temp}}(M)$ and let $\eta \in X^G(\pi)$, then $I^G_P(\eta\sigma) \simeq I^G_P(\sigma)$. By [@Wa03 Proposition III.4.1], there exists $w \in W(M)$ with $w\sigma \simeq \eta\sigma$. Henceforth we take $w \in \bar{W}_\sigma$. Take any $\eta \in L(\sigma)$ whose class modulo $X^M(\sigma)$ equals $\bar{\Gamma}(w)$ (see Proposition \[prop:barGamma\]). Then $\eta$ is of finite order by Proposition \[prop:L-X\^G\], in particular $\eta$ is unitary. For any isomorphism $$i: \eta\sigma \rightiso \tilde{w}\sigma,$$ we deduce an isomorphism $$\begin{aligned} \Ad(i): S^M(\eta\sigma) & \longrightarrow S^M(\tilde{w}\sigma) \\ I & \longmapsto iIi^{-1} =: \Ad(i)I.\end{aligned}$$ By the obvious identifications $S^M(\sigma) = S^M(\eta\sigma) = S^M(\tilde{w}\sigma)$ (without using $i$), one can view $\Ad(i)$ as an automorphism of $S^M(\sigma)$. It leaves $\C^\times$ intact and covers the identity map $X^M(\eta\sigma) \rightiso X^M(\tilde{w}\sigma)$, hence induces a bijection $$\begin{aligned} \Pi_-(S^M(\sigma)) & \longrightarrow \Pi_-(S^M(\sigma)) \\ \rho & \longmapsto \rho \circ \Ad(i).\end{aligned}$$ We shall write $$\tilde{w}\rho := \rho \circ \Ad(i).$$ The puzzling notation will be justified by Lemma \[prop:rho-sigma-correspondence\]. Henceforth, we adopt the following convention: for $I^G_\eta \in S^G(\pi)$, we regard $\Ad(I^G_\eta)$ as an automorphism of $S^M(\sigma)$ via the embedding $S^M(\sigma) \hookrightarrow S^G(\pi)$ provided by Proposition \[prop:S-embedding\]. \[prop:w-eta\] Let $\eta$, $\tilde{w}$ and $i: \eta\sigma \rightiso \tilde{w}\sigma$ be as above. As automorphisms of $S^M(\sigma)$, we have $$\Ad(i) = \Ad(I^G_\eta)$$ for every $I^G_\eta \in S^G(\pi)$ in the preimage of $\eta$. Given $\eta$, the assertion is independent of the choice of $I^G_\eta$. Let us consider the specific choice as follows $$I^G_\eta := r_P(\tilde{w}^{-1}, \tilde{w}\sigma) \circ I^G_P(i) \circ A_\eta: \eta I^G_P(\sigma) \to I^G_P(\sigma).$$ By the definition of the embedding $S^M(\sigma) \hookrightarrow S^G(\pi)$ and that of $A_\eta$, one sees that $\Ad(A_\eta)$ induces the identity map $S^M(\sigma) \rightiso S^M(\eta\sigma)$. Similarly, the functorial properties of $r_P(\tilde{w}^{-1}, \cdot)$ implies that $\Ad(r_P(\tilde{w}^{-1}, \tilde{w}\sigma))$ induces the identity map $S^M(\tilde{w}\sigma) \rightiso S^M(\sigma)$. One readily checks that $\Ad(I^G_P(i))$ induces $\Ad(i): S^M(\eta\sigma) \rightiso S^M(\tilde{w}\sigma)$, hence the assertion follows. Before stating the next result, recall that $\sigma$, $\tilde{w}\sigma$ and $\eta\sigma$ share the same underlying space $V_\sigma$. \[prop:rho-sigma-correspondence\] Let $\rho \in \Pi_-(S^M(\sigma))$ and $\sigma^\sharp \in \Pi_\sigma$. By identifying the groups $S^M(\sigma)$, $S^M(\tilde{w}\sigma)$ and $S^M(\eta\sigma)$, the following are equivalent: 1. $\rho \in \Pi_-(S^M(\sigma))$ corresponds to $\sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$; 2. $\rho \in \Pi_-(S^M(\tilde{w}\sigma))$ corresponds to $\tilde{w}\sigma^\sharp \hookrightarrow \tilde{w}\sigma|_{M^\sharp}$; 3. $\tilde{w}\rho \in \Pi_-(S^M(\eta\sigma))$ corresponds to $\tilde{w}\sigma^\sharp \hookrightarrow \eta\sigma|_{M^\sharp}$; 4. $\tilde{w}\rho \in \Pi_-(S^M(\sigma))$ corresponds to $\tilde{w}\sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$. Recall that $\rho$ corresponds to $\sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$ means that $\rho \boxtimes \sigma^\sharp \hookrightarrow \mathfrak{S}(\sigma)$, where $\mathfrak{S}(\sigma)$ is the $S^M(\sigma) \times M^\sharp(F)$-representation on $V_\sigma$ defined in Theorem \[prop:S-decomp\]. The first two properties are equivalent by a transport of structure via $\Ad(\tilde{w}): M^\sharp \to M^\sharp$. The last two properties are evidently equivalent. Finally, the equivalence between the second and the third properties follows by pulling $\rho$ back via $\Ad(i): S^M(\eta\sigma) \rightiso S^M(\tilde{w}\sigma)$. Now, recall that $Z^M(\sigma)$ is the projection to $X^M(\sigma)$ of the center of $S^M(\sigma)$. Temporarily fix a preimage $I^G_\eta$ for every $\eta \in X^G(\pi)$ and define $$\begin{gathered} \label{eqn:perp} Z^M(\sigma)^\perp := \left\{ \eta \in X^G(\pi) : \forall \omega \in Z^M(\sigma), \; I^G_\eta I^G_\omega = I^G_\omega I^G_\eta \right\} \supset X^M(\sigma).\end{gathered}$$ \[prop:Z-perp\] Let $\sigma$, $w$, $\eta$ as before. For any $\sigma^\sharp \in \Pi_\sigma$ (resp. $\rho \in \Pi_-(S^M(\sigma))$), we have $\tilde{w}\sigma^\sharp \simeq \sigma^\sharp$ (resp. $\tilde{w}\rho \simeq \rho$) if and only if $\eta \in Z^M(\sigma)^\perp$. Let $\rho \in \Pi_-(S^M(\sigma))$ be the representation corresponding to $\sigma^\sharp \in \Pi_\sigma$ by Theorem \[prop:S-decomp\]. By Lemma \[prop:rho-sigma-correspondence\], it suffices to show that $\tilde{w}\rho \simeq \rho$ if and only if $\eta \in Z_M(\sigma)^\perp$. The elements in $\Pi_-(S^M(\sigma))$ are described by a variant of the Stone-von Neumann theorem for the central extension $1 \to \C^\times \to S^M(\sigma) \to X^M(\sigma) \to 1$. Namely, consider the data $(L, \rho_0)$ where - $L$: a maximal abelian subgroup of $S^M(\sigma)$; - $\rho_0$: an irreducible representation of $L$ such that $\rho_0(z)=z$ for all $z \in \C^\times \subset L$. Then $\rho := \text{Ind}^{S^M(\sigma)}_L(\rho_0)$ is an element of $\Pi_-(S^M(\sigma))$. Every $\rho \in \Pi_-(S^M(\sigma))$ arises in this way. Moreover, the isomorphism class of $\rho$ is determined by its central character. These facts are standard consequences of Mackey’s theory. See [@KP84 0.3] and the remark after Proposition \[prop:K\_0-diagram\]. We have $\tilde{w}\rho = \rho \circ \Ad(I^G_\eta)$ by Lemma \[prop:w-eta\]. To conclude the proof, it suffices to show that $\Ad(I^G_\eta)$ fixes the central character of $\rho$ if and only if $\eta \in Z^M(\sigma)^\perp$. This is immediate. \[prop:L-sharp\] Assume $\sigma \in \Pi_{2,\mathrm{temp}}(M)$ and $\sigma^\sharp \in \Pi_\sigma$. Then we have $L(\sigma^\sharp)=Z^M(\sigma)^\perp$, and the map $\Gamma$ in Proposition \[prop:Goldberg\] is an isomorphism $$\begin{aligned} \Gamma: R_{\sigma^\sharp} & \longrightarrow Z^M(\sigma)^\perp/X^M(\sigma) \\ w W^0_{\sigma^\sharp} & \longmapsto \eta X^M(\sigma) \end{aligned}$$ where $w \in W_{\sigma^\sharp}$ and $\eta \in Z^M(\sigma)^\perp$ satisfy the relation $$w\sigma \simeq \eta\sigma.$$ This results immediately from the definition of $L(\sigma^\sharp)$. Cocycles {#sec:cocycle} -------- \[def:obstruction\] Suppose for a moment that $H$ is a finite group and $N$ is a normal subgroup of $H$. Let $\rho$ be an irreducible representation of $N$ and assume that $h\rho := \rho \circ \Ad(h)^{-1} \simeq \rho$ for all $h \in H$. This is a necessary condition for extending $\rho$ to an irreducible representation of $H$, but not sufficient in general. Recall the following construction of an obstruction $\mathbf{c}_\rho \in H^2(H/N, \C^\times)$ for extending $\rho$, where $\C^\times$ is equipped with the trivial $H/N$-action. We can choose intertwining operators $\rho(h) \in \Isom_N(h\rho, \rho)$ for each $h \in H$, such that $$\begin{aligned} \rho(nh) & = \rho(n) \rho(h), \\ \rho(hn) & = \rho(h) \rho(n) \end{aligned}$$ for every $h \in H$, $n \in N$. Note that either of the equations above implies the other. There is a $\C^\times$-valued $2$-cocycle $c_\rho$ characterized by $$\begin{gathered} \label{eqn:obstruction} \rho(h_1 h_2) = c_\rho(h_1, h_2) \rho(h_1) \rho(h_2), \quad h_1, h_2 \in H. \end{gathered}$$ One readily checks that $c_\rho$ factors through $H/N \times H/N$, thus defines a class $\mathbf{c}_\rho \in H^2(H/N, \C^\times)$. This cohomology class only depends on $\rho$ itself. The formalism can also be generalized to the case where $H,N$ are central extensions of finite groups by $\C^\times$, and $\rho(z) = z\cdot\identity$ for all $z \in \C^\times$. Let us return to the formalism of the previous subsection. In particular, we assume $P=MU \subset G$ and $\sigma \in \Pi_{2,\text{temp}}(M)$ with the underlying vector space $V_\sigma$. Set $\pi := I^G_P(\sigma)$ as usual. For every $\eta \in Z^M(\sigma)^\perp$, we fix $w \in W(M)$, a representative $\tilde{w} \in G^\sharp(F)$, and an isomorphism $$i: \eta\sigma \rightiso \tilde{w}\sigma.$$ Let $\rho \in \Pi_-(S^M(\sigma))$ be corresponding to $\sigma^\sharp \in \Pi_\sigma$. Proposition \[prop:Z-perp\] implies that $\tilde{w}\rho \simeq \rho$ for every $\eta$ as above. Equivalently, $\rho \circ \Ad((I^G_\eta)^{-1}) \simeq \rho$ for every $I^G_\eta \in S^G(\pi)$ in the preimage of $\eta$ by Lemma \[prop:w-eta\]. We will use the shorthand $${}^\eta \rho := \rho \circ \Ad((I^G_\eta)^{-1}).$$ As in Definition \[def:obstruction\], one considers the problem of extending $\rho$ to the preimage of $Z^M(\sigma)^\perp$ in $S^G(\pi)$. Recall that $Z^M(\sigma)^\perp/X^M(\sigma) = R_{\sigma^\sharp}$ by Corollary \[prop:L-sharp\]. The goal of this subsection is to describe the obstruction class $\mathbf{c}_\rho \in H^2(R_{\sigma^\sharp}, \C^\times)$ so-obtained. Recall that in Theorem \[prop:S-decomp\], we have defined an $S^M(\sigma) \times M^\sharp(F)$-representation $\mathfrak{S} = \mathfrak{S}(\sigma)$ on $V_\sigma$. Analogously, we define $\mathfrak{S}(\eta\sigma)$ and $\mathfrak{S}(\tilde{w}\sigma)$; all of them are realized on $V_\sigma$. We fix an embedding $\iota: \rho \boxtimes \sigma^\sharp \hookrightarrow \mathfrak{S}(\sigma)$ of $S^M(\sigma) \times M^\sharp(F)$-representations. By Lemma \[prop:rho-sigma-correspondence\], the same map gives $\iota: \rho \boxtimes \tilde{w}\sigma^\sharp \hookrightarrow \mathfrak{S}(\tilde{w}\sigma)$ and $\iota: \rho \boxtimes \sigma^\sharp \hookrightarrow \mathfrak{S}(\eta\sigma)$ with appropriate equivariances. \[prop:existential\] For $\eta$, $\tilde{w}$, $\sigma^\sharp$ fixed as before, we define $\mathfrak{S}'(\eta\sigma)$ to be the $S^M(\eta\sigma) \times M^\sharp(F)$-representation on $V_\sigma$ defined by $$\mathfrak{S}'(\eta\sigma)(I,x) = \mathfrak{S}(\eta\sigma)(\Ad(I^G_\eta)^{-1} I, x), \quad I \in S^M(\eta\sigma), x \in M^\sharp(F).$$ Then the map $\iota$ induces an embedding of $S^M(\eta\sigma) \times M^\sharp(F)$-representations $$\iota: {}^\eta \rho \boxtimes \sigma^\sharp \hookrightarrow \mathfrak{S}'(\eta\sigma)$$ and there exists a unique equivariant isomorphism $$\alpha \boxtimes \sigma^\sharp(\tilde{w})^{-1}: {}^\eta \rho \boxtimes \sigma^\sharp \rightiso \rho \boxtimes \tilde{w}\sigma^\sharp,$$ for some $\alpha \in \Isom_{S^M(\sigma)}({}^\eta \rho, \rho)$ and $\sigma^\sharp(\tilde{w}) \in \Isom_{M^\sharp}(\tilde{w}\sigma^\sharp, \sigma^\sharp)$, which makes the following diagram commutative. $$\xymatrix{ \mathfrak{S}'(\eta\sigma) \ar[rr]^{i}_{\simeq} & & \mathfrak{S}(\tilde{w}\sigma) \\ {}^\eta \rho \boxtimes \sigma^\sharp \ar[u]^{\iota} \ar[rr]_{\alpha \boxtimes \sigma^\sharp(\tilde{w})^{-1}} & & \rho \boxtimes \tilde{w}\sigma^\sharp \ar[u]_{\iota} }$$ Observe that the pair $(\alpha, \sigma^\sharp(\tilde{w})^{-1})$ is unique up to $\{(z,z^{-1}) : z \in \C^\times \}$. The $S^M(\eta\sigma)$-action on $\mathfrak{S}'(\eta\sigma)$ makes $i$ equivariant. The leftmost vertical arrow comes from the original embedding $\iota: \rho \boxtimes \sigma^\sharp \hookrightarrow \mathfrak{S}(\eta\sigma)$ by an $\Ad(I^G_\eta)^{-1}$-twist. The images of the vertical arrows are characterized as the $\sigma^\sharp$ (resp. $\tilde{w}\sigma^\sharp$) -isotypic parts under the $M^\sharp(F)$-action. Proposition \[prop:Z-perp\] implies that $\sigma^\sharp \simeq \tilde{w}\sigma^\sharp$. Therefore there must exist an equivariant isomorphism ${}^\eta \rho \boxtimes \sigma^\sharp \rightiso \rho \boxtimes \tilde{w}\sigma^\sharp$ that makes the diagram commute. Such an isomorphism must be of the form $\alpha \boxtimes \sigma^\sharp(\tilde{w})^{-1}$. \[prop:intertwining\] Write $r_P := r_P(\tilde{w}, \sigma)$ and $r_{P^\sharp} := r_{P^\sharp}(\tilde{w}, \sigma^\sharp)$. There is a commutative diagram $$\xymatrix{ I^G_P(\sigma) \ar[r]^{r_P} & I^G_P(\tilde{w}\sigma) \\ \rho \boxtimes I^{G^\sharp}_{P^\sharp}(\sigma^\sharp) \ar[r]_{\identity \boxtimes r_{P^\sharp}} \ar[u]^{I^{G^\sharp}_{P^\sharp}(\iota)} & \rho \boxtimes I^{G^\sharp}_{P^\sharp}(\tilde{w}\sigma^\sharp) \ar[u]_{I^{G^\sharp}_{P^\sharp}(\iota)} }$$ whose arrows are equivariant for the $S^M(\tilde{w}\sigma) \times G^\sharp(F)$ and $S^M(\sigma) \times G^\sharp(F)$-actions. Without loss of generality we may assume $\rho = \Hom_{M^\sharp}(\sigma^\sharp, \sigma)$, i.e. the multiplicity space. The embedding $\iota: \rho \boxtimes \sigma^\sharp \hookrightarrow \sigma$ can be taken to be $\epsilon \otimes v \mapsto \epsilon(v)$. Then the commutativity of the diagram follows by applying Theorem \[prop:normalizing-invariance\] to each $\epsilon \in \Hom_{M^\sharp}(\sigma^\sharp, \sigma)$. The equivariance of the horizontal arrows results from Theorem \[prop:normalizing-invariance\] and the functorial properties of $r_P(\tilde{w},\cdot)$, $r_{P^\sharp}(\tilde{w}, \cdot)$. \[prop:pull-back\] With the notations of Lemma \[prop:existential\], there is a commutative diagram $$\xymatrix{ I^G_P(\eta\sigma) \ar[rr]^{r_P(\tilde{w}, \sigma)^{-1} \circ I^G_P(i)} & & I^G_P(\sigma) \\ {}^\eta\rho \boxtimes I^{G^\sharp}_{P^\sharp}(\sigma^\sharp) \ar[rr]_{\alpha \boxtimes R_{P^\sharp}(\tilde{w}, \sigma^\sharp)^{-1}} \ar[u]^{I^{G^\sharp}_{P^\sharp}(\iota)} & & \rho \boxtimes I^{G^\sharp}_{P^\sharp}(\sigma^\sharp) \ar[u]_{I^{G^\sharp}_{P^\sharp}(\iota)} }$$ where we set $R_{P^\sharp}(\tilde{w}, \sigma^\sharp) := \sigma^\sharp(\tilde{w}) \circ r_{P^\sharp}(\tilde{w}, \sigma^\sharp)$, by using the pair $(\alpha, \sigma^\sharp(\tilde{w})^{-1})$ of isomorphisms in Lemma \[prop:existential\]. This is the concatenation of the diagram in Lemma \[prop:intertwining\] and that of Lemma \[prop:existential\], after applying $I^{G^\sharp}_{P^\sharp}(\cdot)$. \[prop:cocycle\] Let $\mathbf{c}_\rho$ be the obstruction of extending $\rho$ to the preimage of $Z^M(\sigma)^\perp$ in $S^G(\pi)$, and $\mathbf{c}_{\sigma^\sharp}$ be the class attached to $R_{\sigma^\sharp}$ in . Then we have $$\mathbf{c}_\rho = \mathbf{c}_{\sigma^\sharp}^{-1}$$ in $H^2(R_{\sigma^\sharp}, \C^\times)$. Fix $\iota: \rho \boxtimes \sigma^\sharp \hookrightarrow \sigma$. Also fix a set of representatives $\tilde{w} \in G^\sharp(F)$ for each $w \in R_{\sigma^\sharp}$. For each $\eta$, together with the auxiliary choice $i: \eta\sigma \rightiso \tilde{w}\sigma$, the top row of the diagram in Lemma \[prop:pull-back\] gives an operator $I^G_\eta \circ A_\eta^{-1}: I^G_P(\eta\sigma) \rightiso I^G_P(\sigma)$ for some $I^G_\eta \in S^G(\pi)$. The isomorphism $A_\eta: \eta I^G_P(\sigma) \rightiso I^G_P(\sigma)$ has no effect after restriction. Therefore Lemma \[prop:pull-back\] asserts that $I^G_\eta$ is pulled-back to $\alpha \boxtimes R_{P^\sharp}(\tilde{w}, \sigma^\sharp)^{-1}$ under $I^{G^\sharp}_{P^\sharp}(\iota)$. Now we can forget $i$ and vary $I^G_\eta$ in the preimage of $\eta$ in $S^G(\pi)$, which is a $\C^\times$-torsor. Regard $\alpha = \alpha(I^G_\eta)$ as a function of $I^G_\eta$; it is well-defined once we have pinned down the operator $\sigma^\sharp(\tilde{w})$ coupled with $\alpha$. Suppose that $\eta$ is replaced by $\eta\omega$ where $\omega \in X^M(\sigma)$; accordingly, $I^G_\eta$ is replaced by $I^G_\eta I^G_\omega$ where $I^M_\omega \in S^M(\sigma)$ lies in the preimage of $\omega$ and $I^M_\omega \mapsto I^G_\omega$. This does not affect the chosen data $\tilde{w}$ and $\iota$. On the other hand, the diagram in Lemma \[prop:pull-back\] says that $\alpha \boxtimes R_{P^\sharp}(\tilde{w}, \sigma^\sharp)^{-1}$ is replaced by $$\alpha \circ \rho(I^M_\omega) \boxtimes R_{P^\sharp}(\tilde{w}, \sigma^\sharp)^{-1}.$$ It follows that we can pin down the operators $\sigma^\sharp(\tilde{w})$, and well-define a function $$I^G_\eta \mapsto \alpha(I^G_\eta) \in \Isom_{S^M(\sigma)}({}^\eta \rho, \rho),$$ for every $I^G_\eta$ in the preimage of $\eta \in Z^M(\sigma)^\perp$ in $S^G(\pi)$, such that - $I^G_\eta$ is pulled-back to $\alpha(I^G_\eta) \boxtimes R_{P^\sharp}(\tilde{w}, \sigma^\sharp)^{-1}$ under $I^{G^\sharp}_{P^\sharp}(\iota)$; - $\alpha(I^G_\eta I^G_\omega) = \alpha(I^G_\eta) \rho(I^M_\omega)$ for every $\omega \in X^M(\sigma)$ and $I^M_\omega$ in its preimage. Such a family of intertwining operators meets the requirements of Definition \[def:obstruction\], thus the obstruction can be accounted by the $\C^\times$-valued $2$-cocycle $c_\rho$ given by $$\alpha(I^G_\xi I^G_\eta) = c_\rho(w_\xi, w_\eta) \alpha(I^G_\xi) \alpha(I^G_\eta), \quad \xi, \eta \in Z^M(\sigma)^\perp,$$ where $w_\eta \in R_{\sigma^\sharp}$ denotes the element determined by $\eta$ as in Corollary \[prop:L-sharp\]. Idem for $w_\xi$. On the other hand, write $w_\eta \mapsto \tilde{w}_\eta$ for the map that picks the chosen representative for $w_\eta \in R_{\sigma^\sharp}$. The equation defines $2$-cocycle $c_{\sigma^\sharp}$. For every $\xi, \eta \in Z^M(\sigma)^\perp$ we obtain $$R_{P^\sharp}(\tilde{w}_{\xi\eta}, \sigma^\sharp) = R_{P^\sharp}(\tilde{w}_{\eta\xi}, \sigma^\sharp) = c_{\sigma^\sharp}(w_\eta, w_\xi) R_{P^\sharp}(\tilde{w}_\eta, \sigma^\sharp) R_{P^\sharp}(\tilde{w}_\xi, \sigma^\sharp).$$ All in all, the pull-back of $I^G_\xi I^G_\eta$ by $I^{G^\sharp}_{P^\sharp}(\iota)$ equals $$c_\rho(w_\xi, w_\eta) c_{\sigma^\sharp}(w_\eta, w_\xi)^{-1} \cdot (\text{the pull-back of } I^G_\xi) \circ (\text{the-pull back of } I^G_\eta ).$$ Therefore $c_\rho(w_\xi, w_\eta) = c'_{\sigma^\sharp}(w_\xi, w_\eta) := c_{\sigma^\sharp}(w_\eta, w_\xi)$. It is routine to check that $c'_{\sigma^\sharp}: (R_{\sigma^\sharp})^2 \to \C^\times$ is also a $2$-cocycle. Denote by $\mathbf{c}'_{\sigma^\sharp}$ the cohomology class of $c'_{\sigma^\sharp}$. It remains to show that $\mathbf{c}'_{\sigma^\sharp} = \mathbf{c}^{-1}_{\sigma^\sharp}$. We use the following observation: let $A$ be finite abelian group acting trivially on $\C^\times$, we claim that there is an injective group homomorphism $$\begin{aligned} \text{comm}: H^2(A, \C^\times) & \longrightarrow \Hom\left( \bigwedge^2 A, \C^\times \right), \\ \mathbf{c} & \longmapsto [x \wedge y \mapsto c(y,x)c(x,y)^{-1}] \end{aligned}$$ where $c$ is any $2$-cocycle representing the class $\mathbf{c}$. Indeed, let $$1 \to \C^\times \to \tilde{A} \to A \to 1$$ be the central extension corresponding to $\mathbf{c}$, then $(x,y) \mapsto c(y,x)c(x,y)^{-1}$ is just the commutator pairing of this central extension. The injectivity results from the elementary fact that such an extension splits if and only if $\tilde{A}$ is commutative. Apply this to $A = R_{\sigma^\sharp}$. Since $\text{comm}(\mathbf{c}'_{\sigma^\sharp}) = \text{comm}(\mathbf{c}^{-1}_{\sigma^\sharp})$, we deduce $\mathbf{c}'_{\sigma^\sharp} = \mathbf{c}^{-1}_{\sigma^\sharp}$, as asserted. The inner forms of $\SL(N)$ {#sec:SL} =========================== The groups {#sec:groups} ---------- Fix $N \in \Z_{\geq 1}$ and let $G^* := \GL_F(N)$. Let $A$ be a central simple algebra over $F$ of dimension $N^2$. There exist $n \in \Z_{\geq 1}$ and a central division algebra $D$ over $F$ satisfying $$n^2 \cdot \dim_F D = N^2,$$ such that $A$ is isomorphic to $\End_D(D^n)$. The division $F$-algebra $D$ is uniquely determined by $A$. We put $$\begin{aligned} \Nrd & := \text{ the reduced norm of } A, \\ \GL_D(n) & := A^\times, \\ \SL_D(n) & := \Ker(\Nrd: A^\times \to \Gm).\end{aligned}$$ We can regard $A^\times$ as a reductive $F$-group. It is well-known that $A \mapsto A^\times$ induces a bijection between the central simple $F$-algebras of dimension $N^2$ and the inner forms of $G^*$. Given $A$, or equivalently given $(n,D)$ as above, we shall always write $$G := \GL_D(n).$$ Under an inner twist $\psi: G \times_F \bar{F} \rightiso G^* \times_G \bar{F}$, the determinant map $\det: G^* \to \Gm$ corresponds to $\Nrd: G \to \Gm$. Since the parametrization of the inner forms of an $F$-group $G^*$ only depends on $G^*_\text{AD}$, the map $A \mapsto \SL_D(n)$ establishes a bijection between the central simple $F$-algebras of dimension $N^2$ and the inner forms of $\SL_N(F)$. We write $$G^\sharp := \SL_D(n) = G_\text{der}.$$ Note that $G(F)/G^\sharp(F) = (G/G^\sharp)(F) = F^\times$, since $H^1(F, G^\sharp)$ is trivial by Hasse principle. As mentioned in §\[sec:res-L\], the inner twist $\psi$ gives a correspondence between Levi subgroups: the Levi subgroups of $G$ is of the form $$M = \prod_{i=1}^r \GL_D(n_i), \quad n_1 + \cdots + n_r = n.$$ and the corresponding Levi subgroup of $G^*$, well-defined up to conjugacy, is simply $$M^* = \prod_{i=1}^r \GL_F(n_i \cdot \dim_F D).$$ The L-groups of $G$ and $G^\sharp$ are easily described. We have $$\begin{aligned} \hat{G} = \hat{G^*} & = \GL(N,\C), \\ \widehat{G^\sharp} & = \PGL(N,\C), \\ \hat{G}_\text{SC} = (\widehat{G^\sharp})_\text{SC} & = \SL(N,\C), \\ Z_{\hat{G}_\text{SC}} = Z_{(\widehat{G^\sharp})_\text{SC}} & = \mu_N(\C) \\ & := \{z \in \C^\times : z^N = 1 \}.\end{aligned}$$ These complex groups are endowed with the trivial Galois action, thus $\Lgrp{G} = \hat{G} \times W_F$ and $\Lgrp{G^\sharp} = \widehat{G^\sharp} \times W_F$. The inclusion $G^\sharp \hookrightarrow G$ is dual to the quotient homomorphism $\GL(N,\C) \to \PGL(N,\C)$. It is also possible to describe the characters $\chi_G = \chi_{G^\sharp}$ in explicitly. Observe that $\Gamma_F$ acts trivially on $Z_{\hat{G}_\text{SC}}$, and one can identify the Pontryagin dual of $Z_{\hat{G}_\text{SC}} = \mu_N(\C)$, denoted by $Z_{\hat{G}_\text{SC}}^D$, with $\Z/N\Z$: a class $e \in \Z/N\Z$ corresponds to the character $z \mapsto z^e$. For the inner form $G = \GL_D(n)$ of $G^* = \GL_F(N)$, we have $$\begin{gathered} \label{eqn:chi-G-special} \chi_G \in Z_{\hat{G}_\text{SC}}^D \; \text{ corresponds to } (n \text{ mod } N) \in \Z/N\Z .\end{gathered}$$ Later on, the results of §\[sec:res-2\] will be applied to the tempered representations of $G(F)$. This is justified by the following general result. \[prop:irred\] Let $P=MU$ be a parabolic subgroup of $G$ and $\sigma \in \Pi_{\mathrm{unit}}(M)$, then $I^G_P(\sigma)$ is irreducible. In particular, Hypothesis \[hyp:irred\] is satisfied by $\sigma$. Note that the tempered case is already established in [@DKV84]. Local Langlands correspondences {#sec:LLC} ------------------------------- This subsection is a summary of [@HS12 Chapter 11]. #### Local Langlands correspondence for $\GL_D(n)$ Using the local Langlands correspondence for $G^*$, we can define the notion of $G^*$-generic elements in $\Phi(G)$: a parameter $\phi \in \Phi(G) \subset \Phi(G^*)$ is called $G^*$-generic if it parametrizes a generic representation of $G^*(F)$. This defines a subset $\Phi_{G^*-\text{gen}}(G)$ of $\Phi(G)$. \[prop:LLC-G\] Let $G = \GL_D(n)$ and $G^* = \GL_F(N)$ as in §\[sec:groups\]. There exist a subset $\Pi_{G^*-\mathrm{gen}}(G)$ of $\Pi(G)$ satisfying - $\Pi_{G^*-\mathrm{gen}}(G) \supset \Pi_{\mathrm{temp}}(G)$, - $\Pi_{G^*-\mathrm{gen}}(G)$ is stable under twists by $(G(F)/G^\sharp(F))^D$, and a canonically defined bijection between $\Pi_{G^*-\mathrm{gen}}(G)$ and $\Phi_{G^*-\mathrm{gen}}(G)$, denoted by $\pi \leftrightarrow \phi$, such that $$\xymatrix{ \Pi_{G^*-\mathrm{gen}}(G) \ar@{<->}[r]^{\sim} & \Phi_{G^*-\mathrm{gen}}(G) \\ \Pi_{\mathrm{temp}}(G) \ar@{<->}[r]^{\sim} \ar@{^{(}->}[u] & \Phi_{\mathrm{bdd}}(G) \ar@{^{(}->}[u] \\ \Pi_{2,\mathrm{temp}}(G) \ar@{<->}[r]^{\sim} \ar@{^{(}->}[u] & \Phi_{2,\mathrm{bdd}}(G) \ar@{^{(}->}[u] }$$ The correspondence satisfies the following compatibility properties. 1. When $G=G^*$, the usual Langlands correspondence for $\GL_F(N)$ is recovered. 2. Given $\pi \leftrightarrow \phi$ and $\mathbf{a} \in H^1_{\mathrm{cont}}(W_F, Z_{\hat{G}})$, let $\eta$ be the character of $G(F)$ deduced from $\mathbf{a}$ by local class field theory, then we have $\omega\pi \leftrightarrow a\phi$. 3. Given a Levi subgroup $M = \prod_{i \in I} \GL_D(n_i)$ of $G$ and let $\sigma := \boxtimes_{i \in I} \sigma_i \in \Pi_{\mathrm{temp}}(M)$. Let $\phi_M \in \Phi_{\mathrm{bdd}}(M)$ such that $\sigma \leftrightarrow \phi_M$ and let $\phi$ be the composition of $\phi_M$ with some L-embedding $\Lgrp{M} \hookrightarrow \Lgrp{G}$. Then for any $P \in \mathcal{P}(M)$ we have $$I^G_P(\sigma) \leftrightarrow \phi .$$ Note that in the last assertion, $I^G_P(\sigma)$ is irreducible according to Theorem \[prop:irred\]. The definitions of $\Pi_{G^*-\text{gen}}(G)$ and $\pi \leftrightarrow \phi$ are based upon the local Langlands correspondence for $G^*$ and the Jacquet-Langlands correspondence for essentially square-integrable representations. We refer the reader to [@HS12 §11] for details; the compatibility properties are also implicit therein. Only the tempered/bounded case of the theorem will be used in this article. #### Local Langlands correspondence for $\SL_D(n)$ Let $G = \GL_D(n)$ and $G^\sharp = G_\text{der} = \SL_D(n)$, so that the formalism in §\[sec:res\] is applicable. The idea is to define the packets $\Pi_{\phi^\sharp}$ via restriction, by combining the results in §\[sec:res-rep\] and §\[sec:res-L\]. Let $\Phi_{G^*-\text{gen}}(G^\sharp)$ be the set of $\phi^\sharp \in \Phi(G^\sharp)$ such that $\phi \in \Phi_{G^*-\text{gen}}(G)$ for some lifting $\phi$ of $\phi^\sharp$ (hence for all liftings, since twisting by characters does not affect $G^*$-genericity). For any $\phi^\sharp \in \Phi_{G^*-\text{gen}}(G^\sharp)$, define the corresponding packet by $$\Pi_{\phi^\sharp} := \Pi_\pi, \quad \pi \leftrightarrow \phi \text{ for some lifting } \phi \in \Phi_{G^*-\text{gen}}(G).$$ By Proposition \[prop:res-disjoint\] and Theorem \[prop:lifting-parameter\], the definition of $\Pi_{\phi^\sharp}$ does not depend on the choice of lifting. On the other hand, set $$\Pi_{G^*-\text{gen}}(G^\sharp) = \bigsqcup_{\pi} \Pi_\pi$$ where $\pi$ ranges over the $(G(F)/G^\sharp(F))^D$-orbits in $\Pi_{G^*-\text{gen}}(G)$. Our version of the local Langlands correspondence for $G^\sharp$ is stated as follows. \[prop:LLC-SL\] We have $\Pi_\mathrm{temp}(G^\sharp) \subset \Pi_{G^*-\mathrm{gen}}(G^\sharp)$, and there is a decomposition $$\begin{gathered} \label{eqn:LLC-SL} \Pi_{G^*-\mathrm{gen}}(G^\sharp) = \bigsqcup_{\phi^\sharp \in \Phi_{G^*-\mathrm{gen}}(G^\sharp)} \Pi_{\phi^\sharp} \end{gathered}$$ which restricts to $$\begin{aligned} \Pi_{\mathrm{temp}}(G^\sharp) = \bigsqcup_{\phi^\sharp \in \Phi_{\mathrm{temp}}(G^\sharp)} \Pi_{\phi^\sharp}, \\ \Pi_{2,\mathrm{temp}}(G^\sharp) = \bigsqcup_{\phi^\sharp \in \Phi_{2,\mathrm{temp}}(G^\sharp)} \Pi_{\phi^\sharp}. \end{aligned}$$ This follows from Proposition \[prop:heredity\], Theorem \[prop:LLC-G\], \[prop:lifting-parameter\] and Proposition \[prop:res-disjoint\]. Note that each packet $\Pi_{\phi^\sharp}$ is finite. From the endoscopic point of view, in order to justify the correspondence , one has to explicate 1. the internal structure of the packets $\Pi_{\phi^\sharp}$; 2. their relation to $S$-groups; 3. the endoscopic character identities for $G^\sharp$. We will recall the definition of $S$-groups (or more precisely their component groups, called $\mathscr{S}$-groups...) in the next subsection, then summarize its relation to the internal structure of packets; this is one of the main results in [@HS12]. The character identities will not be used in this article; we refer the interested reader to [@HS12 Theorem 12.7]. #### Normalizing factors Choose a non-trivial additive character $\psi_F: F \to \C^\times$. Now we can exhibit a canonical family of normalizing factors for $G$ and $G^\sharp$ with respect to $\psi_F$. Let us begin with $G$. According to the construction in Remark \[rem:construction-normalization\], it suffices to consider the case of inducing representations $\sigma \in \Pi_{2,\text{temp}}(M)$ where $M$ is a Levi subgroup of $G$. When $D=F$ or equivalently $G=G^*=\GL_F(N)$, the formula in Remark \[rem:normalization\] furnishes a family of normalizing factors in the tempered case, by the Langlands-Shahidi method. To pass to the non-quasisplit case, we use the preservation of $\mu$-functions by Jacquet-Langlands correspondence [@AP05 Theorem 7.2] (up to a harmless constant depending only on $D$ and $n$). From Theorem \[prop:normalizing-invariance\], we deduce a canonical family of normalizing factors for $G^\sharp$, at least for the inducing representations $\sigma^\sharp$ whose central character is unitary. In what follows, the normalized intertwining operators for $G$ and $G^\sharp$ are assumed to be defined with respect to these factors. Identification of $S$-groups {#sec:iden-S-groups} ---------------------------- #### Generalities To begin with, we summarize the definition of the $S$-groups in the non-quasisplit case by following [@Ar06]. Let $G$ be a connected reductive $F$-group. Choose a quasisplit inner twist $\psi: G \times_F \bar{F} \to G^* \times_F \bar{F}$ as well as an $F$-splitting for $G^*(\bar{F})$ to define the L-groups. Let $\phi \in \Phi(G^*)$, we set $$\begin{aligned} S_{\phi, \text{ad}} & := Z_{\hat{G}}(\Im(\phi))/Z^{\Gamma_F}_{\hat{G}} \\ & \rightiso \left( Z_{\hat{G}}(\Im(\phi)) Z_{\hat{G}} \right) / Z_{\hat{G}} \; \subset \hat{G}_\text{AD}, \\ S_{\phi, \text{sc}} & := \text{ the preimage of } S_{\phi, \text{ad}} \text{ in } \hat{G}_\text{SC}, \\ \mathscr{S}_{\phi} & := \pi_0(S_{\phi, \text{ad}}, 1), \\ \tilde{\mathscr{S}}_{\phi} & := \pi_0(S_{\phi, \text{sc}}, 1). \end{aligned}$$ From the central extension $1 \to Z_{\hat{G}_\text{SC}} \to S_{\phi, \text{sc}} \to S_{\phi, \text{ad}} \to 1$, we obtain another central extension $$1 \to \tilde{Z}_\phi \to \tilde{\mathscr{S}}_{\phi} \to \mathscr{S}_\phi \to 1$$ where $$\tilde{Z}_\phi := Z_{\hat{G}_\text{SC}}/(Z_{\hat{G}_\text{SC}} \cap S^0_{\phi, \text{sc}}) = \Im[ Z_{\hat{G}_\text{SC}} \to \tilde{\mathscr{S}}_{\phi} ].$$ When $G$ is an inner form of $\SL(N)$, we recover the definition of the modified $S$-groups in [@HS12]. The relevance condition of L-parameters intervenes in the following result. Recall that we have defined a character $\chi_G$ of $Z^{\Gamma_F}_{\hat{G}_\text{SC}}$ in . \[prop:chi-G-factorization\] If $\phi \in \Phi(G)$, then $\chi_G : Z^{\Gamma_F}_{\hat{G}_\text{SC}} \to \C^\times$ is trivial on $Z_{\hat{G}_\text{SC}} \cap S^0_{\phi, \text{sc}}$. By abuse of notations, the so-obtained character of $Z^{\Gamma_F}_{\hat{G}_\text{SC}}/(Z_{\hat{G}_\text{SC}} \cap S^0_{\phi, \text{sc}}) \subset \tilde{Z}_\phi$ is still denoted by $\chi_G$. Also note that $\chi_G$ depends only on $G_\text{AD}$. Let us reproduce the proof in [@HS12] here. Let $M = M_\phi$ be a minimal Levi subgroup of $G$ through which $\phi$ factorizes (we used the relevance condition here). By the recollections in §\[sec:res-L\], $Z^{\Gamma_F, 0}_{\hat{M}_\text{sc}}$ is a maximal torus in $S^0_{\phi, \text{sc}}$. Therefore $$S^0_{\phi, \text{sc}} \cap Z_{\hat{G}_\text{SC}} = Z^{\Gamma_F, 0}_{\hat{M}_\text{sc}} \cap Z^{\Gamma_F}_{\hat{G}_\text{SC}},$$ and the last group is contained in $\Ker(\chi_G)$ by [@Ar99 Corollary 2.2]. Consider the familiar situation $G_\text{der} \subset G^\sharp \subset G$ (cf. §\[sec:res-L\]), so that we have the $\Gamma_F$-equivariant central extension $$1 \to \hat{Z}^\sharp \to \hat{G} \xrightarrow{\mathbf{pr}} \hat{G^\sharp} \to 1.$$ Let $\phi \in \Phi(G)$ and $\phi^\sharp := \mathbf{pr} \circ \phi \in \Phi(G^\sharp)$. The definitions above pertain to $(G^\sharp, \phi^\sharp)$ as well. Set $$X^G(\phi) := \left\{ \mathbf{a} \in H^1_\text{cont}(W_F, \hat{Z}^\sharp) : \mathbf{a}\phi \sim \phi \right\}.$$ This is a finite abelian group (cf. the proof of Theorem \[prop:lifting-parameter\]). \[prop:S-sequence\] Let $s \in S_{\phi^\sharp, \mathrm{ad}}$, regarded as an element of $\widehat{G^\sharp}/Z^{\Gamma_F}_{\widehat{G^\sharp}}$. Then $s$ determines a class $\mathbf{a} \in H^1_\mathrm{cont}(W_F, \hat{Z}^\sharp)$ characterized by $$\begin{gathered} \label{eqn:s-a-centralize} \tilde{s} \phi(w) \tilde{s}^{-1} = a(w) \phi(w), \quad w \in \WD_F \end{gathered}$$ where - $\tilde{s} \in \hat{G}$ is a lifting of $s$; - $a: W_F \to \hat{Z}^\sharp$ is some $1$-cocycle representing $\mathbf{a}$, inflated to $\WD_F$. This induces an exact sequence $$\mathscr{S}_\phi \to \mathscr{S}_{\phi^\sharp} \to X^G(\phi) \to 1.$$ Choose a lifting $\tilde{s} \in \hat{G}$. Since $s$ centralizes $\phi^\sharp$, there exists a continuous function $a: \WD_F \to \hat{Z}^\sharp$ satisfying . It is straightforward to check that $a$ is inflated from a $1$-cocycle $W_F \to \hat{Z}^\sharp$. The $1$-cocycle $a$ does depend on the choice of $\tilde{s}$, but its class $\mathbf{a} \in H^1_\text{cont}(W_F, \hat{Z}^\sharp)$ is uniquely determined by $s$; it is also obvious that $s \mapsto \mathbf{a}$ is a homomorphism. Conversely, every $s$ that satisfies for some $\tilde{s}, a$ clearly belongs to $S_{\phi^\sharp, \text{ad}}$. Hence the image of $s \mapsto \mathbf{a}$ equals $X^G(\phi)$, by the very definition of $X^G(\phi)$. If $s$ is mapped to the trivial class in $X^G(\phi)$, then we may choose $\tilde{s}$ so that $\tilde{s} \phi \tilde{s}^{-1} = \phi$; therefore $s$ comes from $S_{\phi, \text{ad}}$, and vice versa. Hence we have an exact sequence of locally compact groups $$S_{\phi, \mathrm{ad}} \to S_{\phi^\sharp, \mathrm{ad}} \to X^G(\phi) \to 1 .$$ By a connectedness argument, we may pass from the $S$-groups to the $\mathscr{S}$-groups that gives the asserted exact sequence. #### The case of the inner forms of $\SL(N)$ Let us revert to the situation where $$\begin{aligned} G & = \GL_D(n), \\ G^* & = \GL_F(N), \\ G^\sharp & = \SL_D(n), \\ \chi_G: & Z_{\hat{G}_\text{SC}} = \mu_N(\C) \to \C^\times.\end{aligned}$$ It is well-known that $\mathscr{S}_\phi = \{1\}$ for every $\phi \in \Phi(G^*)$. Indeed, $Z_{\hat{G}}(\Im(\phi))$ is a principal Zariski open subset in some linear subspace of $\text{Mat}_{N \times N}(\C)$, thus is connected; so is its quotient by $Z^{\Gamma_F}_{\hat{G}} = Z_{\hat{G}} = \C^\times$. Let $\phi^\sharp \in \Phi(G^\sharp)$ with a lifting $\phi \in \Phi(G)$. Hence Lemma \[prop:S-sequence\] yields a canonical isomorphism $$\mathscr{S}_{\phi^\sharp} \rightiso X^G(\phi).$$ Assume henceforth that $\phi^\sharp \in \Phi_{G^*-\text{gen}}(G^\sharp)$, so $\phi \in \Phi_{G^*-\text{gen}}(G)$ as well. The local Langlands correspondence for $G$ (Theorem \[prop:LLC-G\]) is thus applicable. Since the local Langlands correspondence is compatible with twisting by characters, we have $X^G(\phi) = X^G(\pi)$ where $\pi \leftrightarrow \phi$. Therefore we deduce the natural isomorphism $$\begin{gathered} \label{eqn:S-X-isom} \mathscr{S}_{\phi^\sharp} \rightiso X^G(\pi), \quad \text{where } \pi \leftrightarrow \phi, \; \phi^\sharp = \mathbf{pr} \circ \phi .\end{gathered}$$ Also observe that $\chi_G$ induces a character of $\tilde{Z}_{\phi^\sharp}$ by Lemma \[prop:chi-G-factorization\], since $\Gamma_F$ acts trivially on $Z_{\hat{G}_\text{SC}}$. \[prop:Lambda\] Let $\phi^\sharp \in \Phi_{G^*-\mathrm{gen}}(G^\sharp)$ with a chosen lifting $\phi \in \Phi_{G^*-\mathrm{gen}}(G)$. Let $\pi \in \Pi_{G^*-\mathrm{gen}}(G)$ such that $\pi \leftrightarrow \phi$ by Theorem \[prop:LLC-G\]. Then there exists a homomorphism $$\Lambda: \tilde{\mathscr{S}}_{\phi^\sharp} \to S^G(\pi)$$ such that the following diagram is commutative with exact rows: $$\xymatrix{ 1 \ar[r] & \tilde{Z}_{\phi^\sharp} \ar[d]_{\chi_G} \ar[r] & \tilde{\mathscr{S}}_{\phi^\sharp} \ar[d]^{\Lambda} \ar[r] & \mathscr{S}_{\phi^\sharp} \ar[d]^{\simeq} \ar[r] & 1 \\ 1 \ar[r] & \C^\times \ar[r] & S^G(\pi) \ar[r] & X^G(\pi) \ar[r] & 1 }$$ where the rightmost vertical arrow is that of . Moreover, $\Lambda$ is unique up to $\Hom(X^G(\pi),\C^\times)$, i.e. up to the automorphisms of the lower central extension, and upon identifying $\mathscr{S}_{\phi^\sharp}$ and $X^G(\pi)$, this diagram is a push-forward of central extensions by $\chi_G$. Note that the assertions about uniqueness and the push-forward are evident; the upshot is the existence of $\Lambda$. Let $\phi^\sharp$, $\phi$, $\pi$ be as above. Put $$\Pi(\tilde{\mathscr{S}}_{\phi^\sharp}, \chi_G) := \left\{ \rho \in \Pi(\tilde{\mathscr{S}}_{\phi^\sharp}) : \forall z \in \tilde{Z}_{\phi^\sharp}, \; \rho(z) = \chi_G(z) \identity \right\}.$$ The homomorphism $\Lambda$ in Theorem \[prop:Lambda\] induces a bijection $\Pi(\tilde{\mathscr{S}}_{\phi^\sharp}, \chi_G) \rightiso \Pi_-(S^G(\pi))$. Recall that in the local Langlands correspondence for $G^\sharp$ (Theorem \[prop:LLC-SL\]), the packet $\Pi_{\phi^\sharp}$ attached to $\phi^\sharp$ is defined as $\Pi_\pi$, the set of irreducible constituents of $\pi|_{G^\sharp}$. Combining Theorem \[prop:S-decomp\] with Theorem \[prop:Lambda\], we arrive at the following description of the packet $\Pi_{\phi^\sharp}$. \[prop:S-S\] Let $\phi^\sharp$, $\phi$, $\pi$ be as above. Let $\Hom(X^G(\pi), \C^\times)$ act on $\Pi_{\phi^\sharp}$ via the canonical isomorphisms $\Pi_{\phi^\sharp} = \Pi_\pi = \Pi_-(S^G(\pi))$. Then there is a bijection $$\Pi(\tilde{\mathscr{S}}_{\phi^\sharp}, \chi_G) \rightiso \Pi_{\phi^\sharp},$$ which is canonical up to the $\Hom(X^G(\pi), \C^\times)$-action on $\Pi_{\phi^\sharp}$. When $G$ is quasisplit, $\chi_G$ will be trivial and $\Pi(\tilde{\mathscr{S}}_{\phi^\sharp}, \chi_G) = \Pi(\mathscr{S}_{\phi^\sharp})$; the bijection in the Corollary can then be normalized by choosing a Whittaker datum for $G^\sharp$ (cf. [@HS12 Chapter 3]). In general, however, there is no reason to expect a canonical choice of the bijection $\Pi(\tilde{\mathscr{S}}_{\phi^\sharp}, \chi_G) \rightiso \Pi_{\phi^\sharp}$. Generalization {#sec:generalization} -------------- Consider the following abstract setting. - Let $M$, $M^\sharp$, $M^\sharp_0$ be connected reductive $F$-groups such that $M$ has an split inner form $M^*$, and $$M_\text{der} \subset M^\sharp_0 \subset M^\sharp \subset M .$$ - For $\pi \in \Pi(M)$, let $S^M(\pi)$ and $X^M(\pi)$ be the groups defined in §\[sec:res-rep\] relative to $M^\sharp$, and denote by $S^M_0(\pi)$, $X^M_0(\pi)$ the groups defined relative to $M^\sharp_0$. - Assume that there are subsets $\Pi_{\text{gen}}(M)$ and $\Phi_{\text{gen}}(M)$ of $\Pi(M)$ and $\Phi(M)$, respectively, together with a “local Langlands correspondence” $\pi \leftrightarrow \phi$ between $\Pi_{\text{gen}}(M)$ and $\Phi_{\text{gen}}(M)$ that is compatible with twist by characters, as in Theorem \[prop:LLC-G\]. We define $\Phi_{\text{gen}}(M^\sharp)$ (resp. $\Phi_{\text{gen}}(M^\sharp_0)$) to be the set of L-parameters that lift to $\Pi_{\text{gen}}(M^\sharp)$ (resp. $\Pi_{\text{gen}}(M^\sharp_0)$) via Theorem \[prop:lifting-parameter\]. - Assume that $\mathscr{S}_\phi = \{1\}$ for every $\phi \in \Phi_{\text{gen}}(M)$. Let $\phi \in \Phi_{\text{gen}}(M)$ and $\pi \in \Pi_\text{gen}(M)$ such that $\pi \leftrightarrow \phi$. As before, we deduce L-parameters $\phi^\sharp \in \Phi_{\text{gen}}(M^\sharp)$ and $\phi^\sharp_0 \in \Phi_{\text{gen}}(M^\sharp_0)$. First of all, let $\bullet$ be one of the subscripts “ad” or “sc”. We have $S_{\phi^\sharp, \bullet} \subset S_{\phi^\sharp_0, \bullet}$ and $S^0_{\phi^\sharp, \bullet} \subset S^0_{\phi^\sharp_0, \bullet}$. In view of the definitions in §\[sec:iden-S-groups\], we deduce natural isomorphisms $\mu$, $\tilde{\mu}$ that fit into the following commutative diagram. $$\begin{gathered} \label{eqn:mu} \xymatrix{ \mathscr{S}_{\phi^\sharp} \ar[rr]^{\mu} & & \mathscr{S}_{\phi^\sharp_0} \\ \tilde{\mathscr{S}}_{\phi^\sharp} \ar[u] \ar[rr]^{\tilde{\mu}} & & \tilde{\mathscr{S}}_{\phi^\sharp_0} \ar[u] \\ \tilde{Z}_{\phi^\sharp} \ar[u] \ar@{->>}[rr]^{\tilde{\mu}} \ar[rd]_{\chi_M} & & \tilde{Z}_{\phi^\sharp_0} \ar[u] \ar[ld]^{\chi_M} \\ & \C^\times & }\end{gathered}$$ Secondly, we have $X^M(\pi) \subset X^M_0(\pi)$ as subgroups of $M(F)^D$. Consequently $S^M(\pi) \subset S^M_0(\pi)$. Iterating the arguments for , we obtain the commutative diagram $$\begin{gathered} \label{eqn:mu-2} \xymatrix{ \mathscr{S}_{\phi^\sharp} \ar[r]^{\simeq} \ar@{^{(}->}[d]_{\mu} & X^M(\pi) \ar@{^{(}->}[d] \\ \mathscr{S}_{\phi^\sharp_0} \ar[r]^{\simeq} & X^M_0(\pi). }\end{gathered}$$ \[prop:generalization\] Let $\phi, \phi^\sharp, \phi^\sharp_0$ and $\pi$ be as above. Assume that there exists a homomorphism $\Lambda_0: \tilde{\mathscr{S}}_{\phi^\sharp_0} \to S^M_0(\pi)$ such that the following diagram is commutative with exact rows: $$\begin{gathered} \label{eqn:comm-diag-0} \xymatrix{ 1 \ar[r] & \tilde{Z}_{\phi^\sharp_0} \ar[d]_{\chi_M} \ar[r] & \tilde{\mathscr{S}}_{\phi^\sharp_0} \ar[d]^{\Lambda_0} \ar[r] & \mathscr{S}_{\phi^\sharp_0} \ar[d]^{\simeq} \ar[r] & 1 \\ 1 \ar[r] & \C^\times \ar[r] & S^M_0(\pi) \ar[r] & X^M_0(\pi) \ar[r] & 1 }\end{gathered}$$ Then by setting $\Lambda := \Lambda_0 \circ \tilde{\mu} : \tilde{\mathscr{S}}_{\phi^\sharp} \to S^M_0(\pi)$ (cf. ), the image of $\Lambda$ lies in $S^M(\pi)$ and the analogous diagram below is commutative $$\begin{gathered} \label{eqn:comm-diag-1} \xymatrix{ 1 \ar[r] & \tilde{Z}_{\phi^\sharp} \ar[d]_{\chi_M} \ar[r] & \tilde{\mathscr{S}}_{\phi^\sharp} \ar[d]^{\Lambda} \ar[r] & \mathscr{S}_{\phi^\sharp} \ar[d]^{\simeq} \ar[r] & 1 \\ 1 \ar[r] & \C^\times \ar[r] & S^M(\pi) \ar[r] & X^M(\pi) \ar[r] & 1 . }\end{gathered}$$ Consequently, there is a bijection $$\Pi(\tilde{\mathscr{S}}_{\phi^\sharp}, \chi_G) \rightiso \Pi_{\phi^\sharp},$$ which is canonical up to the $\Hom(X^G(\pi), \C^\times)$-action on $\Pi_{\phi^\sharp}$. Let $\tilde{s} \in \tilde{\mathscr{S}}_{\phi^\sharp}$, denote by $s$ its image in $\mathscr{S}_{\phi^\sharp}$ and set $s_0 := \mu(s) \in \mathscr{S}_{\phi^\sharp_0}$. Let $\eta \in X^M_0(\pi)$ be the character coming from $\Lambda(\tilde{s}) := \Lambda_0(\tilde{\mu}(\tilde{s})) \in S^M_0(\pi)$. Then by and , $\eta$ is the image of $s_0$ under $\mathscr{S}_{\phi^\sharp_0} \rightiso X^M_0(\pi)$; using , it is also the image of $s$ under $\mathscr{S}_{\phi^\sharp} \rightiso X^M(\pi)$. If we can show $\Lambda(\tilde{s}) \in S^M(\pi)$ for all $\tilde{s}$, then the rightmost square in will commute. Since the square $$\xymatrix{ S^M(\pi) \ar@{^{(}->}[d] \ar@{->>}[r] & X^M(\pi) \ar@{^{(}->}[d] \\ S^M_0(\pi) \ar@{->>}[r] & X^M_0(\pi) \\ }$$ is commutative and cartesian for trivial reasons, it follows that $\Lambda(\tilde{s}) \in S^M(\pi)$. Hence the image of $\Lambda$ lies in $S^M(\pi)$. This also finishes the commutativity of the rightmost square in . Consider the leftmost square in . It follows from that for all $z \in \tilde{Z}_{\phi^\sharp}$, we have $$\Lambda(z) = \Lambda_0(\tilde{\mu}(z)) = \chi_M(\tilde{\mu}(z)) = \chi_M(z).$$ Hence the leftmost square is commutative as well. The bijection $\Pi(\tilde{\mathscr{S}}_{\phi^\sharp}, \chi_G) \rightiso \Pi_{\phi^\sharp}$ follows easily from the previous assertions, as in the proof of Corollary \[prop:S-S\]. The conditions of the Theorem are satisfied if $M$ is a Levi subgroup of $\GL_D(n)$, say of the form $$M = \prod_{i \in I} \GL_D(n_i), \quad \sum_{i \in I} n_i = n$$ and $$M^\sharp_0 := M_\text{der} = \prod_{i \in I} \SL_D(n_i).$$ We simply set $\Pi_{\text{gen}}(M) := \Pi_{M^*-\text{gen}}(M)$ and $\Phi_{\text{gen}}(M) := \Phi_{M^*-\text{gen}}(M)$ by a straightforward generalization of the definitions in §\[sec:LLC\]. The correspondence $\pi \leftrightarrow \phi$ follows from that in Theorem \[prop:LLC-G\], applied to each index $i \in I$. The group $M^\sharp$ can be any intermediate group between $M^\sharp_0$ and $M$, including the important case that $$M^\sharp := M \cap \SL_D(n) = \left\{ (x_i)_{i \in I} \in M : \prod_{i \in I} \Nrd(x_i) = 1 \right\} .$$ Therefore, Theorem \[prop:Lambda\] and Corollary \[prop:S-S\] can be generalized to the Levi subgroups of $\SL_D(n)$. Indeed, it suffices to verify the commutativity of the diagram . Writing $\pi = \boxtimes_{i \in I} \pi_i$ and $\phi = (\phi_i)_{i \in I}$, the results in §\[sec:iden-S-groups\] applied to each $i \in I$ gives a commutative diagram similar to , except that its bottom row is the central extension $$\begin{gathered} \label{eqn:another-bottom-row} 1 \to (\C^\times)^I \to \prod_{i \in I} S^{\GL_D(n_i)}(\pi_i) \to X^G_0(\pi) \to 1, \end{gathered}$$ and that $\chi_M$ is replaced by $$\prod_{i \in I} \chi_{\GL_D(n_i)}: \tilde{Z}_{\phi^\sharp_0} \to (\C^\times)^I .$$ To obtain the desired short exact sequence, it remains to take the push-forward of by the multiplication map $(\C^\times)^I \to \C^\times$. The dual $R$-groups {#sec:dual-R} =================== A commutative diagram --------------------- As in §\[sec:groups\], we take $$\begin{aligned} G & = \GL_D(n), \\ G^* & = \GL_F(N), \\ G^\sharp & = \SL_D(n)\end{aligned}$$ We also fix a Levi subgroup $M$ of $G$ and set $M^\sharp := M \cap G^\sharp$. To define the dual groups $\Lgrp{G}$, $\Lgrp{M}$, etc., we fix a quasisplit inner twist $\psi: G \times_F \bar{F} \rightiso G^* \times_F \bar{F}$ which restricts to a quasisplit inner twist $M \times_F \bar{F} \rightiso M^* \times_F \bar{F}$, as well as an $F$-splitting for $G^*$ that is compatible with $M^*$. Therefore there is a canonical L-embedding $\Lgrp{M} \hookrightarrow \Lgrp{G}$. Idem for $\Lgrp{G^\sharp}$ and $\Lgrp{M^\sharp}$. As usual, the natural projections $\Lgrp{G} \to \Lgrp{G^\sharp}$ and $\Lgrp{M} \to \Lgrp{M^\sharp}$ are denoted by $\mathbf{pr}$. Put $$\begin{aligned} A_{\widehat{M^\sharp}} & := Z_{\widehat{M^\sharp}} = Z^{\Gamma_F, 0}_{\widehat{M^\sharp}} \hookrightarrow \widehat{G^\sharp}.\end{aligned}$$ Consider $\phi_M \in \Phi_{2,\text{bdd}}(M)$. Let $\phi$ be its composition with $\Lgrp{M} \hookrightarrow \Lgrp{G}$. Set $\phi^\sharp_M := \mathbf{pr} \circ \phi_M \in \Phi_{2,\text{bdd}}(M^\sharp)$ and $\phi^\sharp := \mathbf{pr} \circ \phi \in \Phi_{\text{bdd}}(G^\sharp)$. Every $\phi^\sharp \in \Phi_{\text{bdd}}(G^\sharp)$ is obtained in this way (recall Theorem \[prop:lifting-parameter\]). The construction of the dual $R$-group associated to $\phi^\sharp$, denoted by $R_{\phi^\sharp}$, is given as follows. Define $$\begin{aligned} N_{\phi^\sharp, \text{ad}} & := N_{S_{\phi^\sharp, \text{ad}}}(A_{\widehat{M^\sharp}}), \\ \mathfrak{N}_{\phi^\sharp} & := \pi_0(N_{\phi^\sharp, \text{ad}}, 1), \\ W_{\phi^\sharp} & := W(S_{\phi^\sharp, \text{ad}}, A_{\widehat{M^\sharp}}) \hookrightarrow W^{\hat{G}}(\hat{M}), \\ W^0_{\phi^\sharp} & := W(S^0_{\phi^\sharp, \text{ad}}, A_{\widehat{M^\sharp}}) \; \triangleleft W_{\phi^\sharp} , \\ R_{\phi^\sharp} & := W_{\phi^\sharp}/W^0_{\phi^\sharp}.\end{aligned}$$ The meaning of $W(\cdots, \cdots)$ is as follows: for any pair of complex groups $a \subset A$, the symbol $W(A,a)$ denotes the group $N_A(a)/Z_A(a)$. Note that $W^0_{\phi^\sharp}$ is the Weyl group associated to some root system, as $S^0_{\phi^\sharp, \text{ad}}$ is connected and reductive. Since the centralizer of $A_{\widehat{M^\sharp}}$ in the connected reductive group $S^0_{\phi^\sharp, \text{ad}}$ is connected, there exists a canonical injection $W^0_{\phi^\sharp} \hookrightarrow \mathfrak{N}_{\phi^\sharp}$. From the results recalled in §\[sec:res-L\], the torus $A_{\widehat{M^\sharp}}$ is a maximal torus in $S^0_{\phi^\sharp, \text{ad}}$. Using the conjugacy of maximal tori, one sees that the inclusion map $N_{\phi^\sharp, \text{ad}} \hookrightarrow S_{\phi^\sharp, \text{ad}}$ induces a canonical isomorphism $\mathfrak{N}_{\phi^\sharp}/W^0_{\phi^\sharp} \rightiso \mathscr{S}_{\phi^\sharp}$. On the other hand, we also have canonical injections $$\begin{gathered} \mathscr{S}_{\phi^\sharp_M} \hookrightarrow \mathscr{S}_{\phi^\sharp}, \\ \mathscr{S}_{\phi^\sharp_M} \hookrightarrow \mathfrak{N}_{\phi^\sharp}.\end{gathered}$$ The first one follows from the fact that $Z^{\Gamma_F}_{\widehat{M^\sharp}} = Z^{\Gamma_F}_{\widehat{G^\sharp}} Z^{\Gamma_F, 0}_{\widehat{M^\sharp}}$ (see [@Ar99 Lemma 1.1]). The injectivity of the second map follows; moreover, its image is characterized as the elements fixing $A_{\widehat{M^\sharp}}$ pointwise. The relations among these groups are recapitulated in the following result. \[prop:diagram\] The groups above fit into a commutative diagram $$\xymatrix{ & & 1 \ar[d] & \ar[d] 1 & \\ & & W^0_{\phi^\sharp} \ar@{=}[r] \ar[d] & W^0_{\phi^\sharp} \ar[d] & \\ 1 \ar[r] & \mathscr{S}_{\phi^\sharp_M} \ar[r] \ar@{=}[d] & \mathfrak{N}_{\phi^\sharp} \ar[r] \ar[d] & W_{\phi^\sharp} \ar[r] \ar[d] & 1 \\ 1 \ar[r] & \mathscr{S}_{\phi^\sharp_M} \ar[r] & \mathscr{S}_{\phi^\sharp} \ar[r] \ar[d] & R_{\phi^\sharp} \ar[r] \ar[d] & 1 \\ & & 1 & 1 & \\ }$$ whose rows and columns are exact. The arrow $\mathscr{S}_{\phi^\sharp} \to R_{\phi^\sharp}$ is uniquely determined by the other terms in this diagram. Cf. the proof of Lemma \[prop:X-W\] below. The same constructions can be applied to $\phi$ and $\phi_M$. The corresponding objects are denoted by $W_{\phi}$, $W^0_\phi$, etc. Upon identifying $W^{\hat{G}}(\hat{M})$ and $W^G(M)$, we can make $W_{\phi^\sharp}$ act on the tempered L-packet $\Pi_{\phi^\sharp}$. For any $\sigma^\sharp \in \Pi_{\phi^\sharp}$, define $$\begin{aligned} W_{\phi^\sharp, \sigma^\sharp} & := \text{Stab}_{W_{\phi^\sharp}}(\sigma^\sharp), \\ W^0_{\phi^\sharp, \sigma^\sharp} & := \text{Stab}_{W^0_{\phi^\sharp}}(\sigma^\sharp), \\ R_{\phi^\sharp, \sigma^\sharp} & := W_{\phi^\sharp, \sigma^\sharp}/W^0_{\phi^\sharp, \sigma^\sharp}.\end{aligned}$$ The last object $R_{\phi^\sharp, \sigma^\sharp}$, viewed as a subgroup of $R_{\phi^\sharp}$, is what we want to compare with the Knapp-Stein $R$-group $R_{\sigma^\sharp}$. Identification of $R$-groups {#sec:iden-R-groups} ---------------------------- Retain the notations of the previous subsection and fix a parabolic subgroup $P \in \mathcal{P}(M)$. We shall always make the identification $W^G(M)=W^{G^\sharp}(M^\sharp)$. The results in §\[sec:res-2\] are applicable to tempered representations of $M(F)$ by Theorem \[prop:irred\]. Henceforth, let $\sigma \in \Pi_{2,\text{temp}}(M)$ (resp. $\pi \in \Pi_{\text{temp}}(G)$) be the representations corresponding to $\phi_M$ (resp. $\phi$) by Theorem \[prop:LLC-G\]. Then we have $$\pi \simeq I^G_P(\sigma).$$ Recall that we have defined canonical isomorphisms $\mathscr{S}_{\phi^\sharp} \rightiso X^G(\pi)$ and $\mathscr{S}_{\phi^\sharp_M} \rightiso X^M(\sigma)$ in §§\[sec:iden-S-groups\]-\[sec:generalization\]. By inspecting the construction in Lemma \[prop:S-sequence\], we see that these two isomorphisms are compatible with the embeddings $\mathscr{S}_{\phi^\sharp_M} \hookrightarrow \mathscr{S}_{\phi}$ and $X^M(\sigma) \hookrightarrow X^G(\sigma)$. Therefore we obtain $\gamma: X^G(\pi)/X^M(\sigma) \rightiso \mathscr{S}_{\phi^\sharp}/\mathscr{S}_{\phi^\sharp_M}$. \[prop:X-W\] Define an isomorphism $${\hat{\Gamma}}^{-1}: X^G(\pi)/X^M(\sigma) \xrightarrow{\gamma} \mathscr{S}_{\phi^\sharp}/\mathscr{S}_{\phi^\sharp_M} \rightiso W_{\phi^\sharp}/W^0_{\phi^\sharp} =: R_{\phi^\sharp},$$ where the second arrow is given by Proposition \[prop:diagram\]. Then it is characterized by the equation $$\eta\sigma \simeq w\sigma$$ whenever $${\hat{\Gamma}}^{-1}(\eta \text{ mod } X^M(\sigma)) = w \text{ mod } W^0_{\phi^\sharp},$$ for all $\eta \in X^G(\pi)$ and $w \in W_{\phi^\sharp}$. As what the notation suggests, we set $\hat{\Gamma}$ to be the inverse of ${\hat{\Gamma}}^{-1}$. The equation $\eta\sigma \simeq w\sigma$ clearly characterizes ${\hat{\Gamma}}^{-1}$. Let $\eta \in X^G(\pi)$ and $\mathbf{a} \in H^1_{\text{cont}}(W_F, \hat{Z}^\sharp)$ which corresponds to $\eta$, together with a chosen $1$-cocycle $a$ in the cohomology class of $\mathbf{a}$. Since $L(\sigma)=X^G(\pi)$ by Proposition \[prop:L-X\^G\], there exists $w \in W^G(M)$ such that $\eta\sigma \simeq w\sigma$. On the dual side, it implies that there exists $t \in N_{\hat{G}}(\hat{M})$ representing $w$, such that $$t\phi t^{-1} = a\phi.$$ This implies that $t \text{ mod } Z_{\hat{G}}$ belongs to $S_{\phi^\sharp, \text{ad}}$, and its class $[t]$ in $\mathscr{S}_{\phi^\sharp}$ corresponds to $\eta$ (recall the construction in Lemma \[prop:S-sequence\]). On the other hand, we have $t \in N_{\phi^\sharp, \text{ad}}$ and its class $[\mathfrak{t}]$ in $\mathfrak{N}_{\phi^\sharp}$ is mapped to $[t]$ under the arrow $\mathfrak{N}_{\phi^\sharp} \twoheadrightarrow \mathscr{S}_{\phi^\sharp}$ in Proposition \[prop:diagram\]. One can also apply the arrow $\mathfrak{N}_{\phi^\sharp} \twoheadrightarrow W_{\phi^\sharp}$ to $[\mathfrak{t}]$; since $W_{\phi^\sharp}$ is identified as a subgroup of $W^G(M)$, the image is simply $w$. Upon some contemplation of the diagram in Proposition \[prop:diagram\], one can see that the image of $[t]$ under $\mathscr{S}_{\phi^\sharp} \to R_{\phi^\sharp}$ is just $w$ modulo $W^0_{\phi^\sharp}$. This completes the proof. Now recall that we have defined the group $\bar{W}_{\sigma} \subset W^G(M)$. In view of Lemma \[prop:barGamma\] and Proposition \[prop:L-X\^G\], we have a canonical isomorphism $$\Gamma: \bar{W}_\sigma/W_\sigma \rightiso X^G(\pi)/X^M(\sigma).$$ This is to be compared with $\hat{\Gamma}: W_{\phi^\sharp}/W^0_{\phi^\sharp} \rightiso X^G(\pi)/X^M(\sigma)$. \[prop:R-1\] We have 1. $\bar{W}_\sigma = W_{\phi^\sharp}$, 2. $W_\sigma = W^0_{\phi^\sharp}$, 3. $\Gamma = \hat{\Gamma}$. In particular, $R_{\phi^\sharp} \simeq X^G(\pi)/X^M(\sigma)$. The first assertion follows from the definition of $\bar{W}_\sigma$ and Theorem \[prop:lifting-parameter\]. Hence $\Gamma$ and $\hat{\Gamma}$ can be regarded as two surjective homomorphisms from $W_{\phi^\sharp}$ onto $X^G(\pi)/X^M(\sigma)$. However, they admit the same characterization (of the form $\eta\sigma \simeq w\sigma$) by Lemma \[prop:X-W\] and \[prop:barGamma\], hence are equal. This proves the remaining two assertions. \[prop:R-2\] For all $\sigma^\sharp \in \Pi_{\phi^\sharp}$, we have 1. $W_{\sigma^\sharp} = W_{\phi^\sharp, \sigma^\sharp}$, 2. $W^0_{\sigma^\sharp} = W^0_{\phi^\sharp, \sigma^\sharp}$, 3. the restriction of $\hat{\Gamma}$ to $W_{\phi^\sharp, \sigma^\sharp}/W^0_{\phi^\sharp, \sigma^\sharp}$ induces an isomorphism $$W_{\phi^\sharp, \sigma^\sharp}/W^0_{\phi^\sharp, \sigma^\sharp} \rightiso Z^M(\sigma)^\perp/X^M(\sigma).$$ In particular, $R_{\phi^\sharp, \sigma^\sharp} = R_{\sigma^\sharp}$, and the isomorphisms $\Gamma$, $\hat{\Gamma}$ from these $R$-groups onto $Z^M(\sigma)^\perp/X^M(\sigma)$ coincide (recall Proposition \[prop:Goldberg\] and Corollary \[prop:L-sharp\].) Remainder: the group $Z^M(\sigma)^\perp$ above is defined in . Our proof is based on the previous result. The first assertion follows immediately from the disjointness of tempered $L$-packets. By Lemma \[prop:W\^0-equality\] and the fact that $W_\sigma = W^0_\sigma$, we have $$\begin{aligned} W^0_{\phi^\sharp, \sigma^\sharp} & = W^0_{\phi^\sharp} \cap W_{\sigma^\sharp} = W_\sigma \cap W_{\phi^\sharp} \\ & = W^0_{\sigma^\sharp} \cap W_{\sigma^\sharp} = W^0_{\sigma^\sharp}. \end{aligned}$$ The second assertion follows. The third assertion is then immediate from Proposition \[prop:Z-perp\]. Note that the proof for the isomorphism $\hat{\Gamma}: R_{\phi^\sharp, \sigma^\sharp} \rightiso Z^M(\sigma)^\perp/X^M(\sigma)$ is independent of the Knapp-Stein theory. The behaviour of the local Langlands correspondence (Theorem \[prop:LLC-SL\]) for $G^\sharp$ and its Levi subgroups can now be summarized as follows. \[prop:main\] Let $G, G^\sharp$ and $P=MU$, $P^\sharp = M^\sharp U$ be as before. Let $\phi^\sharp_M \in \Phi_{\mathrm{bdd}}(M^\sharp)$ and let $\phi^\sharp \in \Phi_{\mathrm{bdd}}(G^\sharp)$ be the composition of $\phi^\sharp_M$ with $\Lgrp{M^\sharp} \hookrightarrow \Lgrp{G^\sharp}$. 1. For every $\rho \in \Pi(\tilde{\mathscr{S}}_{\phi^\sharp_M}, \chi_M)$ parametrizing an irreducible representation $\sigma^\sharp \in \Pi_{\phi^\sharp_M}$, then $I^{G^\sharp}_{P^\sharp}(\sigma^\sharp)$, regarded as a virtual character of $G^\sharp(F)$, corresponds to that of $\mathrm{Ind}^{\tilde{\mathscr{S}}_{\phi^\sharp}}_{\tilde{\mathscr{S}}_{\phi^\sharp_M}}(\rho)$. 2. For any $\sigma^\sharp$ as above, $I^{G^\sharp}_{P^\sharp}(\sigma^\sharp)$ is irreducible if and only if $Z^M(\sigma)^\perp = X^M(\sigma)$ for some (equivalently, for any) $\sigma \in \Pi_{\mathrm{temp}}(M)$ such that $\sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$. 3. If $\phi^\sharp_M \in \Phi_{2,\mathrm{bdd}}(M^\sharp)$, then we have natural isomorphisms $$\begin{gathered} R_{\phi^\sharp} \simeq X^G(\pi)/X^M(\sigma), \\ R_{\phi^\sharp, \sigma^\sharp} \simeq R_{\sigma^\sharp} \simeq Z^M(\sigma)^\perp/X^M(\sigma) \end{gathered}$$ where we set $\pi := I^G_P(\sigma) \in \Pi_{\mathrm{temp}}(G)$, for any choice of $\sigma \in \Pi_{2,\mathrm{temp}}(M)$ such that $\sigma^\sharp \hookrightarrow \sigma|_{M^\sharp}$. 4. For $\phi^\sharp_M$, $\sigma^\sharp$ and $\rho$ as above, the class $\mathbf{c}_{\sigma^\sharp} \in H^2(R_{\sigma^\sharp}, \C^\times)$ of corresponds to $\mathbf{c}^{-1}_\rho$, where $\mathbf{c}_\rho \in H^2(R_{\phi^\sharp, \sigma^\sharp}, \C^\times)$ is the obstruction for extending $\rho$ to a representation of the preimage in $\tilde{\mathscr{S}}_{\phi^\sharp}$ of $R_{\phi^\sharp, \sigma^\sharp}$ (see Definition \[def:obstruction\]). If $G^\sharp$ is quasisplit, then $Z^M(\sigma)^\perp = X^G(\pi)$ and $\tilde{R}_{\sigma^\sharp} \to R_{\sigma^\sharp}$ splits. As mentioned in the Introduction, this settles Arthur’s conjectures on $R$-groups for $G^\sharp$. The first part is nothing but a special case of Proposition \[prop:K\_0-diagram\]. The second part resuls then from the proof of Proposition \[prop:Z-perp\]; the independence of the choice of $\sigma$ is clear. The third part results from Proposition \[prop:R-1\] and \[prop:R-2\]. The fourth part is the combination of Proposition \[prop:cocycle\] and Theorem \[prop:generalization\]. Finally, $S^G(\pi)$ is commutative when $G^\sharp$ is quasisplit, as $\chi_G=1$. Hence we have $Z^M(\sigma)^\perp = X^G(\pi)$ and $\rho$ can always be extended in that case. The decomposition of $I^{G^\sharp}_{P^\sharp}(\sigma^\sharp)$ depends on $\phi^\sharp_M$, but not on the element $\sigma^\sharp$. This is not expected to hold for other groups. \[rem:nontempered\] We have limited ourselves to the tempered representations. However, if the local Langlands correspondence (Theorem \[prop:LLC-SL\]) and Theorem \[prop:Lambda\] can be extended to Arthur parameters $\psi^\sharp: \WD_F \times \SU(2) \to \Lgrp{G^\sharp}$ (see [@Ar89-unip §6]), then our results should be applicable to Arthur packets $\Pi_{\psi^\sharp}$ as well, except the part concerning the Knapp-Stein $R$-groups $R_{\sigma^\sharp}$. Note that the crucial lifting Theorem \[prop:lifting-parameter\] also holds for Arthur parameters: see [@Lab85 Remarque 8.2]. Examples {#sec:examples} -------- The next example on $R$-groups will be constructed using Steinberg representations, whose definitions are reviewed below. \[def:Steinberg\] For this moment, we assume $G$ to be any connected reductive $F$-group. Fix a minimal parabolic subgroup $P_0$ of $G$. The Steinberg representation $\text{St}_G$ of $G$ is the virtual character of $G(F)$ given by $$\text{St}_G := \sum_{\substack{P \supset P_0 \\ P = M U}} (-1)^{\dim \mathfrak{a}^G_M} \; I^G_P(\delta_P^{-\frac{1}{2}} \mathbbm{1}_M),$$ where the sum ranges over the parabolic subgroups $P$ containing $P_0$ and $\mathbbm{1}_M$ denotes the trivial representation of $M(F)$. The basic fact [@Ca73] is that $\text{St}_G$ comes from a smooth irreducible representation in $\Pi_{2, \text{temp}}(G)$, which we denote by the same symbol $\text{St}_G$. It is clearly independent of the choice of $P_0$. \[prop:St-irred\] For $G$ as in Definition \[def:Steinberg\] and a subgroup $G^\sharp$ satisfying $G_{\mathrm{der}} \subset G^\sharp \subset G$, we have $$\mathrm{St}_G|_{G^\sharp} \simeq \mathrm{St}_{G^\sharp}.$$ In particular, the group $X^G(\mathrm{St}_G)$ defined in §\[sec:res-rep\] is trivial. Recall the bijection $P \mapsto P^\sharp := P \cap G^\sharp$ between the parabolic subgroups of $G$ and $G^\sharp$. Since $(\mathbbm{1}_L)|_{L^\sharp} = \mathbbm{1}_{L^\sharp}$ for any Levi subgroup $L$ of $G$, the first isomorphism follows by comparing the formulas defining $\text{St}_G$ and $\text{St}_{G^\sharp}$, together with Lemma \[prop:res-induction\]. Hence the restriction of $\text{St}_G$ to $G^\sharp$ is irreducible. It follows from Theorem \[prop:S-decomp\] that $X^G(\text{St}_G) = \{1\}$. Let us revert to the setting $G = \GL_D(n)$ and $G^\sharp = \SL_D(n)$. \[ex:nonsplit\] We now set out to construct an example in which $\tilde{R}_{\sigma^\sharp} \to R_{\sigma^\sharp}$ does not split for every $\sigma^\sharp \hookrightarrow \sigma|_M$. First of all, there exists $\GL_D(m)$, for some choice of $D,m$, and a representation $\tau \in \Pi_{2,\text{temp}}(\GL_D(m))$ such that $S^{\GL_D(m)}(\sigma)$ is non-commutative. Indeed, for $m=1$ and $D$ equal to the quaternion algebra over $F$, Arthur exhibits in [@Ar06 p.215] an L-parameter $\phi_\tau \in \Phi_{2,\text{temp}}(D^\times)$ such that - $\tilde{\mathscr{S}}_{\phi^\sharp_\tau}$ is isomorphic to the quaternion group of order $8$; - $\tilde{Z}_{\phi_\tau}$ corresponds to $\{\pm 1\}$. In fact, $\phi_\tau$ factors through a homomorphism $\text{Gal}(K/F) \to \PGL(2,\C)$, where $K$ is a biquadratic extension of $F$, whose image is generated by the elements $$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \in \PGL(2, \C).$$ Since $\chi_{D^\times}$ is injective on $\tilde{Z}_{\phi_\tau}$ by . Theorem \[prop:Lambda\] entails that $S^{D^\times}(\tau)$ is non-commutative. Take $\eta, \omega \in X^{\GL_D(m)}(\tau)$ so that their preimages in $S^{\GL_D(m)}(\tau)$ do not commute. Let $c$ (resp. $d$) be the order of $\eta$ (resp. $\omega$). Put $\text{St} := \text{St}_{\GL_D(m)}$ and take $$\begin{aligned} M & := \GL_D(m) \times \prod_{\substack{1 \leq i \leq c \\ 1 \leq j \leq d}} \GL_D(m), \\ G & := \GL_D(m(cd+1)), \\ \sigma & := \tau \boxtimes \left( \bigboxtimes_{\substack{1 \leq i \leq c \\ 1 \leq j \leq d}} \eta^{i-1} \omega^{j-1} \text{St} \right), \\ \pi & := I^G_P(\sigma) \quad \text{ for some } P \in \mathcal{P}(M). \end{aligned}$$ Here $X^M(\sigma)$ is defined relatively to $M^\sharp := G^\sharp \cap M$ where $G^\sharp = \SL_D(m(cd+1))$. The presence of $\text{St}$ forces $X^M(\sigma)$ to be trivial, by Lemma \[prop:St-irred\]. Hence $\sigma^\sharp := \sigma|_{M^\sharp}$ is irreducible, and it is parametrized by the $1$-dimensional character $\chi_M: \tilde{Z}_{\phi^\sharp_M} \to \C^\times$. According to Theorem \[prop:main\], the central extension $\tilde{R}_{\sigma^\sharp} \twoheadrightarrow R_{\sigma^\sharp}$ splits if and only if $\rho$ can be extended to $\tilde{\mathscr{S}}_{\phi^\sharp}$. This is the case if and only if $S^G(\pi) \twoheadrightarrow X^G(\pi)$ splits, by Theorem \[prop:Lambda\]. Hence it suffices to show the non-commutativity of $S^G(\pi)$. Put $L := \GL_D(m) \times \GL_D(mcd) \in \mathcal{L}^G(M)$ and set $\nu := I^L_{P \cap L}(\sigma)$. We claim $\eta,\omega \in S^L(\nu)$. Indeed, the $\GL_D(m)$-component of $L$ does not cause any problem. As for the $\GL_D(mcd)$-component, take representatives $\tilde{w}_\eta$, $\tilde{w}_\eta$ in $\SL_D(mcd)$ of the cyclic permutations $$\begin{gathered} w_\eta: 1 \to \cdots \to c \to 1, \\ w_\omega: 1 \to \cdots \to d \to 1 \end{gathered}$$ of the indexes $i$ and $j$, respectively. Then the intertwining operators $J_\eta, J_\omega$ are given by the operators in using $\tilde{w}_\eta$, $\tilde{w}_\omega$. Furthermore, $J_\eta$ and $J_\omega$ commute with each other; this follows from and the obvious fact that $\tilde{w}_\eta$ and $\tilde{w}_\omega$ can be chosen to commute. From our choice of $\eta,\omega$, it follows that the preimages of $\eta,\omega$ in $S^L(\nu)$ do not commute. Since $S^L(\nu) \hookrightarrow S^G(\pi)$ by Proposition \[prop:S-embedding\], $S^G(\pi)$ is non-commutative, as required. \[ex:proper-incl\] Now we set out the show that the inclusion $R_{\phi^\sharp, \sigma^\sharp} \subset R_{\phi^\sharp}$ is proper in general. By Theorem \[prop:main\] and the notations therein, it amounts to show that $Z^M(\sigma)^\perp \subsetneq X^G(\pi)$ in general. As in the previous example, we take some $m \geq 1$, a central division $F$-algebra $D$ and $\tau \in \Pi_{2,\text{temp}}(\GL_D(m))$ such that $X^{\GL_D(m)}(\tau)$ contains $\eta, \omega$ with non-commuting preimages in $S^{\GL_D(m)}(\sigma)$. Take another $\tau' \in \Pi_{2,\text{temp}}(\GL_D(m))$ such that $X^{\GL_D(m)}(\tau') = \angles{\eta}$. Denote by $d$ the order of $\omega$. We take $$\begin{aligned} M & := \GL_D(m) \times \prod_{1 \leq j \leq d} \GL_D(m), \\ G & := \GL_D(m(d+1)), \\ \sigma & := \tau \boxtimes \left( \bigboxtimes_{1 \leq j \leq d} \omega^{j-1} \tau' \right), \\ \pi & := I^G_P(\sigma) \quad \text{ for some } P \in \mathcal{P}(M). \end{aligned}$$ Therefore $X^M(\sigma) = \angles{\eta}$ (defined relative to $M^\sharp = M \cap G^\sharp$ with $G^\sharp := \SL_D(m(d+1))$ as before), and $S^M(\sigma)$ is commutative. In particular $Z^M(\sigma)=X^M(\sigma)=\angles{\eta}$. On the other hand, a variant of the arguments in the previous example show that $\omega, \eta \in X^G(\pi)$ with non-commuting preimages in $S^G(\pi)$. Hence $\omega \in X^G(\pi)$ and $\omega \notin Z^M(\sigma)^\perp$, as required. Note that such $\tau$, $\tau'$ do exist when $D$ is the quaternion algebra over $F$ and $m=1$; in that case $\eta, \omega$ are identified with quadratic characters of $F^\times$. Indeed, a candidate of $\tau$ is given in the previous example. On the other hand, to construct $\tau'$ for a given $\eta$, we are reduced to construct $\tau'' \in \Pi_{2,\text{temp}}(\GL_F(2))$ with $X^{\GL_F(2)}(\tau'') = \{1, \eta\}$ and then take $\tau$ to be the Jacquet-Langlands transfer of $\tau''$. To finish the construction, let $E$ be the quadratic extension of $F$ determined by $\eta$ and $\theta: E^\times \to \C^\times$ be a continuous character. Set $\tau'' := \Ind_{E/F}(\theta)$ (the local automorphic induction, cf. [@JL70 Theorem 4.6]), then $\eta \tau'' = \tau''$. From [@LL79 pp.738-739], one sees that $\tau''$ is cuspidal and $|X^{\GL_F(2)}(\tau'')|=2$ for general $\theta$, which suffices to conclude. Kuok Fai Chao\ Institute of Mathematics,\ Academy of Mathematics and Systems Science,\ Chinese Academy of Sciences,\ 55, Zhongguancun East Road, 100190 Beijing\ People’s Republic of China.\ E-mail address: `kchao@amss.ac.cn` Wen-Wei Li\ Morningside Center of Mathematics,\ Academy of Mathematics and Systems Science,\ Chinese Academy of Sciences,\ 55, Zhongguancun East Road, 100190 Beijing\ People’s Republic of China.\ E-mail address: `wwli@math.ac.cn`

--- abstract: 'The 1–loop effective Lagrangian for a massive scalar field on an arbitrary causality violating spacetime is calculated using the methods of Euclidean quantum field theory in curved spacetime. Fields of spin ${1\over2}$, spin 1 and twisted field configurations are also considered. In general, we find that the Lagrangian diverges to minus infinity at each of the $n$th polarised hypersurfaces of the spacetime with a structure governed by a DeWitt–Schwinger type expansion.' author: - | Michael J. Cassidy\ Department of Applied Mathematics and Theoretical Physics\ University of Cambridge, Silver St., Cambridge CB3 9EW, England date: | \ DAMTP/R-97/24 title: Divergences in the Effective Action for Acausal Spacetimes --- =6.2 truein =8 truein Introduction {#intro} ============ If one attempts to quantise fields on acausal spacetimes, one inevitably runs into awkward problems of interpretation. Quantities that are well defined in globally hyperbolic spacetimes can become ambiguous in geometries where strong causality is violated. For example, the first attempt to construct an interacting quantum field theory on a Morris, Thorne, Yurtsever [@thorne] type wormhole spacetime found that the $S$ matrix was nonunitary when the state evolved through the region of closed timelike curves (CTCs) [@fps]. One also encounters problems with the definition of a suitable Green function. Consider the (normally well defined) commutator of two free field operators $iD(x,y) = [\phi(x),\phi(y)]$. In an acausal spacetime, even if $x$ and $y$ are locally spacelike separated, it is not clear that $D=0$ because there may be a large timelike loop connecting the two points, due to the nontrivial homotopy of the spacetime. Various attempts to circumvent these problems in a consistent way have been suggested [@2; @3; @4; @5; @6; @7]. This paper is concerned with the Euclidean approach, proposed in a recent paper by Hawking [@2]. Motivation for this proposal comes from the simple observation that in Euclidean space, there are no CTCs and in particular, no closed or self–intersecting null geodesics. Therefore, if one considers a Euclidean space $M_E$ which has some acausal Lorentzian section $M_L$, then the appropriate analytic continuation of quantities that are well defined on $M_E$ should give unambiguous results valid on the chronology violating section. The object of this paper is to apply the methods of Euclidean quantum field theory in curved spacetime to derive a 1–loop effective action for fields of arbitrary mass and spin in a typical causality violating spacetime. Accordingly, section \[geom\] is devoted to a discussion of multiply connected Euclidean spaces and their universal covering spaces. Section \[heat\] reviews all the necessary theory of the heat operator, most notably its divergence structure and asymptotic expansion for multiply connected spacetimes. This expansion is then used in section \[act\] to derive the 1–loop effective Lagrangian for a massive scalar field, renormalised by the point splitting method. The corresponding expressions for twisted configurations and fields of spin ${1\over2}$ and spin 1 are also obtained. In section \[egs\], these results are applied to a number of interesting chronology violating spacetimes, including Grant’s generalisation of Misner space [@grant] and the wormhole spacetime studied by Kim and Thorne [@kth]. The relevance of these results for chronology protection [@cpc] is discussed in section \[conc\]. Multiply connected Euclidean spaces {#geom} =================================== Consider an arbitrary multiply connected Euclidean spacetime, $M_E$. This spacetime is just the quotient space $$M_E = {{\overline M}_E\over\Gamma}$$ where ${\overline M}_E$ is the simply connected universal covering space and $\Gamma$ is a properly discontinuous, discrete group of isometries of ${\overline M}_E$. $\Gamma$ is isomorphic to the fundamental group of $M_E$, $\pi_1(M_E)$ and $M_E$ is obtained from ${\overline M}_E$ by identifying points equivalent under $\Gamma$. If $\pi_1(M_E)=Z_\infty$, then the fundamental domains ([*i.e.*]{} the copies of $M_E$ in ${\overline M}_E$) can be labelled by a single integer $n$, usually interpreted as a winding number. Copies of the point $p\in M_E$ in the covering space are labelled ${\overline p}_n\in {\overline M}_E$, where the points ${\overline p}_n$ are obtained by repeated application of $\Gamma$ to the right, [*i.e.*]{} ${\overline p}_n={\overline p}_0 \gamma_n$. It will be useful to have a concrete example to refer to throughout the paper. Therefore, we shall consider the Euclidean section of Grant space [@grant]. Grant space is just flat Minkowski space with points identified under a combined boost in the $(x,t)$ plane and translation in the $y$ direction. It is the universal covering space of the Gott spacetime [@gott], which describes two cosmic strings passing each other with a constant velocity. The appropriate Euclidean section is flat Euclidean space with points identified under a combined rotation plus a translation in the orthogonal direction, as before. In other words, for an arbitrary point ${\overline q}=(\tau, r,\theta,z)$, the effect of acting on ${\overline q}$ by $\gamma_n$ is just $${\overline q}_n={\overline q}\gamma_n=(\tau+n\beta, r, \theta + n\alpha,z)\space.$$ One recovers the Lorentzian Grant space by analytically continuing the rotation parameter to a boost ($\alpha\rightarrow a=i\alpha$). A fundamental quantity of interest is $\sigma_n(p,\lbrace\beta_E\rbrace)=\sigma({\overline p}, {\overline p}_n)$, which gives the squared distance along the spacelike geodesic connecting $p\in M_E$ to itself with winding number $n$. $\lbrace\beta_E\rbrace$ collectively denotes various metric parameters, which relate equivalent points in the covering space. On the Euclidean section of Grant space, one would have $$\sigma({\overline q},{\overline q}_n{}')= (\tau-\tau'-n\beta)^2 + r^2 + r'^2 - 2rr'\cos(\theta -\theta'-n\alpha) + (z-z')^2$$ so that $$\label{mis} \sigma_n(q,\lbrace\alpha,\beta\rbrace)= 2r^2\biggl(1-\cos(n\alpha)\biggr) + n^2 \beta^2\space.$$ On the Euclidean section $M_E$, provided the parameters $\lbrace\beta_E\rbrace$ are nonzero, the equation $$\label{hyp} \sigma_n(p,\lbrace\beta_E\rbrace)=0$$ can (in general) only be satisfied if $n=0$. However, if one considers analytically continuing any of the metric parameters to imaginary values, thus obtaining a Lorentzian spacetime $M_L$ with parameters $\lbrace\beta_L\rbrace$, then one may be able to find solutions to the equivalent of (\[hyp\]) for all values of $n$. In that case, the point $p$ would be joined to itself by a (self–intersecting) null geodesic with winding number $n$. One can define the $n$th polarised hypersurface as the set of points $\lbrace p\in M_L: \sigma_n(p,\lbrace\beta_L\rbrace)=0\rbrace$. The Cauchy horizon is just the limit of this family of surfaces as $n\rightarrow\infty$. One may easily verify that by setting equation (\[mis\]) equal to zero and then analytically continuing $\alpha\rightarrow a=i\alpha$, one obtains the criterion for polarised hypersurfaces in Grant space (see [@grant]). The Heat Operator {#heat} ================= As a first step towards obtaining the effective Lagrangian, we need to examine the structure of the heat operator defined on $M_E$. Heat operators on Riemannian manifolds have been extensively studied, so here we only review the most relevant properties. For further technical details, the reader is referred to the paper by Wald [@wald] and its associated references. Quantum field theory on multiply connected spacetimes is discussed in Dowker [@dowk1] and Dowker and Banach [@dowk2]. We begin by considering the ‘wave operator’ ${\cal{A}} =-\nabla^2 + m^2$, defined on the dense domain $C_0^\infty(M_E)$ of smooth ($C^\infty$) functions of compact support. $\cal{A}$ is a symmetric operator on $L^2(M_E)$, the Hilbert space of square integrable functions on $M_E$. However, to do quantum theory we need a self–adjoint operator so we must extend this domain of definition in an appropriate way. Since $\cal{A}$ is positive on its initial domain, standard theory states that positive self–adjoint extensions must exist. The only problem is that $\cal{A}$ is unbounded, so there may be more than one possible extension. However, if $M_E$ is a complete manifold without boundary, then Gaffney [@gaff] has shown that $\cal{A}$ has a unique self–adjoint extension, defined as the closure of $\cal{A}$ and denoted by $A$. The domain of $A$ is just the Cauchy completion of dom($\cal{A}$) in the norm $\parallel\psi\parallel^2 + \parallel {\cal A}\psi\parallel^2$, for $\psi\in L^2(M_E)$. This property of the wave operator is known as essential self–adjointness and also holds for some incomplete manifolds, such as Euclidean space with a point removed. It does not hold for manifolds with boundaries or most manifolds with singularities. In this paper, we shall always assume that $\cal{A}$ is essentially self–adjoint. The heat operator is defined as $$e^{-\tau A} = \int e^{-\tau\lambda} dE_\lambda$$ where $E_\lambda$ is the spectral family of $A$. Once $e^{-\tau A}$ has been constructed, one can apply the functional calculus of self–adjoint operators [@reed] to obtain mathematically well–defined expressions for quantities of physical interest. If one considers the 1–parameter family of integrals $$\label{gen} K(s)=\int^\infty_0e^{-\tau A} \tau^{s-1} d\tau,$$ then $K(1)$ and $K(0)$ are particularly interesting. $K(1)=A^{-1}$ defines the Feynman propagator and $K(0)$ is related to $\ln A$ (the effective Lagrangian) by $$\label{efflag} \ln A=\lim_{\epsilon\rightarrow 0}\left( -\int^\infty_\epsilon e^{-\tau A} {d\tau\over\tau} + (\gamma -\ln\epsilon)I\right)$$ where $I$ is the identity operator and Euler’s constant $\gamma = \int_0^\infty e^{-x}\ln x dx$. It is well known that for $\tau>0$, the heat operator is given by a smooth integral kernel $H(\tau,x,x')$. Thus the only possible divergences in (\[gen\]) which could prevent $K(s)$ from being given by a smooth integral kernel $K(s,x,x')$ are those which could arise as $\tau\rightarrow\infty$ (infra–red divergences) or $\tau\rightarrow 0$ (ultra–violet). Infra–red divergences can only occur if the field mass $m=0$ and $M_E$ is noncompact. Since we are considering the massive scalar field, we shall not worry about these divergences. We shall be more concerned with the ultra–violet divergence structure, which is completely determined by the asymptotic expansion of $H(\tau,x,x')$ about $\tau=0$ $$\label{expan} H(\tau,x,x')= {\Delta^{1\over2}(x,x')\over (4\pi\tau)^{d\over2}} e^{-m^2\tau} e^{-{\sigma(x,x')\over4\tau}} \sum_{j=0}^N a_j(x,x') \tau^j$$ The coefficients $a_j(x,x')$ are recursively obtained and depend on local geometric quantities. $\sigma(x,x')$ was defined earlier as the square of the geodesic distance between $x$ and $x'$ and $d={\rm dim}(M)$. $\Delta(x,x')=-{\rm det}(-\sigma_{;\mu\nu'})$ is the Van–Vleck determinant. One can see that if $x\ne x'$, the factor $e^{-{\sigma(x,x')\over4\tau}}$ ensures that $H$ vanishes as $\tau\rightarrow 0$ faster than any power of $\tau$. Therefore, provided there are no infra–red divergences, $K(s)$ is given by an integral kernel $K(s,x,x')$ which can only be singular when $x=x'$. In a multiply connected spacetime, one can express $H(\tau,x,x')$ in terms of ${\overline H}$, the heat kernel defined on the universal covering space. The most general relation is $$H(\tau,x,x')=\sum_\gamma a(\gamma){\overline H}(\tau, x, x'\gamma)$$ where $a(\gamma)$ is a unitary, 1–dimensional representation of $\Gamma$ ([*i.e.*]{} $a(\gamma_1)a(\gamma_2)=a(\gamma_2 \gamma_1)$). Note that from now on, points in the covering space have no bars on them as the distinction between $x\in M_E$ and $x\in {\overline M}_E$ should be clear. If $\Gamma=\pi_1(M_E)=Z_\infty$, one can write $$H(\tau,x,x')=\sum_{n=-\infty}^\infty a(\gamma_n) {\overline H}(\tau, x, x_0'\gamma_n)$$ where $a(\gamma_n)=e^{2\pi in\delta}$ and $0\le\delta\le{1\over2}$. In general, for real fields $a(\gamma_n)$ must be real also, so that $a(\gamma_n)$ can only take the values $\pm 1$, where the negative value would correspond to a twisted field configuration [@isham; @dewish]. For the moment, we take $a(\gamma_n)=1$ for simplicity, and note that this decomposition can be suitably extended to the family of integrals $K(s)$, so that $$K(s,x,x')=\sum_{n=-\infty}^\infty {\overline K}(s, x,x_0'\gamma_n)$$ The Effective Action {#act} ==================== The effective action of the quantum field, $S$, is related to the operator $A$ by $$e^{-S}=({\rm det}A)^{-{1\over2}}=e^{-{1\over2}{\rm tr}(\ln A)}.$$ One would like to represent $\ln A$ by an integral kernel $L(x,x')$, so that $$S={1\over2}\int L(x,x) g^{1\over2} d^4x\space.$$ One could then obtain the energy–momentum tensor by functionally differentiating the effective Lagrangian ${\cal L}(x)={1\over2}L(x,x)$ with respect to the metric $g_{\mu\nu}$. However, from the above discussion of ultra–violet divergences in $K(s,x,x')$, it is clear that $\ln A$ will be singular at $x=x'$. We must therefore adopt some renormalisation prescription. In the point–splitting approach [@chris1; @chris2], one first considers the quantity $$L(x,x')=-\int_0^\infty H(\tau,x,x') {d\tau\over\tau}$$ which is well defined for $x\ne x'$. In 4 dimensions, the divergences which occur as the limit $x'\rightarrow x$ is taken are governed entirely by the first 3 terms of the asymptotic expansion (\[expan\]). We therefore obtain a finite, renormalised $L(x,x')$ by subtracting this divergent part from $L(x,x')$ before taking the coincidence limit. For a multiply connected spacetime $M_E$, we have $$L(x,x')=-\sum_{n=-\infty}^\infty\int_0^\infty {\overline H}(\tau,x,x_n'){d\tau\over\tau}.$$ The $\tau$ integration is performed with the help of the definite integral [@grad] $$\int_0^\infty x^{\nu -1} e^{-{\beta\over x} -\gamma x}dx = 2\left(\beta\over\gamma\right)^{\nu\over2} K_\nu\left(2\sqrt{\beta\gamma}\right)$$ where $K_\nu$ is the modified Hankel function [@grad]. The contribution to $L(x,x')$ from the first 3 terms in the series is $$\begin{aligned} L(x,x')=&-&{1\over(4\pi)^2} \sum_{n=-\infty}^\infty \Delta^{1\over2}\left(x,x_n'\right)\Biggl[ 8 \left(m^2\over\sigma(x, x_n')\right)K_2\left(m\sqrt{\sigma ( x, x_n')}\right)\Biggr.\nonumber \\ &+& \Biggl. 4a_{1}(x,x'_n)\left(m^2\over\sigma_n\right)^{1\over2} K_1\left(m\sqrt{\sigma_n}\right) + a_{2n}\int_0^\mu e^{-m^2\tau-{\sigma_n\over4\tau}}{d\tau\over\tau} + O(\sigma_n) \Biggr]\end{aligned}$$ where the cutoff $\mu$ is to prevent infra–red divergences in the massless limit. The first coefficient $a_0=1$ for scalar fields. Recalling the discussion in section \[geom\], the only value of $n$ for which $\sigma( x,x_n)\rightarrow0$ as $x'\rightarrow x$ is $n=0$. To renormalise therefore, one drops the $n=0$ term in $L(x,x')$ and takes the coincidence limit to obtain $$\begin{aligned} \label{final} -2{\cal L}(x)&=&-L(x,x)= {1\over(4\pi)^2} \sum_{\scriptstyle n=-\infty\atop\scriptstyle n\ne0}^\infty \Delta^{1\over2}(x,x_n)\Biggl[ 8\left(m^2\over\sigma_n\right)K_2\left(m\sqrt{\sigma_n}\right)\Biggr. \nonumber \\ &+&\Biggl. 4a_{1n}\left(m^2\over\sigma_n\right)^{1\over2} K_1\left(m\sqrt{\sigma_n}\right) +a_{2n}\int_0^\mu e^{-m^2\tau-{\sigma_n\over4\tau}}{d\tau\over\tau} + O(\sigma_n). \Biggr]\end{aligned}$$ It is now clear what happens if one can analytically continue any of the metric parameters to obtain an acausal Lorentzian section. The quantity $\sigma( x, x_n)$ goes to zero at each of the $n$th polarised hypersurfaces and hence the renormalised effective Lagrangian diverges. The first point to note is that equation (\[final\]) should be understood as a formal expression only. For the purposes of practical calculation, the first term gives the Gaussian approximation [@bek] to the effective action which is only exact in special cases, namely flat space and the Einstein universe. In a more general spacetime, one has to consider the coefficients $a_i$ for $i>0$. The covariant expansion of these coefficients in terms of the geodetic interval can be found in the papers by Christensen [@chris1; @chris2]. Consider the expansion of the first nontrivial coefficient $a_1$ for a scalar field $$a_1(x,x')= \left({1\over6}-\xi\right)R - {1\over2}\left({1\over6}-\xi\right)R_{;\alpha}\sigma^{\alpha} + \biggl(\dots\biggr)_{\alpha\beta}\sigma^{\alpha}\sigma^{\beta} +\dots$$ where $\sigma_\alpha = \partial_\alpha \sigma$. Normally, one would have $\sigma^{\alpha}\rightarrow 0$ as the coincidence limit is taken, which leaves a simple finite expression for the coefficient of interest. However, if identifications have been made, the quantity $\sigma^{\alpha}$ does not necessarily go to zero as the points are brought together. Hence we are left with an infinite number of terms which may or may not converge. However, for most purposes one would only be interested in the strongest divergence, which is given by the Gaussian approximation $$-{\cal L}(x)={a_0\over2\pi^2}\sum_{n=1}^\infty \left(m^2\over\sigma_n\right) \Delta_n{}^{1\over2}K_2\Bigl(m\sqrt{\sigma_n}\Bigr)\space.$$ A twisted real scalar field configuration can be considered by including a factor of $(-1)^n$ in the Lagrangian. In this case, the contributions from twisted and untwisted fields cancel at odd numbered polarised hypersurfaces, but reinforce at even numbered ones. For higher spin fields, the only factor which changes in this expression is the coefficient $a_0$. For spin ${1\over2}$, $a_0$ is given by the unit spinor, whose trace is just the number of spinor components ([*i.e.*]{} the dimension of the gamma matrices used). For spin 1 fields, $a_0$ has four components and is just the metric tensor $g_{\mu\nu}$ (in the Feynman gauge). The ghost Lagrangian is given by minus twice the scalar Lagrangian, because one would have to consider two anticommuting scalar ghost fields. Here, the ghost contribution would cancel with two of the vector field components so overall, the spin ${1\over2}$ and spin 1 Lagrangians would still diverge to minus infinity at the polarised hypersurfaces. Examples {#egs} ======== In flat space, we can obtain an exact result. The Van–Vleck determinant $\Delta(x,x')=1$ and the only nonzero coefficient is $a_0=1$, so for Euclidean space identified under a combined rotation and orthogonal translation, one obtains $$\label{masgr} -{\cal L}(x)={1\over2\pi^2}\sum_{n=1}^\infty \left(m^2\over\sigma_n\right)K_2\Bigl(m\sqrt{\sigma_n}\Bigr)$$ which in the massless limit becomes $$\label{malgr} -{\cal L}(x)= {1\over\pi^2}\sum_{n=1}^\infty {1\over\left(2r^2\Bigl(1-\cos(n\alpha)\Bigr) + n^2\beta^2\right)^2}.$$ Analytically continuing $\alpha\rightarrow a=i\alpha$ yields the Grant space result which as stated above, diverges at each of its polarised hypersurfaces. The Gaussian approximation is also known to be exact in the Einstein universe, which has topology $R\times S^3$. If one identifies points on the Euclidean section under a combined rotation plus translation, then the effective Lagrangian can be calculated as before. In this case, however, one must also sum over contributions from geodesics which loop around the three–sphere more than once, so one has to sum over two winding numbers $n$ and $m$. If the metric is written as $$ds^2= d\tau^2 + r^2\Bigl( d\chi^2 + \sin^2\chi\left(d\theta^2 + \sin^2\theta d\phi^2\right)\Bigr)$$ and the points $(\tau,\chi,\theta,\phi)$ and $(\tau+m\beta,\chi,\theta,\phi+m\alpha)$ are identified, then the geodetic interval is given by $$\sigma_{nm}(x,x')= (\tau-\tau'-m\beta)^2 + (s_m + 2\pi nr)^2$$ where $$\cos\left(s_m\over r\right) = \cos\chi\cos\chi'+ \sin\chi\sin\chi' (\cos\theta\cos\theta' + \sin\theta\sin\theta'\cos(\phi-\phi'-m\alpha))\space.$$ One therefore obtains $$-L(x,x')= {1\over2\pi^2} \sum_{\scriptstyle m=-\infty\atop\scriptstyle m\ne0}^\infty \sum_{n=-\infty}^\infty {{s_m\over r}+2\pi n \over\sin\left(s_m\over r\right)}{1\over\Bigl((\tau-\tau'-m\beta)^2 + (s_m + 2\pi nr)^2\Bigr)^2}$$ for the integral kernel in the massless limit, where the factor $\left({s_m\over r} + 2\pi n\right)/\sin\left(s_m\over r\right)$ is the Van–Vleck determinant for this spacetime. This expression can be written in an alternative form by combining terms of positive and negative $n$. $${1\over2\pi^2 r} \sum_{\scriptstyle m=-\infty\atop\scriptstyle m\ne0}^\infty \sum_{n=-\infty}^\infty {s_m\over\sin\left(s_m\over r\right)}{y^4 + 2y^2(x+n)(x-n) + (x^2 + 3n^2)(x+n)(x-n) \over16\pi^4 r^4(n+z_1)^2(n-z_1)^2(n+z_1^*)^2(n-z_1^*)^2}$$ where the complex quantity $$z_1=x+iy= {s_m + i(\tau-\tau'-m\beta)\over2\pi r}$$ The sum over $n$ can be evaluated using the method of residues to obtain finally $$-L(x,x')={1\over4\pi^2 r^4}\sum_{\scriptstyle m=-\infty\atop\scriptstyle m\ne0}^\infty {\left(\tau-\tau'-m\beta\over r\right)^{-1}\sinh\left(\tau-\tau'-m\beta\over r\right)\over\left(\cosh\left(\tau-\tau'-m\beta\over r\right)- \cos\left(s_m\over r\right)\right)^2}$$ If one analytically continues the parameter $\alpha\rightarrow a=i\alpha$ in this case, the spacetime that one obtains is the product of three dimensional de Sitter space and the real line, periodically identified under a combined boost and translation. The condition for polarised hypersurfaces in this spacetime is given by $$\cosh\left(m\beta\over r\right) -1 + \sin^2\chi\sin^2\theta\Bigl(1-\cosh(ma)\Bigr)=0 \space.$$ This criterion and the Lagrangian both reduce to the Grant space expressions in the limit as $r\rightarrow\infty$, if one defines a new radial coordinate by $r'=r\sin\chi\sin\theta$. One can also try to calculate the effective Lagrangian for the Anti–de Sitter analogue of Grant space. The Gaussian approximation is exact for conformally invariant fields in this case also, due to the fact that Anti–de Sitter space can be conformally mapped into half of the Einstein static universe. The Euclidean section of Anti–de Sitter space can be realised as the 4–dimensional hyperboloid $$-\left(\omega^0\right)^2 +\left(\omega^1\right)^2 + \left(\omega^2\right)^2 + \left(\omega^3\right)^2 + \left(\omega^4\right)^2=r^2$$ in the 5–dimensional space with metric $$ds^2=-\left(d\omega^0\right)^2+\left(d\omega^1\right)^2 + \left(d\omega^2\right)^2 + \left(d\omega^3\right)^2 + \left(d\omega^4\right)^2.$$ If one defines $$\begin{aligned} \omega^0&=&{1\over r}\cosh\tau\sec\rho \nonumber \\ \omega^1&=&{1\over r}\tan\rho\cos\theta \nonumber \\ \omega^2&=&{1\over r}\tan\rho\sin\theta\cos\phi \nonumber \\ \omega^3&=&{1\over r}\tan\rho\sin\theta\sin\phi \nonumber \\ \omega^4&=&{1\over r}\sinh\tau\sec\rho\end{aligned}$$ then the metric takes the form $$ds^2={\sec^2\rho\over r^2}\left(d\tau^2 + d\rho^2 + \sin^2\rho\Bigl(d\theta^2 + \sin^2\theta d\phi^2\Bigr)\right).$$ Once again, we identify the points $(\tau,\rho,\theta,\phi)$ and $(\tau+n\beta,\rho,\theta,\phi+n\alpha)$. In Anti–de Sitter space, the chief problem encountered when trying to construct quantum field theoretic quantities comes from the fact that information can be lost to, or gained from, spatial infinity in a finite coordinate time. Appropriate boundary conditions need to be imposed at infinity, so that the field (or its gradient) vanishes there [@ads]. If one thinks of the Einstein universe as a cylinder, then Anti–de Sitter spatial infinity is the timelike surface at $\chi={\pi\over2}$ obtained by slicing the cylinder with a vertical plane wave. Thus, the Anti–de Sitter Lagrangian is obtained from the Einstein expression by adding in the image charge at the antipodal point and inserting the appropriate conformal weighting factor, to obtain $$\begin{aligned} -L(x,x')&=&{\cos^2\rho\cos^2\rho'\over4\pi^2}\sum_{\scriptstyle m=- \infty\atop\scriptstyle m\ne0}^\infty\left[ {\left(\tau-\tau'-m\beta\over r\right)^{-1} \sinh\left(\tau-\tau'-m\beta\over r\right)\over\left(\cosh\left(\tau-\tau'-m\beta\over r\right)-\cos\left(s_m\over r\right)\right)^2}\right. \nonumber \\ &\pm&\Biggl. (\pi-\rho', \pi-\theta', \pi+\phi')\hbox{ image charge}\Biggr]\space,\end{aligned}$$ where the upper (lower) sign refers to Dirichlet (Neumann) boundary conditions. As a final example, we consider a massless scalar field in the wormhole spacetime originally studied by Kim and Thorne [@kth], who calculated the (divergent) behaviour of its renormalised energy–momentum tensor. One constructs this spacetime by removing two 3–spheres of radius $b$ from Minkowski space and identifying the resulting world tubes which form when one sets the right hand mouth moving towards the left with speed $\beta$. Initially the two mouths are separated by a shortest distance $D$. Kim and Thorne have calculated the Van–Vleck determinant and geodetic interval for this spacetime. Combining their results with our expression, one immediately obtains $$-{\cal L}(x)={1\over\pi^2}\sum_{n=1}^\infty {1\over D}\left(b\over2D\right)^{n-1} {\xi^{2n}\left(1-\xi\right)\over 1-\xi^n}\left({1\over\lambda(x,x')}+{1\over\lambda(x',x)}\right)^2\space,$$ for the Lagrangian, where $\xi=\left({1-\beta\over 1+\beta}\right)^{1\over2}$ is the inverse Doppler blueshift suffered by a ray passing along the $X$ axis and through the wormhole, and the quantity $\lambda$ is defined by $$\lambda(x,x')=2\left(b-\sqrt{b^2-\rho^2}\right) +X-T-(X'-T')\xi^n$$ where a point $x$ has coordinates $(T,X,Y,Z)$ and $\rho=\sqrt{Y^2+Z^2}$ measures the transverse shift of $x$ from the axis of symmetry. Discussion {#conc} ========== One cannot dispute the fact that in many causality violating spacetimes, the renormalised expectation value $\langle T_{\mu\nu}\rangle$ diverges as the Cauchy horizon is approached. Indeed, the original chronology protection conjecture [@cpc] was motivated heavily by this fact, and it was therefore proposed that the back reaction induced by this divergent energy–momentum would distort the spacetime geometry sufficiently to prevent the formation of CTCs. Recently, however, examples have been presented which lead one to question the universal validity of this basic mechanism and it is now known that $\langle T_{\mu\nu}\rangle$ does not necessarily diverge for all initial quantum states as the Cauchy horizon is approached. Sushkov, who considered automorphic fields on four dimensional Misner space [@sush], gave an example of a Hadamard state for which $\langle T_{\mu\nu}\rangle$ vanishes everywhere on the initially globally hyperbolic region (see also Krasnikov [@kras]). Actually, one does not even need to consider automorphic fields, as one can readily find a simple counterexample from inspecting the closed form of the scalar field energy–momentum tensor on Misner space, obtained by Euclidean methods. Recall that Misner space is just Minkowski space with points identified under a boost in the $x$ direction. The appropriate Euclidean section of this spacetime, therefore, is flat Euclidean space identified under a rotation, $\alpha$. This space also happens to be the analytic continuation of the Lorentzian spacetime produced by an infinitely long cosmic string. The energy–momentum tensor for a massless conformally coupled scalar field in the cosmic string spacetime is well known, and is given on the Euclidean section (in ($\tau,r,\theta,z$) coordinates) by $$\langle T^\mu{}_\nu\rangle={1\over1440\pi^2 r^4}\left(\left(2\pi\over\alpha\right)^4 -1\right) {\rm diag}\Bigl(1,1,-3,1\Bigr)\space.$$ Clearly, if one analytically continues the parameter $\alpha\rightarrow a=i\alpha$ in this case, then the energy–momentum tensor vanishes everywhere if $a=2\pi$, so there will be no divergence in this case. $\langle T_{\mu\nu}\rangle$ has also been shown to be bounded at the Cauchy horizon for (sufficiently) massive fields in Gott space [@boul] and Grant space [@tanhis]. Cramer and Kay [@ckay] have replied to all of these examples by demonstrating that even though there is no divergence, $\langle T_{\mu\nu}\rangle$ must always be ill defined on the Cauchy horizon itself. However, one is still left with the feeling that $\langle T_{\mu\nu}\rangle$ does not quite tell the whole story. In this paper, we have offered a new viewpoint by focusing on the effective Lagrangian and a general expression for the leading order divergence at the polarised hypersurfaces of a typical causality violating spacetime has been obtained. Immediately one can apply this result to the examples outlined in the preceding paragraph. In four dimensional Misner space, a quick inspection of (\[malgr\]) with the parameter $\beta=0$ shows that even though $\langle T_{\mu\nu}\rangle$ can remain finite, ${\cal L}$ always diverges to minus infinity at the Cauchy horizon $r=0$. Similarly, (\[masgr\]) implies that ${\cal L}$ diverges at the Cauchy horizon in Grant space (and therefore Gott space), even though $\langle T_{\mu\nu}\rangle$ can remain finite for massive fields at the Cauchy horizon. Finally, consider the behaviour of a Euclidean path integral of the form $$\Psi=\int {\cal D}[g]{\cal D}[\phi] e^{-S[g,\phi]}\space,$$ where $S$ is obtained from a Lagrangian appropriate for some causality violating spacetime. If the metric parameters are adjusted so as to introduce CTCs into the spacetime, then we have already shown that the action diverges to minus infinity. If one now interprets this path integral according to the no–boundary proposal, then it seems that causality violations will be strongly enhanced, rather then suppressed. However, as one might expect, there is a subtlety involved. We shall leave a full discussion of this problem to a future paper, but conclude with a few brief remarks. Basically, one is interested in constructing the density of states, or microcanonical partition function, as the squared amplitude $\Psi^2$. The problem is that the microcanonical partition function should be defined as a function of definite conserved quantities, such as energy and angular momentum. The amplitude $\Psi$ above, however, is generally given as a function of the metric parameters which relate equivalent points in the universal covering space, which could be inverse temperature or angular velocity, for example. In order to achieve the correct result, one must project the amplitude $\Psi$ on to states of definite ‘charge’ rather than the ‘potentials’ before constructing the microcanonical partition function. If one does this, then one finds that the corrected $\Psi^2$ tends to zero, rather than infinity, as the CTCs are introduced. The situation is rather similar to that encountered when trying to calculate the rate of pair production of electrically and magnetically charged black holes. In that case, the introduction of a projection operator is necessary to ensure that the pair production rates for both types of black holes are equal, as one would expect [@hawross]. In this case, it means that causality violating amplitudes are strongly suppressed, in accordance with the chronology protection conjecture. Acknowledgements ================ The author would like to thank Stephen Hawking for valuable discussions. This work was completed with the financial support of EPSRC. [10]{} M. S. Morris, K. S. Thorne and U. Yurtsever, Phys. Rev. Lett. [**61**]{}, 1446 (1988) J. L. Friedman, N. J. Papastamatiou and J. Z. Simon, Phys. Rev. [**D46**]{}, 4456 (1992). S. W. Hawking, Phys. Rev. [**D52**]{}, 5681 (1995) J. B. Hartle, Phys. Rev. [**D49**]{}, 6543 (1994) H. D. Politzer, Phys. Rev. [**D49**]{}, 3981 (1994) D. Deutsch, Phys. Rev. [**D44**]{}, 3197 (1991) C. J. Fewster and C. G. Wells, Phys. Rev. [**D52**]{}, 5773 (1995) A. Anderson, Phys. Rev. [**D51**]{}, 5707 (1995) J. D. E. Grant, Phys. Rev. [**D47**]{}, 2388 (1993) J. R. Gott III, Phys. Rev. Lett. [**66**]{}, 1126 (1991) S–W Kim and K. S. Thorne, Phys. Rev. [**D43**]{}, 3929 (1991) S. W. Hawking, Phys. Rev. [**D46**]{}, 603 (1992) J. S. Dowker, J. Phys. [**A5**]{}, 936 (1972) R. Banach and J. S. Dowker, J. Phys. [**A11**]{}, 2255 (1978) R. M. Wald, Comm. Math. Phys. [**70**]{}, 221 (1979) C. J. Isham, Proc. Roy. Soc. London [**A362**]{}, 383 (1978) B. S. DeWitt, C. F. Hart and C. J. Isham, Physica [**A96**]{}, 197 (1979) M. P. Gaffney, Ann. Math. [**60**]{}, 140 (1954) M. Reed and B. Simon, Functional Analysis, New York, Academic Press (1972) I. S. Gradshteyn and I. W. Ryzhik, Tables of Integrals, Series and Products, New York, Academic Press (1965) S. M. Christensen, Phys. Rev. [**D14**]{}, 2490 (1976) S. M. Christensen, Phys. Rev. [**D17**]{}, 946 (1978) J. D. Bekenstein and L. Parker, Phys. Rev. [**D23**]{}, 2850 (1981) S. J. Avis, C. J. Isham and D. Storey, Phys. Rev. [**D18**]{}, 3565 (1978) S. V. Sushkov, Class. Qu. Grav. [**14**]{}, 523 (1997) S. V. Krasnikov, Phys. Rev. [**D54**]{}, 7322 (1996) T. Tanaka and W. A. Hiscock, Phys. Rev. [**D52**]{}, 4503 (1995) D. G. Boulware, Phys. Rev. [**D46**]{}, 4421 (1992) C. R. Cramer and B. S. Kay, Class. Qu. Grav. [**13**]{}, L143 (1996) S. W. Hawking and S. F. Ross, Phys. Rev. [**D52**]{}, 5865 (1995)

--- bibliography: - 'main.bib' ---

--- abstract: 'First order perturbations for the fields with spin on the background metric of the gravitational shock waves are discussed. Applying the Newman – Penrose formalism, exact solutions of the perturbation equations are obtained. For particle physics, this would be one approach to the problem of scattering paticle at Planck energy.' --- October 1995\ [**Newman - Penrose Formalism for Gravitational Shock Waves**]{}\ Koichi HAYASHI${}^{1}$ and Toshiharu SAMURA${}^{2}$\ ${}^{1}$[*Department of Mathematics and Physics,*]{}\ [*Faculty of Science and Technology,*]{}\ [*Kinki University, Higashi-Osaka,*]{}\ [*Osaka 577, Japan*]{}\ ${}^{2}$[*The Graduate School of Science and Technology*]{}\ [*Kobe University, Nada, Kobe 657, Japan*]{}\ Introduction ============ The gravitational shock wave (GSW) of a black hole is a solution that is obtained by the black hole moving at the limit of light velocity. Recently, we have calculated the metrics both for Schwarzschild and Kerr black holes [@HS94]. Comparing with the GSW metric derived by Aichelburg and Sexl [@AS71](below AS), our metrics are interesting for the point that the mass of the black hole is finite. As physical applications of these metrics, two cases have been considered: the gravitational waves emission when two black holes collide [@DP92], and the scattering of particles at Planck scale as a model of quantum gravity [@QG]. In these papers, these applications had been calculated using only the AS metric. We have also investigated these using our metric [@HS94b]. In this paper, we apply the Newman – Penrose (NP) formalism for GSW metrics and calculate various physical quantities in perturbations. The usual approach to obtain perturbed solution is perturbing the metric directly and solving for the resulting perturbed field equation. However, it is in general very difficult to obtain the direct solution for metric perturbation equation except for simple cases. On the other hand, the approach of the NP formalism provides soluble perturbation equations for much wider cases. The perturbations of the complicated metrics like the Kerr black hole or Reissner – Nordström one are successful by this approach. In the NP formalism, the perturbation equations for Weyl tensors decouple to independent equations, for which the partial wave analysis are posible using suitable radial functions and spin weighted spherical harmonics. When perturbed solutions are obtained, we can deal with astrophysical applications: e.g. stability of a black hole, tidal friction effects, superradiant scatterings, and gravitational wave processes. The NP formalism of the GSW metrics derived in this paper is not similary another mathematical method to obtain the results by ordinary procedures, but the motivations which would shed new lights on physics exist. Especially, this approach is suitable to treat the scattering problem at the Plankian energies. Moreover, we can challenge to several unsolved problems by this method: the scattering of particles with spin, the collision of two GSW’s, the scattering of a particle moving near a black hole, the spontaneous emission of radiation, and so on. In this paper, we show the first step to these approaches. In section 2, we choose a simple NP tetrad of GSW metric, then, the spin coefficients, the Wely tensors, the Ricci tensors and the scalar tensor are calculated. The field equation of a scalar field in flat space–time is given by the Klein – Gordon equation, $(\Box + m^2) \phi=0$. In the curved space–time, this is modified to $( \nabla_\alpha \nabla^\alpha + m^2) \phi =0$, using covariant derivatives. For the metric of GSW, the solution of this equation is given by ’t Hooft [@QG], and we can see the effects of GSW metric on the behavior of the scalar field. How about the other fields with spin? This is the main subject of this paper. We calculate the effects of GSW metric on various fields with spin by perturbations. In section 3, we derive the differential equations for a test nertrino field (spin $= 1/2$), a test electromagnetic field (spin $= 1$) and a gravitational perturbation (spin $= 2$) in the background metric of GSW, Exact solutions of these equations are given. Section 4 is devoted to conclusious and discussions. Newman – Penrose Formalism ========================== In the Newman-Penrose (NP) formalism, a set of null tetrad [$\ell$]{}, [$n$]{}, [$m$]{} and [${\overline{m}}$]{} is to be introdeced, where [$\ell$]{}, [$n$]{} are real, and [$m$]{}, [${\overline{m}}$]{} are complex conjugates of each other. These must satisfy the orthogonality conditions: $$\begin{aligned} \mbox{\boldmath$\ell$}\cdot\mbox{\boldmath$m$}~=~ \mbox{\boldmath$\ell$}\cdot \mbox{\boldmath${\overline{m}}$} {}~=~\mbox{\boldmath$n$}\cdot\mbox{\boldmath$m$}~=~ \mbox{\boldmath$n$}\cdot \mbox{\boldmath${\overline{m}}$} {}~=~0~~, \label{eqn:1}\end{aligned}$$ null conditions: $$\begin{aligned} \mbox{\boldmath$\ell$}\cdot\mbox{\boldmath$\ell$}~=~ \mbox{\boldmath$n$}\cdot\mbox{\boldmath$n$} {}~=~\mbox{\boldmath$m$}\cdot\mbox{\boldmath$m$}~=~ \mbox{\boldmath${\overline{m}}$}\cdot \mbox{\boldmath${\overline{m}}$} {}~=~0~~, \label{eqn:2}\end{aligned}$$ and the normalization conditions: $$\begin{aligned} \mbox{\boldmath$\ell$}\cdot\mbox{\boldmath$n$}~=~1~~ {\rm{and}}~~ \mbox{\boldmath$m$}\cdot\mbox{\boldmath${\overline{m}}$}~=~-1~~. \label{eqn:3}\end{aligned}$$ Now, we will investigate the NP formalism of the gravitational shock wave (GSW) metrics which are derived by us [@HS94]. For the construction of a null – tetrad frame for the NP formalism, we must find the null tangent vectors for geodesics. The GSW metric is generally written by: $$\begin{aligned} ds^2~=~du~dv-d\rho^2-\rho^2d\varphi^2-A(\rho)~\delta\left(u\right)~du^2~~, \label{eqn:4}\end{aligned}$$ where $u=t-z, v=t+z$, and the concrete forms of $A(\rho)$ are given in the previous paper. The null geodesic of (\[eqn:4\]) satisfy: $$\begin{aligned} \frac{1}{2} \dot{u}~&=&~{\rm{constant}}~\left( =\frac{1}{2} \right) \nonumber\\ \frac{1}{2} \dot{v}+A(\rho)~\delta(u)~\dot{u} ~&=&~{\rm{constant}}~\left( =0 \right) \nonumber\\ \rho^2 {\dot{\varphi}}~&=&~{\rm{constant}}~\left( =0 \right) \nonumber\\ \ddot{\rho}~=~\rho {\dot{\varphi}}^2&-&\frac{1}{2}A' \left( \rho \right) \delta \left( u \right) \dot{u}^2~~, \label{eqn:5}\end{aligned}$$ where dot and dash denotes the derivative with respect to the affine parameter $\lambda$, and $\rho$, respectively. With no loss of generality, we take the constants of the RHS of (\[eqn:5\]) to the numbers in the parenthesis. With this choice, $\dot{u}=1$, so that $u$ can be identified as the affine parameter $\lambda$ itself. The equations in (\[eqn:5\]) are rewritten as $$\begin{aligned} \dot{u}~&=&~1\nonumber\\ \dot{v}~&=&~-2A\delta\nonumber\\ {\dot{\varphi}}~&=&~0\nonumber\\ \dot{\rho}~&=&~-\frac{1}{2}A'\theta~~, \label{eqn:5-1}\end{aligned}$$ where $\theta$ is the step function. For simplicity we put $A(\rho)$, $\delta(u)$, $\theta\left(u\right)$, and $\left.A' \left( \rho \right) \right|_{\rho=\rho_0}$ as $A$, $\delta$, $\theta$ and $A'$, respectively. Then since the null tangent vector is $$\begin{aligned} v^i~=~\left(\dot{u},~\dot{v},~\dot{\rho},~\dot{\varphi}\right) {}~=~\left(1,~-2A\delta,~-\frac{1}{2}A'\theta,~0 \right)~~, \label{eqn:6}\end{aligned}$$ the vector [$\ell$]{} of the NP formalism is taken as $v^i$ itself: $$\begin{aligned} \ell^i~&=&~\left(\ell^u,~\ell^v,~\ell^{\rho},~\ell^{\varphi} \right) {}~=~\left(1,~-2A\delta,~-\frac{1}{2}A'\theta,~0 \right)~~,\nonumber\\ \ell_i~&=&~\left(0,~\frac{1}{2},~\frac{1}{2}A'\theta,~0 \right)~~. \label{eqn:7}\end{aligned}$$ It is to be noted that $\ell^i \ell_i~=~-A\delta-(\dot\rho)^2~=~0$, since the derivative of the equation ${\dot{\rho}}^2=-A\delta$ with respect to $\lambda$ gives exactly the last equations in (\[eqn:5\]). For the null vector [$n$]{} and [$m$]{} we will take $$\begin{aligned} n^i~&=&~\left(a~,b~,c~,d~\right)~~,\nonumber\\ m^i~&=&~\left(0~,\alpha~,\beta~,{\rm{i}}\gamma~\right)~~. \label{eqn:8}\end{aligned}$$ From conditions of (1)(2)(3), $m^i$ is determined uniquely: $$\begin{aligned} \alpha~=~-\frac{1}{\sqrt{2}}A'\theta,~~\beta~=~\frac{1}{\sqrt{2}}, {}~~\gamma~=~\frac{1}{\sqrt{2}\rho}~~, \label{eqn:9}\end{aligned}$$ while for $n^i$, the following equations are obtained: $$\begin{aligned} &&b+A'\theta c~=~2 \nonumber\\ &&A'\theta a+2 c~=~0 \nonumber\\ &&c^2~=~ab+A\delta a^2~~. \label{eqn:10}\end{aligned}$$ The simplest solution is $$\begin{aligned} a~=~0,~~b~=~2,~~c~=~0~~. \label{eqn:11}\end{aligned}$$ Let us summarize the null vectors of NP formalism which we have chosen: $$\begin{aligned} \ell^i~&=&~\left(1,~-2A\delta,~-\frac{1}{2}A'\theta,~0 \right)~~, \nonumber\\ n^i~&=&~\left( 0,~2~,0~,~0 \right)~~,\nonumber\\ m^i~&=&~\left(0,~-\frac{1}{\sqrt{2}}A'\theta,\frac{1}{\sqrt{2}}, {}~\frac{\rm{i}}{\sqrt{2}\rho} \right)~~. \label{eqn:12}\end{aligned}$$ and corresponding covariant vectors are $$\begin{aligned} \ell_i~&=&~\left(0,\frac{1}{2},~\frac{1}{2}A'\theta,~0 \right)~~, \nonumber\\ n_i~&=&~\left( 1,~0~,0~,0~ \right)~~,\nonumber\\ m_i~&=&~\left(-\frac{1}{\sqrt{2}}A'\theta,~0,~-\frac{1}{\sqrt{2}}, {}~-\frac{\rm{i\rho}}{\sqrt{2}} \right)~~. \label{eqn:13}\end{aligned}$$ The nonvanising spin coefficients (for example, see (286) in Chapter 1 of Chandrasekhar [@Chandra83]) are $$\begin{aligned} \kappa~=~\frac{1}{2\sqrt{2}}A'\delta,~~\rho~=~-\sigma~=~ \frac{1}{4\rho}A'\theta, {}~~\alpha~=~-\beta~=~-\frac{1}{2\sqrt{2}\rho}~~. \label{eqn:14}\end{aligned}$$ Here, note that $A'$ is a constant. Finally, we can calculate the components of the Wely tensor, $\Psi_i$, Ricci tensor, $\Phi_{ij}$, and scalar tensor, $\Lambda$ (see (294) and (300) in Chap.1 of [@Chandra83]). Then we see all components of these tensors vanish: $$\begin{aligned} \Psi_i~=~\Phi_{ij}~=~\Lambda~=0~~\left( i,j=0~\sim~4 \right) {}~~\left( {\rm{For}}~u\neq0 \right). \label{eqn:15}\end{aligned}$$ Since the space – time of GSW is empty except for $u=0$, this is rather a natural result. The whole space – time is the patchwork of two flat Minkowskii spaces pasted at $u=0$, where the continuity is destroyed. Because of this discontinuity, (\[eqn:14\]) is obtained and physics are not trivial even with (\[eqn:15\]). In the next section we consider the perturbation over this background space – time. Although this is almost flat, it reflects this discontinuity through (\[eqn:14\]). Perturbations on the Gravitational Shock Waves Background ========================================================= The NP equations are system of first – order differential equations linking the tetrad, the spin coefficients, Wely tensors, Ricci tensors and the scalar curvature. The perturbed geometries in NP formalism are specified by: $$\begin{aligned} \mbox{\boldmath$\ell$}~=~\mbox{\boldmath$\ell$}^u+ \mbox{\boldmath$\ell$}^p,~~ \mbox{\boldmath$n$}~=~\mbox{\boldmath$n$}^u+ \mbox{\boldmath$n$}^p,~~ \mbox{\boldmath$m$}~=~\mbox{\boldmath$m$}^u+ \mbox{\boldmath$m$}^p, \label{eqn:16}\end{aligned}$$ where the supersuffix $u$ and $p$ means “unperturbed” and “perturbed”, respectively. All the NP quantities can be written in this form. The spin coefficients $\kappa$, $\rho$, $\sigma$, $\alpha$ and $\beta$ have both the unperturbed quantities and the perturbed ones. The other spin coefficients, all Wely tensors, Ricci tensors and the scalar curvature have only the unperturbed quantities. The complete set of perturbation equations are obtained from the NP equations by keeping perturbed terms only to first order. In this section, we will derive the source free perturbation equations for two component neutrino fields, electromagnetic fields, and gravitational fields on the GSW background. Neutrino equations ------------------ In the NP formalism for the GSW space – time, the Dirac equations of the massless particle are written by the following equations (see (108) in Chap.10 of [@Chandra83]): $$\begin{aligned} \left( D-\rho \right) F_1+\left( \delta ^\ast-\alpha \right) F_2~=~0 \label{eqn:17}\\ \left( \delta-\alpha \right) F_1+\Delta F_2~=~0~~, \label{eqn:18}\end{aligned}$$ where $F_1$ and $F_2$ are 2–spinors. When we consider the realistic problem of the scattering of a massless neutrino off the GSW background, the $F$’s can be treated as the first order test fields. $D$, $\delta$ and $\Delta$ are directional derivatives along the basis null vectors defined by: $$\begin{aligned} DF~=~F_{;\mu}~\ell^\mu~~,~\Delta F ~=~F_{;\mu}~n^\mu~~, \delta F~=~F_{;\mu} n^\mu~~. \label{eqn:19}\end{aligned}$$ Eliminating $F_2$ from (18) and (\[eqn:18\]), we obtain the equation for $F_1$: $$\begin{aligned} \left[ \Delta \left( D - \rho \right) - \left( \delta^\ast - \alpha \right) \left( \delta - \alpha \right) \right] F_1~=~0~~, \label{eqn:20}\end{aligned}$$ where we have used the relation, $\Delta \left( \delta^\ast - \alpha \right) - \left( \delta^\ast - \alpha \right) \Delta =0$. We want to find the solution for $F_i$ which have the form: $$\begin{aligned} F_i~=~{\rm{e}}^{ {\rm{i}} \left( k_u u +k_v v+ m \varphi\right)} f_i \left( \rho \right)~~. \label{eqn:22}\end{aligned}$$ where $k_u$ and $k_v$ are constants and m is an integer, General solutions are to be obtained by superpositions of them. When $u>0$, the equation of $F_1$ is given by: $$\begin{aligned} f_{1,\rho,\rho}+\frac{1}{\rho} f_{1,\rho} + \left[ \left( 4 k_u k_v - k_v^2 A^{'2} \right) - \frac{\left(m - 1/2 \right)^2}{\rho^2} \right] f_1 {}~=~0~~, \label{eqn:23}\end{aligned}$$ and the solution of it is: $$\begin{aligned} f_1 \left( \rho \right)~=~J_{m-1/2} \left( y \right)~~~ \left( \rm{for}~ u>0 \right)~~, \label{eqn:24}\end{aligned}$$ where $J$ is the Bessel function, $y=\Omega \rho$ and $\Omega = \sqrt{ 4 k_u k_v - k_v^2 A'}$. For $u<0$, $f_1(\rho)$ is similarly obtained: $$\begin{aligned} f_1\left( \rho \right)~=~J_{m-1/2} \left( y' \right)~~ \left( {\rm{for}}~u<0 \right)~~, \label{eqn:25}\end{aligned}$$ where $y'=\Omega'\rho$ and $\Omega'=\sqrt{4k_uk_v}$. Next, we will give the $F_2$. From (\[eqn:18\]) and (\[eqn:22\]), $f_2$, which is the $\rho$–direction field of $F_2$, is represented by $f_1$: $$\begin{aligned} 2 \sqrt{2}{\rm{i}}k_v f_2 ~=~-f_{1,\rho}+{\rm{i}}A'k_vf_1 + \frac{m-1/2}{\rho}f_1~~\left( {\rm{for}}~u>0 \right)~~. \label{eqn:26}\end{aligned}$$ Using the formula, $J'_\nu(z) =-J_{\nu+1}(z)+ \nu/zJ_\nu(z)$, $f_2$ for $u>0$ is given by: $$\begin{aligned} f_2 \left( \rho \right) ~=~ \frac{\Omega}{2\sqrt{2}{\rm{i}}k_v} \left\{J_{m+1/2} \left( y \right) +\frac{{\rm{i}}k_vA'}{\Omega}J_{m-1/2} \left( y \right) \right\}~~ \left( {\rm{for}}~u>0 \right)~~, \label{eqn:27}\end{aligned}$$ while for $u<0$, $$\begin{aligned} f_2 \left( \rho \right) ~=~ \frac{\Omega'}{2\sqrt{2}{\rm{i}}k_v} J_{m+1/2} \left( y' \right) ~~ \left( {\rm{for}}~u<0 \right)~~. \label{eqn:27-2}\end{aligned}$$ It is to be noted that these solutions are the exact solutions for the neutrino fields. Electromagnetic and Gravitational Equations ------------------------------------------- Maxwell equations in the NP formalism of the GSW geometry can be written by (see (330)–(333) in Chap.1 of [@Chandra83]): $$\begin{aligned} \left( D-\rho \right) \phi_2-\delta^* \phi_1 ~&=&~0 \label{eqn:28}\\ \left( \delta-2\alpha \right) \phi_2-\Delta\phi_1 ~&=&~0 \label{eqn:29}\\ \delta \phi_1 -\Delta \phi_0 +\sigma \phi_2 ~&=&~0 \label{eqn:30}\\ \left( \delta-2\rho \right) \phi_1-\left( \delta^* -2\alpha \right) \phi_0 ~&=&~0~~, \label{eqn:31}\end{aligned}$$ where $\phi$’s are Maxwell field strengths. With $\Delta \times$(29)$-\delta^*\times$(30), the equation for $\phi_2$ is given by the following equation: $$\begin{aligned} \left[ \Delta \left( \Delta - \rho \right) - \delta^\ast \left( \delta - 2\alpha \right) \right] \phi_2~=~0~~, \label{eqn:32}\end{aligned}$$ As in the previous subsection, we put $$\begin{aligned} \phi_i~=~{\it{e}}^{ {\rm{i}} \left( k_u u +k_v v+ m \varphi\right)} R_i \left( \rho \right)~~. \label{eqn:33}\end{aligned}$$ Then (\[eqn:32\]) is written by the following differential equation for $u>0$: $$\begin{aligned} R_{2,\rho,\rho}+\frac{1}{\rho} R_{2,\rho} + \left[ \left( 4 k_u k_v - k_v^2 A^{'2} \right) - \frac{\left(m - 1 \right)^2}{\rho^2} \right] R_2 {}~=~0~~. \label{eqn:34}\end{aligned}$$ The equation is easily solved: $$\begin{aligned} R_2 \left( \rho \right)~=~J_{m-1} \left( y \right)~~~ \left( \rm{for}~ u>0 \right)~~, \label{eqn:35}\end{aligned}$$ as before. On the other hand, the solution for $u<0$ is $$\begin{aligned} R_2 \left( \rho \right)~=~J_{m-1} \left( y' \right)~~~ \left( \rm{for}~ u<0 \right)~~, \label{eqn:36}\end{aligned}$$ where $y$ and $y'$ are given by (\[eqn:24\]) and (\[eqn:25\]). As $\phi_1$ and $\phi_2$ are related by (30), $R_1$ is given by $$\begin{aligned} R_1 \left( \rho \right) ~=~ \left( - \frac{\Omega} {2\sqrt{2}{\rm{i}}k_v} \right) \left\{ J_{m} \left( y \right) +\frac{{\rm{i}}k_vA'}{\Omega}J_{m-1} \left( y \right) \right\}~~ \left( {\rm{for}}~u>0 \right)~~, \label{eqn:37}\end{aligned}$$ while for $u<0$, it is: $$\begin{aligned} R_1 \left( \rho \right) ~=~ \left( -\frac{\Omega'} {2\sqrt{2}{\rm{i}}k_v} \right) J_{m} \left( y' \right) ~~ \left( {\rm{for}}~u<0 \right)~~. \label{eqn:38}\end{aligned}$$ Finally, $R_0$ is given from (\[eqn:30\]) $$\begin{aligned} R_0 \left( \rho \right) ~&=&~\left( - \frac{\Omega} {2\sqrt{2}{\rm{i}}k_v} \right)^2 \left\{ J_{m+1} \left( y \right)+\frac{2{\rm{i}}k_vA'}{\Omega}J_{m} \left( y \right) \right. \nonumber\\ &&~~~~~~~~~~~~~~~\left. +\left( \frac{{\rm{i}}k_vA'} {\Omega} \right)^2 J_{m-1}\left( y \right) \right\}~~\left( {\rm{for}}~u>0 \right)~~\\ \label{eqn:39} R_0 \left( \rho \right) ~&=&~\left( - \frac{\Omega'} {2\sqrt{2}{\rm{i}}k_v} \right)^2 J_{m+1} \left( y' \right) ~~ \left( {\rm{for}}~u<0 \right)~~. \label{eqn:40}\end{aligned}$$ Now we want to discuss about the first – order perturbations of the gravitational field itself. We find that four of the eight NP Bianchi identities provide the following linear homogeneous equations to first order in the perturbations (see (321) in Chap.1 of [@Chandra83]): $$\begin{aligned} \left(\delta^* + 2\alpha \right) \Psi_3 - \left( D -\rho \right) \Psi_4~=~0 \\ \label{eqn:41} -\Delta \Psi_0 +\left( \delta -2\beta \right) \Psi_1 +3\sigma \Psi_2~=~0 \\ \label{eqn:42} -\Delta \Psi_1 + \delta \Psi_2 +2\sigma \Psi_3~=~0\\ \label{eqn:43} -\Delta \Psi_2 +\left( \delta+2\beta \right) \Psi_3 +\sigma \Psi_4 ~=~0\\ \label{eqn:44} -\Delta \Psi_3 +\left( \delta +4\beta \right) \Psi_4~=~0 \label{eqn:45}\end{aligned}$$ where $\Psi$’s are the perturbed Wely tensors. We can calculate $\Psi$’s by the same procedure as neutrino and electromagnetic cases, and the final forms of $\Psi$’s are: $$\begin{aligned} \Psi _n &=& \exp(ik_u u+ik_v v+im\varphi)~B^{4-n}~J_{m-n+2} (y')~~\nonumber\\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm{for}}~u<0)~~,\\ \label{eqn:46} \Psi _n &=& \exp(ik_u u+ik_v v+im\varphi)~ B^{4-n}\sum^{5-n}_{l=1} {}_4{\rm{C}}_l D^{l-1} J_{m-n+3-l} (y)~~\nonumber\\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({\rm{for}}~u>0)~~, \label{eqn:47}\end{aligned}$$ where $$\begin{aligned} B = -\frac{\Omega'}{2\sqrt{2} i k_v}~~, D = \frac{ik_v A'}{\Omega}~~, \nonumber\\\end{aligned}$$ and ${}_4{\rm{C}}_l$ represents the binomial coefficient. Conclusions =========== We have derived the first order perturbations for fields with spin on the background metric of GSW. Exact solutions of fields are obtained. From these, we can find the behavior of the field crossing the discontinuity of GSW at $u=0$. Especially, we can find, for example, the refraction angles and scattering cross sections of various fields near the black hole which is moving with a relativistic speed. For particle physics, this would be one approach to the physics at Planck energy. These physics are now under investigations. [0]{} K. Hayashi and T. Samura, Phys. Rev. [**[D50]{}**]{}, 3666(1994). P. C. Aichelburg and R. U. Sexl, Gen. Relativ. Gravit. [**[2]{}**]{}, 303(1971). P. D. D’Eath, Phys. Rev. [**[D18]{}**]{}, 990(1778); P. D. D’Eath, and P. N. Payne, ibid, [**[46]{}**]{}, 658(1992); [**[46]{}**]{}, 675(1992); [**[46]{}**]{}, 694(1992). For example see, G.’t Hooft, Phys. Lett. [**[B198]{}**]{}, 61(1987); ibid, Nucl. Phys. [**[B355]{}**]{}, 138 (1990); D. Amati et.al., Phys. Lett. [**[B197]{}**]{}, 81(1987); C.O.Loustó and N.Sánchez, Phys. Lett. [**[B232]{}**]{}, 462(1989); ibid, Int. J. Mod. Phys. [**[A5]{}**]{}, 915 (1990); ibid, Nucl. Phys. [**[B355]{}**]{}, 231(1991); ibid, Nucl. Phys. [**[B383]{}**]{}, 377 (1992); H. Verlinde and E. Verlinde, Nucl. Phys. [**[B371]{}**]{}, 246 (1992); M. Fabbrichesi et. al., Nucl. Phys. [**[B419]{}**]{}, 147 (1994); V. Ferrari and M. Martellini, Nucl. Phys. [**[B385]{}**]{}, 604 (1992). K. Hayashi and T. Samura, Planckian Scatterings of Massive Particles and Gravitational Shock Waves, preprint KOBE-FHD-94-05 and hep-th/9405013 (1994). S Chandrasekhar, [*[The Mathematical Theory of Black Holes]{}*]{} (Clarendon Press, Oxford, 1983).

--- abstract: 'We study generalized Bayesian inference under misspecification, i.e. when the model is ‘wrong but useful’. Generalized Bayes equips the likelihood with a learning rate $\eta$. We show that for generalized linear models (GLMs), $\eta$-generalized Bayes concentrates around the best approximation of the truth within the model for specific $\eta \neq 1$, even under severely misspecified noise, as long as the tails of the true distribution are exponential. We then derive MCMC samplers for generalized Bayesian lasso and logistic regression, and give examples of both simulated and real-world data in which generalized Bayes outperforms standard Bayes by a vast margin.' bibliography: - 'safebayesbib.bib' --- Introduction ============ Over the last ten years it has become abundantly clear that Bayesian inference can behave quite badly under misspecification, i.e. if the model ${\ensuremath{\mathcal F}}$ under consideration is ‘wrong but useful’ [@GrunwaldL07; @ErvenGR07; @muller2013risk; @syring2017calibrating; @yao2018using; @holmes2017assigning; @grunwald2017inconsistency]. For example, @GrunwaldL07 exhibit a simple nonparametric classification setting in which, even though the prior puts positive mass on the unique distribution in ${\ensuremath{\mathcal F}}$ that is closest in KL divergence to the data generating distribution $P$, the posterior never concentrates around this distribution. @grunwald2017inconsistency give a simple linear regression problem in which standard Bayesian ridge regression, model selection and model averaging severely overfit the data (the prior does not regularize nearly enough) for small samples. They also propose a remedy for this problem: equip the likelihood with an exponent or [ *learning rate*]{} $\eta$ (see  below). Such a [*generalized Bayesian*]{} (also known as [*fractional*]{} or [ *tempered*]{} Bayesian) approach was considered earlier by e.g. [@barron1991minimum; @WalkerH02; @zhang2006information]. In practice, $\eta$ will usually (but not always — see Section \[sec:lassoandhs\] below) be chosen smaller than one, making the prior have a stronger regularizing influence. @grunwald2017inconsistency show that for Bayesian ridge regression and model selection/averaging, this results in excellent performance, being competitive with standard Bayes if the model is correct and very significantly outperforming standard Bayes if it is not. Extending Zhang’s ([-@zhang2006epsilon; -@zhang2006information]) earlier work, @grunwald2016fast (GM from now on) show that, under what was earlier called the $\bar\eta$-[*central condition*]{} (Definition \[def:central\] below), generalized Bayes with a specific finite learning rate $\bar\eta$ (usually $\neq 1$) will indeed, with high probability, concentrate in the neighborhood of the ‘best’ $f \in {\ensuremath{\mathcal F}}$, closest in KL divergence to $P$. Yet three important parts of the story are missing in this existing line of work: (1) Can Grünwald-Van Ommen-type examples, showing failure of standard Bayes $(\eta = 1)$ and empirical success of generalized Bayes with the right $\eta$, be given more generally, for different priors $\pi$ (say of lasso-type ($\pi(f) \propto \exp(- \lambda \| f\|_1)$) rather than ridge-type $\pi(f) \propto \exp(-\lambda \| f \|_2^2)$), and for different models, say for [*generalized*]{} linear models (GLMs)? (2) Can we find examples of generalized Bayes outperforming standard Bayes with real-world data rather than with toy problems such as those considered by Grünwald and Van Ommen? (3) Under what circumstances does the central condition — which allows for good theoretical behavior of generalized Bayes — hold for GLMs? Here we show that the answer to all three questions is affirmative:[^1] in Section \[sec:genglm\] in Section \[sec:bad\] below, we give a toy example on which the Bayesian lasso and the Horseshoe estimator fail; later in the paper, in Section \[sec:examples\] we also give a toy example on which standard Bayes logistic regression fails, and two real-world data sets on which standard Bayesian lasso and Horseshoe regression fail; in all cases generalized Bayes with the right $\eta$ shows much better performance. In Section \[sec:genglm\], we show that for GLMs, even if the noise is severely misspecified, as long as the distribution of the predictor variable $Y$ has exponentially small tails (which is automatically the case in classification, where the domain of $Y$ is finite), the central condition holds for some $\eta > 0$, so GM’s theoretical results suggest that generalized Bayes with this $\eta$ should lead to good results — this is corroborated by our experimental results in Section \[sec:examples\]. These findings are not obvious: one might for example think that the sparsity-inducing prior used by Bayesian lasso regression circumvents the need for the additional regularization induced by taking an $\eta < 1$. Indeed in the original setting of Grünwald and Van Ommen, the standard Bayesian lasso $(\eta =1)$ succeeds; yet Example \[ex:basic\] below shows that under a modification of their example, it fails after all. In order to demonstrate the failure of standard Bayes and the success of generalized Bayes, we devise (in Section \[sec:sampling\]) MCMC algorithms for generalized Bayes posterior sampling for Bayesian lasso and logistic regression; these algorithms are an additional contribution of this paper. Below we first define our setting more precisely, we then (Section \[sec:bad\]) give a first example of bad standard-Bayesian behavior. Section \[sec:consistency\] recalls a theorem from GM indicating that under the $\bar\eta$-central condition, generalized Bayes for $\eta < \bar\eta$ should perform well. We then present new theoretical results, showing that, if $|Y|$ has exponential tails and the noise (not the regression function) is misspecified, then the central condition does hold for GLMs. We next (Section \[sec:sampling\]), present our algorithms for generalized Bayesian posterior sampling, and we continue (Section \[sec:examples\]) to empirically demonstrate how generalized Bayes outperforms standard Bayes under misspecification. All proofs are in Appendix \[app:proofs\]. The setting =========== A *learning problem* can be characterized by a tuple $(P, {\ensuremath{\ell}}, {\ensuremath{\mathcal F}})$, where ${\ensuremath{\mathcal F}}$ is a set of predictors, also referred to as [*model*]{}, $P$ is a distribution on sample space ${\ensuremath{\mathcal Z}}$ and ${\ensuremath{\ell}}: {\ensuremath{\mathcal F}}\times {\ensuremath{\mathcal Z}}\rightarrow {{\mathbb R}}\cup \{\infty \}$ is a loss function. We denote by ${\ensuremath{\ell}}_f(z) \coloneqq {\ensuremath{\ell}}(f,z)$ the loss of predictor $f \in {\ensuremath{\mathcal F}}$ under outcome $z \in {\ensuremath{\mathcal Z}}$. If $Z \sim P$, we abbreviate ${\ensuremath{\ell}}_f(Z)$ to ${\ensuremath{\ell}}_f$. In all our examples, ${\ensuremath{\mathcal Z}}= {\ensuremath{\mathcal X}}\times {\ensuremath{\mathcal Y}}$. We obtain e.g. standard (random-design) regression with squared loss by taking ${\ensuremath{\mathcal Y}}= {{\mathbb R}}$ and ${\ensuremath{\mathcal F}}$ to be some subset of the class of all functions $f: {\ensuremath{\mathcal X}}\rightarrow {{\mathbb R}}$ and, for $z=(x,y)$, ${\ensuremath{\ell}}_f(x,y) = (y -f(x))^2$; logistic regression is obtained by taking ${\ensuremath{\mathcal F}}$ as before, ${\ensuremath{\mathcal Y}}= \{-1,1\}$ and ${\ensuremath{\ell}}_f(x,y) = \log (1 + \exp(- y f(x))$. We get conditional density estimation by taking $\{p_f(Y \mid X) :f \in {\ensuremath{\mathcal F}}\}$ to be a family of conditional probability mass or density functions (defined relative to some measure $\mu$), extended to $n$ outcomes by the i.i.d. assumption, and taking conditional log-loss $\ell_f(x,y) \coloneqq - \log p_f(y \mid x)$. We are given an i.i.d. sample $Z^n \coloneqq Z_1, Z_2, \ldots, Z_n \sim P$ where each $Z_i$ takes values in ${\ensuremath{\mathcal Z}}$, and we consider, as our learning algorithm, the [ *generalized Bayesian posterior*]{}, also known as the [*Gibbs posterior*]{}, $\Pi_n$ on ${\ensuremath{\mathcal F}}$, defined by its density $$\label{eq:bayesgenpost} \pi_n (f) \coloneqq \frac{\exp\left(-\eta \sum_{i=1 }^{n} {\ensuremath{\ell}}_f(z_i) \right) \cdot \pi_0(f)}{\int_{\mathcal{F}}\exp\left(-\eta \sum_{i= 1 }^{n} {\ensuremath{\ell}}_f(z_i)\right) \cdot \pi_0(f) \text{d} \rho(f)},$$ where $\eta>0$ is the *learning rate*, and $\pi_0$ is the density of some prior distribution $\Pi_0$ on ${\ensuremath{\mathcal F}}$ relative to an underlying measure $\rho$. Note that, in the conditional log-loss setting, we get that $$\label{eq:genbayeslik} \pi_n (f) \propto \prod_{i=1}^n (p_f(y_i \mid x_i))^{\eta} \pi_0(f),$$ which, if further $\eta=1$, reduces to standard Bayesian inference. While GM’s result (quoted as Theorem \[thm:metric\] below) works for arbitrary loss functions, in our novel results of this paper, Theorem \[thm:expfam-exptails\] and our empirical simulations, we concentrate on (generalized) linear models, in which has two equivalent interpretations: first, in terms of the original loss functions $\ell_f$; second, in terms of the conditional likelihood $p_f$. For example, for regression, with fixed $\eta$, with ${\ensuremath{\ell}}_f(x,y) = (y -f(x))^2$, (\[eq:bayesgenpost\]) induces the same posterior distribution $\pi_n(f)$ over ${\ensuremath{\mathcal F}}$ as does (\[eq:genbayeslik\]) with the conditional distributions $p_f(y|x) \propto \exp(-(y-f(x))^2$, which is again the same with (\[eq:bayesgenpost\]) with $\ell_f$ replaced by the conditional log-loss $\ell' _f(x,y) \coloneqq - \log p_f(y|x)$, giving a likelihood corresponding to Gaussian errors with a particular fixed variance; an analogous statement holds for logistic regression. Thus, all our examples can be interpreted in terms of (\[eq:genbayeslik\]) for a model which is misspecified, i.e. the density of $P(Y|X)$ is not equal to $p_f$ for any $f \in {\ensuremath{\mathcal F}}$. As is customary (see, e.g. GM or @bartlett2005local), we assume throughout that there exists an optimal ${{f^*}}\in {\ensuremath{\mathcal F}}$ that achieves the smallest *risk* (expected loss) ${\operatorname{\mathbf{E}}}[{\ensuremath{\ell}}_{{{f^*}}}(Z)] = \inf_{f \in {\ensuremath{\mathcal F}}} {\operatorname{\mathbf{E}}}[{\ensuremath{\ell}}_f(Z)]$. If ${\ensuremath{\mathcal F}}$ is a GLM, the risk minimizer again has additional interpretations: first, $f^*$ minimizes, among all $f \in {\ensuremath{\mathcal F}}$, the conditional KL divergence ${\bf E}_{(X,Y) \sim P} [\log \left( p(Y|X)/ p_f(Y|X) \right)]$ to the true distribution $P$. Second, if there is an $f \in {\ensuremath{\mathcal F}}$ with ${\bf E}_{X,Y \sim P}[Y \mid X] = f(X)$ (i.e. ${\ensuremath{\mathcal F}}$ contains the [*true regression function*]{}, or equivalently, [*true conditional mean*]{}), then the risk minimizer satisfies $f^* = f$. Bad Behavior of Standard Bayes {#sec:bad} ------------------------------ \[ex:basic\][ We consider a Bayesian lasso regression setting [@Park] with random design, with a Fourier basis. We sample data $Z_i = (X_i, Y_i)$ i.i.d. $\sim P$, where $P$ is defined as follows: we first sample [*preliminary*]{} $(X'_i,Y'_i)$ with $X'_i \overset{i.i.d.}{\sim}$ Uniform$[-1,1]$; the dependent variable $Y'_i$ is set to $Y'_i = 0 + \epsilon$, with $\epsilon \sim {\ensuremath{\mathcal N}}(0, \sigma^2)$ for some fixed value of $\sigma$, independently of $X'_i$. In other words: the true distribution for $(X'_i, Y'_i)$ is ‘zero with Gaussian noise’. Now we toss a fair coin for each $i$. If the coin lands heads, we set the actual $(X_i,Y_i) \coloneqq (X'_i,Y'_i)$, i.e. we keep the $(X'_i, Y'_i)$ as they are, and if the coin lands tails, we put the pair to zero: $(X_i, Y_i) \coloneqq (0,0)$. We model the relationship between $X$ and $Y$ with a $p$th order Fourier basis. Thus, ${\ensuremath{\mathcal F}}= \{ f_{\beta}: \beta \in {{\mathbb R}}^{2p+1} \}$, with $f_{\beta}(x)$ given by $$\left\langle \beta, \frac{1}{\pi} \cdot \left[2^{-1/2}, \cos(x), \sin(x), \cos(2x), {\hbox to 1em{.\hss.\hss.}}, \sin(px)\right] \right\rangle,$$ and the $\eta$-posterior is defined by (\[eq:bayesgenpost\]) with $\ell_{f_{\beta}}(x,y) = (y- f_{\beta}(x))^2$; the prior is the Bayes lasso prior whose definition we recall in Section \[sec:lassosamp\]. Since our ‘true’ regression function ${\bf E}[Y_i \mid X_i]$ is $0$, in an actual sample around $50\%$ of points will be noiseless, *easy* points, lying on the true regression function. The actual sample of $(X_i,Y_i)$ having less noise then the original sample $(X'_i, Y'_i)$, we would thus expect our Baysian lasso regression to learn the correct regression function, but as we see in the blue line in Figure \[fig:ex1\], it overfits and learns the noise instead (later on (Figure \[fig:risk-simulaties resultaten\] in Section \[sec:lassoandhs\]) we shall see that, not surprisingly, this results in terrible predictive behavior). By removing the noise in half of the datapoints, we misspecified the model: we made the noise heteroscedastic, whereas the model assumes homoscedastic noise. Thus, in this experiment the [ *model is wrong*]{}. Still, the distribution in ${\ensuremath{\mathcal F}}$ closest to the true $P$, both in KL divergence and in terms of minimizing the squared error risk, is given by (the conditional distribution corresponding to) $Y_i = 0 + \epsilon_i$, where $\epsilon_i$ is i.i.d. $\sim {\ensuremath{\mathcal N}}(0,\sigma^2)$. While this element of ${\ensuremath{\mathcal F}}$ is in fact [*favored*]{} by the prior (the lasso prior prefers $\beta$ with small $\|\beta\|_1$), nevertheless, for small samples, the standard Bayesian posterior puts most if its mass at $f$ with many nonzero coefficients. In contrast, the generalized posterior with $\eta = 0.25$ gives excellent results here. To learn this $\eta$ from the data, we can use the Safe-Bayesian algorithm of @Grunwald12. The result is depicted as the red line in Figure \[fig:ex1\]. Implementation details are in Section \[sec:lassosamp\] and Appendix \[app:impl\]; the details of the figure are in Appendix \[ap:detailsfigures\]. ]{} ![‘True’ distribution (green) and predictions of standard Bayes (blue) and SafeBayes (red), $n=50$. \[fig:ex1\]](Eersteplaatje-eps-converted-to.pdf){width="50.00000%"} The example is similar to that of @grunwald2017inconsistency, who use multidimensional $X$ and a ridge (normal) prior on $\|\beta\|$; in their original example standard Bayes succeeds when equiped with a lasso prior; by using a trigonometric basis we can let it ‘fail’ after all. @grunwald2017inconsistency relate the potential for the overfitting-type of behavior of standard Bayes, as well as the potential for full (i.e. even holding as $n \rightarrow \infty$) inconsistency as noted by @GrunwaldL07 to properties of the Bayes predictive distribution ${\bar{p}(Y _{n+1} \mid X_{n+1}, Z^n)} \coloneqq {\int p_f(Y_{n+1} \mid X_{n+1}) \pi_n(f \mid Z^n) \text{d} \rho(f)}$. Being a mixture of $f \in {\ensuremath{\mathcal F}}$, $\bar{p}(Y_{n+1} \mid X_n+1)$, is a member of the convex hull of densities ${\ensuremath{\mathcal F}}$ but not necessarily of ${\ensuremath{\mathcal F}}$ itself. As explained by Grünwald and Van Ommen, severe overfitting may take place if $\bar{p}(Y_{n+1} \mid X_{n+1}, Z^n)$ is ‘far’ from any of the distributions in ${\ensuremath{\mathcal F}}$. It turns out that this is exactly what happens in the lasso example above, as we see from Figure \[fig:predvar\] (details in Appendix \[ap:detailsfigures\]). In this figure we plotted the datapoints as $(X_i, 0)$ to indicate their location, from which we see that the predictive variance of standard Bayes fluctuates, and is small around the data points and large elsewhere, whereas obviously, for every density $p_f$ in our model ${\ensuremath{\mathcal F}}$, the variance is fixed independently of $X$ — and thus $\bar{p}(Y_{n+1} \mid X_{n+1}, Z^n)$ is indeed very far from any particular $p_f$ with $f \in {\ensuremath{\mathcal F}}$. In contrast, for the generalized Bayesian lasso with $\eta = 0.25$, the corresponding predictive variance is almost constant, thus at the level $\eta = 0.25$, the predictive distribution is almost ‘in-model’ (in machine learning terminology, we may say that $\bar{p}$ is ‘proper’ [@shalev2014understanding]; and then the overfitting behavior does not occur anymore. ![Variance of Predictive Distribution $\bar{p}(Y_{n+1} \mid X_{n+1}, Z^n)$ for a single run with $n=50$.[]{data-label="fig:predvar"}](predvar-eps-converted-to.pdf){width="50.00000%" height="4.5cm"} When Generalized Bayes Concentrates {#sec:consistency} ----------------------------------- Having just seen bad behavior for $\eta=1$, we now recall some results from GM, who show that, under some conditions, generalized Bayes, for appropriately chosen $\eta$, does concentrate at fast rates even under misspecification. We first recall a (very special case) of the asymptotic behavior under misspecification theorem of GM. GM bound (a) the [*misspecification metric*]{} ${{d}}_{\bar\eta}$ in terms of (b) the [*information complexity*]{}. The bound (c) holds under a simple condition on the learning problem that was termed the [*central condition*]{} by @erven2015fast. Before presenting the theorem we explain (a)-(c). As to (a), we define the [ *misspecification metric*]{} ${{d}}_{\bar\eta}$ in terms of its square by $$\begin{aligned} {{d}}^2_{\bar\eta}(f,f') \coloneqq \frac{2}{\bar{\eta}} \left(1- \int \sqrt{p_{f,\bar\eta}(z) p_{f',\bar\eta}(z)} \text{d} \mu(z) \right)\end{aligned}$$ which is the squared Hellinger distance between $p_{f,\bar\eta}$ and $p_{f',\bar\eta}$ defined as $$\nonumber p_{f,\bar\eta}(z) \coloneqq p(z) \frac{\exp(-\bar\eta {L_{f}}(z))}{{\operatorname{\mathbf{E}}}[\exp(-\bar\eta {L_{f}}(Z))]},$$ where ${L_{f}} = {\ensuremath{\ell}}_f - {\ensuremath{\ell}}_{{{f^*}}}$ is the [*excess loss*]{} of $f$. GM show that ${{d}}_{\bar\eta}$ defines a metric for all $\bar\eta > 0$. If $\bar\eta=1$, $\ell$ is log-loss, and the model is well-specified, then it is straightforward to check $p_{f,\bar\eta} = p_f$ and ${{d}}_{\bar\eta}$ becomes the standard Hellinger metric. As to (b), we denote by ${\mathrm{IC}}_{n, \eta}(\Pi_0)$ the information complexity, defined as: $$\begin{aligned} {\mathrm{IC}}_{n,\eta}(\Pi_0) \coloneqq {\ensuremath{\mathbf E}}_{{\underline{f}} \sim {\ensuremath{\Pi}}_n} \left[ \frac{1}{n} \sum_{i=1}^n {L_{{\underline{f}}}(Z_i)} \right] + \frac{{\text{\sc KL}}( {\ensuremath{\Pi}}_n \operatorname*{\|}{\ensuremath{\Pi_0}})}{\eta \cdot n } = \nonumber \hspace*{-0.5 cm} \\ \ \hspace*{-2 cm} \ \label{eq:bayesmarginal} - \frac{1}{ \eta n} \log \int \pi_0(f) e^{- \eta \sum_{i=1}^n \ell_f(Z_i)} \text{d} \rho(f) - \sum_{i=1}^n \ell_{f^* }(Z_i),\end{aligned}$$ where ${\underline{f}}$ denotes the predictor sampled from the posterior $\Pi_n$; we suppress dependency of ${\mathrm{IC}}$ on ${{f^*}}$ in the notation. The fact that both lines above are equal (noticed by, among others, @zhang2006information; GM give an explicit proof) allows us to write the information complexity in terms of a generalized Bayes predictive density which is also known as [ *extended stochastic complexity*]{} [@Yamanishi98]. It also plays a central role in the field of prediction with expert advice as the [*mix-loss*]{} [@erven2015fast; @CesaBianchiL06] and coincides with the minus log of the standard Bayes predictive density if $\eta=1$ and $\ell$ is log-loss. It can be thought of as a complexity measure analogous to VC dimension and Rademacher complexity. As to (c), GM’s result holds under the *central condition* ([@li1999estimation]; name due to [@erven2015fast]) which expresses that, for some fixed $\bar\eta > 0$, for all fixed $f$, the probability that its loss exceeds that of the optimal $f^*$ by $a/\bar\eta$ is exponentially small in $a$: \[def:central\] Let $\bar{\eta} > 0$. We say that ${({P},{\ensuremath{\ell}},{\mathcal{F}})}$ satisfies the *$\bar\eta$-strong central condition* if for all $f \in {\ensuremath{\mathcal F}}$: ${\operatorname{\mathbf{E}}}\left[e^{-\bar\eta {L_{f}}} \right] \leq 1$. As straightforward rewriting shows, this condition holds [ *automatically*]{}, for any $\bar\eta \leq 1$ in the density estimation setting, if the model is correct; @erven2015fast provide some other cases in which it holds, and show that many other conditions on $\ell$ and $P$ that allow fast rate convergence that have been considered before in the statistical and on-line learning literature, such as [*exp-concavity*]{} [@CesaBianchiL06], the [*Tsybakov*]{} and [*Bernstein*]{} conditions [@bartlett2005local; @Tsybakov04] and several others, can be viewed as special cases of the central condition; yet they don’t discuss GLMs. Here is GM’s result: \[thm:metric\] Suppose that the $\bar{\eta}$-strong central condition holds. Then for any $0 < \eta < \bar\eta$, the metric ${{d}}_{\bar\eta}$ satisfies $$\begin{aligned} {\operatorname{\mathbf{E}}}_{Z^n \sim P} {\operatorname{\mathbf{E}}}_{{\underline{f}} \sim {\ensuremath{\Pi}}_n} \left[ {{d}}^2_{\bar\eta}({{f^*}}, {\underline{f}}) \right] \leq C_{\eta} \cdot {\operatorname{\mathbf{E}}}_{Z^n \sim P}\left[ {\mathrm{IC}}_{n,\eta}(\Pi_0) \right] \end{aligned}$$ with $C_{\eta} = \eta / (\bar\eta- \eta)$. In particular, $C_{\eta} < \infty$ for $0 < \eta < \bar\eta$, and $C_{\eta} = 1$ for $\eta = \bar\eta/2$. Thus, we expect the posterior to concentrate at a rate dictated by ${\bf E}[{\mathrm{IC}}_{n,\eta}]$ in neighborhoods of the best (risk-minimizing, KL optimal, or even true regression function) $f^*$. The misspecification metric ${{d}}^2_{\bar\eta}$ on the left hand side is a weak metric, however, in Appendix \[sec:witness\] we show that we can replace it by stronger notions such as KL-divergence, squared error or logistic loss. Theorem \[thm:metric\] is a generalization to the the misspecified setting of previous results (e.g. @zhang2006epsilon [@zhang2006information]). In the well-specified case, Zhang, as well as several other authors [@WalkerH02; @martin2017empirical], state a result that holds for any $\eta < 1$ but not $\eta=1$. This shows that there is an advantage to taking $\eta$ slightly smaller than one even when the model is well-specified (for more details see @zhang2006epsilon). To make the theorem work for GLMs under misspecification, we must verify (a) that the central condition still holds (which is in general not guaranteed) and that (b) the information complexity is sufficiently small. As to (a), in the following section we show that the central condition holds (with $\bar\eta$ usually $\neq 1$) for $1$-dimensional exponential families and high-dimensional generalized linear models (GLMs) if the noise is misspecified, as long as $P$ has exponentially small tails, and we relate $\bar\eta$ to the variance of $P$. As to (b), if the model is correct (the conditional distribution $P(Y \mid X)$ has density $f$ equal to $p_f$ with $f \in {\ensuremath{\mathcal F}}$), where ${\ensuremath{\mathcal F}}$ represents a $d$-dimensional GLM, then it is known (see e.g. @zhang2006information) that, for any prior $\Pi_0$ with continuous, strictly positive density on ${\ensuremath{\mathcal F}}$, the information complexity satisfies $$\label{eq:bic} {\bf E}_{Z^n \sim P} \left[ {\mathrm{IC}}_{n,\eta}(\Pi_0)\right] = O \left( \frac{d}{n} \cdot \log n \right),$$ which leads to bounds within a log-factor of the minimax optimal rate (among all possible estimators, Bayesian or not), which is $O(d/n)$. While such results were only established for the well-specified case, in Proposition \[prop:tenerife\] below, we show that, for GLMs, they continue to hold for the misspecified case. Generalized GLM Bayes {#sec:genglm} ===================== Below we first show that the central condition holds for natural univariate exponential families; we then extend this result to the GLM case, and establish bounds in information complexity of GLMs. Let the class ${\ensuremath{\mathcal F}}= \{ p_{\theta}: \theta \in \Theta\}$ be a univariate natural exponential family of distributions on ${\ensuremath{\mathcal Z}}= {\ensuremath{\mathcal Y}}$, represented by their densities, indexed by natural parameter $\theta \in \Theta \subset {{\mathbb R}}$ [@BarndorffNielsen78]. The elements of this restricted family have probability density functions $$\begin{aligned} \label{eq:expfamdef} p_\theta(y) \coloneqq \exp( \theta y - F(\theta) + r(y) ) ,\end{aligned}$$ for log-normalizer $F$ and carrier measure $r$. We denote the corresponding distribution as $P_{\theta}$. In the first part of the theorem below we assume that $\Theta$ is restricted to an arbitrary closed interval $[\underline\theta,\bar\theta]$ with $\underline\theta < \bar\theta$ that resides in the interior of the natural parameter space $\bar{\Theta} = \{ \theta: F(\theta) < \infty\}$. Such $\Theta$ allow for a simplified analysis because within $\Theta$ the log-normalizer $F$ as well as all its derivatives are uniformly bounded from above and below; see in Appendix \[app:proofs\]. As is well-known (see e.g. @BarndorffNielsen78), exponential families can equivalently be parameterized in terms of the mean-value parameterization: there exists a $1$-to-$1$ strictly increasing function $\mu: \bar{\Theta} \rightarrow {{\mathbb R}}$ such that ${\operatorname{\mathbf{E}}}_{Y \sim P_{\theta}}[Y] = \mu(\theta)$. As is also well-known, the density $p_{{{f^*}}} \equiv p_{\theta^*}$ within ${\ensuremath{\mathcal F}}$ minimizing KL divergence to the true distribution $P$ satisfies $\mu(\theta^*) = {\bf E}_{Y\sim P}[Y]$, whenever the latter quantity is contained in $\mu(\Theta)$ [@Grunwald07]. In words, the best approximation to $P$ in ${\ensuremath{\mathcal F}}$ in terms of KL divergence has the same mean of $Y$ as $P$. \[thm:expfam-exptails\] Consider a learning problem ${({P},{\ensuremath{\ell}},{\mathcal{F}})}$ with ${\ensuremath{\ell}}_{\theta}(y) = - \log p_{\theta}(y)$ the log loss and ${\ensuremath{\mathcal F}}= \{p_{\theta}: \theta \in \Theta\}$ a univariate exponential family as above.\ (1). Suppose that $\Theta = [\underline\theta,\bar\theta]$ is compact as above and that $\theta^* = \arg\min_{\theta \in \bar\Theta} D(P\| P_{\theta})$ lies in $\Theta$. Let $\sigma^2> 0$ be the true variance ${\operatorname{\mathbf{E}}}_{Y \sim P}(Y - E[Y])^2$ and let $(\sigma^*)^2$ be the variance ${\operatorname{\mathbf{E}}}_{Y \sim P_{\theta^*}}(Y - E[Y])^2$ according to $\theta^*$. Then (i) for all $\bar\eta > (\sigma^*)^2/\sigma^2$, the $\bar\eta$-central condition does [ *not*]{} hold. (ii) Suppose there exists $\eta^{\circ} > 0$ such that [$\bar{C} \coloneqq {\operatorname{\mathbf{E}}}_P[\exp(\eta^{\circ} |Y|)] < \infty$.]{} Then there exists $\bar\eta > 0$, depending only on $\eta^{\circ}$, $\bar{C}, \underline{\theta}$ and $\overline{\theta}$ such that the $\bar\eta$-central condition holds. Moreover, (iii), for all $\delta > 0$, there is an $\epsilon > 0$ such that, for all $\bar\eta \leq (\sigma^*)^2/\sigma^2 - \delta$, the $\bar{\eta}$-central condition holds relative to the restricted model ${\ensuremath{\mathcal F}}_{\epsilon} = \{p_{\theta}: \theta \in [\theta^*- \epsilon,\theta^*+\epsilon]$}. [*(2). Suppose that $P$ is Gaussian with variance $\sigma^2> 0$ and that ${\ensuremath{\mathcal F}}$ indexes a full Gaussian location family. Then $\bar\eta$-central holds iff $\bar\eta \leq (\sigma^*)^2/\sigma^2$.* ]{} We provide (iii) just to give insight — ‘locally’, i.e. in restricted models that are small neighborhoods around the best-approximating $\theta^*$, the smallest $\bar\eta$ for which central holds is determined by a ratio of variances. The final part shows that for the Gaussian family, the same holds not just locally but globally (note that we do not make the compactness assumption on $\Theta$ there); we warn the reader though that the standard posterior ($\eta=1$) based on a model with fixed variance $\sigma^*$ is quite different from the generalized posterior with $\eta = (\sigma^*)^2/\sigma^2$ and a model with variance $\sigma^2$ [@grunwald2017inconsistency]. Finally, while in practical cases we often find $\bar\eta < 1$ (suggesting that Bayes may only succeed if we learn ‘slower’ than with the standard $\eta =1$, i.e. the prior becomes more important), the result shows that we can also very well have $\bar\eta > 1$; a practical example is given at the end of Section \[sec:examples\]. Theorem \[thm:expfam-exptails\] is new and supplements the various examples of ${\ensuremath{\mathcal F}}$ which satisfy the central condition given by . In the proposition we require that both tails of $Y$ have exponentially small probability. #### Central Condition: GLMs Let ${\ensuremath{\mathcal F}}$ be the generalized linear model [@McCullaghN89] (GLM) indexed by parameter $\beta \in {\ensuremath{\mathcal B}}\subset {{\mathbb R}}^d$ with link function $g: {{\mathbb R}}\rightarrow {{\mathbb R}}$. By definition this means that there exists a set ${\ensuremath{\mathcal X}}\subset {{\mathbb R}}^{d}$ and a univariate exponential family ${\ensuremath{\mathcal Q}}= \{ p_{\theta} : \theta \in \bar\Theta \}$ on ${\ensuremath{\mathcal Y}}$ of the form such that the conditional distribution of $Y$ given $X=x$ is, for all possible values of $x \in {\ensuremath{\mathcal X}}$, a member of the family ${\ensuremath{\mathcal Q}}$, with mean-value parameter $g^{-1}(\langle \beta, x \rangle)$. Then the class ${\ensuremath{\mathcal F}}$ can be written as ${\ensuremath{\mathcal F}}= \{ p_\beta: \beta \in \mathcal{B}\}$, a set of conditional probability density functions such that $$\begin{aligned} p_\beta(y \mid x) \coloneqq \exp \bigl( \theta_x(\beta) y - F(\theta_x(\beta)) + r(y) \bigr), \label{eq:toulon}\end{aligned}$$ where $\theta_x(\beta) \coloneqq \mu^{-1}(g^{-1}(\langle \beta, x \rangle))$, and $\mu^{-1}$, the inverse of $\mu$ defined above, sends mean parameters to natural parameters. We then have ${\operatorname{\mathbf{E}}}_{P_\beta} [ Y \mid X ] = g^{-1}(\langle \beta, X \rangle)$, as required. \[prop:tenerife\] Under the following three assumptions, the learning problem ${({P},{\ensuremath{\ell}},{\mathcal{F}})}$ with ${\ensuremath{\mathcal F}}$ as above satisfies the $\bar\eta$-central condition for some $\bar\eta>0$ depending only on the parameters of the problem: 1. (Conditions on $g$): the inverse link function $g^{-1}$ has bounded derivative on the domain ${\ensuremath{\mathcal B}}\times {\ensuremath{\mathcal X}}$, and the image of the inverse link on the same domain is a bounded interval in the interior of the mean-value parameter space $\{ \mu\in {{\mathbb R}}: \mu = {\operatorname{\mathbf{E}}}_{Y \sim q}[Y]\; :\; q \in {\ensuremath{\mathcal Q}}\}$ (for all standard link functions, this can be enforced by restricting ${\ensuremath{\mathcal B}}$ and ${\ensuremath{\mathcal X}}$ to an (arbitrarily large but still) compact domain). 2. (Condition on ‘true’ $P$): for some $\eta > 0$ we have\ ${\sup}_{x \in {\ensuremath{\mathcal X}}} {\operatorname{\mathbf{E}}}_{Y \sim P}[\exp(\eta | Y|) \mid X=x] < \infty$. 3. (Well-specification of conditional mean): there exists $\beta^{\circ} \in {\ensuremath{\mathcal B}}$ such that ${\operatorname{\mathbf{E}}}[ Y \mid X ] = g^{-1}(\langle \beta^{\circ}, X \rangle)$. A simple argument (differentiation with respect to $\beta$) shows that under the third condition, it must be the case that $\beta^{\circ} = \beta^*$, where $\beta^* \in {\ensuremath{\mathcal B}}$ is the index corresponding to the density $p_{{{f^*}}} \equiv p_{\beta^*}$ within ${\ensuremath{\mathcal F}}$ that minimizes KL divergence to the true distribution $P$. Thus, our conditions imply that ${\ensuremath{\mathcal F}}$ contains a $\beta^*$ which correctly captures the conditional mean (and this will then be the risk minimizer); thus, as is indeed the case in Example \[ex:basic\], the regression function must be well-specified but the noise can be severely misspecified. We stress that the three conditions have very different statuses. The first is mathematically convenient; it can be enforced by truncating parameters and data, which is awkward but may not lead to substantial deterioriation in practice. Whether it is even really needed or not is not clear (and may in fact depend on the chosen exponential family). The second condition is really necessary — as can immediately be seen from Definition \[def:central\], the strong central condition cannot hold if $Y$ has polynomial tails and for some $f$ and $x$, ${\ensuremath{\ell}}_f(x,Y)$ increases polynomially in $Y$ (in Section 6 of their paper, GM consider weakenings of the central condition that still work in such situations). For the third condition, however, we suspect that there are many cases in which it does not hold yet still the strong central condition holds; so then the GM convergence result would still be applicable under ‘full misspecification’; investigating this will be the subject of future work. #### GLM Information Complexity To apply Theorem \[thm:metric\] to get convergence bounds for exponential families and GLMs, we need to verify that the central condition holds (which we just did) and we need to bound the information complexity, which we proceed to do now. It turns out that the bound on ${\mathrm{IC}}_{n,\eta}$ of $O( (d/n) \log n)$ of (\[eq:bic\]) continues to hold unchanged under misspecification, as is an immediate corollary of applying the following proposition to the definition of ${\mathrm{IC}}_{n,\eta}$ given above (\[eq:bayesmarginal\]): \[prop:entroboundb\] Let ${({P},{\ensuremath{\ell}},{\mathcal{F}})}$ be a learning problem with ${\ensuremath{\mathcal F}}$ a GLM satisfying Conditions 1–3 above. Then for all $f \in {\ensuremath{\mathcal F}}$, $ {\operatorname{\mathbf{E}}}_{X,Y \sim P}[L_f] = {\operatorname{\mathbf{E}}}_{X,Y \sim P_{{{f^*}}}} [L_f]. $ This result follows almost immediately from the ‘robustness property of exponential families’ (Chapter 19 of @Grunwald07); for convenience we provide a proof in Appendix \[app:proofs\]. The result implies that any bound in ${\mathrm{IC}}_{n,\eta}(\Pi_0)$ for a particular prior in the well-specified GLM case, in particular (\[eq:bic\]), immediately transfers to the same bound for the misspecified case, as long as our regularity conditions hold, allowing us to apply Theorem \[thm:metric\] to obtain the parameteric rate for GLMs under misspecification. MCMC Sampling {#sec:sampling} ============= Below we devise MCMC algorithms for obtaining samples from the $\eta$-generalized posterior distribution for two problems: regression and classification. In the regression context we consider one of the most commonly used sparse parameter estimation techniques, the lasso. For classification we use the logistic regression model. In our experiments in Section \[sec:examples\], we compare the performance of generalized Bayesian lasso with Horseshoe regression [@horseshoe]. The derivations of samplers are given in Appendix \[app:impl\]. Bayesian lasso regression {#sec:lassosamp} ------------------------- Consider the regression model $Y= X \beta+{\varepsilon}$, where $\beta\in{\mathbb{R}}^p$ is the vector of parameters of interest, , , and is a noise vector. The Least Absolute Shrinkage and Selection Operator (LASSO) of @Lasso is a regularization method used in regression problems for shrinkage and selection of features. The lasso estimator is defined as $$\begin{aligned} \label{eq:betalasso} {\hat{\beta}}_{\text{lasso}} \coloneqq \operatorname*{arg\,min}_{{\beta}} \| Y - X{\beta}\|_2^2 + \lambda \|\beta\|_1\,.\end{aligned}$$ where $\|\cdot\|_1,\|\cdot\|_2$ are $l_1$ and $l_2$ norms correspondingly. It can be interpreted as a Bayesian posterior mode (MAP) estimate when the priors on $\beta$ are given by independent Laplace distributions. As discovered by @Park, the same posterior on $\beta$ is also obtained by the following Gibbs sampling scheme: set $\eta = 1$ and denote $ {D_\tau}\coloneqq{{\text{diag}}}(\tau_1,\dots,\tau_n).$ Also, let $a\coloneqq\frac{\eta}{2}(n-1) + \frac p2 + \alpha$ and $b_\tau\coloneqq\frac{\eta}{2}( {{Y}-X\beta})^T ( {{Y}-X\beta}) + \frac12 {\beta}^T {D_\tau}^{-1} {\beta} + \gamma$, where $\alpha,\gamma>0$ are hyperparameters. Then the Gibbs sampler is constructed as follows. $$\begin{aligned} \begin{split} {\beta} \sim &\,{\ensuremath{\mathcal N}}\left( \eta M_\tau {X}^T {{Y}}, \sigma^2M_\tau \right), \end{split} \\ \begin{split} \sigma^2 \sim &\,\text{Inv-Gamma} \left( a, b_\tau \right), \end{split} \quad {\tau^{-2}_j} \sim \,\text{IG}\left(\displaystyle \sqrt{{\lambda^2\sigma^2}/{\beta_j^2}}, \lambda^2\right),\end{aligned}$$ where IG is the inverse Gaussian distribution and $M_\tau\coloneqq(\eta {X}^T {X} + {D_\tau}^{-1})^{-1}$. Following @Park, we put a Gamma prior on the shrinkage parameter $\lambda$. Now, in their paper @Park only give the scheme for $\eta = 1$ but as is straightforward to derive from their paper, the scheme above actually gives the $\eta$-[*generalized*]{} posterior corresponding to the lasso prior for general $\eta$ (more details in Appendix \[app:impl\]). We will use the Safe-Bayesian algorithm for choosing the optimal $\eta$ developed by @grunwald2017inconsistency (see Appendix \[ap:safebayesimpl\]). #### Horseshoe estimator The Horseshoe prior is the state-of-the-art global-local shrinkage prior for tackling high-dimensional regularization, introduced by @horseshoe. Unlike the Baysian lasso it has flat Cauchy-like tails, which allow strong signals to remain unshrunk a posteriori. For completeness we include the horseshoe in our regression comparison, using the implementation of @vanderpas2016horseshoe. Bayesian logistic regression ---------------------------- Consider the standard logistic regression model $\{ f_{\beta}: \beta \in {{\mathbb R}}^{p}\}$, the data $Y_1,\dots, Y_n\in\{0,1\}$ are independent binary random variables observed at the points $X_1,\dots,X_n\in \mathbb{R}^p$ with $$\nonumber P_{f_{\beta}}(Y_i=1 \mid X_i) \coloneqq p_{f_{\beta}}(1 \mid X_i) \coloneqq \frac{e^{X_i^T\beta}}{1+e^{X_i^T\beta}}.$$ The standard Bayesian approach involves putting a Gaussian prior on the parameter $\beta\sim {\ensuremath{\mathcal N}}(b,B)$ with mean $b\in{\mathbb{R}}^p$ and the covariance matrix $B\in{\mathbb{R}}^{p\times p}$. To sample from the $\eta$-generalised posterior we modify a P[ó]{}lya–Gamma latent variable scheme sampler described in @polson2013bayesian. Let $\kappa\coloneqq(Y_1-1/2,\dots, Y_n-1/2)^T$, ${V_\omega\coloneqq(X^T\Omega X+B^{-1})^{-1}}$, and $ {m_\omega\coloneqq V_\omega(\eta X^T\kappa+B^{-1}b)}$. Then the Gibbs sampler for $\eta$-generalized posterior is given by $$\begin{aligned} \omega_i\sim \text{PG}(\eta, X_i^T\beta), \quad \beta\sim {\ensuremath{\mathcal N}}(m_\omega, V_\omega), \end{aligned}$$ where PG is the P[ò]{}lya-Gamma distribution. Experiments {#sec:examples} =========== Below we present the results of experiments that compare the performance of the derived Gibbs samplers with their standard counterparts. More details/experiments are in Appendix \[ap:detailsfigures\]. Simulated data {#sec:lassoandhs} -------------- #### Regression In our experiments we focus on prediction, and we run simulations to determine the *square-risk* (expected squared error loss) of our estimate relative to the underlying distribution $P$: , where $X\beta$ would be the conditional expectation, and thus the square-risk minimizer, if $\beta$ would be the true parameter (vector). Consider the data generated as described in Example \[ex:basic\]. We study the performance of the $\eta$-generalised Bayesian lasso with $\eta$ chosen by the SafeBayes algorithm (we call it the Safe-Bayesian lasso) in comparison with two popular estimation procedures for this context: the Bayesian lasso (which corresponds to $\eta$=1), and the Horseshoe method. In Figure \[fig:risk-simulaties resultaten\] the empirical square-risk is plotted as a function of the sample size for all three methods. We average over enough samples so that the graph appears to be smooth ($25$ iterations for SafeBayes, $1000$ for the two standard Bayesian methods). It shows that both the standard Bayesian lasso and the Horseshoe perform significantly worse than the Safe-Bayesian lasso. Moreover we see that the risks for the standard methods initially grows with the sample size (additional experiments not reported here suggest that Bayes will ‘recover’ at very large $n$). ![Empirical squared error risk with respect to $P$ as function of sample size for the *wrong-model* experiments of Section \[sec:lassoandhs\] according to the posterior predictive distribution of the standard Bayesian lasso (green, solid), the Safe-Bayesian lasso (red, dotted), both with standard improper priors, and the Horseshoe (blue, dashed); and $201$ Fourier basis functions. \[fig:risk-simulaties resultaten\]](RegressionRiskPlot.pdf){width="45.00000%"} #### Classification {#sec:logisticregression} We focus on finding coefficients $ \beta$ for prediction, and our error measure is the expected logarithmic loss, which we call *log-risk*: , where $ \text{Li}_\beta(Y{\, |\, }X)\coloneqq {e^{YX^T\beta}}/({1+e^{X^T\beta}})$. We start with an example that is very similar to the previous one. We generate a $n\times p$ matrix of independent standard normal random variables with $p=25$. For every feature vector $ X_i$ we sample a corresponding $Z_i \sim {\ensuremath{\mathcal N}}(0, \sigma^2)$, as before, and we misspecify the model by putting approximately half of the $Z_i$ and the corresponding $X_{i,1}$ to zero. Next, we sample the labels $Y_i \sim \text{Binom}(\exp(Z_i) / (1 + \exp(Z_i))$. We compare standard Bayesian logistic regression ($\eta=1$) to a generalized version ($\eta = 0.125$). In Figure \[fig:logregrisk\] we plot the empirical log-risk as a function of the sample size. As in the regression case, the risk for standard Bayesian logistic regression ($\eta=1$) is substantially worse than the one for generalized Bayes ($\eta=0.125$). Even for generalized Bayes, the risk initially goes up a little bit, the reason being that the prior is [*too good*]{}: it is strongly concentrated around the risk-optimal $\beta^* =0$. Thus, the first prediction made by the Bayes predictive distribution coincides with the optimal $(\beta=0)$ prediction, and in the beginning, due to noise in the data, predictions will first get slightly worse. This is a phenomenon that also applies to standard Bayes with well-specified models; see for example [@GrunwaldH04 Example 3.1]. ![Empirical logistic risk as function of sample size for the *wrong-model* experiments of Section \[sec:logisticregression\] according to the posterior predictive distribution of standard Bayesian logistic regression (green, solid), and generalized Bayes ($\eta=0.125$, red, dotted) with $25$ noise dimensions. \[fig:logregrisk\]](Logit.pdf){width="40.00000%"} Even for the well-specified case it can be beneficial to use $\eta\neq 1$. It is easy to see that the maximum [*a posteriori*]{} for the generalized logistic regression corresponds to the ridge logistic regression method (which penalises large $\|\beta\|_2$) with the shrinkage parameter $\lambda=\eta^{-1}$. However, when the the prior mean is zero but the risk minimizer $\beta^*$ is far from zero, penalising large norms of $\beta$ is inefficient, and we find that the best performance is achieved with $\eta>1$. Real World Data {#sec:real} --------------- We present two examples with real world data to demonstrate that bad behavior under misspecification also occurs in practice. For these datasets, we compare the performance of Safe-Bayesian lasso and standard Bayesian lasso. As the first example we consider the data of the daily maximum temperatures at Seattle Airport as a function of the time and date (source: $\texttt{R}$-package $\texttt{weatherData}$, also available at $\texttt{www.wunderground.com}$). A second example is London air pollution data (source: $\texttt{R}$-package $\texttt{Openair}$, for more details see @openair1 [@openair2]). Here the quantity of interest is the concentration of nitrogen dioxide (NO$_2$), again as a function of time and date. In both settings we divide the data in a training set and a test set, and focus on the prediction. In both examples, SafeBayes picks an $\hat\eta$ strictly smaller than one. Also, for both data sets the SafeBayes lasso substantially outperforms the standard Bayesian lasso and the Horseshoe in terms of mean square prediction error, as seen from Table \[table:seattleandlondon\] (details in Appendix \[ap:detailsfigures\]). [0.48]{}[| l | X | X | X |]{} & Horse-shoe & Bayesian lasso & SafeBayes lasso\ MSE ($(^{\circ} \text{C})^2$) & $6.53$ & $6.16$ & $6.04$\ MSE ($(\text{ppm})^2$) & $1169$ & $1201$ & $1142$\ Future work =========== We provide both theoretical and empirical evidence that $\eta$-generalized Bayes can significantly outperform standard Bayes for GLMs. However, the empirical examples are only given for Bayesian lasso linear regression and logistic regression. In future work we would like to devise generalized posteriors samplers for other GLMs and speed up the sampler for the generalized Bayesian logistic regression, since our current implementation is slow, and (unlike our linear regression implementation) cannot deal with high-dimensional (and thus, real-world) data yet. Furthermore, the SafeBayes algorithm of [@Grunwald12], used to learn $\eta$, enjoys good theoretical performance but is computationally very slow. Since learning $\eta$ for which the central condition holds (preferably the largest possible value, since small values of $\eta$ mean slower learning) is essential for using generalized Bayes in practice, there is a necessity for speeding up SafeBayes or finding an alternative. A potential solution might be using cross-validation to learn $\eta$, but its theoretical properties (e.g.satisfying central condition) are yet to be established. \[3\][\#3]{} \[3\][\#2]{} Outline {#app:outline} ======= The appendix of this paper is organized as follows: - Appendix \[app:proofs\] provides the proofs for Section \[sec:genglm\]. - Appendix \[sec:witness\] shows how we can replace ${{d}}^2_{\bar\eta}$ in Theorem \[thm:metric\] by stronger notions. - Appendix \[app:impl\] provides (implementation) details on the $\eta$-generalized Bayesian lasso and logistic regression; and the Safe-Bayesian algorithm. - Appendix \[ap:detailsfigures\] contains details for the experiments and figures in the main text, and provides additional figures. Proofs {#app:proofs} ====== Proof of Theorem \[thm:expfam-exptails\] ---------------------------------------- The second part of the theorem about the Gaussian location family is a straightforward calculation, which we omit. As to the first part (Part (i)—(iii)), we will repeatedly use the following fact: for every $\Theta$ that is a nonempty compact subset of the interior of $\bar{\Theta}$, in particular for $\Theta = [\underline{\theta},\bar{\theta}]$ with $\underline\theta < \bar\theta$ both in the interior of $\overline{\Theta}$, we have: $$\label{eq:ineccsi} \begin{aligned} - \infty < \inf_{\theta \in \Theta} F(\theta) &< \sup_{\theta \in \Theta} F(\theta) < \infty \\ - \infty < \inf_{\theta \in \Theta} F'(\theta) &< \sup_{\theta \in \Theta} F'(\theta) < \infty \\ 0 < \inf_{\theta \in \Theta} F''(\theta) &< \sup_{\theta \in \Theta} F''(\theta) < \infty. \end{aligned}$$ Now, let $\theta, \theta^* \in \Theta$. We can write $$\begin{aligned} \label{eq:basecamp} {\operatorname{\mathbf{E}}}\left[ e^{-\eta ({{\ensuremath{\ell}}_\theta - {\ensuremath{\ell}}_{\theta^*}})} \right] = {\operatorname{\mathbf{E}}}_{Y \sim P} \left[ \left( \frac{p_\theta(Y)}{p_{\theta^*}(Y)} \right)^\eta \right] = \exp \left (-G(\eta(\theta- \theta^*)) + \eta F(\theta^*) - \eta F(\theta) \right). \end{aligned}$$ where $G(\lambda) = - \log {\operatorname{\mathbf{E}}}_{Y \sim P} \left[ \exp( \lambda Y)\right]$. If this quantity is $- \infty$ for all $\eta > 0$, then (i) holds trivially. If not, then (i) is implied by the following statement: $$\label{eq:limitvariance} \limsup_{\epsilon \rightarrow 0} \left\lbrace \eta: \text{for all $\theta \in [\theta^*- \epsilon,\theta^* + \epsilon]$}, \ {\operatorname{\mathbf{E}}}[\exp(\eta {L_{p_{\theta}}})] \leq 1 \right\rbrace = \frac{(\sigma^*)^2}{\sigma^2}. $$ Clearly, this statement also implies (iii). To prove (i), (ii) and (iii), it is thus sufficient to prove (ii) and (\[eq:limitvariance\]). We prove both by a second-order Taylor expansion (around $\theta^*$) of the right-hand side of . [*Preliminary Facts*]{}. By our assumption there is a $ \eta^{\circ}> 0 $ such that ${\operatorname{\mathbf{E}}}[\exp(\eta^\circ |Y|)] = \bar{C} < \infty$. Since $\theta^* \in \Theta = [\underline\theta,\overline\theta]$ we must have for every $0 < \eta < \eta^{\circ}/(2 |\overline\theta - \underline\theta|)$, every $\theta \in \Theta$, $$\label{eq:finite} \begin{aligned} {\operatorname{\mathbf{E}}}[ \exp(2 \eta (\theta - \theta^*) \cdot Y)] \leq {\operatorname{\mathbf{E}}}[ \exp(2 \eta | \theta - \theta^*| \cdot |Y|)] \leq {\operatorname{\mathbf{E}}}[ \exp(\eta^{\circ} (| \theta - \theta^*|/|\overline\theta - \underline\theta|) \cdot |Y|)] \leq \bar{C} < \infty. \end{aligned}$$ The first derivative of the right of is: $$\label{eq:ffirst} \eta {\operatorname{\mathbf{E}}}\left[ (Y - F'(\theta)) \exp \Bigl( \eta \bigl( (\theta - \theta^*) Y + F(\theta^*) - F(\theta) \bigr) \Bigr) \right] .$$ The second derivative is: $$\label{eq:fsecond} {\operatorname{\mathbf{E}}}\left[ \left( -\eta F''(\theta) + \eta^2 (Y - F'(\theta))^2 \right) \cdot \exp \Bigl( \eta \bigl( (\theta - \theta^*) Y + F(\theta^*) - F(\theta) \bigr) \Bigr) \right] .$$ We will also use the standard result [@Grunwald07; @BarndorffNielsen78] that, since we assume $\theta^* \in \Theta$, $$\label{eq:expfacts} \begin{aligned} {\operatorname{\mathbf{E}}}[Y] = {\operatorname{\mathbf{E}}}_{Y \sim P_{\theta^*}}[Y] = \mu(\theta^*); \qquad \text{for all $\theta \in \bar{\Theta}$:} \ F'(\theta) = \mu(\theta); \qquad F''(\theta) = {\operatorname{\mathbf{E}}}_{Y \sim P_{\theta}}(Y- E(Y))^2, \end{aligned}$$ the latter two following because $F$ is the cumulant generating function. [*Part (ii)*]{}. We use an exact second-order Taylor expansion via the Lagrange form of the remainder. We already showed there exist $\eta' > 0$ such that, for all $0 < \eta \leq \eta'$, all $\theta \in {\Theta}$, ${\operatorname{\mathbf{E}}}[\exp(2\eta (\theta- \theta^*) Y)] < \infty$. Fix any such $\eta$. For some $\theta' \in \left\{ (1 - \alpha) \theta + \alpha \theta^* \colon \alpha \in [0, 1] \right\}$, the (exact) expansion is: $$\nonumber \begin{multlined} {\operatorname{\mathbf{E}}}\left[ e^{-\eta ({{\ensuremath{\ell}}_\theta - {\ensuremath{\ell}}_{\theta^*}})} \right] = 1 + \eta (\theta - \theta^*) {\operatorname{\mathbf{E}}}\left[ Y - F'(\theta^*) \right] \label{eqn:d-cgf-is-mean} - \frac{\eta}{2} (\theta - \theta^*)^2 F''(\theta') \cdot {\operatorname{\mathbf{E}}}\left[ \exp \Bigl( \eta \bigl( (\theta' - \theta^*) Y + F(\theta^*) - F(\theta') \bigr) \Bigr) \right] \nonumber \\ + \frac{\eta^2}{2} (\theta - \theta^*)^2 {\operatorname{\mathbf{E}}}\left[ (Y - F'(\theta'))^2 \right. \cdot \left. \exp \Bigl( \eta \bigl( (\theta' - \theta^*) Y + F(\theta^*) - F(\theta') \bigr) \Bigr) \right] . \nonumber \end{multlined}$$ Defining $\Delta = \theta' - \theta$, and since $F'(\theta^*) = {\operatorname{\mathbf{E}}}[ Y ]$ (see ), we see that the central condition is equivalent to the inequality: $$\begin{aligned} \eta {\operatorname{\mathbf{E}}}\left[ (Y - F'(\theta'))^2 e^{\eta \Delta Y} \right] \leq F''(\theta') {\operatorname{\mathbf{E}}}\left[ e^{\eta \Delta Y} \right] . \end{aligned}$$ From Cauchy-Schwarz, to show that the $\eta$-central condition holds it is sufficient to show that $$\begin{aligned} \eta \left \| (Y - F'(\theta'))^2 \right\|_{L_2(P)} \left \| e^{\eta \Delta Y} \right\|_{L_2(P)} \leq F''(\theta') {\operatorname{\mathbf{E}}}\left[ e^{\eta \Delta Y} \right] , \end{aligned}$$ which is equivalent to $$\label{eqn:eta-expfam-upper-bound} \begin{aligned} \eta \leq \frac{F''(\theta') {\operatorname{\mathbf{E}}}\left[ e^{\eta \Delta Y} \right]} { \sqrt{{\operatorname{\mathbf{E}}}\left [ (Y - F'(\theta'))^4 \right] {\operatorname{\mathbf{E}}}\left[e^{2 \eta \Delta Y} \right]} } . \end{aligned}$$ We proceed to lower bound the RHS by lower bounding each of the terms in the numerator and upper bounding each of the terms in the denominator. We begin with the numerator. $F'(\theta)$ is bounded by . Next, by Jensen’s inequality, $${\operatorname{\mathbf{E}}}\left[ \exp({\eta \Delta Y}) \right] \geq \exp( {\operatorname{\mathbf{E}}}[\eta \Delta \cdot Y ])\geq \exp(- \eta^{\circ} |\overline\theta - \underline\theta| | \mu( \theta^*)| )$$ is lower bounded by a positive constant. It remains to upper bound the denominator. Note that the second factor is upper bounded by the constant $\bar{C}$ in . The first factor is bounded by a fixed multiple of ${\operatorname{\mathbf{E}}}|Y|^4 + {\operatorname{\mathbf{E}}}[F'(\theta)^4]$. The second term is bounded by , so it remains to bound the first term. By assumption ${\operatorname{\mathbf{E}}}[\exp(\eta^{\circ} |Y|)] \leq \bar{C}$ and this implies that ${\operatorname{\mathbf{E}}}|Y^4| \leq a^4 + \bar{C}$ for any $a \geq e$ such that $a^4 \leq \exp(\eta^{\circ} a)$; such an $a$ clearly exists and only depends on $\eta^{\circ}$. We have thus shown that the RHS of is upper bounded by a quantity that only depends on $\bar{C}, \eta^{\circ}$ and the values of the extrema in , which is what we had to show. [*Proof of (iii)*]{}. We now use the asymptotic form of Taylor’s theorem. Fix any $\eta > 0$, and pick any $\theta$ close enough to $\theta^*$ so that is finite for all $\theta'$ in between $\theta$ and $\theta^*$; such a $\theta \neq \theta^*$ must exist since for any $\delta > 0$, if $|\theta - \theta^*| \leq \delta$, then by assumption must be finite for all $\eta \leq \eta^\circ/\delta$. Evaluating the first and second derivative and at $\theta = \theta^*$ gives: $$\begin{multlined} {\operatorname{\mathbf{E}}}\left[ e^{-\eta ({{\ensuremath{\ell}}_\theta - {\ensuremath{\ell}}_{\theta^*}})} \right] = 1 + \eta (\theta - \theta^*) {\operatorname{\mathbf{E}}}\left[ Y - F'(\theta^*) \right] - \left( \frac{\eta}{2} (\theta - \theta^*)^2 F''(\theta^*) - \frac{\eta^2}{2} (\theta - \theta^*)^2 \cdot {\operatorname{\mathbf{E}}}\left[ (Y - F'(\theta^*))^2 \right] \right) \\ + h(\theta)(\theta - \theta^*)^2 = 1 - \frac{\eta}{2} (\theta - \theta^*)^2 F''(\theta^*) + \frac{\eta^2}{2} (\theta - \theta^*)^2 {\operatorname{\mathbf{E}}}\left[ (Y - F'(\theta^*))^2 \right] + h(\theta) (\theta - \theta^*)^2, \end{multlined}$$ where $h(\theta)$ is a function satisfying $\lim_{\theta \rightarrow \theta^*} h(\theta) = 0$, where we again used , i.e. that $F'(\theta^*) = {\operatorname{\mathbf{E}}}\left[Y\right]$. Using further that $\sigma^2 = {\operatorname{\mathbf{E}}}\left[ (Y - F'(\theta^*))^2 \right]$ and $F''(\theta^*) = (\sigma^*)^2$, we find that ${\operatorname{\mathbf{E}}}\left[ e^{-\eta ({{\ensuremath{\ell}}_\theta - {\ensuremath{\ell}}_{\theta^*}})} \right] \leq 1$ iff $$- \frac{\eta}{2} (\theta - \theta^*)^2 (\sigma^*)^2 + \frac{\eta^2}{2} (\theta - \theta^*)^2 \sigma^2 + h(\theta) (\theta - \theta^*)^2 \leq 0.$$ It follows that for all $\delta > 0$, there is an $\epsilon > 0$ such that for all $\theta \in [\theta^*- \epsilon,\theta^* + \epsilon]$, all $\eta > 0$, $$\begin{aligned} \frac{\eta^2}{2} \sigma^2 \leq \frac{\eta}{2} (\sigma^*)^2 - \delta & \Rightarrow {\operatorname{\mathbf{E}}}\left[ e^{-\eta ({{\ensuremath{\ell}}_\theta - {\ensuremath{\ell}}_{\theta^*}})} \right] \leq 1 \label{eq:goodcondition} \\ \label{eq:badcondition} \frac{\eta^2}{2} \sigma^2 \geq \frac{\eta}{2} (\sigma^*)^2 + \delta & \Rightarrow {\operatorname{\mathbf{E}}}\left[ e^{-\eta ({{\ensuremath{\ell}}_\theta - {\ensuremath{\ell}}_{\theta^*}})} \right] \geq 1\end{aligned}$$ The condition in is implied if: $$0 < \eta \leq \frac{(\sigma^*)^2}{\sigma^2} - \frac{2 \delta}{\eta \sigma^2}.$$ Setting $C = 4 \sigma^2/(\sigma^*)^4$ and $\eta_{\delta} = (1- C {\delta}) (\sigma^*)^2/\sigma^2$ we find that for any $\delta < (\sigma^*)^4/(8 \sigma^2)$, we have $1- C {\delta} \geq 1/2$ and thus $\eta_{\delta} > 0$ so that in particular the premise in is satisfied for $\eta_{\delta}$. Thus, for all small enough $\delta$, both the premise and the conclusion in hold for $\eta_{\delta} > 0$; since $\lim_{\delta \downarrow 0} \eta_{\delta} = (\sigma^*)^2/\sigma^2$, it follows that there is an increasing sequence $\eta_{(1)}, \eta_{(2)}, \ldots$ converging to $(\sigma^*)^2/\sigma^2$ such that for each $\eta_{(j)}$, there is $\epsilon_{(j)} > 0$ such that for all $\theta \in [\theta^*- \epsilon_{(j)}, \theta^*+\epsilon_{(j)}]$, ${\operatorname{\mathbf{E}}}\left[ e^{-\eta_{(j)} ({{\ensuremath{\ell}}_\theta - {\ensuremath{\ell}}_{\theta^*}})} \right] \leq 1$. It follows that the $\lim \sup$ in is at least $(\sigma^*)^2/\sigma^2$. A similar argument (details omitted) using shows that the $\lim \sup$ is at most this value; the result follows. Proof of Proposition \[prop:entroboundb\] ----------------------------------------- For arbitrary conditional densities $p'(y \mid x)$ with corresponding distribution $P'\mid X$ for which $$\label{eq:daling} {\operatorname{\mathbf{E}}}_{P'}[Y |X] = g^{-1}(\langle \beta,X),$$ and densities $p_{{{f^*}}} = p_{\beta^*}$ and $p_\beta$ with $\beta^*,\beta \in {\ensuremath{\mathcal B}}$, we can write: $$\begin{aligned} {\operatorname{\mathbf{E}}}_{X\sim P} {\operatorname{\mathbf{E}}}_{Y \sim P'\mid X}\left[ \log \frac{p_{\beta^*}(Y \mid X)}{p_{\beta}(Y | X)}\right] &= {\operatorname{\mathbf{E}}}{\operatorname{\mathbf{E}}}\left[ (\theta_X(\beta^*) - \theta_X(\beta) ) Y - \log \frac{F(\theta_X(\beta^*))}{F(\theta_X(\beta))} \mid X \right]\\ &= {\operatorname{\mathbf{E}}}_{X \sim P} \left[ (\theta_X(\beta^*) - \theta_X(\beta) )g^{-1}(\langle \beta, X\rfloor_d \rangle - \log F(\theta_X(\beta^*)) + \log F(\theta_X(\beta)) \mid X \right], \end{aligned}$$ where the latter equation follows by . The result now follows because both holds for the ‘true’ $P$ and for $P_{{{f^*}}}$. Proof of Proposition \[prop:tenerife\] -------------------------------------- The fact that under the three imposed conditions the $\bar\eta$-central condition holds for some $\bar\eta > 0$ is a simple consequence of Theorem \[thm:expfam-exptails\]: Condition 1 implies that there is some compact $\Theta$ such that for all $x \in {\ensuremath{\mathcal X}}$, $\beta \in {\ensuremath{\mathcal B}}$, $\theta_x(\beta) \in \Theta$. Condition 3 then ensures that $\theta_x(\beta)$ lies in the interior of this $\Theta$. And Condition 2 implies that $\bar\eta$ in Theorem \[thm:expfam-exptails\] can be chosen uniformly for all $x \in {\ensuremath{\mathcal X}}$. Excess risk and KL divergence instead of generalized Hellinger distance {#sec:witness} ======================================================================= The misspecification metric/generalized Hellinger distance ${{d}}_{\bar\eta}$ appearing in Theorem \[thm:metric\] is rather weak (it is ‘easy’ for two distributions to be close) and lacks a clear interpretation for general, non-logarithmic loss functions. Motivated by these facts, GM study in depth under what additional conditions the (square of this) metric can be replaced by a stronger and more readily interpretable divergence measure. They come up with a new, surprisingly weak condition, the [*witness condition*]{}, under which ${{d}}_{\bar\eta}$ can be replaced by the [*excess risk*]{} ${\bf E}_P[L_f]$, which is the additional risk incurred by $f$ as compared to the optimal $f^*$. For example, with the squared error loss, this is the additional mean square error of $f$ compared to $f^*$; and with (conditional) log-loss, it is the well-known [*generalized KL divergence*]{} ${\bf E}_{X,Y \sim P}[\log \frac{p_{f^*}(Y \mid X)}{p_f(Y|X)}]$, coinciding with standard KL divergence if the model is correctly specified. Bounding the excess risk is a standard goal in statistical learning theory; see for example [@bartlett2005local; @erven2015fast]. The following definition appears (with substantial explanation including the reason for its name) as Definition 12 in GM: \[def:witness\] We say that ${({P},{\ensuremath{\ell}},{\mathcal{F}})}$ satisfies the $(u,c)$-*empirical witness of badness condition* (or *witness condition*) for constants $u > 0$ and $c \in (0,1]$ if for all $f \in {\ensuremath{\mathcal F}}$ $$\begin{aligned} {\operatorname{\mathbf{E}}}\left[ ({{\ensuremath{\ell}}_{f} - {\ensuremath{\ell}}_{{{f^*}}}}) \cdot {\mathop{{\bf 1}_{\{{{\ensuremath{\ell}}_{f} - {\ensuremath{\ell}}_{{{f^*}}}} \leq u\}}}} \right] \geq c {{\operatorname{\mathbf{E}}}[ {{\ensuremath{\ell}}_{f} - {\ensuremath{\ell}}_{{{f^*}}}} ]} . \end{aligned}$$ More generally, for a function $\tau: {{\mathbb R}}^+ \to [1,\infty)$ and constant $c \in (0,1)$ we say that ${({P},{\ensuremath{\ell}},{\mathcal{F}})}$ satisfies the *$(\tau,c)$-witness condition* if for all $f \in {\ensuremath{\mathcal F}}$, ${{\operatorname{\mathbf{E}}}[ {{\ensuremath{\ell}}_{f} - {\ensuremath{\ell}}_{{{f^*}}}} ]} < \infty$ and $$\begin{aligned} {\operatorname{\mathbf{E}}}\left[ ({{\ensuremath{\ell}}_{f} - {\ensuremath{\ell}}_{{{f^*}}}}) \cdot {\mathop{{\bf 1}_{\{{{\ensuremath{\ell}}_{f} - {\ensuremath{\ell}}_{{{f^*}}}} \leq \tau({{\operatorname{\mathbf{E}}}[ {{\ensuremath{\ell}}_{f} - {\ensuremath{\ell}}_{{{f^*}}}} ]})\}}}} \right] \geq c {{\operatorname{\mathbf{E}}}[ {{\ensuremath{\ell}}_{f} - {\ensuremath{\ell}}_{{{f^*}}}} ]} . \end{aligned}$$ It turns out that the $(\tau,c)$-witness condition holds in many practical situations, including our GLM-under-misspecification setting. Before elaborating on this, let us review (a special case of) Theorem 12 of GM, which is the analogue of Theorem \[thm:metric\] but with the misspecification metric replaced by the excess risk. First, let, for arbitrary $0 < \eta < \bar\eta$, $c_u \coloneqq \frac{1}{c} \frac{\eta u + 1}{1 - \frac{\eta}{\bar{\eta}}}$. Note that for large $u$, $c_u$ is approximately linear in $u/c$. [**\[Specialization of Theorem 12 of GM\]**]{} Consider a learning problem ${({P},{\ensuremath{\ell}},{\mathcal{F}})}$. Suppose that the $\bar\eta$-strong central condition holds. If the $(u,c)$-witness condition holds, then for any $\eta \in (0,\bar\eta)$, $$\begin{aligned} \label{eq:same} {\bf E}_{Z^n \sim P} {\operatorname{\mathbf{E}}}_{{\underline{f}} \sim {\ensuremath{\Pi}}_n}\left[{{\operatorname{\mathbf{E}}}[ {L_{f}} ]}\right] \leq c_u \cdot {\bf E}_{Z^n \sim P} \left[ {\mathrm{IC}}_{n,\eta} \left( \Pi_0 \right) \right], \end{aligned}$$ with $c_u$ as above. If instead the $(\tau,c)$-witness condition holds for some nonincreasing function $\tau$ as above, then for any $\lambda > 0$, $$ {\bf E}_{Z^n \sim P} {\operatorname{\mathbf{E}}}_{{\underline{f}} \sim {\ensuremath{\Pi}}_n} \left[{{\operatorname{\mathbf{E}}}[ {L_{f}} ]} \right] \ \leq \ \lambda + c_{\tau(\lambda)} \cdot {\bf E}_{Z^n \sim P} \left[ {\mathrm{IC}}_{n,\eta} \left( \Pi_0\right)\right]. \nonumber$$ The actual theorem given by GM generalizes this to an in-probability statement for general (not just generalized Bayesian) learning methods. If the $(u,c)$-witness condition holds, then, as is obvious from (\[eq:same\]) and Theorem \[thm:metric\], the same rates can be obtained for the excess risk as for the squared misspecification metric. For the $(\tau,c)$-witness condition things are a bit more complicated; the following lemma (Lemma 16 of GM) says that, under an exponential tail condition, $(\tau,c)$-witness holds for a sufficiently ‘nice’ function $\tau$, for which we loose at most a logarithmic factor: \[lem:kl-hell-exp-tails\] Define $M_{\kappa} \coloneqq \sup_{f \in {\ensuremath{\mathcal F}}} {\operatorname{\mathbf{E}}}\left[ e^{\kappa {L_{f}}} \right]$ and assume that the excess loss $L_f$ has a uniformly exponential upper tail, i.e. $M_{\kappa} < \infty$. Then, for the map $\tau: x \mapsto 1 \operatorname*{\vee}\kappa^{-1} {\log \frac{2 M_\kappa }{\kappa x}} = O(1 \operatorname*{\vee}\log (1/x))$, the $(\tau,c)$-witness condition holds with $c = \nicefrac{1}{2}$. As an immediate consequence of this lemma, GM’s theorem above gives that for any $\eta \in (0,\bar\eta)$, (using $\lambda= 1/n$), there is $C_{\eta} < \infty$ such that $$\begin{aligned} \label{eqn:kl-hell-exp-tails-2} {\bf E}_{Z^n \sim P} {\operatorname{\mathbf{E}}}_{{\underline{f}} \sim {\ensuremath{\Pi}}_n} \left[ {{\operatorname{\mathbf{E}}}[ {L_{{\underline{f}}}} ]} \right] \leq \frac{1}{n} + C_{\eta} \cdot (\log n) \cdot {\bf E}_{Z^n \sim P} \left[{\mathrm{IC}}_{\eta,n} \left( {{f^*}}\operatorname*{\|}{{\ensuremath{\Pi}}_|}\right) \right],\end{aligned}$$ so our excess risk bound is only a log factor worse than the bound that can be obtained for the squared misspecifation metric in Theorem \[thm:metric\]. We now apply this to the misspecified GLM setting: #### Generalized Linear Models and Witness Recall that the central condition holds for generalized linear models under the three assumptions made in Proposition \[prop:tenerife\]. Let ${\ensuremath{\ell}}_\beta \coloneqq {\ensuremath{\ell}}_\beta(X,Y) = -\log p_\beta(Y \mid X)$ be the loss of action $\beta \in {\ensuremath{\mathcal B}}$ on random outcome $(X,Y) \sim P$, and let $\beta^*$ denote the risk minimizer over ${\ensuremath{\mathcal B}}$. The first two assumptions taken together imply, via , that there is a $\kappa > 0$ such that $$\begin{aligned} \sup_{\beta \in B} {\operatorname{\mathbf{E}}}_{X,Y \sim P}\left[ e^{\kappa ({\ensuremath{\ell}}_\beta - {\ensuremath{\ell}}_{\beta^*})} \right] &\leq \sup_{\beta \in {\ensuremath{\mathcal B}}, x \in {\ensuremath{\mathcal X}}} {\operatorname{\mathbf{E}}}_{Y \sim P \mid X=x}\left[ e^{\kappa ({\ensuremath{\ell}}_\beta - {\ensuremath{\ell}}_{\beta^*})} \right] \\ &= \sup_{\beta \in {\ensuremath{\mathcal B}}, x \in {\ensuremath{\mathcal X}}} \left( \frac{F_{\theta_x(\beta)}}{F_{\theta_x(\beta^*)}}\right)^{\kappa} \cdot {\operatorname{\mathbf{E}}}_{Y \sim P \mid X=x}\left[ e^{\kappa | Y| }\right] < \infty. \end{aligned}$$ The conditions of Lemma \[lem:kl-hell-exp-tails\] are thus satisfied, and so the $(\tau,c)$-witness condition holds for the $\tau$ and $c$ in that lemma. From we now see that we get an $O((\log n)^2/n)$ bound on the expected excess risk, which is equal to the parametric (minimax) rate up to a $(\log n)^2$ factor. Thus, fast learning rates in terms of excess risks and KL divergence under misspecification with GLMs are possible under the conditions of Proposition \[prop:tenerife\]. MCMC Sampling {#app:impl} ============= The $\eta$-generalized Bayesian lasso ------------------------------------- Here, following @Park we consider a slightly more general version of the regression problem: $$Y=\mu+ X \beta+{\varepsilon},$$ where $\mu\in{\mathbb{R}}^n$ is the overall mean, $\beta\in{\mathbb{R}}^p$ is the vector of parameters of interest, $y\in{\mathbb{R}}^n$, $X\in{\mathbb{R}}^{n\times p}$, and ${\varepsilon}\sim N(0,\sigma^2 I_n)$ is a noise vector. For a given shrinkage parameter $\lambda>0$ the Bayesian lasso of @Park can be represented as follows. $$\begin{aligned} \label{eq:hiermodelbayesianlassopark} {Y} \rvert \mu, {X}, {\beta}, \sigma^2 &\sim N(\mu + {X\beta}, \sigma^2 {I_n}) \, , \\ \nonumber {\beta} \rvert \tau_1^2, \ldots, \tau_p^2, \sigma^2 &\sim N( {0}, \sigma^2 {D}_\tau), \, \, \, \, {D}_\tau = {{\text{diag}}}(\tau_1^2, \ldots, \tau_p^2) \, , \\ \nonumber \tau_1^2, \ldots, \tau_p^2 &\sim \prod_{j=1}^p \frac{\lambda^2}{2} e^{-\lambda^2\tau_j^2 /2} d\tau_j^2, \, \, \, \, \tau_1^2, \ldots, \tau_p^2 > 0 \, , \\ \nonumber \sigma^2 &\sim \pi(\sigma^2) \, d\sigma^2 \, .\end{aligned}$$ In this model formulation the $\mu$ on which the outcome variables $Y$ depend, is the overall mean, from which ${X\beta}$ are deviations. The parameter $\mu$ can be given a flat prior and subsequently integrated out, as we do in the coming sections. We will use the typical inverse gamma prior distribution on $\sigma^2$, i.e. for $\sigma^2>0$ $$\pi(\sigma^2)=\frac{\gamma^\alpha}{\Gamma(\alpha)}\sigma^{-2\alpha-2}e^{-\gamma/\sigma^2},$$ where $\alpha,\gamma>0$ are hyperparameters. With the hierarchy of the joint density for the posterior with the likelihood to the power $\eta$ becomes $$\begin{gathered} \label{eq:joint generalized density} (f({Y} \rvert \mu, {\beta}, \sigma^2))^\eta \, \pi(\sigma^2) \, \pi(\mu) \prod_{j=1}^p \, \pi( {\beta}_j \rvert \tau_j^2, \sigma^2 ) \, \pi(\tau_j^2) = \\ =\left( \frac{1}{(2\pi\sigma^2)^{n/2}} \, e^{\frac{1}{2\sigma^2} ( {Y} -\mu {1}_n - {X \beta})^T ( {Y} -\mu {1}_n - {X \beta}) } \right) ^\eta \frac{\gamma^\alpha}{\Gamma(\alpha)}\sigma^{-2\alpha-2}e^{-\frac{\gamma}{\sigma^2}} \prod_{j=1}^p \frac{1}{(2\sigma^2\tau_j^2)^{1/2}} \, e^{- \frac{1}{2\sigma^2\tau_j^2} {\beta}_j^2} \frac{\lambda^2}{2} \, e^{- \lambda^2\tau^2_j /2} \,.\end{gathered}$$ Let $ {\tilde{Y}}$ be $ {Y - \overline{Y}}$. If we integrate out $\mu$, the joint density marginal over $\mu$ is proportional to $$\begin{aligned} \label{eq:joint generalized density marginalized over mu} \sigma^{-\eta(n-1)} \, e^{-\frac{\eta}{2\sigma^2} ( {\tilde{Y}-X\beta})^T ( {\tilde{Y}-X\beta}) } \, \sigma^{-2\alpha-2} \, e^{-\frac{\gamma}{\sigma^2}} \prod_{j=1}^p \frac{1}{(\sigma^2\tau_j^2)^{1/2}} \, e^{- \frac{1}{2\sigma^2\tau_j^2} {\beta}_j^2} \, e^{- \lambda^2\tau^2_j /2} .\end{aligned}$$ First, observe that the full conditional for $ \beta$ is multivariate normal: the exponent terms involving $ {\beta}$ in are $$-\frac{\eta}{2\sigma^2} ( {\tilde{Y}-X\beta})^T ( {\tilde{Y}-X\beta}) - \frac{1}{2\sigma^2} {\beta}^T {D_\tau}^{-1} {\beta}= -\frac{1}{2\sigma^2} \left\lbrace ( {\beta}^T (\eta {X}^T {X} + {D_\tau}^{-1} ) {\beta} - 2\eta {\tilde{Y}X\beta} + \eta {\tilde{Y}}^T {\tilde{Y}} ) \right\rbrace.$$ If we now write $M_\tau = (\eta {X}^T {X} + {D_\tau}^{-1})^{-1}$ and complete the square, we arrive at $$\begin{aligned} &-\frac{1}{2\sigma^2} \left\lbrace ( {\beta} - \eta M_\tau {X}^T {\tilde{Y}})^T {M_\tau^{-1}} \, ( {\beta} - \eta {M_\tau} {X}^T {\tilde{Y}}) + {\tilde{Y}}^T (\eta {I}_n - \eta^2 {X}^{-1}M_\tau {X}^T) {\tilde{Y}} \right\rbrace .\end{aligned}$$ Accordingly we can see that $ {\beta}$ is conditionally multivariate normal with mean $\eta {M_\tau} {X}^T {\tilde{Y}}$ and variance $\sigma^2 {M_\tau}$.\ The terms in that involve $\sigma^2$ are: $$(\sigma^2)^{ \{ -\eta(n-1)/2 - p/2 - \alpha - 1 \} } \exp \Big\{ -\frac{\eta}{2\sigma^2}( {\tilde{Y}-X\beta})^T ( {\tilde{Y}-X\beta}) - \frac{1}{2\sigma^2} {\beta}^T {D_\tau}^{-1} {\beta} - \frac{\gamma}{\sigma^2} \Big\} .$$ We can conclude that $\sigma^2$ is conditionally inverse gamma with shape parameter $\displaystyle \eta \, \frac{n-1}{2} + \frac{p}{2} + \alpha$ and scale parameter $\displaystyle \frac{\eta}{2}( {\tilde{Y}-X\beta})^T ( {\tilde{Y}-X\beta}) + {\beta}^T {D_\tau}^{-1} {\beta} /2 + \gamma$.\ \ Since $\tau_j^2$ is not involved in the likelihood, we need not modify the implementation of it and follow @Park: $$\frac{1}{\tau^2_j} \sim \text{IG}\left(\displaystyle \sqrt{{\lambda^2\sigma^2}/{\beta_j^2}}, \, \lambda^2\right).$$ Summarizing, we can implement a Gibbs sampler with the following distributions: $$\begin{aligned} \label{Blasso:condbetagen} {\beta} &\sim \text{N} \left( \eta(\eta {X}^T {X} + {D_\tau}^{-1})^{-1} {X}^T {\tilde{Y}}, \, \sigma^2(\eta {X}^T {X} + {D_\tau}^{-1})^{-1} \right) \, , \\ \label{Blasso:condsigmagen} \sigma^2 & \sim \text{Inv-Gamma} \big( \frac{\eta}{2}(n-1) + p/2 + \alpha,\, \frac{\eta}{2}( {\tilde{Y}-X\beta})^T ( {\tilde{Y}-X\beta}) + {\beta}^T {D_\tau}^{-1} {\beta}/2 + \gamma \big) \, , \\ \label{Blasso:condtausqgen} \frac{1}{\tau^2_j} &\sim \text{IG}\left(\displaystyle \sqrt{{\lambda^2\sigma^2}/{\beta_j^2}}, \, \lambda^2\right) \, .\end{aligned}$$ There are several ways to deal with the shrinkage parameter $\lambda$. We follow the hierarchical Bayesian approach and place a hyperprior on the parameter. In our implementation we provide three ways to do so: a point mass (resulting in a fixed $\lambda$), a gamma prior on $\lambda^2$ following @Park and a beta prior following @deloscampos, details about the implementation of the latter two priors can be found in those papers respectively. The $\eta$-generalized Bayesian logistic regression {#ap:logreg} --------------------------------------------------- We follow the construction of the P[ó]{}lya–Gamma latent variable scheme for constructing a Bayesian estimator in the logistic regression context described in [@polson2013bayesian]. First, for $b>0$ consider the density function of a P[ó]{}lya-Gamma random variable $PG(b,0)$ $$p(x{\, |\, }b,0)=\frac{2^{b-1}}{\Gamma(b)}\sum_{n=1}^\infty (-1)^n\frac{\Gamma(n+b)}{\Gamma(n+1)}\frac{(2n+b)}{\sqrt{2\pi x^3}}e^{-\frac{(2n+b)^2}{8x}}.$$ The general class $PG(b,c)$ ($b,c>0$) is defined through an exponential tilting of the $PG(b,0)$ and has the density function $$p(x{\, |\, }b,c)=\frac{e^{-\frac{c^2x}2}p(x|b,0)}{{{\mathbb{E}}}e^{-\frac{c^2\omega}2}},$$ where $\omega\sim PG(b,0)$. To derive our Gibbs sampler we use the following result from [@polson2013bayesian]. \[PG\_thm\] Let $p_{b,0}(\omega)$ denote the density of $PG(b,0)$. Then for all $a\in {\mathbb{R}}$ $$\frac{(e^\psi)^a}{(1+e^\psi)^b}=2^{-b}e^{\kappa \psi}\int_0^\infty e^{-\omega\psi^2/2}p_{b,0}(\omega)d\omega,$$ where $\kappa=a-b/2.$ According to Theorem \[PG\_thm\] the likelihood contribution of the observation $i$ taken to the power $\eta$ can be written as $$L_{i,\eta}(\beta)=\left[\frac{(e^{X_i^T\beta)^{y_i }}}{1+e^{X_i^T\beta}}\right]^\eta\propto e^{\eta\kappa_iX_i^T\beta}\int_0^\infty e^{-\omega_i\frac{(X_i^T\beta)^2}2}p(\omega_i{\, |\, }\eta,0),$$ where $\kappa_i\coloneqq y_i-1/2$ and $p(\omega_i{\, |\, }\eta,0)$ is the density function of $PG(\eta,0)$. Let $$\begin{gathered} X\coloneqq (X_1,\dots, X_n)^T, \quad Y\coloneqq (Y_1,\dots,Y_n)^T, \quad \kappa\coloneqq (\kappa_1,\dots,\kappa_n)^T, \\ \omega\coloneqq (\omega_1,\dots,\omega_n)^T, \quad \Omega\coloneqq {{\text{diag}}}(\omega_1,\dots,\omega_n).\end{gathered}$$ Also, denote the density of the prior on $\beta$ by $\pi(\beta)$. Then the conditional posterior of $\beta$ given $\omega$ is $$p(\beta{\, |\, }\omega,Y)\propto \pi(\beta) \prod_{i=1}^n L_{i,\eta}(\beta{\, |\, }\omega_i)=\pi(\beta)\prod_{i=1}^ne^{\eta\kappa_iX_i^T\beta-\omega_i\frac{(X_i^T\beta)^2}2}\propto \pi(\beta)e^{-\frac12 (z-X\beta)^T\Omega(z-X\beta)},$$ where $z\coloneqq \eta(\frac{\kappa_1}{\omega_1},\dots,\frac{\kappa_n}{\omega_n})$. Observe that the likelihood part is conditionally Gaussian in $\beta$. Since the prior on $\beta$ is Gaussian, a simple linear-model calculation leads to the following Gibbs sampler. To sample from the the $\eta$-generalized posterior one has to iterate these two steps $$\begin{aligned} \omega_i{\, |\, }\beta\sim& PG(\eta, X_i^T\beta),\\ \beta{\, |\, }Y,\omega\sim& {\ensuremath{\mathcal N}}(m_\omega, V_\omega), \end{aligned}$$ where $$\begin{aligned} V_\omega\coloneqq &(X^T\Omega X+B^{-1})^{-1}, \\ m_\omega\coloneqq &V_\omega(\eta X^T\kappa+B^{-1}b).\end{aligned}$$ To sample from the P[ólya]{}-Gamma distribution $PG(b,c)$ we adopt a method from [@PG2014], which is based on the following representation result. According to [@polson2013bayesian] a random variable $\omega\sim PG(b,c)$ admits the following representation $$\omega {\overset{\mathrm{d}}{=}}\sum_{n=0}^\infty \frac{g_n}{d_n},$$ where $g_n\sim Ga(b,1)$ are independent Gamma distributed random variables, and $$d_n\coloneqq 2\pi^2(n+\frac12)^2+2c^2.$$ Therefore, we approximate the PG random variable by a truncated sum of weighted Gamma random variables. [@PG2014] shows that the approximation method performs well with the truncation level $N=300$. Furthermore, we performed our own comparison of the sampler with the STAN implementation for Bayesian logistic regression, which showed no difference between the methods (for $\eta=1$). The Safe-Bayesian Algorithms {#ap:safebayesimpl} ---------------------------- The version of the Safe-Bayesian algorithm we are using for the experiments is called *R-log-SafeBayes*, more details and other versions can be found in @grunwald2017inconsistency. The $\hat{\eta}$ is chosen from a grid of learning rates $\eta$ that minimizes the *cumulative Posterior-Expected Posterior-Randomized log-loss*: $$\begin{aligned} \sum_{i=1}^n \, {\operatorname{\mathbf{E}}}_{\beta, \sigma^2 \sim \Pi \rvert z^{i-1}, \eta} \left[-\log f(Y_i \rvert X_i, \beta, \sigma^2) \right].\end{aligned}$$ Minimizing this comes down to minimizing $$\begin{aligned} \sum_{i=1}^{n-1} \, \, \textsc{av} \left[ \frac{1}{2} \log 2 \pi \sigma^2_{i,\eta} + \frac{1}{2} \frac{(Y_{i+1} - X_{i+1}\beta_{i, \eta})^2}{\sigma^2_{i,\eta}} \right].\end{aligned}$$ The loss between the brackets is averaged over many draws of $(\beta_{i, \eta}, \sigma^2_{i,\eta})$ from the posterior, where $\beta_{i, \eta}$ (or $\sigma^2_{i,\eta}$) denotes one random draw from the conditional $\eta$-generalized posterior based on data points $z^{i}$. For the sake of completeness we present the algorithm below.\ $\mathcal{S}_n \coloneqq \{1, 2^{-\mathcal{K}_{\text{STEP}}}, 2^{-2\mathcal{K}_{\text{STEP}}}, 2^{-3\mathcal{K}_{\text{STEP}}}, \ldots, 2^{-\mathcal{K}_{\text{MAX}}}, \} $ Choose $\hat{\eta} \coloneqq \operatorname*{arg\,min}_{\eta \in \mathcal{S}_n} \{ s_\eta \}$ (if min achieved for several $\eta \in \mathcal{S}_n$, pick largest) Details for the experiments and figures {#ap:detailsfigures} ======================================= Below we present the results of additional simulation experiments for Section \[sec:lassoandhs\] (Appendix \[ap:simulations\]) and the description of experiments with real-world data (Appendix \[ap:realworld\]). We also give details for Figure \[fig:predvar\] in Appendix \[ap:delete\]. Additional Figures for Section \[sec:lassoandhs\] {#ap:simulations} ------------------------------------------------- Consider the regression context described in Section \[sec:lassoandhs\]. Here, we explore different choices of the number of Fourier basis functions, showing that regardless of the choice Safe-Baysian lasso outperforms its standard counterpart. In Figures \[fig:b vs sb\] and \[fig:1\] we see conditional expectations ${\operatorname{\mathbf{E}}}\left[ Y \mid X \right]$ according to the posteriors of the standard Bayesian lasso (blue) and the Safe-Bayesian lasso (red, $\hat\eta = 0.5$) for the *wrong-model* experiment described in Section \[sec:lassoandhs\], with $100$ data points. We take $201$ and $25$ Fourier basis functions respectively. Now we consider logistic regression setting and show that even for some well-specified problems it is beneficial to choose $\eta\neq 1$. In Figure \[fig:logregriskwell\] we see a comparison of the log-risk for $\eta=1$ and $\eta=3$ in the well-specified logistic regression case (described in Section \[sec:logisticregression\]). Here $p=1$ and $\beta=4$. ![Prediction of standard Bayesian lasso (blue) and Safe-Bayesian lasso (red, $\eta = 0.5$) with $n=200$, $p=100$.\[fig:b vs sb\]](Plaatje_B_vs_SB-eps-converted-to.pdf){width="50.00000%"} ![Prediction of standard Bayesian lasso (blue) and Safe-Bayesian lasso (red, $\eta = 0.5$) with $n=200$, $p=12$.\[fig:1\]](Bayes_vs_SB_dim25-eps-converted-to.pdf){width="50.00000%"} ![The empirical logistic risk as a function of the sample size for the *correct-model* experiments described in Section \[sec:logisticregression\] according to the posterior predictive distribution of standard Bayesian logistic regression ($\eta=1$), and generalized Bayes ($\eta=3$). \[fig:logregriskwell\]](LogitWell.pdf){width="45.00000%"} Real-world data {#ap:realworld} --------------- #### Seattle Weather Data {#ap:seattle} The $\texttt{R}$-package $\texttt{weatherData}$ [@weatherData] loads weather data available online from $\texttt{www.wunderground.com}$. Besides data from many thousands of personal weather stations and government agencies, the website provides access to data from Automated Surface Observation Systems (ASOS) stations located at airports in the US, owned and maintained by the Federal Aviation Administration. Among them is a weather station at Seattle Tacoma International Airport, Washington (WMO ID $72793$). From this station we collected the data for this experiment. The training data are the maximum temperatures for each day of the year 2011 at Seattle airport. We divided the data randomly in a training set (300 measurements) and a test set (65 measurements). First, we sampled the posterior of the standard Bayesian lasso with a 201-dimensional Fourier basis and standard improper priors on the training set, and we did the same for the Horseshoe. Next, we sampled the generalized posterior with the learning rate $\hat\eta$ learnt by the Safe-Bayesian algorithm, with the same model and priors on the same training set. The grid of $\eta$’s we used was $1, 0.9, 0.8, 0.7, 0.6, 0.5$. We compare the performance of the standard Bayesian lasso and Horseshoe and the Safe-Bayesian versions of the lasso (SB) in terms of mean square error. In all experiments performed with different partitions, priors and number of iterations, SafeBayes never picked $\hat\eta = 1$. We averaged over 10 runs. Moreover, whichever learning rate was chosen by SafeBayes, it always outperformed standard Bayes (with $\eta =1$) in an unchanged set-up. Experiments with different priors for $\lambda$ yielded similar results. #### London Air Pollution Data {#ap:air} As training set we use the following data. We start with the first four weeks of the year $2013$, starting at Monday January $7$ at midnight. We have a measurement for (almost) every hour until Sunday February $3^\text{rd}$, $23.00$. We also have data for the first four weeks of $2014$, starting at Monday January $6$ at midnight, until Sunday February $2^\text{nd}$, $23.00$. For each hour in the four weeks we randomly pick a data point from either $2013$ or $2014$. We remove the missing values. We predict for the same time of year in $2015$: starting at Monday January $5$ at midnight, until Sunday February $1^\text{st}$ at $23.00$. We do this with a (Safe-)Bayesian lasso and Horseshoe with a $201$-dimensional Fourier basis and standard improper priors. The grid of $\eta$’s we used for the Safe-Bayesian algorithm was again $1, 0.9, 0.8, 0.7, 0.6, 0.5$. We look at the mean square prediction errors, and average the errors over $20$ runs of the generalized Bayesian lasso with the $\eta$ learnt by SafeBayes, and the standard Bayesian lasso and Horseshoe. Again we find that SafeBayes performs substantially better than standard Bayes. Details for Figure \[fig:predvar\] {#ap:delete} ---------------------------------- Here we sampled the posteriors of the standard and generalized Bayesian lasso ($\eta = 0.25$) on $50$ model-wrong data points (approximately half easy points) with $101$ Fourier basis functions, and estimated the predictive variance on a grid of new data points $X_{\text{new}} = \{ -1.00, -0.99, \ldots, 1.00 \}$ with the Monte Carlo estimate: $$\hat{\textsc{var}}(Y_{\text{new}} \mid X_{\text{new}}, Z_{\text{old}}) = \operatorname{\textbf{E}}_{\theta \mid Z_{\text{old}}} \left[ \textsc{var}(Y_{\text{new}} \mid \theta) \right] + \hat{\textsc{var}} \left[ \operatorname{\textbf{E}}(Y_{\text{new}} \mid \theta) \right],$$ where $$\begin{gathered} \operatorname{\textbf{E}}_{\theta \mid Z_{\text{old}}} \left[ \textsc{var}(Y_{\text{new}} \mid \theta) \right] = \frac{1}{m} \sum_{k=1}^m \sigma^{2 \left[k \right]} = \,\overline{\sigma^2},\\ \hat{\textsc{var}} \left[ \operatorname{\textbf{E}}(Y_{\text{new}} \mid \theta) \right] = \,\hat{\textsc{var}} \left[X_{\text{new}}\beta \right]= \frac{1}{m} \sum_{k=1}^m \left( X_{\text{new}}\beta^{\left[ k \right]} \right)^2 - \left(X_{\text{new}}\overline{\beta}\right)^2.\end{gathered}$$ Here $\overline{\beta}$ is the posterior mean of the parameter for the coefficients and $\overline{\sigma^2}$ is the posterior mean of the variance. [^1]: Some of the results in Section \[sec:genglm\] have been derived some time ago, and can also be found in the initial arxiv versions of GM. However, after consultation with the associate journal editor, it was decided that they are best placed in another paper; thus, in version v4 of the arxiv paper and the upcoming journal publication of GM, they have been removed.

--- abstract: | The observed spectra and X-ray luminosities of millisecond pulsars in 47 Tuc can be interpreted in the context of theoretical models based on strong, small scale multipole fields on the neutron star surface. For multipole fields that are relatively strong as compared to the large scale dipole field, the emitted X-rays are thermal and likely result from polar cap heating associated with the return current from the polar gap. On the other hand, for weak multipole fields, the emission is nonthermal and results from synchrotron radiation of $e^{\pm}$ pairs created by curvature radiation. The X-ray luminosity, $L_x$, is related to the spin down power, $L_{sd}$, expressed in the form $L_x \propto L^{\beta}_{sd}$ with $\beta \sim 0.5$ and $\sim 1$ for strong and weak multipole fields respectively. If the polar cap size is of the order of the length scale of the multipole field, $s$, the polar cap temperature is $\sim 3 \times 10^6 K \left(\frac{L_{sd}}{10^{34}erg s^{-1}}\right)^{1/8} \left(\frac{s}{3\times 10^4 cm}\right)^{-1/2}$. A comparison of the X-ray properties of millisecond pulsars in globular clusters and in the Galactic field suggests that the emergence of relatively strong small scale multipole fields from the neutron star interior may be correlated with the age and evolutionary history of the underlying neutron star. author: - 'K. S. Cheng and Ronald E. Taam' title: 'On the Origin of X-ray Emission From Millisecond Pulsars in 47 Tuc' --- Accepted for publication in ApJ. INTRODUCTION ============ The discovery of millisecond pulsars (MSPs) as a class of rapidly rotating ($P < 10$ ms), weakly magnetized ($B \lta 10^{10}$ G) neutron stars has stimulated considerable interest in the fundamental properties of these objects. The detailed observational study of these sources over periods of time have provided insights into their origin and evolution in close binary systems (see for example, Phinney & Kulkarni 1994). The hypothesis that MSPs are neutron stars recycled in a spin up phase during which angular momentum and mass are accreted from a companion star (Radhakrishnan & Srinivasan 1982; Alpar et al. 1982) has been dramatically confirmed with the observational detection of the four millisecond accreting X-ray pulsars J1808.4-3658 (Wijnands & van Klis 1998), J1751-305 (Markwardt et al. 2002), J0929-314 (Galloway et al. 2002), and J1807-294 (Markwardt, Smith, & Swank 2003). Their combination of short spin period and low dipole magnetic field strengths have, furthermore, provided important clues on the temporal evolution of magnetic fields in neutron stars in low mass X-ray binary systems (van den Heuvel, van Paradijs, & Taam 1986; see also Bhattacharya 2002 for a recent review). Insights into the nature of the emission mechanisms have been facilitated by observational investigations over broad spectral regions. As an example, the early X-ray studies of MSPs using the ROSAT satellite revealed that the MSPs in the Galactic field appear to have a non thermal character (see Becker & Trümper 1997, 1999) with a power law photon index ranging from $\sim -2$ to -2.4. On the other hand, the recent X-ray studies with the Chandra satellite by Grindlay et al. (2002) indicate that the MSPs in 47 Tuc appear to be consistent with a thermal black body spectrum characterized by a temperature corresponding to an energy of 0.2 - 0.3 keV. Additional evidence supporting the apparent difference between the MSPs in the Galactic field and in 47 Tuc and, hence difference in their fundamental properties, can be gleaned from the relation between the X-ray luminosity, $L_x$, and the spin down power, $L_{sd}$, expressed in the form $L_x \propto L_{sd}^{\beta}$. Using ROSAT data Becker & Trümper (1997, 1999) found that $\beta \sim 1$ for MSPs in the Galactic field, whereas there are hints that the dependence is shallower ($\beta \sim 0.5$) for the MSPs in 47 Tuc (see Grindlay et al. 2002). An existence of a correlation between these two quantities provides strong evidence for relating the energy source of the X-ray emission to the rotational energy of the underlying neutron star. We shall, for convenience, group the MSPs with properties similar to the Galactic field as Type I and those similar to the MSPs in 47 Tuc as Type II, even though the nearest MSP J0437-4715 has an X-ray spectrum consisting of two thermal components and one non thermal component (Zavlin et al. 2002). The conversion of rotational energy to X-ray radiation in these MSPs is likely produced by electromagnetic processes in the neutron star’s magnetosphere (e.g., Halpern & Ruderman 1993) rather than by frictional processes in its interior (Alpar et al. 1984; Shibazaki & Lamb 1988). In this case, the emission can take place either at the magnetic poles (Daugherty & Harding 1996) or in the outer magnetosphere (Cheng, Ho, & Ruderman 1986). Specifically, it has been argued that the non thermal X-ray emission of rotation powered pulsars results from the synchrotron radiation of $e^{\pm}$ pairs created in the magnetosphere near the neutron star surface by curvature photons. Such photons are emitted by charged particles on their way from the outer magnetospheric gap to the neutron star surface (Cheng, Gil & Zhang 1998; Cheng & Zhang 1999). The non thermal X-ray luminosity is roughly about 0.1% of the spin-down power. We note, however, that the presence of a complicated surface magnetic field can change the character of the emission since the open field lines, where the outer magnetospheric gap is located, can curve upward. In this case $e^{\pm}$ production and outflow can occur on all open field lines and, hence, quench the outer magnetospheric gap (Ruderman & Cheng 1988). Observational evidence in support of emission taking place in the magnetosphere is suggested by the existence of pulsed emission in both the radio and soft X-ray region of the 5.75 ms pulsar J0437-4715 (Becker & Trümper 1999). On the other hand, thermal X-ray radiation can be produced by either neutron star cooling (Tsuruta 1998) or polar cap heating (Arons 1981; Harding, Ozernoy & Usov 1993). Since MSPs are extremely old pulsars, the internal heating mechanisms lead to surface temperatures $\lta 10^5$ K (Alpar et al. 1984; Shibazaki  Lamb 1989; Cheng et al. 1992). Hence, the blackbody thermal emission observed from the MSPs in 47 Tuc should be attributed to polar cap heating associated with the impact of the return current of high energy electrons, perhaps produced in the inner or outer gaps of the magnetosphere (see Cheng, Gil, & Zhang 1997; Cheng & Zhang 1999), on the neutron star surface. However, although the X-ray emission from MSPs in the Galactic disk is dominated by non thermal emission, the pulsed fraction, in cases that can be determined, is usually less than 50%. For example, the pulsed-fraction of PSR J2124-3358 is 55% in ASCA energy range and 33% in ROSAT energy range respectively (Sakurai et al. 2001; Becker & Trümper 1999). Furthermore, Stappers et al. (2003) have reported an X-ray nebula associated with PSR 1957 +20. Therefore it is possible that a significant fraction of non thermal X-ray emission may come from an unresolved nebular component around the pulsar. According to the observed results of Stappers et al. (2003), this unresolved X-ray emission likely represents the shock where the winds of the pulsar and its companion collide. Grindlay et al. (2002) have also suggested that the MSP in NGC6397 may have such a contribution as well. On the other hand, many MSPs in 47 Tuc have a binary companion, but their X-ray emissions are still dominated by a thermal spectrum. Furthermore, Tennant et al. (2001) have detected X-ray emission from the Crab pulsar at the pulse minimum. This indicates that some unpulsed-fraction can originate from the pulsar magnetosphere. We believe that although it is possible that the nebula may contribute to the non thermal emission, perhaps resulting in a spectral difference between MSPs in the disk and in 47 Tuc, the observed results have not yet provided compelling evidence to support this conjecture. Since the spin period, binary period, X-ray luminosity, and estimated dipole magnetic field of the two groups of MSPs overlap, other differences in properties must be sought to explain the dichotomy. Recently, Grindlay et al. (2002) suggested that their differences may be related to either the existence of high order multipole fields on the neutron star surface or to the formation of higher mass neutron stars in the dense cluster environment of 47 Tuc. The small radius of curvature associated with high order fields can facilitate the production of $e^{\pm}$ pair formation close to the neutron star surface and to an increased efficiency of polar cap heating with a corresponding increase in the level of thermal X-ray emission. Higher mass neutron stars are more compact and can prolong the effectiveness of the inverse Compton scattering of thermal photons from the neutron star surface in facilitiating pair production (see Harding, Muslimov, & Zhang 2002) for MSPs with spin down ages $\gta 10^8$ yr. Such a scattering process can lead to the relation $L_x \propto L_{sd}^{1/2}$, however, the emission resulting from this latter process is distinctly non thermal. We suggest, in this paper, that a small scale, strong surface magnetic field may play a crucial role in determining the X-ray emission properties of MSPs. The existence of such a magnetic field may sensitively depend upon the formation history of MSPs possibly providing an explanation for the differences between the MSPs in the field and in globular clusters. In the next section we examine the hints provided by the observed features of MSPs. The generic features of polar cap heating models related to the characteristic properties of the X-ray spectrum and to the relation between its X-ray luminosity and spin down power are presented in §3. In §4, we compare the observed properties of MSPs and suggest that a possible factor differentiating these two types of MSPs is their age. The origin and evolution of multipole magnetic fields in neutron stars is discussed within the context of the emission models in §5. Finally, we summarize and discuss the implications of our study in the concluding section. HINTS FROM OBSERVED FEATURES OF MSPS ==================================== In the past decade, there has been significant progress in detecting and understanding X-ray emission from rotation powered pulsars. The X-ray data obtained from the ROSAT, ASCA, RXTE, BeppoSAX, Chandra and XMM-Newton satellite observatories have provided very important constraints on theoretical models. For example, Becker & Trümper (1999) presented results of soft X-ray emission from 10 MSPs in a reanalysis of archival ROSAT data, concluding that the close correlation between the pulsar’s spin-down power and the observed X-ray luminosity suggested rotation as the energy source for the bulk of the observed non thermal X-rays. The linear relation between the X-ray and spin-down luminosity among MSPs ($L_x \propto L_{sd}$) is consistent with that found in normal radio pulsars (Becker & Trümper 1997). The non thermal spectral features of some MSPs have also been reported by Saito et al. (1997) and Takahashi et al. (2001) based on ASCA observations and by Mineo et al. (2000) based on BeppoSax observations. Although the X-ray luminosity is dominated by the non thermal component, composite spectra (power law plus black body with temperature around a few million degrees) clearly give a better fit for the observed spectrum. However, the exact contribution of the thermal component to the X-ray luminosity is difficult to determine. Recently, Grindlay et al. (2002) presented a homogeneous data set of MSPs in 47 Tuc observed with Chandra. This data provides a good estimate of the X-ray luminosities and color temperatures of MSPs in 47 Tuc because these pulsars are located at a common distance and therefore have a common interstellar column density. This is in contrast to the field where the uncertainties are greater. Although the MSPs in globular clusters share these common quantities, the gravitational acceleration of the globular cluster on the MSPs contaminates the measurement of the period derivative, $\dot{P}$. For example, about half of the MSPs in 47 Tuc have negative $\dot{P}$ (Freire et al. 2001). While it is possible to obtain the intrinsic $\dot{P}$ after subtraction of the gravitational effect of the cluster by numerical modelling (see Grindlay et al. 2002) the uncertainties in the intrinsic $\dot{P}$ of an individual MSP can be large compared to the uncertainties in the average intrinsic value of all the MSPs in the cluster. Here, we show that useful information can still be gleaned from the observed data based upon general considerations. In particular, the mean spin-down power of MSPs in 47 Tuc can be estimated from the typical age of these MSPs. An upper limit to their age is given by the age the cluster, which is estimated to be $\sim$ 11 - 13 Gyr (Schiavon et al. 2002) based on spectroscopy and the cluster’s color-magnitude diagram. A more realistic age estimate could be derived from the age of their white dwarf companions. Recently, Hansen, Kalogera & Rasio (2003) suggest that the typical age of helium white dwarfs in 47 Tuc should be $<$ 2.7 Gyr. This age gives the mean spin-down power as $<L_{sd}> \sim \frac{I <\Omega^2>}{2}/2.7 G yr \sim 2\times 10^{34}$ erg s$^{-1}$, where $I$ is the moment of inertia taken to be equal to $10^{45}$ g cm$^2$ and $<\Omega>$ is the average rotational angular frequency of the MSPs. The mean observed X-ray luminosity of MSPs in 47 Tuc is $<L_x> = 1.95\times 10^{30}$ erg s$^{-1}$ (cf. Table 1 of Grindlay et al. 2002) and the ratio of these quantities is $\sim 10^{-4}$. Becker & Trümper (1997, 1999) found that this ratio for normal radio pulsars as well as for MSPs, but not including MSPs in 47 Tuc is significantly larger ($\sim 10^{-3}$). If $L_x$ is assumed to correlate with $L^{\beta}_{sd}$, it implies that $L_x \propto B^{2\beta}P^{-4\beta}$ where $B$ is the dipolar magnetic field strength and $P$ is the spin period. We note that the dependence of the X-ray luminosity is more sensitive to the spin period than to the magnetic field. In Figure 1, $L_x$ is illustrated as a function of $1/P^2$. The circles are MSPs in 47 Tuc, the triangles are MSPs excluding those in 47 Tuc, and the squares are normal radio pulsars. By fitting these three sets of data by linear regression, the slopes are $0.49\pm 0.21$, $2.16\pm0.82$ and $1.82\pm0.45$ with correlation coefficients of 0.55(15), 0.71(9) and 0.71(18), where the value within the parentheses corresponds to the number of degrees of freedom. The correlation coefficients imply that the chances of probability are $0.0335$, $0.03282$, and $8.7\times10^{-4}$ respectively. Obviously, the data is very scattered due to the variation of magnetic field and, hence, the chances of probability are not very significant. However, the slopes of normal radio pulsars and MSPs in the field are consistent with each other, whereas the slope of MSPs in 47 Tuc is clearly different. The expected polar cap temperature of MSPs in 47 Tuc can be estimated as $T_{exp} \sim \left(\frac{<L_x>}{\sigma_B <A_p>}\right)^{1/4} < 10^6 K$ where $<A_p> = \pi \frac{R^3<\Omega>}{c}$ is the mean polar cap area inferred for a dipolar magnetic field. Here, $R$ is the radius of the neutron star. The inferred color temperature is $\sim 3\times 10^6$ K, implying that the polar cap area is about a hundred times less than the expected value. The presence of a much smaller scale magnetic field on the neutron star surface could be consistent with this result. Finally, we re-emphasize that $L_x$ is dominated by the non thermal component for normal radio pulsars as well as for MSPs in the field whereas $L_x$ is dominated by a black body thermal component for MSPs in 47 Tuc. Taken as an aggregate, these four distinguishing features suggest that the MSPs in 47 Tuc are very distinct from their Galactic field counterparts. In next section, we review several model predictions for the X-ray luminosities, which must depend on the pulsar parameters, i.e. spin period and dipolar magnetic field strength. Although the fields of the MSPs in 47 Tuc are somewhat uncertain due to the gravitational effect of the cluster, one can roughly estimate their values. For example, Grindlay et al. (2002) have used the King model to subtract the gravitational effect of the cluster to obtain an estimate of the dipolar field of each MSP in 47 Tuc. The errors in $\dot P$ are estimated to be +0.3/-0.1 in the log, which provides the error estimates in the magnetic fields used for the calculation of the X-ray luminosities shown in Table 1 (see §3). Friere et al. (2001) has adopted a more conservative approach and has provided upper limits of $B$ for each MSP. In Figure 2, the observed $L_x$ versus $BP^{-2}$ is illustrated. The circles use the field estimates of Grindlay et al (2002) and the dots use the upper limit of Friere et al. (2001). The slopes of these two set of data are $0.92\pm 0.20$, and $0.89\pm0.18$ with correlation coefficients of 0.78(15) and 0.81(15), which imply that the chances of probability are $5.8\times 10^{-4}$ and $2.8\times 10^{-4}$ respectively. Considering Figure 1 and Figure 2 together, it appears that $L_x$ is likely proportional to $P^{-1}$ to $P^{-2}$ but the latter is more favorable because the probability of such a chance occurrence from an ensemble of systems is low. GENERIC FEATURES OF POLAR CAP HEATING MODELS ============================================ There is a clear indication from the observed data that the spectrum of MSPs in 47 Tuc is thermal. However, thermal emission resulting from residual heat or frictional processes in the interior of old neutron stars is insufficient. The primary mechanism for MSPs, therefore, very likely involves polar cap heating. In this section, we review the various polar cap heating models and compare their general features with the observed data. Among the great number of models that have been developed to explain the pulsar radio emission, a large fraction involve an acceleration region located near the polar cap known as the polar gap or inner gap. Charged particles are accelerated to relativistic energies in the polar gap, whose potential drop is limited by pair creation. Coherent radio emission could result from the two stream instability of the faster primary charged particle beam and the slower secondary pair beam (for a general review, see Michel 1991). Some of these pairs created inside the polar gap can be separated by the electric field, resulting in a backflow current. In general, the polar cap heating can result from this backflow current, $J_{b}$, striking the polar caps. The X-ray luminosity is, therefore, simply given by $$L_x = J_{b}V_{gap}$$ where $V_{gap}$ is the potential difference of the polar gap. Although the exact backflow current to each polar cap is not known, it should be of the order of the Goldreich-Julian current (Goldreich & Julian 1969), which can be written as $$J_{GJ} = 1.35 \times 10^{30}B_{12}P^{-2} e s^{-1}$$ where $e$ is the charge of electron and $B_{12}$ is the magnetic field in units of $10^{12}$ G. In other words, $$J_{b} = \alpha J_{GJ}$$ where $\alpha$ is a model dependent parameter. For a uniform pair production situation inside the polar gap, $\alpha \sim \frac{1}{2}$. However, this factor could be further reduced if the current is concentrated in sparks rather than uniformly over the polar cap (Cheng & Ruderman 1980; Gil & Sendyk 2000) or the electric field in the pair creation region is actually weaker than in other regions (Arons 1981), which is supported by statistical analysis (e.g. Fan, Cheng & Manchester 2001). The predicted luminosity of the thermal X-ray radiation is rather model-dependent since there exist a wide range of models for the polar gap potential difference. Here, we discuss two classes of polar gap models which depend upon whether the polar gap is sensitive or insensitive to the pulsar parameters (viz., spin period and dipolar magnetic field). Models representative of the first group are those described by Arons (1981) and Harding & Muslimov (2001). Specifically, Arons (1981) assumed the free emission of electrons (outflow) from the stellar surface with the polar cap heating resulting from the trapped positrons (inflow) in the acceleration zone (the polar gap) bombarding the stellar surface. The X-ray luminosity caused by this bombardment is estimated as $$L^A_x \sim 2\times 10^{26}B_{12}P^{-27/8}f_p^{-1/4} erg ~s^{-1}$$ where $f_p = 921 P^{1/2}s_5^{-1}$ is the ratio of the dipole radius of curvature to the actual radius of curvature $s_5$ (in units of $10^5$ cm). Harding & Muslimov (2001) have included the frame dragging effect in the emission of electron polar gap models (Scharlemann, Arons  Fawley 1978; Arons & Scharlemann 1979) and predict the thermal X-ray luminosity from the polar cap as $$L^{HM}_x(R) = 10^{-5} L_{sd}P^{-1/2}{\tau}_6^{3/2}$$ if resonant inverse Compton scattering mechanism is dominant otherwise $$L^{HM}_x(NR) = 10^{-5} L_{sd}P^{1/2}{\tau}_6$$ where the spin down power of the pulsar is given by $$L_{sd} = 3.8 \times 10^{31} B_{12}^2P^{-4} erg ~s^{-1},$$ and $\tau_6$ is the characteristic age of the pulsar $\frac{P}{2\dot{P}}$ in units of $10^6$ yr. Another class of models predicts thermal X-ray luminosities similar to each other because the polar gap potentials in these models are insensitive to the pulsar parameters. For example, in the situation where ions are bound to the polar cap surface (Ruderman & Sutherland 1975) $$V_{RS} = 10^{12} B_{12}^{-1/7}P^{-1/7}s_6^{4/7} \rm{volts}.$$ where $s_6 $ is the radius of curvature of the surface magnetic field in units of $10^6$ cm. In the presence of a strong surface magnetic field, Gil and his co-workers ( Gil & Mitra 2001; Gil & Melikidze 2002) show that the Ruderman-Sutherland potential should be modified as $$V'_{RS} = \zeta^{1/7} b^{-1/7} V_{RS} \rm{volts}.$$ where $\zeta$ is the general relativistic correction factor, which is about 0.85 for typical neutron star parameters; $b = B_s/B_d$, where $B_s$ is the surface magnetic field and $B_d$ is the inferred dipolar field from the observed spin period and period derivative. In the superstrong magnetic field B $> 0.1B_q \approx 5 \times 10^{12}$G, the high energy photons with energy $E_{\gamma}$ will produce electron and positron pairs at or near the kinetic threshold (Daugherty & Harding 1983). Here $E_{\gamma} = 2mc^2/sin\theta$ and $sin\theta = l_{ph}/s$ where $l_{ph}$ is the photon mean free path for pair formation. Cheng and Zhang (1999) argued that if the surface magnetic field is sufficiently localized then $sin\theta = l_{ph}/s \sim 1$ and the minimum condition for the magnetic pair production is simply $E_{\gamma} > 2mc^2$ instead of $ \frac{E_{\gamma}B}{2mc^2B_q} > \frac{1}{15}$ (Ruderman & Sutherland 1975). This assumption yields $$V_{CZ} = 1.6 \times 10^{11} s_6^{1/3} \rm{volts}.$$ The Goldreich-Julian current will be dominated by the ion flow when the polar cap temperature $T$ is higher than the critical temperature $$T_{i} = 10^5 \eta B_{12}^{\delta} K,$$ where the coefficients $\eta$ and $\delta$ are model dependent parameters. Jones (1986) obtained $\eta = 0.7$ and $\delta = 0.7$ respectively, whereas Abrahams & Shapiro (1991) and Usov & Melrose (1995) gave $\eta = 3.5$ and $\delta = 0.73$ respectively. In this case, the potential of a warm polar cap was suggested to be determined by the space charge limited flow of ions due to the finite inertia of ions (Cheng & Ruderman 1977; Arons & Scharlemann 1979). However, ions stream out from a warm neutron star surface (kT $<$ 10KeV), and the photoejection of the most tightly bound electrons of ions (Jones 1980) acts like the electron and positron creation mechanism to reduce the potential of the polar gap to $$V_{J} = \gamma (A/Z) 10^9 volts,$$ where $\gamma$ is the Lorentz factor of ions and A/Z is the ratio of atomic weight and atomic number. Typically $\gamma \sim$ 10 is required to photoeject the innermost electrons. It should be noted that all these potentials (eqs.8, 9, 10 and 12) are insensitive to the pulsar parameters. If the return current is proportional to the Goldreich-Julian current, the functional dependence of the model luminosities on pulsar parameters predicted by these potentials is close to $B_{12}P^{-2}$. Cheng & Ruderman (1980) argued that although the return current is difficult to determine, the ion flow depends on the surface temperature exponentially. It is possible that the return current and ion flow can adjust the surface temperature so that it is always near the critical temperature $T_i$ shown in equation (11). They estimated the X-ray luminosity as $L_x = \sigma_B T_i^4 A_p$ where $A_p$ is the polar cap area. If $A_p$ is the dipolar area, the model X-ray luminosity is $$L_x^{CR} = 3.7 \times 10^{24} \eta^4 B_{12}^{4\delta}P^{-1} erg ~s^{-1}.$$ In Table 1, we compare the model predicted X-ray luminosities and the observed data. Unless the estimates of dipolar magnetic fields are totally incorrect, models $L_X^{RS}$, $L_X^{CR}$, $L_X^{HM}(R)$ and $L_X^{HM}(NR)$ are inconsistent with the observed data. The model X-ray luminosities are higher (lower) than the observed values by more than an order of magnitude. We have adopted $s \sim 500$ m, which is the typical dimension of the stellar crust for realistic equations of state (Cheng & Dai 1997) rather than the dipolar radius of curvature. This choice is motivated by the work of Arons (1993), who suggested that the surface magnetic field should be a superposition of clumps covering the entire surface of the neutron star. Further support for such a choice is suggested by the study of Ruderman (1991a,b), who argued that the surface magnetic field of pulsars should have a sunspot-like clump structure. Because the core of the neutron star is liquid, it is natural that the size of these clumps should be about the thickness of the solid crust. In the model developed by Jones (1980), the radius of curvature does not enter explicitly in the potential, but implicitly through the Lorentz factor. In this case $$\gamma \approx \frac{10KeV}{3kT_{cap}(1-cos\theta_x)},$$ where $T_{cap}$ is the polar cap temperature, which is $\sim 2.6\times 10^6$K (Grindlay et al. 2002) and $\theta_x \sim \frac{h}{s}$ is the angle between the X-ray photon and the local magnetic field with $h$ corresponding to the height of the polar gap. If the radius of curvature is the dipolar value, then $\gamma \sim 10^3$, and this will not be an important mechanism to limit the potential of the polar gap. However, assuming that $s$ is small and hence $(1-cos\theta_x) \sim 1$, the potential is limited by the polar cap temperature. In this approximation, we choose $\gamma \approx \frac{10KeV}{3kT} \approx 20$ and $A/Z \approx 2.$ for the model calculations. Actually $L_x^{CR}$ and $L_x^{HM}(R)$ can be consistent with the observed values for a much stronger dipolar surface magnetic field ($B_{12} > 1$). However, the predicted surface temperature $T_i$ of model $L_x^{CR}$ is still significantly lower than the observed value by an order of magnitude. $L_x^{RS}$ can be also consistent with the observed values if the return current is significantly lower than the Goldreich-Julian current. In fact, all models with the predicted $L_x^{model}$ consistent with $L_X^{obs}$ require the existence of a strong multipole field on the stellar surface including $L_x^A$. In calculating $L_x^A$, we have assumed the actual radius of curvature $s \sim 500m$. If $s$ is dipolar, $L_x^A$ will increase by a factor of $\sim 10$, which makes the model predicted luminosity higher than the observed luminosity by an order of magnitude. With the exception of $L_x^A$, all model X-ray luminosities discussed here depend on $1/P^{\delta}$ with $\delta$ between 1 and 2, which approximately reproduces the observed data. However, if the magnetic field dependence is included only models $L_x^{RS}, L_x^{CZ}$ and $L_x^{J}$ give the correct dependence on $\sim B/P^2$. These three models require small scale, strong surface magnetic fields $> 10^{12}$G, at least near the polar cap but not necessarily over the entire stellar surface. This strong surface magnetic field also implies that the polar cap area is determined by the length dimension of the surface field instead of the dipolar area. For a length scale of the multipole field $s$, the polar cap temperature is $\sim 3\times 10^6 K \left(\frac{L_{sd}}{10^{34}erg s^{-1}}\right)^{1/8} \left(\frac{s}{3\times 10^4 cm} \right)^{-1/2}$, which is relatively insensitive to the spin down power and consistent with the observed data. OBSERVATIONAL PROPERTIES OF MSPS ================================ As described previously, the Type I and Type II MSPs are distinct in their X-ray characteristics, but their overall timing properties are common. For example, the spin periods of the Type I X-ray sampled MSPs range from 1.56 ms - 5.26 ms (Grindlay et al. 2002) and those of the Type II X-ray sampled MSPs range from 2.1 ms - 7.59 ms (Camilo et al. 2000). The orbital periods of those MSPs in binary systems also span a common range as well lying between 0.38 d - 5.74 d for Type I (see Taam, King, & Ritter 2000) and 0.12 d - 2.36 d for Type II (Camilo et al. 2000). Although the determination of the spin period derivative of Type II MSPs is contaminated by accelerations in the globular cluster gravitational potential, the upper limits for the surface dipole magnetic fields of $\lta 10^9$ G (Freire et al. 2001) are similar in magnitude to Type I MSPs (with B in the range from $10^8 - 10^9$ G). Thus, upon comparison of these observed and inferred properties there are no apparently distinguishing characteristics to differentiate the X-ray properties of these two groups. The masses of the neutron star could be different between the two groups, but the metallicity of 47 Tuc (with \[Fe/H\] = -0.7) does not significantly differ from that of the Galactic disk to affect the properties of the neutron star (see Woosley, Heger, & Weaver 2002). On the other hand, tidal capture (Bailyn & Grindlay 1987) and exchange collisions (Rasio, Pfahl, & Rappaport 2000) followed by a common envelope phase can provide additional channels for the formation of binary MSPs in globular clusters. However, the various evolutionary scenarios for the formation of MSPs in the Galactic field (Taam et al. 2000) do not necessarily lead to systematically different neutron star masses in the Type I group in every single case. In fact, the fundamental issue of whether the neutron star’s mass is significantly increased during the LMXB phase is inconclusive since the amount of matter accreted can depend on whether mass loss from the system is significant during a phase when the accretion disk surrounding the neutron star is thermally unstable (cf., Li 2002; Podsiadlowski, Rappaport, & Pfahl 2002). Given that there are no clear distinguishing characteristics between the neutron stars in the MSPs in the Type I and Type II groups, we consider the hypothesis that age is a possible discriminating factor. Neutron stars in globular clusters are likely to be older than their counterparts in the Galactic field. MSP formation in globular clusters is likely to differ from that in the Galactic field since the long period primordial binaries (which are the prime progenitors of short binary period MSPs in the Galactic field) are soft in the cluster environment and, hence, can be disrupted by stellar encounters (Heggie 1975; Hut 1984; Taam & Lin 1992). Therefore, the neutron stars that are present as MSPs in a globular cluster are likely to have been isolated for as long as several Gyr, after which they underwent an exchange collision or tidal capture to form a close interacting binary system. The subsequent spin up evolution during an accretion phase and the spin down evolution during the post accretion phase is not likely to be dissimilar to those neutron stars in MSPs in the Galactic field. Since the MSPs in the Type I group form from primordial binaries, in contrast to those in Type II, the neutron stars in short orbital period systems that are the primary focus of this study are likely to have a relatively short pre accretion phase determined by the main sequence turnoff timescale of its binary companion. If we assume that the short period binary MSPs are formed via the common envelope phase (for a review, see Taam & Sandquist 2000) directly from a progenitor system containing an intermediate mass companion (Podsiadlowski et al. 2002), then an upper limit on the duration of the pre accretion phase for MSPs of Type I can be estimated, for example, to be $\lta 10^8$ yr for $\sim 3 \msun$ companion. The age of the neutron star as a MSP, however, corresponds to the time since the accretion phase ceased and can be estimated from the characteristic pulsar age given by the spin down timescale, ${P \over 2 \dot P}$ for a braking index equal to 3. We note that this age is only an upper limit since the spin period at the cessation of the accretion phase may not significantly differ from its present day spin period. For the MSPs in 47 Tuc, the uncertainties of their characteristic ages are large and not as reliable as those inferred for the MSPs in the Galactic field. An observational clue for differentiating the MSPs in the two groups is provided by the existence of 3.05 ms MSP B1821-34 in the globular cluster M28 (Lyne et al. 1987). In contrast to the MSPs in 47 Tuc, the X-ray emission from B1821-34 is distinctly non thermal (Becker & Trümper 1997) and $L_x \propto L_{sd}$. However, its short spin down age amounting to less than $3 \times 10^7$ years distinguishes it from the other MSPs in the Type II group. Since its spin down timescale is much less than any of the spin down timescales estimated for the MSPs in 47 Tuc (see Grindlay et al. 2002), it is highly suggestive that the neutron star age as a MSP is one factor which may help to distinguish MSPs in Type I from those in Type II. The 3.65 ms pulsar PSR J1740-5340 in the globular cluster NGC 6397 (D’Amico et al. 2001) can also provide a similar constraint, however, it is unclear whether it belongs to either of these two groups since its X-ray spectrum is non thermal, but yet it appears that $L_x \propto L_{sd}^{1/2}$. This source is unique in that it is an eclipsing MSP and the observations of nonvanishing emission during eclipse (Grindlay et al. 2002) suggest that the emission region is extended. As a result, the X-ray emission may not solely reflect processes taking place in the immediate vicinity of the neutron star surface and the inferences drawn from J1740-5340 are inconclusive. The pulse timing properties MSPs in the Type I group reveal that the upper limit of the characteristic pulsar ages is in the range from $\sim 10^9 - 3 \times 10^{10}$ yr. In principle, a better estimate of their age can be obtained from the determination of the cooling timescales of the white dwarf companion of those MSPs in binary systems. Amongst the MSPs in the Type I group with detectable X-ray emission, the cooling age of the white dwarf companion in J1012+5307 has been estimated to be less than $8 \times 10^8$ yrs (Hansen & Phinney 1998), significantly less than its pulsar characteristic age of $>5.4 \times 10^9$ yr (Lorimer et al. 1995). However, uncertainties exist in such cooling ages since they sensitively depend on the thickness of the hydrogen-rich envelope especially for the cooling of low mass helium white dwarfs. For such white dwarfs hydrogen burning in the nondegenerate envelope can significantly prolong its lifetime (see, for example, Schönberner, Driebe, & Blöcker 2000) provided that thermally unstable shell flashes do not remove the outer layers (via Roche lobe overflow) to reduce the effectiveness of nuclear burning (Ergma, Sarna, & Gerskevits-Antipova 2001). Given that the available observational evidence does not discriminate between the pulsar ages of the MSPs in Type I from those in Type II on a case by case basis, the differences in the duration of the post accretion phase are not well constrained. On the other hand, it is likely that the ages of the neutron stars in 47 Tuc are comparable to the age of the cluster corresponding to 11 - 13 Gyr (Schiavon et al. 2002) and are older than the neutron stars in the Type I group. ORIGIN AND EVOLUTION OF MAGNETIC FIELDS ======================================= The hypothesis of strong small scale multipole fields at the neutron star surface provides a consistent interpretative framework for understanding the X-ray emission properties of the MSP of Type II. The existence of such fields in MSP is not new, however, since they had been suggested as possibly responsible for the complex profiles of MSPs (Krolik 1991). Furthermore, such fields have been hypothesized for facilitating copious production of $e^{\pm}$ pairs required for pulsar emission in the seminal papers of Ruderman & Sutherland (1975) and Arons & Scharlemann (1979). Thus, their presence appears to be a common component in the phenomenology of pulsars. The creation of such fields may arise from thermal effects occurring in the thin layers of the crust (e.g., Blandford, Applegate, & Hernquist 1983; Arons 1993), from the coupling between various field components via the Hall effect during their evolution (Shalybkov & Urpin 1997), or from rearrangment of the field, anchored in the core, due to crustal movements (Ruderman 1991a, 1991b). Although an understanding of the origin and evolution of neutron star magnetic fields remains far from complete, observational evidence suggests that little field decay takes place during the active lifetime of a pulsar (i.e., before the pulsar reaches the death line) on timescales less than about a few times $10^8$ yrs (Bhattacharya et al. 1992). Significant evolution of the magnetic field can take place, on the other hand, during a longer pre accretion phase or during the spin up phase associated with the accretion of matter (Konar & Bhattacharya 1997). Since the electrical conductivity in the neutron star core is very high, the evolutionary timescale for the magnetic field threading the core exceeds the Hubble timescale (Baym, Pethick, & Pines 1969). The magnetic field decay, therefore, reflects processes taking place in the crust. In this paper, we adopt the hypothesis that the magnetic field decay results from ohmic dissipation, rather than screening (Bisnovatyi-Kogan & Komberg 1974; Taam & van den Heuvel 1986; Romani 1990) which is susceptible to magnetic instabilities (Bhattacharya 2002; see also, Litwin, Brown & Rosner 2001). For a review on the current status of spontaneous and accretion induced magnetic evolution in neutron stars see the recent article by Bhattacharya (2002). Since the pre accretion phase of neutron stars in globular clusters is likely to be significantly longer than the neutron stars in binaries in the Galactic field, it is likely that a crustal field can decay on timescales $\sim 10^9$ yr due to the finite electrical conductivity associated with the electron phonon scattering process in the deep crustal layers (Konar & Bhattacharya 1997). In this phase, the magnetic field can diffuse through the crust, perhaps embedding the complex field topology of the crustal layers into the denser regions, providing for a range of initial conditions for the field evolution that takes place in the following accretion phase. When the neutron star accretes matter from its binary companion (via Roche lobe overflow) the magnetic field decay process can accelerate. This occurs after an isolated neutron star has acquired a companion as a result of an exchange collision with a primordial binary or via tidal capture in a globular cluster or as a result of the evolution of the binary itself in the Galactic field. The increase in the interior temperatures of the neutron star resulting from the compressional heating and nuclear burning lead to reduced electrical conductivities associated with the electron phonon scattering process. In contrast to the pre accretion phase, the ion electron scattering process can play a role and be more important than the electron phonon scattering process since the impurity content in the denser layers from nuclear burning can be increased (Schatz et al 1999), although the nuclear processing to Fe group nuclei that takes place during a superburst (Schatz, Bildsten, & Cumming 2003) may limit its importance. As a result a more rapid decay of the magnetic field takes place. The decay process, however, is limited by the depth to which matter is accreted since the electrical conductivity increases in the denser layers which itself tends to decelerate the field decay process. The overall trend found by Konar & Bhattacharya (1997) reveals that, for a given initial dipole magnetic field, the fields decay to lower values for lower mass accretion rates. We note that the evolution of higher order magnetic fields has not been calculated for the accretion phase (which may, in part, be generated at the expense of low order fields by magnetic instabilities), although the evolution of a pre existing multipole field has been found to be similar to the dipole field, but on a shorter timescale, for the non accreting phase (Mitra, Konar, & Bhattacharya 1999). After the accretion phase has ceased, the recycled neutron star re-enters the pulsar phase as a MSP with its magnetic field primarily residing in its core. Any further field evolution depends on flux expulsion from the core to the inner crustal regions as the neutron star slows down (Srinivasan et al. 1990; Bhattacharya & Srinivasan 1995). This field may reflect its pre existing topology, the diffusion of the field into the core during the pre accretion phase or the displacement of the crustal field into the core accompanying the replacement of the crust during accretion. However, the pulsar characteristic lifetime of several of the X-ray emitting MSPs discussed above can be long and the pulsar may not have spun down significantly suggesting that further field decay may be minimal. Hence, a picture emerges whereby the field decay depends on the evolutionary history of the MSP through its age, the amount of matter accreted by the neutron star, and the timescale on which the accretion takes place (e.g., Bhattacharya 2002). Based on these rudimentary models for the magnetic field evolution, we hypothesize that the relative importance of the multipole fields is related to the long duration of the pre accretion phase in globular clusters. This hypothesis may be necessary, but is not sufficient to explain the disparity between the MSPs in Type I and Type II because of the existence of the MSPs in the globular clusters M28 and NGC 6397. If we assume that the neutron star in the MSP in M28 is formed in the same way as those in 47 Tuc and did not form recently as a result of an accretion induced collapse, perhaps leading to different initial field configurations, its existence would suggest that the emergence of the multipole field from the core at a stage when the neutron star is a MSP is delayed from the time that the accretion phase ceased. In other words, sufficient time must elapse for the multipole component of the core field to rediffuse through the crust to the surface. Since the core cools significantly after the accretion phase and the degree of impurities in the deep crust may be small, the timescale for rediffusion may exceed several Gyr. As described in the previous section, the existence of the MSP in NGC 6397 does not lend itself to a straightforward interpretation unless it belongs to the Type I group similar to the MSP in M28, except with a longer pulsar characteristic age. On the other hand, if it falls under the Type II category, then the multipole field must reemerge on timescales of $\lta 3 - 6 \times 10^8$ yrs. Recognizing that the pulsar ages are only upper limits, it is possible that some neutron stars in the Type I group have a longer MSP phase than the MSP in NGC 6397. This would present difficulties within the above framework and could imply that some MSP are so old that their multipole field has decayed during the post accretion phase. However, this may be considered unlikely since the field decay is ultimately determined by flux expulsion from the core during this phase. Because the spin down timescale is so long for the very old pulsars, little field decay is expected giving preference to the categorization for the MSP in NGC 6397 as Type I. CONCLUSIONS =========== It has been shown that the X-ray properties of MSPs in 47 Tuc are distinct from other MSPs and normal radio pulsars. In particular, the X-ray spectra can be described by a black body model. For the inferred temperature and luminosity the emitting region is found to be significantly smaller than the polar cap X-ray area deduced from a dipolar magnetic field. In addition, the ratio of X-ray luminosity to spin down luminosity is abnormally low (about an order of magnitude lower), and the X-ray luminosity appears to exhibit a shallower dependence on spin period power. We suggest that the black body X-ray emission (kT $\sim 0.3$ keV), from very old MSPs (age $> 10^9$ yr) results from polar cap heating associated with the return current from the polar gap. Such temperatures result from models where the potential difference of the polar gap is insensitive to the observed pulsar global parameters, i.e., the rotation period and the dipolar magnetic field. A prediction of such models is that the thermal X-ray luminosity of the pulsar is roughly proportional to the square root of its own spin-down power. These models share one similarity, namely, the existence of a very strong surface magnetic field ($> 10^{12}$ G) of very small scale ($< 10^5$ cm). Such field strengths can follow from flux conservation arguments provided that the dipolar field lines and the surface field lines are connected, and the frame dragging effect (Asseo & Khechinashvili 2003) is important. Specifically, for a polar cap area $\sim s^2$, the surface field is $\sim 3\times10^{10}G \left(\frac{s}{3\times 10^4cm}\right)^{-2} \left(\frac{B_d}{3\times 10^8G}\right) \left(\frac{P}{3\times 10^{-3}s}\right)^{-1}$. Furthermore, the general relativistic effect can amplify the surface magnetic field by a factor $>30$ for multipole components of sufficiently high order ($\sim 5$) (see Asseo & Khechinashvili 2003). If this interpretation is correct it imposes a very strong hint/constraint on the evolution of MSPs. The hypothesis of multipole fields has also been invoked as a possible explanation for the existence of PSR J2144-3933 beyond the pulsar death line (see Young, Manchester, & Johnston 1999; Gil & Mitra 2001). In addition, Becker et al. (2003) have recently found marginal evidence of an emission line centered at 3.3 keV from MSP PSR B1821-24. If this is identified as an electron cyclotron line, it implies a magnetic field, at least one hundred times stronger than its dipolar field. We have suggested that most of the field MSPs with very old spin-down ages are probably young MSPs (e.g., PSR B1957+20, PSR J0751+1807). Therefore, the strong multipole field may not have had sufficient time to diffuse to the stellar surface in these pulsars. On the other hand, some field MSPs could be very old (e.g., PSR J1012+5307, PSR J1024-0719, PSR J1744-1134), but they may still lack a strong surface magnetic field. We speculate that this may relate to their actual age and/or the amount mass accreted from their companions. If the mass accreted is small, the multipole field may not anchor deeply inside the crust so it decays on a short time scale. On the other hand, the true age of these MSPs may not be as old as those MSPs in 47 Tuc. Confirmation of their actual age will provide further insight into the dependence of field evolution on their different formation histories (as hence initial magnetic field initial conditions) and their different pre-accretion, accretion, and post accretion phases. We thank Drs. W. Becker, F. Camilo, J. Gil, W. Lewin, R.N. Manchester, E.S. Phinney, and N. Shibazaki for helpful comments. We are also grateful for the useful comments of the anonymous referee. This research was supported in part by a RGC grant of the Hong Kong Government to KSC and by NASA under grant NAG5-7011 and by the National Science Foundation under Grant No. AST-0200876 to RT. Abrahams, A. M., & Shapiro, S. L. 1991, , 382, 233 Alpar, M. A., Cheng, A. F., Ruderman, M. A., & Shaham, J. 1982, , 300, 728 Alpar, M. A., Pines, D., Anderson, P. W., & Shaham, J. 1984, , 276, 325 Arons, J. 1981, , 248, 1099 Arons, J. 1993, , 408, 160 Arons, J., & Scharlemann, E. T. 1979, , 231, 854 Asseo E., & Khechinashvili, D. 2003, , 334, 743 Bailyn, C. D., & Grindlay, J. E. 1987, , 316, L25 Baym, G., Pethick, C., & Pines, D. 1969, , 224, 675 Becker, W. et al. 2003, , in press Becker, W. & Trümper, J. 1997, , 326, 682 Becker, W. & Trümper, J. 1999, , 341, 803 Bhattacharya, D. 2002, JAA, 22. 67 Bhattacharya, D., & Srinivasan, G. 1995, X-ray Binaries, ed. W. H. G. Lewin, J. A. van Paradijs, & E. P. J. van den Heuvel (Cambridge: Cambridge University Press), 495 Bhattacharya, D., Wijers, R. A. M. J., Hartman, J. W., & Verbunt, F. 1992, , 254, 198 Bisnovatyi-Kogan, G. S., & Komberg, B. V. 1974, Sov. Astr., 18, 217 Blandford, R. D., Applegate, J. H., & Hernquist, L. 1983, , 204, 1025 Camilo, F., Lorimer, D. R., Freire, P., Lyne, A. G., Manchester, R. N. 2000, , 535, 975 Cheng, A. F.,& Ruderman, M. A. 1980, , 235, 576 Cheng, K. S., Chau, W. Y., Zhang, J. L., & Chau, H. F. 1992, , 396, 135 Cheng, K. S., & Dai, Z. G. 1997, , 476, L39 Cheng, K. S., Gil, J. & Zhang, L. 1997, , 493, L35 Cheng, K. S., & Zhang, L. 1999, , 515, 337 Cheng, K. S., Ho, C., & Ruderman, M. A. 1986, , 300, 500 D’Amico, N., Possenti, A., Manchester, R. N., Sarkissian, J., Lyne, A. G., & Camilo, F. 2001, , 561, L89 Daugherty, J. K., & Harding, A. K. 1996, , 458, 278 Ergma, E., Sarna, M. J., & Gerskevits-Antipova, J. 2001, , 321, 71 Fan, G. L., Cheng, K. S., & Manchester, R. N. 2001, , 557, 297 Freire, P. C., Camilo, F., Lorimer, D. R., Lyne, A. G., Manchester, R. N., & D’Amico, N. 2001, , 326, 901 Freire, P. C., Camilo, F., Kramer, M., Lorimer, D. R., Lyne, A. G., Manchester, R. N., & D’Amico, N. 2003, , 340, 1359 Galloway, D. K., Chakrabarty, D., Morgan, E. H., & Remillard, R. A. 2002, , 576, L137 Gil, J., & Melikidze, G. K. 2002, , 577, 909 Gil, J., & Mitra, D. 2001, , 550, 383 Gil, J., & Sendyk, M. 2000, , 541, 351 Goldreich, P. & Julian, W. H. 1969, , 157 Grindlay, J. E., Camilo, F., Heinke, C. O., Edmonds, P. D., Cohn, H., & Lugger, P. 2002, , 581, 470 Halpern, J. P., & Ruderman, M. 1993, , 415, 286 Hansen, B. M. S., Kalogera, V. & Rasio, F. A. 2003, , 415, 286 Hansen, B. M. S., & Phinney, E. S. 1998, , 294, 569 Harding, A. K., Muslimov, A. G., & Zhang, B. 2002, , 576, 376 Harding, A. K., Ozernoy, L. M., Usov, V. V. 1993, , 265, 921 Heggie, D. C. 1975, , 173, 729 Hut, P. 1984, , 55, 11 Jones, P. B. 1980, , 192, 847 Jones, P. B. 1986, , 222, 577 Konar, S., & Bhattacharya, D. 1997, , 284, 311 Krolik, J. 1991, , 373, L69 Li, X. D. 2002, , 564, 930 Litwin, C., Brown, E.F., & Rosner, R. 2001, , 553, 788 Lorimer, D. R., Lyne, A. G., Festin, L., & Nicastro, L. 1995, , 376, 393 Lyne, A. G., Brinklow, A., Middleditch, J., Kulkarni, S. R., Backer, D. C., & Clifton, T. R. 1987, , 328, 399 Markwardt, C. B., Smith, E., & Swank, J. H. 2003, IAU Circ. No. 8080 Markwardt, C., Swank, J. H., Strohmayer, T. E., Zand, J. J. M. in’t, & Marshall, F. E. 2002, , 575, L21 Michel, F. C. 1991, , 383, 808 Mineo, T. et al. 2000, , 355, 1053 Mitra, D., Konar, S., & Bhattacharya, D. 1999, , 307, 459 Phinney, E. S. & Kulkarni, S. R. 1994, , 32, 591 Podsiadlowski, Ph., Rappaport, S., & Pfahl, E. D. 2002, , 565, 1107 Radhakrishnan, V. & Srinivasan, G. 1982, Curr. Sci., 51, 1096 Rasio, F., Pfahl, E. D., & Rappaport, S. 2000, , 532, L47 Romani, R. 1990, , 347, 741 Ruderman, M. 1991a, , 366, 261 Ruderman, M. 1991b, , 382, 576 Ruderman, M. & Cheng, K. S. 1988, , 335, 306 Ruderman, M. A., & Sutherland, P. G. 1975, , 196, 51 Saito, Y. et al. 1997, , 477, L37 Sakurai, I. et al. 2001, , 53, 535 Scharlemann, E. T., Arons, J., & Fawley, W. M. 1978, , 222, 297 Schatz, H., Bildsten, L, & Cumming, A. 2003, , 583, L87 Schatz, H., Bildsten, L, Cumming, A., & Wiescher, M. 1999, , 524, 1014 Schiavon, R. P., Faber, S. M., Rose, J. A., & Castilho, B. V. 2002, , 580, 873 Schönberner, D., Driebe, T., & Blöcker, T. 2000, , 356, 929 Shalybkov, D. A., & Urpin, V. A. 1997, , 321, 685 Shibazaki, N., & Lamb, F. K. 1988, , 346, 808 Srinivasan, G. et al. 1990, Curr. Sci., 59, 31 Stappers, B.W., Gaensler, B.M., Kaspi, V.M., van der Klis, M., & Lewin, W. H. G. 2003, Science, 299, 1372 Taam, R. E., King, A. R., & Ritter, H. 2000, , 541, 329 Taam, R. E., & Lin, D. N. C. 1992, , 390, 440 Taam, R. E., & Sandquist, E. L. 2000, , 38, 113 Taam, R. E., & van den Heuvel, E. P. J. 1986, , 305, 235 Takahashi, M. et al. 2001, , 554, 316 Tennant, A.F. et. al. 2001, , 554, 173 Tsuruta, S. 1998, Phy. Rep., 292, 1 Usov, V. V., & Melrose, D. B. 1995, AuJPh., 48, 571 van den Heuvel, E. P., van Paradijs, J. A., & Taam, R. E. 1986, , 322, 153 Wijnands, R., & van der Klis, M. 1998, , 394, 344 Young, M. D., Manchester, R. N., & Johnston, S. 1999, , 400, 848 Woosley, S.E., Heger, A., & Weaver, T.A. 2002, Rev. Mod. Phys., 74, 1015 Zavlin, v. E., Pavlov, G. G., Sanwal, D., Manchester, R. N., & Trümper, J. 2002, , 569, 894 [lrrrrrrrrr]{}\ & & & & &\ & & & & &\ & & & & &\ \ & & & & & & & & &\ \ 47Tuc-C & $3.98$ & $1.28 \left(^{1.72 }_{1.16 }\right)$ & $2.19 $ & $5.19 \left(^{7.33 }_{4.62 }\right)$ & $9.69 $ & $2.79 \left(^{3.95 }_{2.49 }\right)$ & $5.22 $ & $5.32 \left(^{7.52 }_{4.74 }\right)$ & $9.95 $\ 47Tuc-D & $19.95$ & $2.16 \left(^{2.90 }_{1.96 }\right)$ & $4.62 $ & $9.22 \left(^{13.03 }_{8.22 }\right)$ & $22.39 $ & $4.97 \left(^{7.02 }_{4.43 }\right)$ & $12.06 $ & $10.54 \left(^{14.89 }_{9.40 }\right)$ & $25.59 $\ 47Tuc-E & $39.81$ & $5.68 \left(^{7.64 }_{5.15 }\right)$ & $11.25 $ & $23.17 \left(^{32.73 }_{20.65 }\right)$ & $51.38 $ & $12.48 \left(^{17.63 }_{11.12 }\right)$ & $27.67 $ & $49.38 \left(^{69.76 }_{44.01 }\right)$ & $109.51 $\ 47Tuc-F & $31.62$ & $5.85 \left(^{7.87 }_{5.30 }\right)$ & $19.02 $ & $20.65 \left(^{29.17 }_{18.40 }\right)$ & $81.68 $ & $11.12 \left(^{15.71 }_{9.91 }\right)$ & $43.99 $ & $68.88 \left(^{97.29 }_{61.39 }\right)$ & $272.44 $\ 47Tuc-G & $12.59$ & $2.44 \left(^{3.28 }_{2.21 }\right)$ & $7.54 $ & $9.22 \left(^{13.03 }_{8.22 }\right)$ & $34.44 $ & $4.97 \left(^{7.02 }_{4.43 }\right)$ & $18.55 $ & $16.10 \left(^{22.74 }_{14.35 }\right)$ & $60.11 $\ 47Tuc-H & $12.59$ & $1.64 \left(^{2.21 }_{1.49 }\right)$ & $9.25 $ & $5.19 \left(^{7.33 }_{4.62 }\right)$ & $38.96 $ & $2.79 \left(^{3.95 }_{2.49 }\right)$ & $20.99 $ & $12.78 \left(^{18.05 }_{11.39 }\right)$ & $96.02 $\ 47Tuc-I & $15.85$ & $3.16 \left(^{4.25 }_{2.87 }\right)$ & $9.07 $ & $11.61 \left(^{16.40 }_{10.35 }\right)$ & $39.68 $ & $6.25 \left(^{8.83 }_{5.57 }\right)$ & $21.37 $ & $25.30 \left(^{35.74 }_{22.55 }\right)$ & $86.45 $\ 47Tuc-J & $19.95$ & $6.44 \left(^{8.66 }_{5.83 }\right)$ & $10.47 $ & $20.65 \left(^{29.17 }_{18.40 }\right)$ & $36.40 $ & $11.12 \left(^{15.71 }_{9.91 }\right)$ & $19.61 $ & $96.14 \left(^{135.80 }_{85.68 }\right)$ & $169.48 $\ 47Tuc-L & $25.12$ & $3.87 \left(^{5.20 }_{3.51 }\right)$ & $6.45 $ & $16.40 \left(^{23.17 }_{14.62 }\right)$ & $29.76 $ & $8.83 \left(^{12.48 }_{7.87 }\right)$ & $16.03 $ & $25.66 \left(^{36.25 }_{22.87 }\right)$ & $46.56 $\ 47Tuc-M & $12.59$ & $2.08 \left(^{2.80 }_{1.89 }\right)$ & $4.46 $ & $7.33 \left(^{10.35 }_{6.53 }\right)$ & $17.82 $ & $3.95 \left(^{5.57 }_{3.52 }\right)$ & $9.60 $ & $14.73 \left(^{20.81 }_{13.13 }\right)$ & $35.84 $\ 47Tuc-N & $15.85$ & $4.08 \left(^{5.48 }_{3.70 }\right)$ & $10.30 $ & $14.62 \left(^{20.65 }_{13.03 }\right)$ & $43.06 $ & $7.87 \left(^{11.12 }_{7.02 }\right)$ & $23.19 $ & $38.83 \left(^{54.84 }_{34.60 }\right)$ & $114.35 $\ 47Tuc-O & $39.81$ & $5.83 \left(^{7.84 }_{5.29 }\right)$ & $18.72 $ & $20.65 \left(^{29.17 }_{18.40 }\right)$ & $80.46 $ & $11.12 \left(^{15.71 }_{9.91 }\right)$ & $43.34 $ & $68.11 \left(^{96.20 }_{60.70 }\right)$ & $265.38 $\ 47Tuc-Q & $12.59$ & $4.41 \left(^{5.93 }_{4.00 }\right)$ & $7.57 $ & $18.40 \left(^{26.00 }_{16.40 }\right)$ & $34.56 $ & $9.91 \left(^{14.00 }_{8.83 }\right)$ & $18.62 $ & $32.21 \left(^{45.49 }_{28.70 }\right)$ & $60.48 $\ 47Tuc-T & $10.00$ & $2.05 \left(^{2.76 }_{1.86 }\right)$ & $5.21 $ & $10.35 \left(^{14.62 }_{9.22 }\right)$ & $30.68 $ & $5.57 \left(^{7.87 }_{4.97 }\right)$ & $16.53 $ & $7.02 \left(^{9.91 }_{6.25 }\right)$ & $20.81 $\ 47Tuc-U & $19.95$ & $3.87 \left(^{5.20 }_{3.51 }\right)$ & $8.01 $ & $16.40 \left(^{23.17 }_{14.62 }\right)$ & $38.33 $ & $8.83 \left(^{12.48 }_{7.87 }\right)$ & $20.64 $ & $25.69 \left(^{36.29 }_{22.90 }\right)$ & $60.03 $\ \ \ \ \ \ \ & & & & &\ & & & & &\ & & & & &\ \ & & & & & & & & &\ \ 47Tuc-C& $3.98$ & $1.87 \left(^{5.14 }_{1.34 }\right)$ & $11.64 $ & $26.74 \left(^{18.93 }_{30.01 }\right)$ & $14.31 $ & $1.44 \left(^{1.44 }_{1.44 }\right)$ & $1.44 $\ 47Tuc-D& $19.95$ & $7.11 \left(^{19.49 }_{5.08 }\right)$ & $94.68 $ & $19.34 \left(^{13.69 }_{21.70 }\right)$ & $7.97 $ & $1.61 \left(^{1.61 }_{1.61 }\right)$ & $1.61 $\ 47Tuc-E& $39.81$ & $14.02 \left(^{38.43 }_{10.02 }\right)$ & $143.44 $ & $32.95 \left(^{23.33 }_{36.97 }\right)$ & $14.86 $ & $3.00 \left(^{3.00 }_{3.00 }\right)$ & $3.00 $\ 47Tuc-F& $31.62$ & $2.36 \left(^{6.47 }_{1.69 }\right)$ & $130.91 $ & $105.12 \left(^{74.42 }_{117.95 }\right)$ & $26.58 $ & $4.69 \left(^{4.69 }_{4.69 }\right)$ & $4.69 $\ 47Tuc-G& $12.59$ & $1.81 \left(^{4.97 }_{1.30 }\right)$ & $85.01 $ & $51.92 \left(^{36.76 }_{58.26 }\right)$ & $13.91 $ & $2.45 \left(^{2.45 }_{2.45 }\right)$ & $2.45 $\ 47Tuc-H& $12.59$ & $0.11 \left(^{0.30 }_{0.08 }\right)$ & $40.05 $ & $206.49 \left(^{146.19 }_{231.69 }\right)$ & $27.49 $ & $3.46 \left(^{3.46 }_{3.46 }\right)$ & $3.46 $\ 47Tuc-I& $15.85$ & $1.74 \left(^{4.76 }_{1.24 }\right)$ & $62.83 $ & $69.20 \left(^{48.99 }_{77.65 }\right)$ & $20.25 $ & $3.06 \left(^{3.06 }_{3.06 }\right)$ & $3.06 $\ 47Tuc-J& $19.95$ & $0.81 \left(^{2.21 }_{0.58 }\right)$ & $4.22 $ & $228.88 \left(^{162.03 }_{256.81 }\right)$ & $129.83 $ & $6.54 \left(^{6.54 }_{6.54 }\right)$ & $6.54 $\ 47Tuc-L& $25.12$ & $13.87 \left(^{38.03 }_{9.91 }\right)$ & $79.03 $ & $22.62 \left(^{16.01 }_{25.38 }\right)$ & $12.46 $ & $2.20 \left(^{2.20 }_{2.20 }\right)$ & $2.20 $\ 47Tuc-M& $12.59$ & $0.59 \left(^{1.61 }_{0.42 }\right)$ & $7.87 $ & $90.94 \left(^{64.38 }_{102.04 }\right)$ & $37.38 $ & $2.83 \left(^{2.83 }_{2.83 }\right)$ & $2.83 $\ 47Tuc-N& $15.85$ & $1.80 \left(^{4.93 }_{1.28 }\right)$ & $42.10 $ & $87.26 \left(^{61.78 }_{97.91 }\right)$ & $29.63 $ & $3.73 \left(^{3.73 }_{3.73 }\right)$ & $3.73 $\ 47Tuc-O& $39.81$ & $2.45 \left(^{6.71 }_{1.75 }\right)$ & $129.93 $ & $102.40 \left(^{72.49 }_{114.89 }\right)$ & $26.28 $ & $4.64 \left(^{4.64 }_{4.64 }\right)$ & $4.64 $\ 47Tuc-Q& $12.59$ & $13.52 \left(^{37.07 }_{9.66 }\right)$ & $85.16 $ & $26.19 \left(^{18.54 }_{29.38 }\right)$ & $13.94 $ & $2.46 \left(^{2.46 }_{2.46 }\right)$ & $2.46 $\ 47Tuc-T& $10.00$ & $53.66 \left(^{147.11 }_{38.34 }\right)$ & $1282.06 $ & $5.10 \left(^{3.61 }_{5.72 }\right)$ & $1.72 $ & $0.95 \left(^{0.95 }_{0.95 }\right)$ & $0.95 $\ 47Tuc-U& $19.95$ & $13.82 \left(^{37.89 }_{9.87 }\right)$ & $164.74 $ & $22.68 \left(^{16.06 }_{25.45 }\right)$ & $9.71 $ & $2.20 \left(^{2.20 }_{2.20 }\right)$ & $2.20 $\ \ \

--- abstract: 'We present a systematic theory of Coulomb-induced correlation effects in the nonlinear optical processes within the strong-coupling regime. In this paper we shall set a dynamics controlled truncation scheme [@Axt; @Stahl] microscopic treatment of nonlinear parametric processes in SMCs including the electromagnetic field quantization. It represents the starting point for the microscopic approach to quantum optics experiments in the strong coupling regime without any assumption on the quantum statistics of electronic excitations (excitons) involved. We exploit a previous technique, used in the semiclassical context, which, once applied to four-wave mixing in quantum wells, allowed to understand a wide range of observed phenomena [@Sham; @PRL95]. We end up with dynamical equations for exciton and photon operators which extend the usual semiclassical description of Coulomb interaction effects, in terms of a mean-field term plus a genuine non-instantaneous four-particle correlation, to quantum optical effects.' author: - 'S. Portolan$^{1,3,}$[^1], O. Di Stefano$^2$, S. Savasta$^2$, F. Rossi$^3$, and R. Girlanda$^2$' title: 'Dynamics-Controlled Truncation Scheme for Quantum Optics and Nonlinear Dynamics in Semiconductor Microcavities' --- Introduction ============ Since the early Seventies [@Esaki; @Tsu] researchers have been exploring the possible realization of semiconductor-based heterostructures, devised according to the principles of quantum mechanics. The development of sophisticated growth techniques started a revolution in semiconductor physics, determined by the possibility of confining electrons in practical structures. In addition, the increasing ability in controlling fabrication processes has enabled the manipulation of the interaction between light and semiconductors by engineering, in addition to the electronic wave functions, the light modes. Entanglement is one of the key features of quantum information and communication technology [@Nielsen-Chuang] and a hot topic in quantum optics too. Parametric down-conversion is the most frequently used method to generate highly entangled pairs of photons for quantum-optics applications, such as quantum cryptography and quantum teleportation. Rapid development in the field of quantum information requires monolithic, compact sources of nonclassical photon states enabling efficient coupling into optical fibres and possibly electrical injection. Semiconductor-based sources of entangled photons would therefore be advantageous for practical quantum technologies. The strong light-matter interaction in these systems gives rise to cavity polaritons which are hybrid quasiparticles consisting of a superposition of cavity photons and quantum well (QW) excitons [@Weisbuch-Houdre]. Demonstrations of parametric amplification and parametric emission in semiconductor microcavities (SMCs) with embedded QWs[@Baumberg; @Erland; @Langbein; @PRB2004], together with the possibility of ultrafast optical manipulation and ease of integration of these microdevices, have increased the interest on the possible realization of nonclassical cavity-polariton states [@squeezing; @Quattropani; @CiutiBE; @Savasta; @PRL2005; @LosannaCC; @SSC; @Savasta]. In 2004 squeezed light generation in SMCs in the strong coupling regime has been demonstrated [@Giacobino]. In 2005 an experiment probing quantum correlations of (parametrically emitted) cavity polaritons by exploiting quantum complementarity has been proposed and realized [@Savasta; @PRL2005]. Specifically, it has been shown that polaritons in two distinct idler modes interfere if and only if they share the same signal mode so that which-way information cannot be gathered, according to Bohr’s quantum complementarity principle. Laser spectroscopy in semiconductors and in semiconductor quantum structures has been greatly used because exciting with ultrashort optical pulses in general results in the creation of coherent superpositions of many-particle states. Thus it constitutes a very promising powerful tool for the study of correlation and an ideal arena for semiconductor cavity quantum electrodynamics (cavity QED) experiments as well as coherent control, manipulation, creation and measurement of non-classical states [@AxtKuhn; @Sham; @PRL95; @Nature; @CuCl; @Savasta; @PRL2005]. The analysis of nonclassical correlations in semiconductors constitutes a challenging problem, where the physics of interacting electrons must be added to quantum optics and should include properly the effects of noise and dephasing induced by the electron-phonon interaction and the other environment channels [@Kuhn-Rossi; @PRB; @2005]. The nonlinear optical properties of exciton-cavity system play a key role in driving the quantum correlations and the nonclassical optical phenomena. The crucial role of many-particle Coulomb correlations in semiconductors marks a profound difference from the nonlinear optics of dilute atomic systems, where the optical response is well described by independent transitions between atomic levels, and the nonlinear dynamics is governed only by saturation effects mainly due to the balance of populations between different levels. The Dynamics Controlled Truncation Scheme (DCTS) provides a (widely adopted) starting point for the microscopic theory of the light-matter interaction effects beyond mean-field [@AxtKuhn], supplying a consistent and precise way to stop the infinite hierarchy of higher-order correlations which always appears in the microscopic approaches of many-body interacting systems without need to resort to any assumption on the quantum statistics of the quasi-particle arising in due course. By exploting this scheme, it was possible to express nonlinearities originating from the Coulomb interaction as an instantaneous mean-field exciton-exciton interaction plus a noninstantaneous term where four-particle correlation effects beyond menfield are contained entirely in a retarded memory function [@Sham; @PRL95]. In 1996 the DCTS was extended in order to include in the description the quantization of the electromagnetic field and polariton effects [@Savasta; @PRL96]. This extension has been applied to the study of quantum optical phenomena in semiconductors and it was exploited to predict polariton entanglement [@SSC; @Savasta]. The obtained equations showed that quantum optical correlations (as nonlinear optical effects) arise from both saturation effects (phase-space filling) and Coulomb induced correlations due to four-particle states (including both bound and unbound biexciton states). The dynamical equations included explicitly biexciton states. The structure of those equations didn’t allow the useful separation of Coulomb interaction in terms of a mean-field interaction term plus a noninstantaneous correlation term performed in the semiclassical description. In this paper we shall set a DCTS microscopic treatment of nonlinear parametric processes in SMCs including the light-field quantization. It represents the starting point for the microscopic approach to quantum optics experiments in the strong coupling regime. For this purpose we shall exploit a previous technique [@Sham; @PRL95] which, once applied to four-wave mixing in QWs, it allowed to understand a wide range of observed phenomena. Here all the ingredients contributing to the dynamics are introduced and commented. We shall give in great details the manipulations required in order to provide an effective description of the nonlinear parametric contributions beyond mean-field in an exciton-exciton correlation fashion. In particular we derive the coupled equations of motion for the excitonic polarization and the intracavity field. It shows a close analogy to the corresponding equation describing the semiclassical (quantized electron system, classical light field) coherent $\chi^{(3)}$ response in a QW [@Sham; @PRL95], the main difference being that here the (intracavity) light field is regarded not as a driving external source but as a dynamical field [@Savasta; @PRL2003]. This correspondence is a consequence of the linearization of quantum fluctuations in the nonlinear source term here adopted, namely the standard linearization procedure of quantum correlations adopted for large systems [@Walls]. However the present approach includes the light field quantization and can thus be applied to the description of quantum optical phenomena. Indeed, striking differences between the semiclassical and the full quantum descriptions emerge when considering expectation values of exciton and photon numbers or even higher order correlators, key quantities for the investigation of coherence properties of quantum light [@Savasta; @PRL2005]. This is the main motivation for the derivation of fully operatorial dynamical equations, within such lowest order nonlinear coherent response, we address in the last section. The results here presented provide a microscopic theoretical starting point for the description of quantum optical effects in interacting electron systems with the great accuracy accomplished for the description of the nonlinear optical response in such many-body systems, see e.g. [@Sham; @PRL95; @Savasta; @PRL2003; @Savasta; @PRB2001; @Buck; @AxtKuhn] and references therein. The proper inclusion of the detrimental environmental interaction, an important and compelling issue, is left for a detailed analysis in another paper of ours [@nostro; @PRB]. In Section \[1\] the generality of the coupled system taken into account are exposed, here all the ingredients contributing to the dynamics are introduced and commented. The linear and the lowest nonlinear dynamics is the subject of Sec. \[2\], whereas in Sec. \[3\] we shall give in great details the manipulations required in order to provide an effective description of the nonlinear parametric contributions beyond mean-field in an exciton-exciton correlation fashion. In Sec. \[4\] the operatorial equations of motion for exciton and intracavity photon operators are derived. The Coupled System {#1} ================== The system we have in mind is a semiconductor QW grown inside a semiconductor planar Fabry-Perot resonator. In the following we consider a zinc-blende-like semiconductor band structure. The valence band is made from $p$-like ($l=1$) orbital states which, after spin-orbit coupling, give rise to $j = 3/2$ and $j=1/2$ decoupled states. In materials like GaAs, the upper valence band is fourfold degenerate ($j=3/2$), whereas in GaAs-based QWs the valence subbands with $j=3/2$ are energy splitted into two-fold degenerate heavy valence subbands with $j_z=\pm 3/2$ and light lower energy subbands with $j_z= \pm 1/2$. The conduction band, arising from an $s$-like orbital state (l=0), gives rise to $j=1/2$ twofold states. In the following we will consider for the sake of simplicity only twofold states from the upper valence and lowest conduction subbands. As a consequence electrons in a conduction band as well as holes have an additional spin-like degree of freedom as electrons in free space. When necessary both heavy and light hole valence bands or subbands can be included in the present semiconductor model. Only electron-hole ([*[eh]{}*]{}) pairs with total projection of angular momentum $\sigma = \pm 1$ are dipole active in optical interband transitions. In GaAs QWs photons with circular polarizations $\sigma = -$($+$) excite electrons with $j_z^{\it e}=+1/2$ ($j_z^{\it e}=-1/2$) and holes with $j_z^{\it h}=-3/2$ ($j_z^{\it h}=3/2$). We label optically active [*[eh]{}*]{} pairs with the same polarization label of light generating them; e.g. $\sigma = +1$ indicates an [*[eh]{}*]{} pair with $j_z^{\it e}=-1/2$ and $j_z^{\it h}=3/2$. We start from the usual model for the electronic Hamiltonian of semiconductors [@Haugh; @AxtKuhn]. It is obtained from the many-body Hamiltonian of the interacting electron system in a lattice, keeping explicitly only those terms in the Coulomb interaction preserving the number of electrons in a given band, see Appendix \[Npair states\]. The system Hamiltonian can be rewritten as $$\label{Ham electron} \hat{H}_e = \hat{H}_0 + \hat{V}_{\text{Coul}}= \sum_{N \alpha} E_{N \alpha} {\mid{N \alpha}\rangle} {\langle{N \alpha}\mid}\, ,$$ where the eigenstates of $\hat{H}_e$, with energies $E_{N \alpha} =\hbar \omega_{N \alpha}$, have been labelled according to the number N of [*eh*]{} pairs. The state ${\mid{N=0}\rangle}$ is the electronic ground state, the $N=1$ subspace is the exciton subspace with the additional collective quantum number $\alpha$ denoting the exciton energy level $n$, the in-plane wave vector ${\bf k}$ and the spin index $\sigma$. When needed we will adopt the following notation: $\alpha\equiv (n,k)$ with $k\equiv ({\bf k}, \sigma)$. In QWs, light and heavy holes in valence band are split off in energy. Assuming that this splitting is much larger than kinetic energies of all the involved particles and, as well, much larger than the interaction between them, we shall consider only heavy hole states as occupied. On the contrary to the bulk case, in a QW single particle states experience confinement along the growth direction and subbands appear, anyway in the other two orthogonal directions translational invariance is preserved and the in-plane exciton wave vector remains a good quantum number. Typically, the energy difference between the lowest QW subband level and the first excited one is larger than the Coulomb interaction between particles, and we will consider excitonic states arising from electrons and heavy holes in the lowest subbands. Eigenstates of the model Hamiltonian with N=1 (called excitons) can be created from the ground state by applying the exciton creation operator: $$\label{exciton def} \bigl| 1 n \sigma {\bf k}\bigr>=\hat{B}^\dagger_{n \sigma {\bf k}} \bigl| N=0\bigr>\, ,$$ which can be written in terms of electrons and holes operators as $$\hat{B}^\dagger_{n \sigma {\bf k}} = \sum_{{\bf k}'} \Phi^{\bf k}_{n \sigma {\bf k}'} \hat{c}^\dagger_{\sigma, {\bf k}' + \eta_e{\bf k}/2}\hat{d}^\dagger_{\sigma, -{\bf k}' + \eta_h{\bf k}/2}\, , \label{Bdag}$$ here $\Phi^{\bf k}_{n \sigma {\bf k}'}$ is the exciton wave function, being ${\bf k}$ the total wave vector ${\bf k} = {\bf k}_e + {\bf k}_h$, and ${\bf k}' = \eta_e{\bf k}_e - \eta_h{\bf k}_h$ with $\eta_{(e,h)} = m_{(e,h)}/(m_{(e)} + m_{(h)})$ ($m_{e}$ and $m_{h}$ are the electron and hole effective masses). These exciton eigenstates can be obtained by requiring the general one [*eh*]{} pair states to be eigenstates of $\hat{H}_e$: $$\hat{H}_e \bigl| 1 n \sigma {\bf k}\bigr> = \hbar \omega_{1 n \sigma {\bf k}} \bigl| 1 n \sigma {\bf k}\bigr>\, , \label{secular}$$ and projecting this secular equation onto the set of product ([*eh*]{}) states $\bigl|k_e,k_h\bigr> = \hat{c}^\dag_{k_e} \hat{d}^\dag_{k_h} {\mid0\rangle}$, (see Appendix \[Npair states\] for details): $$\label{Schrodinger} \sum_{k_e,{k}_h} ( {\langlek'_e,k'_h\mid}\hat{H}_c {\midk_e ,k_h\rangle} - \hbar \omega_{n \sigma {\bf k}} \delta_{k'_e k'_h,k_e k_h} ) \bigl< k_e,k_h \bigl| 1 n \sigma {\bf k}\bigr> = 0\, .$$ Thus, having expressed the correlated exciton state as a superposition of uncorrelated product states, $$\bigl| 1 n \sigma {\bf k}\bigr> = \sum_{k_e,k_h} {\Bigg (}\bigl< k_e,k_h \bigl| 1 n \sigma{\bf k}\bigr> {\Bigg )}\bigl|k_e,k_h\bigr> \, ,$$ the scalar products, coefficients of this expansion, represent nothing but the envelope function $\Phi^{\bf k}_{n, \sigma, {\bf k}'}$ of the excitonic aggregate being the solution of the corresponding Schrödinger equation (\[Schrodinger\]). It describes the correlated [*eh*]{} relative motion in k-space. In order to simplify a bit the notation, the spin convention in Eq. (\[Bdag\]) has been changed by using the same label for the exciton spin quantum number and for the spin projections of the electron and hole states forming the exciton. The next relevant subspace ($N=2$) is the biexciton one, spanning all the states with 2 [*eh*]{} pairs. It seems worth noting that the above description of [*eh*]{} complexes arises from the properties of quantum states and, once fixed the system Hamiltonian, no approximations have been introduced insofar. Indeed such a property hold for any N [*eh*]{} pairs aggregate and we will give a full account of it in Appendix \[Npair states\]. The eigenstates of the Hamiltonian $\hat{H}_{c}$ of the cavity modes can be written as ${\midn, \lambda\rangle}$ where $n$ stands for the total number of photons in the state and $\lambda = ({\bf k}_1, \sigma_1; ... ;{\bf k}_n, \sigma_n)$ specifies wave vector and polarization $\sigma$ of each photon. Here we shall neglect the longitudinal- transverse splitting of polaritons [@Kavokin] originating mainly from the corresponding splitting of cavity modes. It is more relevant at quite high in-plane wave vectors and often it results to be smaller than the polariton linewidths. The present description can be easily extended to include it. We shall treat the cavity field in the quasi-mode approximation, that is to say we shall quantize the field as the mirrors were perfect and subsequently we shall couple the cavity with a statistical reservoir of a continuum of external modes. This coupling is able to provide the cavity losses as well as the feeding of the coherent external impinging pump beam. The cavity mode Hamiltonian, thus, reads $$\label{Ham cavity} \hat{H}_{c} = \sum_k \hbar \omega^c_k \hat{a}_k ^{\dag} \hat{a}_k\, ,$$ with the operator $\hat{a}^\dag_k$ which creates a photon state with energy $\hbar \omega^c_k =\hbar (\omega^2_{\text{exc}} + v^2 |{\bf k}|^2)^{1/2}$, $v$ being the velocity of light inside the cavity and $k = (\sigma,{\bf k})$. The coupling between the electron system and the cavity modes is given in the usual rotating wave approximation [@Savasta; @PRL96; @HRS; @Savasta] $$\label{Ham inter cav-exc} \hat{H}_{I} = - \sum_{n k} V^*_{n k} \hat{a}_k ^{\dag} \hat{B}_{n k} + H.c.\, ,$$ $V_{n,k }$ is the photon-exciton coupling coefficient enhanced by the presence of the cavity [@Savona; @Quattropani; @SSC] set as $V_{n,k} = \tilde V_{\sigma} \sqrt{A} \phi^*_{n,\sigma}({\bf x}=0)$, the latter being the real-space exciton envelope function calculated in the origin whereas $A$ is the in-plane quantization surface, $\tilde V_{\sigma}$ is proportional to the interband dipole matrix element. Modeling the loss through the cavity mirrors within the quasi-mode picture means we are dealing with an ensemble of external modes, generally without a particular phase relation among themselves. An input light beam impinging on one of the two cavity mirrors is an external field as well and it must belong to the family of modes of the corresponding side (i.e. left or right). Being coherent, it will be the non zero expectation value of the ensemble. It can be shown [@Savasta; @PRL96; @nostro; @PRB] that for a coherent input beam, the driving of the cavity modes may be described by the model Hamiltonian [@Savasta; @PRL96; @nostro; @PRB] $$\label{H quasi modi} \hat{H}_p = i\, t_c \sum_{\bf k} ({E}_{\bf k} \hat{a}^\dag_{\bf k} - {E}^{*}_{\bf k} \hat{a}_{\bf k})\, ,$$ where ${E}_{\bf k}$ (${E}^*_{\bf k}$) is a $\mathbb{C}$-number describing the positive (negative) frequency part of the coherent input light field amplitude. Linear and Nonlinear Dynamics {#2} ============================= The idea is not to use a density matrix approach, but to derive directly expectation values of all the quantities at play. The dynamics is described by “transition" operators (known as generalized Hubbard operators): $$\begin{aligned} \label{Hubbard} \hat{X}_{N,\alpha;M,\beta} = {\mid{N,\alpha}\rangle} {\langle{M,\beta}\mid} \nonumber \\ \hat{Y}_{n,\lambda;m,\mu}= {\midn,\lambda\rangle} {\langlem,\mu\mid}\, .\end{aligned}$$ The fundamental point in the whole analysis is that, thanks to the form of the interaction Hamiltonian $\hat{H}_I$ and thanks to the quasiparticle conservation the free Hamiltonians possess, we can use the so-called dynamics controlled truncation scheme, stating that we are facing a rather special model where the correlation have their origin only in the action of the electromagnetic field and thus the general theorem due to Axt and Stahl [@Axt; @Stahl] holds. For our purpose we will need its generalization in order to include the quantization of the electromagnetic field [@Savasta; @PRL96], it reads: $$\begin{aligned} \label{DCTS} \langle \hat{X}_{N,\alpha;M,\beta} \hat{Y}_{n,\lambda;m,\mu} \rangle = \sum_{i=0}^{i_0} \langle \hat{X}_{N,\alpha;M,\beta} \hat{Y}_{n,\lambda;m,\mu} \rangle ^{(N+M+n+m+2i)} \nonumber \\+ \mathcal{O}(E^{(N+M+n+m+2i_0+2)})\, ,\end{aligned}$$ i.e. the expectation value of a **zero to N-pair** transition is at least of order N in the external electromagnetic field. There are only even powers because of the spatial inversion symmetry which is present. Once a perturbative order in the external coherent fields is chosen, Eq. (\[DCTS\]) limits the expectation values to take into account, thus providing a precise way to truncate the hierarchy of equations of motions. The exciton and photon operators can be expressed as $$\begin{aligned} \label{a,B} \hat{a}_k &=& \hat{Y}_{0;1 k} + \sum_{n \geq 1} \sqrt{n_k+1} \hat{Y}_{n_k k; (n_k+1) k} \nonumber \\ \hat{B}_{n k} &=& \hat{X}_{0;1 n k} + \sum_{N \geq 1, \alpha \beta} {\langleN \alpha\mid}\hat{B}_{n k} {\mid(N+1) \beta\rangle} \hat{X}_{N \alpha;(N+1) \beta}\, , \end{aligned}$$ where in writing the photon expansion we omitted all the states not belonging to the $k$-th mode which add up giving the identity in every Fock sector [@detail]. The equation of motion for the generic quantity of interest $\hat{X}_{N,\alpha;M,\beta} \hat{Y}_{n,\lambda;m,\mu}$ is reported in Appendix \[gen eq\]. In the Heisenberg picture we start considering the equation of motion for the photon and exciton operators, once taken the expectation values we exploit theorem (\[DCTS\]) retaining only the linear terms. With the help of the generalized Hubbard opertors all this procedure may be done by inspection. The linear dynamics for $ \left<\right. \hat{a}_k \left.\right>^{(1)} = \left<\right. \hat{Y}_{0;1 n k} \left.\right>^{(1)}$ and $ \left<\right. \hat{B}_{n k} \left.\right>^{(1)} = \left<\right. \hat{X}_{0;1 n k} \left.\right>^{(1)}$ reads: $$\begin{aligned} \label{lin order} && \frac{d}{dt}\left<\right. \hat{a}_k \left.\right>^{(1)} = -i \bar{\omega}^c_k \left<\right. \hat{a}_k \left.\right>^{(1)} +i \sum_{n} \frac{V^*_{n k}}{\hbar} \left<\right. \hat{B}_{n k} \left.\right>^{(1)} + t_c \frac{E_k}{\hbar} \\ && \frac{d}{dt}\left<\right. \hat{B}_{n k} \left.\right>^{(1)} = -i \bar{\omega}_{1 n k} \left<\right. \hat{B}_{n k} \left.\right>^{(1)} + i \frac{V_{n k}}{\hbar} \left<\right. \hat{a}_{k} \left.\right>^{(1)} \, . \end{aligned}$$ In these equations $\bar{\omega}^c_k = \omega^c_k-i\gamma_k$, where $\gamma_k$ is the cavity damping, analogously $\bar{\omega}_{1 n k} = \omega_{1 n k} -i\Gamma_{\text{x}}$ and $\bar{\omega}_{2 \beta} = \omega_{2 \beta}-i\Gamma_{\text{xx}}$. The dynamics up to the third order is a little bit more complex, we shall make extensively use of (\[dt gen op\]) (in the following the suffix $^{+(n)}$ stands for “up to" $n$-th order terms in the external electromagnetic exciting field). With Eq. (\[a,B\]) the exciton and the photon expectation values can be expanded as follows: $$\label{B up to 3} \left<\right. \hat{B}_{n k} \left.\right>^{+(3)} = \left<\right. \hat{X}_{0;1 n k} \left.\right>^{+(3)} + \sum_{\alpha \beta}{\langle1 \alpha\mid} \hat{B}_{n k} {\mid2 \beta\rangle} \left<\right. \hat{X}_{1 \alpha;2 \beta} \left.\right>^{(3)}\, ,$$ $$\label{a up to 3} \left<\right. \hat{a}_{k} \left.\right>^{+(3)} = \left<\right. \hat{Y}_{0;1 k} \left.\right>^{+(3)} + \sqrt{2} \left<\right. \hat{Y}_{1 k;2 k} \left.\right>^{(3)}\, .$$ With a bit of algebra we obtain $$\label{dt a up to 3} \frac{d}{dt}\left<\right. \hat{a}_k \left.\right>^{+(3)} = -i \bar{\omega}^c_k \left<\right. \hat{a}_k \left.\right>^{+(3)} + i \sum_{n} \frac{V^*_{n k}}{\hbar} \left<\right. \hat{B}_{n k} \left.\right>^{+(3)} + t_c \frac{E_k}{\hbar}\, ,$$ $$\begin{aligned} \label{dt B up to 3} &&\frac{d}{dt}\left<\right. \hat{B}_{n k} \left.\right>^{+(3)} = -i \bar{\omega}_{1 n k} \left<\right. \hat{B}_{n k} \left.\right>^{+(3)} + i \frac{V_{n k}}{\hbar} \left<\right. \hat{a}_{k} \left.\right>^{+(3)} + \nonumber \\ && \hspace{2.0cm} + \sum_{\tilde{n} \tilde{k}} {\Bigg [} \frac{i}{\hbar} \sum_{n' k', \alpha} V_{n' k'} {\langle1 \tilde{n} \tilde{k}\mid} [\hat{B}_{n k}, \hat{B}^\dag_{n' k'} ] - \delta_{(n' k');(n k)} {\mid1 \alpha\rangle} \langle \hat{X}_{1 \tilde{n} \tilde{k}; 1 \alpha} \hat{Y}_{0; 1 k'} \left.\right>^{(3)} - \nonumber \\ && \hspace{3.0cm} - i \sum_{\beta}(\omega_{2 \beta} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k}) {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} {\mid2 \beta\rangle} \langle \hat{X}_{1 \tilde n \tilde k; 2 \beta} \hat{Y}_{0;0} \left.\right>^{(3)} {\Bigg ]}\, ,\end{aligned}$$ in analogy with the eqs [@Savasta; @PRL96] (see also Ref. [@Sham; @PRL95]). The resulting equation of motion for the lowest order biexciton amplitude is $$\begin{aligned} \label{X02} && \frac{d}{dt}\left<\right. \hat{X}_{0;2 \beta} \left.\right>^{(2)} = -i \bar{\omega}_{2 \beta} \left<\right. \hat{X}_{0;2 \beta} \left.\right>^{(2)} + \nonumber \\ && \hspace{3.0cm} + \frac{i}{\hbar} \sum_{n' k';n'' k''} V_{n' k'}{\langle2 \beta\mid} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \left<\right. \hat{X}_{0,1 n'' k''} \hat{Y}_{0,1 k'} \left.\right>^{(2)}\, .\end{aligned}$$ Coherent Response {#3} ================= Thanks to the fact we want to treat coherent optical processes it is possible to manipulate further the parametric contributions under two assumptions. We are addressing a coherent optical response, thus we may consider that a coherent pumping mainly generates *coherent* nonlinear processes, as a consequence the dominant contribution of the biexciton sector on the third-order nonlinear response can be calculated considering the system quantum state as a pure state, which means the nonlinear term is regarded as originating mainly from coherent contributions. Moreover nonclassical correlations are taken into account up to the lowest order. The first assumption results in the factorizations $\langle \hat{X}_{1 \tilde n \tilde k; 2 \beta} \hat{Y}_{0;0} \left.\right>^{(3)} \simeq \langle \hat{X}_{1 \tilde n \tilde k; 0} \rangle^{(1)} \langle \hat{X}_{0; 2 \beta} \left.\right>^{(2)}$ and $\langle \hat{X}_{1 \tilde{n} \tilde{k}; 1 \beta} \hat{Y}_{0; 1 k'} \left.\right>^{(3)} \simeq \langle \hat{X}_{1 \tilde{n} \tilde{k};0}\rangle^{(1)} \langle \hat{X}_{0; 1 \beta} \hat{Y}_{0; 1 k'} \left.\right>^{(2)}$. The second implies $\langle \hat{X}_{0; 1 \beta} \hat{Y}_{0; 1 k'} \left.\right>^{(2)} \simeq \langle \hat{X}_{0;1 \beta}\rangle^{(1)} \langle \hat{Y}_{0; 1 k'} \left.\right>^{(1)}$, in the nonlinear source term, namely the standard linearization procedure of quantum correlations adopted for large systems [@Walls]. Of course these two approximations can be avoided at the cost of enlarging the set of coupled equations in order to include the equation of motions for the resulting correlation functions. It neglects higher order quantum optical correlation effects between the electron system and the cavity modes leading to a renormalization of the biexciton dynamics with intriguing physical perspectives. However for extended systems, like QWs in planar microcavities, these are effects in most cases of negligible impact, on the contrary in fully confined geometries such as cavity embedded quantum dots they could give significant contributions. In the end, within such a *coherent limit*, we are able to describe the biexciton contribution effectively as an exciton-exciton correlation [@Sham; @PRL95]. The resulting equations for the coupled exciton an cavity-field expectation values coincide with those obtained within a semiclassical theory (quantized electron-system and classical cavity field). Nevertheless completely different results can be obtained for exciton or photon number expectation values or for higher order correlation function [@SSC; @Savasta; @HRS; @Savasta]. In the next section we will derive operator equations useful for the calculation of such correlation functions. After the two approximations described above (linearization of quantum fluctuations and coherent limit), Eqs (\[dt B up to 3\]) becomes $$\label{dt B up to 3fin} \frac{d}{dt}\left<\right. \hat{B}_{n k} \left.\right>^{+(3)} = -i \bar{\omega}_{1 n k} \left<\right. \hat{B}_{n k} \left.\right>^{+(3)} +i \frac{V_{n k}}{\hbar} \left<\right. \hat{a}_{k} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\tilde{n} \tilde{k}} \langle \hat{B}_{\tilde n \tilde k} \rangle^{*(1)} R^{(2)}_{n k;\tilde n \tilde k}\, ,$$ where \[R nl\] &&R\^[(2)]{}\_[n k;n k]{} = Q\^[(2)]{}\_[n k;n k]{} + Q\^[(2)]{}\_[n k;n k]{}\ &&Q\^[(2)]{}\_[n k;n k]{} = \_[n’ k’, n” k”]{} C\^[n’ k’,n” k”]{}\_[n k,n k]{} \_[n” k”]{} \^[(1)]{} \_[k’]{} \^[(1)]{}\ &&Q\^[(2)]{}\_[n k;n k]{} = \_c\^[(1)]{}\_[n k;n k;]{} \_[0; 2 ]{} \^[(2)]{} , with $$\begin{aligned} \label{coeffs} && C^{n' k',n'' k''}_{\tilde n \tilde k,n k} = V_{n' k'} {\langle1 \tilde{n} \tilde{k}\mid} \delta_{(n' k');(n k)} - [\hat{B}_{n k}, \hat{B}^\dag_{n' k'} ] {\mid1 n'' k''\rangle} \\ && c^{(1)}_{n k;\tilde n \tilde k;\beta} = \hbar (\omega_{2 \beta} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k}) {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} {\mid2 \beta\rangle}\, .\end{aligned}$$ This equation is analogous to the corresponding equation describing the semiclassical (quantized electron system, classical light field) coherent $\chi^{(3)}$ response in a QW [@Sham; @PRL95], the main difference being that here the (intracavity) light field is regarded not as a driving external source but as a dynamical field [@Savasta; @PRL2003]. This close correspondence for the dynamics of expectation values of the exciton operators, is a consequence of the linearization of quantum fluctuations. However the present approach includes the light field quantization and can thus be applied to the description of quantum optical phenomena. By explicit calculation it is easy to see that the first term in Eq. (\[coeffs\]) is zero unless all the involved polarization labels $\sigma$ coincide. In order to manipulate the last term we follow the procedure of Ref. [@Sham; @PRL95] which succeeded in reformulating the nonlinear term coming from the Coulomb interaction as an exciton-exciton (X-X) mean-field contribution plus a correlation term driven by a two-exciton correlation function. Even if we are about to perform more or less the same steps of Ref. [@Sham; @PRL95] we shall provide a detailed account of all the key points of the present derivation. A clear comprehension of these details will be essential for the extension to operatorial dynamical equations of the next section. In performing this we shall need the two identities: $$\begin{aligned} && c^{(1)}_{n k;\tilde n \tilde k;\beta} = \hbar (\omega_{2 \beta} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k}) {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} {\mid2 \beta\rangle} = \nonumber \\ && \hspace{2.5cm} = \hbar {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) {\mid2 \beta\rangle} \end{aligned}$$ and $$\begin{aligned} \label{Sham identity} && \frac{d}{dt} \Bigg( \langle \hat{B}_{n' k'} \left.\right>^{(1)} \langle \hat{B}_{n'' k''} \left.\right>^{(1)} e^{-i \Omega(u-t)} \bigg) = \\ && \hspace{2.5cm} = +\frac{i}{\hbar} \bigg( V_{n' k'} \langle \hat{a}_{k'} \rangle^{(1)} \langle \hat{B}_{n'' k''} \rangle^{(1)} + V_{n'' k''} \langle \hat{a}_{k''} \rangle^{(1)} \langle \hat{B}_{n' k'} \rangle \bigg)e^{-i \Omega(t-t')} \nonumber\, , \end{aligned}$$ or $$\label{Sham identity2} \frac{1}{2}\ \frac{d}{dt} \sum_{n' k';n'' k''} \Bigg( \langle \hat{B}_{n' k'} \left.\right>^{(1)} \langle \hat{B}_{n'' k''} \left.\right>^{(1)} e^{-i \Omega(t-t')} \bigg) = +\frac{i}{\hbar} \sum_{n' k';n'' k''} V_{n' k'} \langle \hat{a}_{k'} \rangle^{(1)} \langle \hat{B}_{n'' k''} \rangle^{(1)} e^{-i \Omega(t-t')}\, ,$$ where $\Omega \doteq \omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}}\, .$ Employing the forma solution of the biexciton amplitude Eq. (\[X02\]) we have: $$\begin{aligned} \label{conti c} && \hspace{-1.0cm} \sum_{\beta}c^{(1)}_{n k;\tilde n \tilde k;\beta} \langle \hat{X}_{0; 2 \beta} \rangle^{(2)} = \hbar \sum_{\beta} {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) {\mid2 \beta\rangle} \cdot \\ && \hspace{0.0cm} i \sum_{n' k';n'' k''} \frac{V_{n' k'}}{\hbar} {\langle2 \beta\mid} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \int_{-\infty}^t dt' e^{-i (\omega_{2 \beta} - i \Gamma_{\text{xx}})(t-t')} \langle \hat{a}_{k'} \rangle^{(1)}(t') \langle \hat{B}_{n'' k''} \rangle^{(1)}(t') \nonumber\, .\end{aligned}$$ We observe that the matrix elements entering the nonlinear source terms are largely independent on the wave vectors for the range of wave vectors of interest in the optical response. Neglecting such dependence we can thus exploit the identity (\[Sham identity2\]), obtaining $$\begin{aligned} \label{conti c 2} && \hspace{-1.0cm} = \hbar \sum_{\beta} {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) {\mid2 \beta\rangle} \int_{-\infty}^t dt' e^{-i (\omega_{2 \beta} - i \Gamma_{\text{xx}})(t-t')} \nonumber \\ && \hspace{0.0cm} \sum_{n' k';n'' k''} {\langle2 \beta\mid} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \frac{1}{2}\ \frac{d}{dt'} \Bigg( \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \Bigg) e^{+i \Omega(t-t')} = \nonumber\\ && \hspace{-1.0cm} = \hbar \sum_{n' k';n'' k''} \int_{-\infty}^t dt' {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} e^{- \Gamma_{\text{xx}}(t-t')} \nonumber \\ && \hspace{0.0cm} \frac{1}{2}\ \frac{d}{dt'} \Bigg( \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \Bigg) e^{+i \Omega(t-t')}\, ,\end{aligned}$$ where in the last lines we have resummed all the biexciton subspace by virtue of its completeness. By performing an integration by part, Eq.(\[conti c 2\]) can be rewritten as $$\begin{aligned} \label{conti c 3} && \hspace{-1.0cm} = \frac{1}{2}\hbar \!\!\! \sum_{n' k';n'' k''}\!\! \Bigg[ \Bigg\{ e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')}\! \hat{B}^\dag_{n' k'}\!\! {\mid1 n'' k''\rangle} \nonumber \\ && \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \Bigg\}^t_{-\infty}- \nonumber \\ && - \int_{-\infty}^t dt' \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \frac{d}{dt'} \Bigg\{ e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} \nonumber \\ && {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \Bigg\} \Bigg]=\end{aligned}$$ $$\begin{aligned} \label{conti c 4} && \hspace{-1.0cm} = \frac{1}{2} \hbar \!\!\!\! \sum_{n' k';n'' k''}\!\! \Bigg\{ {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) \! \hat{B}^\dag_{n' k'}\!\! {\mid1 n'' k''\rangle} \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t) \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t) - \nonumber \\ && - \int_{-\infty}^t dt' \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \frac{d}{dt'} \Bigg\{ e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} \nonumber \\ && {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \Bigg\}\, .\end{aligned}$$ The first and the second term can be expressed in terms of a double commutator structure: $$\label{doppio comm 1} {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) = {\langle0\mid} [\hat{B}_{\tilde n \tilde k},[\hat{B}_{n k},\hat{H}_{c}]] \doteq {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k}\, ,$$ where a *force* operator $\hat{D}$ is defined [@Sham; @PRL95] and $$\begin{aligned} \label{doppio comm 2} && \frac{d}{dt'} \Bigg\{ e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} \nonumber \\ && {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \Bigg\} = \nonumber \\ && = \frac{d}{dt'} \Bigg\{ {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k} e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} \Bigg\} = \nonumber \\ && = {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k} e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} i \Big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 n' k'} - \omega_{1 n'' k''} - i (\Gamma_{\text{xx}} - 2 \Gamma_{\text{x}}) \Big) \nonumber \\ && \hspace{1.0cm} \hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} = \nonumber\\ && = e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} i F^{n'' k'', n' k'}_{\tilde n \tilde k,n k}(t-t') + \\ && + (\Gamma_{\text{xx}} - 2 \Gamma_{\text{x}})e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k}(t-t')\hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} \nonumber\, , \end{aligned}$$ where the memory kernel reads $$\label{F} F^{n'' k'', n' k'}_{\tilde n \tilde k,n k}(t-t') = {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k}(t-t') \hat{D}^\dag_{n'' k'',n' k'} {\mid0\rangle}\, .$$ The usual time dependence in the Heisenberg picture is given by $ \hat{D}(\tau) = e^{i (\hat{H}_{c}/\hbar) \tau}\hat{D}e^{-i (\hat{H}_{c}/\hbar) \tau}$. Altogether, the nonlinear term originating from Coulomb interaction can be written as $$\begin{aligned} \label{H-F} && \hspace{-1.0cm} Q^{\text{COUL}(2)}_{n k;\tilde n \tilde k} = \sum_{\beta}c^{(1)}_{n k;\tilde n \tilde k;\beta} \langle \hat{X}_{0; 2 \beta} \rangle^{(2)} = \nonumber \\ && \hspace{-1.0cm} \frac{1}{2} \hbar \sum_{n' k';n'' k''} \Bigg\{ {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k} \hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t) \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t) - \nonumber \\ && - i \int_{-\infty}^t dt' F^{n'' k'', n' k'}_{\tilde n \tilde k,n k}(t-t') \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{- \Gamma_{\text{xx}}(t-t')} \Bigg\} - \\ && \hspace{-1.0cm} - \frac{\hbar}{2}(\Gamma_{\text{xx}} - 2 \Gamma_{\text{x}}) \hspace{-0.1cm} \sum_{\substack{ n' k'\\n'' k''}} \hspace{-0.1cm} \int_{-\infty}^t dt' {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k}(t-t')\hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') \nonumber \, .\end{aligned}$$ For later purpose we are interested in the optical response dominate by the 1S exciton sector, with $\Gamma_{\text{xx}} \simeq 2 \Gamma_{\text{x}}$ in the cases of counter- and co-circularly polarized waves. Specifying to this case the Coulomb-induced term with Eq. (\[H-F\]) becomes $$\begin{aligned} \label{Coul 1S} && \frac{d}{dt}\left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}\Biggl|_{\text{COUL}}= -i \bar{\omega}_{\bf k} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}-\frac{i}{\hbar} \sum_{\tilde{\sigma} {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{*(1)} Q^{\text{COUL}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} = \\ && =-i \bar{\omega}_{\bf k} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}-\frac{i}{\hbar} \sum_{\bf k' k'' \tilde{k}} \delta_{\bf k+\tilde k;k'+k''} V_{\text{xx}} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{*(1)}(t) \langle \hat{B}_{\pm {\bf k'}} \rangle^{(1)}(t) \langle \hat{B}_{\pm {\bf k''}}\rangle^{(1)}(t) + \nonumber \\ && -\frac{1}{\hbar} \sum_{\substack{\sigma' \sigma'' \tilde{\sigma}\\ \bf k' k'' \tilde{k}}} \delta_{\bf k+\tilde k;k'+k''} \delta_{\pm+\tilde \sigma;\sigma'+\sigma''} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{*(1)}(t) \nonumber \\ && \hspace{3.0cm} \int_{-\infty}^t dt' F^{\sigma' \sigma''}(t-t') \langle \hat{B}_{\sigma' {\bf k}'} \left.\right>^{(1)}(t') \langle \hat{B}_{\sigma'' {\bf k}''} \left.\right>^{(1)}(t') e^{- \Gamma_{\text{xx}}(t-t')}\, , \nonumber\end{aligned}$$ where, in order to lighten the notation, we dropped the two spin indexes $\sigma$ and $\tilde \sigma$ in the four-particle kernel function $F$ defined in Eq. (\[F\]) for they are already univocally determined once chosen the others (i.e. $\sigma'$ and $\sigma''$) as soon as their selection rule ($\delta_{\sigma+\tilde \sigma;\sigma'+\sigma''}$) is applied. Moreover, the $\hbar/2$ has been reabsorbed in the Coulomb nonlinear coefficients $V_{\text{xx}}$ and $F^{\sigma' \sigma''}(t-t')$. A detail microscopic account for the mean-field $V_{\text{xx}}$, for the $F$’s and their selection rules are considered in [@Takayama; @EPJ; @Kwong-Binder; @PRB; @2001]. For the range of ${\bf k}$-space of interest, i.e. $|{\bf k}| \ll \frac{\pi}{a_{\text{x}}}$ (much lower than the inverse of the exciton Bohr radius) they are largely independent on the center of mass wave vectors. While $V_{\text{xx}}$ and $F^{\pm \pm}(t-t')$ (i.e. co-circularly polarized waves) conserve the polarizations, $F^{\pm \mp}(t-t')$ and $F^{\mp \pm }(t-t')$ (counter-circular polarization) give rise to a mixing between the two circularly polarizations. The physical origin of the three terms in Eq. (\[H-F\]) can be easily understood: the first is the Hartee-Fock or mean-field term representing the first order treatment in the Coulomb interaction between excitons, the second term is a pure biexciton (four-particle correlations) contribution. This coherent memory may be thought as a non-Markovian process involving the two-particle (excitons) states interacting with a bath of four-particle correlations [@Sham; @PRL95]. Equation (\[H-F\]) even if formally similar to that of Ref. [@Sham; @PRL95], represents its extension including polaritonic effects due to the presence of the cavity . It has been possible thanks to the inclusion of the dynamics of the cavity modes whereas in Ref. [@Sham; @PRL95] the electromagnetic field entered as a parameter only. Former analogous extensions have been obtained within a semiclassical model [@Takayama; @EPJ; @Kwong-Binder; @PRB; @2001; @Savasta; @PRL2003]. The strong exciton-photon coupling does not modify the memory kernel because four-particle correlations do not couple directly to cavity photons. As pointed out clearly in Ref. [@Savasta; @PRL2003], cavity effects alter the phase dynamics of two-particle states during collisions, indeed, the phase of two-particle states in SMCs oscillates with a frequency which is modified respect to that of excitons in bare QWs, thus producing a modification of the integral in Eq. (\[H-F\]). In this way the exciton-photon coupling $V_{n k}$ affects the exciton-exciton collisions that govern the polariton amplification process. Ref. [@Savasta; @PRL2003] considers the first (mean-field) and the second (four-particle correlation) terms in the particular case of cocircularly polarized waves, calling them without indexes as $V_{\text{xx}}$ and $F(t)$ respectively. In Fig. 1 they show ${\cal F}(\omega)$, the Fourier transform of $F(t)$ plus the mean-field term $V_{\text{xx}}$, $${\cal F}(\omega) = V_{\text{xx}} -i \int^\infty_{- \infty} dt F(t) e^{i \omega t}\, .$$ Its imaginary part is responsible for the frequency dependent excitation induced dephasing, it reflects the density of the states of two-exciton pair coherences. Towards the negative detuning region the dispersive part Re$({\cal F})$ increases whereas the absorptive part Im$({\cal F})$ goes to zero. The former comprises the mean-field contribution effectively reduced by the four-particle contribution. Indeed, the figure shows the case with a binding energy of 13.5 meV, it gives $V_{\text{xx}} n_{\text{sat}} \simeq 11.39$ meV which clearly is an upper bound for Re$({\cal F})$ for negative detuning. The contribution carried by $F(t)$ determines an effective reduction of the mean-field interaction (through its imaginary part which adds up to $V_{\text{xx}}$) and an excitation induced dephasing. It has been shown [@Savasta; @PRL2003] that both effects depends on the sum of the energies of the scattered polariton pairs. The third term in Eq. (\[H-F\]) can be thought as a reminder of the mismatch between the picture of a biexciton as a composite pair of exciton. In the following we will set $\Gamma_{\text{xx}} \simeq 2 \Gamma_{\text{x}}$. The other nonlinear source term in Eq. (\[R nl\]) depends directly on the exciton wave function and reads $$\sum_{\tilde{n} \tilde{k}} \langle \hat{B}_{\tilde n \tilde k} \rangle^{*(1)} \sum_{n' k',n'' k''}C^{n' k',n'' k''}_{\tilde n \tilde k,n k} \langle \hat{a}_{k'} \rangle^{(1)} \langle \hat{B}_{n'' k''} \rangle^{(1)}\, .$$ It represents a phase-space filling (PSF) contribution, due to the Pauli blocking of electrons. It can be developed as follows, $$\begin{aligned} \label{PSFconti} && C^{n' k',n'' k''}_{\tilde n \tilde k,n k} = V_{n' k'} {\langle1 \tilde{n} \tilde \sigma {\bf \tilde{k}}\mid} \delta_{(n' k');(n k)} - [\hat{B}_{n \sigma {\bf k}}, \hat{B}^\dag_{n' \sigma' {\bf k'}} ] {\mid1 n'' \sigma'' {\bf k''}\rangle} = \nonumber \\ && = V_{n' k'} \delta_{\sigma,\sigma'} \Biggl\{ \sum_{\bf q} \Phi^{{\bf k}\, *}_{n \sigma {\bf q}} \Phi^{\bf k'}_{n' \sigma' ({\bf q}+\eta_h({\bf k'}-{\bf k}))} {\langle1 \tilde{n} \tilde \sigma {\bf \tilde{k}}\mid} \hat{c}^\dag_{\sigma',{\bf q}+\eta_h({\bf k'}-{\bf k})+\eta_e{\bf k'}} c_{\sigma,{\bf q}+\eta_e{\bf k}} {\mid1 n'' \sigma'' {\bf k''}\rangle} + \nonumber \\ && \sum_{\bf q} \Phi^{{\bf k}\, *}_{n \sigma {\bf q}} \Phi^{\bf k'}_{n' \sigma' ({\bf q}-\eta_e({\bf k'}-{\bf k}))} {\langle1 \tilde{n} \tilde \sigma {\bf \tilde{k}}\mid} \hat{d}^\dag_{\sigma',-{\bf q}+\eta_e({\bf k'}-{\bf k})+\eta_h{\bf k'}} d_{\sigma,-{\bf q}+\eta_h{\bf k}} {\mid1 n'' \sigma'' {\bf k''}\rangle} \Biggr\} = \nonumber \\ && = V_{n' k'} \delta_{\sigma,\sigma'} \delta_{\bf k+\tilde k;k'+k''} \Biggl\{ \sum_{\bf q} \Phi^{{\bf k}\, *}_{n \sigma {\bf q}} \Phi^{\bf k'}_{n' \sigma' {\bf q}_0} \Phi^{{\bf \tilde k}\, *}_{{\tilde{n} \tilde \sigma} {\bf q}_1} \Phi^{\bf k''}_{n'' \sigma'' {\bf q}_2} + \nonumber \\ && \sum_{\bf q} \Phi^{{\bf k}\, *}_{n \sigma {\bf q}} \Phi^{\bf k'}_{n' \sigma' {\bf q}_3} \Phi^{{\bf \tilde k}\, *}_{{\tilde{n} \tilde \sigma} {\bf q}_4} \Phi^{\bf k''}_{n'' \sigma'' {\bf q}_5} \Biggr\} \, , \end{aligned}$$ the explicit expressions of the ${\bf q}$’s are given in [@q's]. Thus, the nonlinear dynamics of Eq. (\[dt B up to 3fin\]) driven by $\hat{H}_I$ can be written $$\begin{aligned} \label{dt B H_I} &&\frac{d}{dt}\left<\right. \hat{B}_{n \sigma {\bf k}} \left.\right>^{+(3)}\Bigl|_{\hat{H}_I} = +i \frac{V_{n \sigma {\bf k}}}{\hbar} \left<\right. \hat{a}_{\sigma {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\substack{n' n'' \tilde{n}\\ \bf k' k'' \tilde{k}}} \delta_{\bf k+\tilde k;k'+k''} \langle \hat{B}_{\tilde n \sigma {\bf \tilde k}} \rangle^{*(1)} \nonumber \\ && \hspace{1.0cm} \langle \hat{a}_{\sigma {\bf k'}} \rangle^{(1)} \langle \hat{B}_{n'' \sigma {\bf k''}}\rangle^{(1)} \tilde V^*_{\sigma} \Bigl[ \sum_{\bf q} \Phi^{{\bf k}\, *}_{n \sigma {\bf q}} \Phi^{{\bf \tilde k}\, *}_{{\tilde{n} \sigma} {\bf q}_1} \Phi^{\bf k''}_{n'' \sigma {\bf q}_2} + \sum_{\bf q} \Phi^{{\bf k}\, *}_{n \sigma {\bf q}} \Phi^{{\bf \tilde k}\, *}_{{\tilde{n} \sigma} {\bf q}_4} \Phi^{\bf k''}_{n'' \sigma {\bf q}_5} \Bigr] \, .\end{aligned}$$ We are interested in studying polaritonic effects in SMCs where the optical response involves mainly excitons belonging to the 1S band with wave vectors close to normal incidence, i.e. $|{\bf k}| \ll \frac{\pi}{a_{\text{x}}}$ (much lower than the inverse of the exciton Bohr radius). In this case the exciton relative wave functions are independent on spins as well as on the center of mass wave vector. They are such that $\sum_{\bf q=-\infty}^\infty |\Phi_{\bf q}|^2 = 1$, i.e. $\Phi_{\bf q} = \frac{1}{\sqrt{A}} \frac{\sqrt{2 \pi} 2 a_{\text{x}}}{(1+(a_{\text{x}}|{\bf q}|)^2)^{3/2}}$, $a_{\text{x}}$ is the exciton Bohr radius. From now on whenever no excitonic level is specified the 1S label is understood. It yields $$\begin{aligned} \label{dt B H_I 2} &&\frac{d}{dt}\left<\right. \hat{B}_{\sigma {\bf k}} \left.\right>^{+(3)}\Bigl|_{\hat{H}_I} = +i \frac{V_{\sigma {\bf k}}}{\hbar} \left<\right. \hat{a}_{\sigma {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\bf k' k'' \tilde{k}} \delta_{\bf k+\tilde k;k'+k''} \nonumber \\ && \hspace{1.0cm} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{*(1)} \langle \hat{a}_{\sigma {\bf k'}} \rangle^{(1)} \langle \hat{B}_{\sigma {\bf k''}}\rangle^{(1)} 2 \tilde V^*_{\sigma} O^{\text{PSF}}\, ,\end{aligned}$$ where the overlap $O^{\text{PSF}}$ has been calculated in the case of zero center of mass wave vector, namely $$\label{O^PSF} O^{\text{PSF}} = \sum_{\bf q} \Phi^{*}_{{\bf q}} \Phi^{*}_{\bf q} \Phi_{\bf q}\nonumber \, .$$ In SMCs a measured parameter is the so-called vacuum Rabi splitting $V_{n \sigma {\bf k}}$ [@Baumberg] of the 1S excitonic resonance, for the range of ${\bf k}$-space of interest essentially constant. Defining $V \doteq V_{\sigma} = \tilde V_{\sigma} \sqrt{A} \phi^*(0)$ $$\tilde V^*_{\sigma} O^{\text{PSF}} = \frac{V}{\sqrt{A}\phi^*(0)} O^{\text{PSF}} = \frac{8}{7} \frac{\pi a^2_{\text{x}}}{A}V = \frac{1}{2} \frac{V}{n_\text{sat}}\, ,$$ where we have set $n_{\text{sat}} \doteq (7/16)\!\! \cdot \!\! (A/\pi a^2_{\text{x}})$, called saturation density. In terms of the two circular polarizations the dynamics induced by $\hat{H}_I$ finally reads $$\label{dt B H_I fin} \frac{d}{dt}\left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}\Bigl|_{\hat{H}_I} = +i \frac{V}{\hbar} \left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\bf \tilde{k}} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{*(1)} Q^{\text{PSF}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}}\, ,$$ where $$\label{PSF term} \sum_{\bf \tilde{k}} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{*(1)} Q^{\text{PSF}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} = \frac{V}{n_\text{sat}} \sum_{\bf k' k'' \tilde{k}} \delta_{\bf k+\tilde k;k'+k''} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{*(1)} \langle \hat{a}_{\pm {\bf k'}} \rangle^{(1)} \langle \hat{B}_{\pm {\bf k''}}\rangle^{(1)}\, .$$ The same lines of argument can be followed for computing the Coulomb-induced interactions $Q^{\text{COUL}(2)}$ [@Takayama; @EPJ; @Kwong-Binder; @PRB; @2001]. We are lead to introduce the saturation density for two main reasons. The most obvious is our interest to refer this work to the literature where $n_{\text{sat}}$ is extensively used [@Langbein; @PRB2004; @Savasta; @PRL2003; @Ciuti; @SST; @Savasta; @PRB2001]. The other most interesting reason is that we can directly compute this quantity. Indeed, the equation of motion for the exciton operator reads $$\begin{aligned} \label{dt B PSF } && \frac{d}{dt}\left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}= -i \bar{\omega}_{\bf k} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)} +i \frac{V}{\hbar} \left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\tilde{\sigma}=\pm {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{*(1)} Q^{\text{COUL}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} \nonumber \\ && \hspace{1.0cm} -\frac{i}{\hbar} 2 \frac{V}{\sqrt{A}\phi^*(0)} O^{\text{PSF}} \sum_{\bf k' k'' \tilde{k}} \delta_{\bf k+\tilde k;k'+k''} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{*(1)} \langle \hat{a}_{\pm {\bf k'}} \rangle^{(1)} \langle \hat{B}_{\pm {\bf k''}}\rangle^{(1)} \nonumber\, . \end{aligned}$$ Leaving apart the discrepancy between the order in the DCTS we can compute the so-called *oscillator strength* (OS), defined as what multiplies the photon expectation values $\langle \hat{a}_{\pm {\bf k}=0} \rangle$, $$\begin{aligned} \label{OS} OS =\!\! i\frac{V}{\hbar} {\Bigg (} 1 - \frac{2}{\sqrt{A}\phi^*(0)} O^{\text{PSF}}\Big[ \langle \hat{B}_{\pm 0} \rangle^{*(1)} \langle \hat{B}_{\pm 0} \rangle^{(1)} \Big] {\Bigg )}\, .\end{aligned}$$ The saturation density may be defined as the exciton density that makes the oscillator strength to be zero. We obtain $$\label{nsat value} n_{\text{sat}} = \Biggl( \frac{2}{\sqrt{A}\phi^*(0)} O^{\text{PSF}} \Biggr)^{-1} = \frac{A}{\pi a^2_{\text{x}}} \ \frac{7}{16}\, .$$ Eventually, the lowest order ($\chi^{(3)}$) nonlinear optical response in SMCs are described by the following set of coupled equations: $$\label{dt a} \frac{d}{dt}\left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} = -i \bar{\omega}^c_{\bf k} \left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} + i \frac{V}{\hbar} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)} + t_c \frac{E_{\pm {\bf k}}}{\hbar}\, ,$$ $$\label{dt B} \frac{d}{dt}\left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)} = -i \bar{\omega}_{\bf k} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)} +i \frac{V}{\hbar} \left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\tilde{\sigma} {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{*(1)} R^{(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}}\, ,$$ with $\sum_{\tilde{\sigma} {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{*(1)} R^{(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} = \sum_{\tilde{\sigma} {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{*(1)} Q^{\text{COUL}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} + \sum_{\bf \tilde{k}} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{*(1)} Q^{\text{PSF}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}}$, with the first of the two addenda originating from Coulomb interaction, Eq. (\[Coul 1S\]), whereas the second represents the phase-space filling contribution written in Eqs. (\[PSF term\]). Starting from here, in the strong coupling case, it might be useful to transform the description into a polariton basis. The proper inclusion of dephasing/relaxation and the application of these equations to parametric processes, in the strong coupling regime, is described in another paper of ours [@nostro; @PRB]. Equations (\[dt a\]) and (\[dt B\]) is exact to the third order in the exciting field. While a systematic treatment of higher-order optical nonlinearities would require an extension of the equations of motions (see e.g. Appendix), a restricted class of higher-order effects can be obtained from solving equations (\[dt a\]) and (\[dt B\]) self-consistently up to arbitrary order as it is usually employed in standard nonlinear optics. This can be simply accomplished by replacing, in the nonlinear sources, the linear excitonic polarization and light fields with the total fields [@Sham; @PRL95; @Savasta; @PRL2003; @Buck]. Multiple-scattering processes are expected to be very effective in cavity-embedded QW’s due to multiple reflections at the Bragg mirrors. Parametric Photoluminescence: Towards Semiconductor Quantum Optics {#4} ================================================================== Entanglement is one of the key features of quantum information and communication technology [@Nielsen-Chuang]. Parametric down-conversion is the most frequently used method to generate highly entangled pairs of photons for quantum-optics applications, such as quantum cryptography and quantum teleportation. This $\chi^{(3)}$ optical nonlinear process consists of the scattering of two polaritons generated by a coherent pump beam into two final polariton modes. The total energy and momentum of the final pairs equal that of pump polariton pairs. The scattering can be spontaneous (parametric emission) or stimulated by a probe beam resonantly exciting one of the two final polariton modes. In 2005 an experiment probing quantum correlations of (parametrically emitted) cavity polaritons by exploiting quantum complementarity has been proposed and realized [@Savasta; @PRL2005]. The most common set-up for parametric emission is the one where a single coherent pump feed resonantly excites the structure at a given energy and wave vector, $\bf{k}_p$. Within the DCTS we shall employ Eqs. (\[dt a up to 3\]), (\[dt B up to 3\]) and Eq. (\[X02\]) in operatorial form, provided all the equations to become fully significant as soon as the expectation value quantities we shall work out would lie within the consistent perturbative DCTS order we set from the beginning [@HRS; @Savasta]. In order to be more *specific* we shall derive explicitly the case of input light beams activating only the $1 S$ exciton sector with all the same circularly (e.g. $\sigma^+$) polarization, thus excluding the coherent excitation of bound two-pair coherences (biexciton) mainly responsible for polarization-mixing [@Sham; @PRL95]. Equations involving polariton pairs with opposite polarization can be derived in complete analogy following the same steps. Starting from the Heisenberg equations for the exciton and photon operators and keeping only terms providing lowest order nonlinear response (in the input light field) we obtain, $$\label{dt Y up to 3} \frac{d}{dt} \hat{a}_{k} = -i \omega^c_k \hat{a}_{k} + i \frac{V^*_{k}}{\hbar} \hat{B}_{k} + t_c \frac{E_k}{\hbar}\, ,$$ $$\begin{aligned} \label{dt X up to 3} &&\frac{d}{dt} \hat{B}_{k} = - i \omega_{k} \hat{B}_{k} +i \frac{V_{k}}{\hbar}\ \hat{a}_{k} + \nonumber \\ && \hspace{2.0cm} + \frac{i}{\hbar} \sum_{\tilde{k}, k', \alpha} V_{k'} {\langle1 \tilde{k}\mid} [\hat{B}_{k}, \hat{B}^\dag_{k'} ] - \delta_{(k'),(k)} {\mid1 \alpha\rangle} \hat{X}_{1 \tilde{k},0} \hat{X}_{0,1 \alpha} \hat{Y}_{0; 1 k'} - \nonumber \\ && \hspace{3.0cm} - \frac{i}{\hbar} \sum_{\tilde{k} \beta}(\omega_{2 \beta} - \omega_{1 \tilde k} - \omega_{1 k}) {\langle1 \tilde k\mid} \hat{B}_{k} {\mid2 \beta\rangle} \hat{X}_{1 \tilde{k},0} \hat{X}_{0, 2 \beta} \, .\end{aligned}$$ In the following we will assume that the pump polaritons driven by a quite strong coherent input field consists of a classical ($\mathbb{C}$-number) field. This approximation is in close resemblance to the two approximations performed in the previous section (linearization of fluctuations and coherent nonlinear processes). We shall show that under this approximation, we may perform the same manipulations ending up to a set of coupled equations analogous to Eqs. (\[dt a\]) and (\[dt B\]). In addition, having a precise set-up chosen, we will be able to specialize our equations and give an explicit account of the parametric contributions as well as the shifts the lowest order nonlinear dynamics provides. We shall retain only those terms containing the semiclassical pump amplitude at $k_p$ twice, thus focusing on the “direct" pump-induced nonlinear parametric scattering processes. It reads $$\begin{aligned} \label{dt X con shift} &&\frac{d}{dt} \hat{B}_{\pm {\bf k}} = - \omega_{\bf k} \hat{B}_{\pm {\bf k}} + i \frac{V}{\hbar}\ \hat{a}_{\pm {\bf k}} - \\ && \hspace{0.0cm} - \frac{i}{\hbar} \frac{V}{n_{\text{sat}}} \sum_{\bf \tilde{k}, k', k''} \delta_{\bf k+\tilde{k},k'+k''} \hat{X}_{1 \pm {\bf \tilde{k}},0} \hat{X}_{0,1 \pm{\bf k''}} \hat{Y}_{0; 1 \pm {\bf k'}} ( \delta_{{\bf k''},{\bf k}_p} \delta_{{\bf k'},{\bf k}_p} + \delta_{{\bf \tilde{k}},{\bf k}_p} \delta_{{\bf k''},{\bf k}_p} + \delta_{{\bf \tilde{k}},{\bf k}_p} \delta_{{\bf k'},{\bf k}_p} ) - \nonumber \\ && \hspace{0.0cm} - \frac{i}{\hbar} \sum_{\tilde \sigma {\bf \tilde{k}}, \sigma_{\beta} {\bf k}_\beta} (\omega_{2 {\bf k}_\beta} - \omega_{1 {\bf \tilde k}} - \omega_{1 {\bf k}}) {\langle1 {\bf \tilde \sigma \tilde k}\mid} \hat{B}_{\pm {\bf k}} {\mid2 \sigma_{\beta} {\bf k}_\beta\rangle} \hat{X}_{1 \tilde \sigma {\bf \tilde{k}},0} \hat{X}_{0, 2 \sigma_{\beta} {\bf k}_\beta} (\delta_{{k}_\beta,2 {k}_p} + \delta_{{\tilde{k}},{k}_p}\delta_{{k}_\beta,{k}+{k}_p} )\nonumber\, ,\end{aligned}$$ where we have already manipulated the phase-space filling matrix element. Here in brackets the first addendum of each line would be responsible for the parametric contribution, whereas the others will give the shifts. It is understood, from now on, that the pump-driven terms (e.g. the $X$ and $Y$ at $k_p$) are ${\mathbb{C}}$-numbers coherent amplitudes like the semiclassical electromagnetic pump field, we will make such distinction in marking with a “hat" the operators only. We need some care in manipulating the Coulomb-induced terms, the last line. Written explicitly it is $$\begin{aligned} \label{X01 Coulomb} &&\frac{d}{dt} \hat{B}_{\pm {\bf k}}{\Bigg |}_{\text{Coul}} = \nonumber \\ && \hspace{0.5cm} - \frac{i}{\hbar} \sum_{\tilde \sigma {\bf \tilde{k}}, \sigma_{\beta} {\bf k}_\beta} (\omega_{2 {\bf k}_\beta} - \omega_{1 {\bf \tilde k}} - \omega_{1 {\bf k}}) {\langle1 \tilde \sigma {\bf \tilde k}\mid} \hat{B}_{\pm {\bf k}} {\mid2 \sigma_{\beta} {\bf k}_\beta\rangle} \hat{X}_{1 \tilde \sigma {\bf \tilde{k}},0} {X}_{0, 2 \sigma_p {\bf k}_p} + \nonumber \\ && \hspace{0.5cm} - \frac{i}{\hbar} \sum_{\tilde \sigma {\bf \tilde{k}}, \sigma_{\beta} {\bf k}_\beta} (\omega_{2 {\bf k}_\beta} - \omega_{1 {\bf \tilde k}} - \omega_{1 {\bf k}}) {\langle1 \tilde \sigma {\bf \tilde k}\mid} \hat{B}_{\pm {\bf k}} {\mid2 \sigma_{\beta} {\bf k}_\beta\rangle} X_{1 \sigma_p {\bf k}_p,0} \hat{X}_{0, 2 \sigma_{{\bf k}+{\bf k}_p}({\bf k}+{\bf k}_p)} \,\end{aligned}$$ As for the term containing ${X}_{0, 2 k_p}$, we are facing a ${\mathbb{C}}$-number which gives no problem in performing the very same procedure of the previous chapter. As for the other we would exploit the formal biexciton solution $$\begin{aligned} \label{X02 sol} && \hat{X}_{0;2 (k+k_p)} (t) = \int_{-\infty}^t dt' e^{-i \omega_{2 (k+k_p)}(t-t')} \frac{i}{\hbar} {\Bigg (} V_{k_p} {\langle2 (k+k_p)\mid} \hat{B}^\dag_{k_p} {\mid1 k\rangle} \hat{X}_{0,1 k} Y_{0,1 k_p} + \nonumber \\ && \hspace{0.5cm} V_{k} {\langle2 (k+k_p)\mid} \hat{B}^\dag_{k} {\mid1 k_p\rangle} X_{0,1 k_p} \hat{Y}_{0,1 k} {\Bigg )} \, ,\end{aligned}$$ where, for the sake of consistence, we are neglecting $\hat{X}_{0;2 (k+k_p)} (-\infty)$ because the biexciton, within the present approximations, is always generated by an operator at $k$ times a classical amplitude at $k_p$ which is always zero before the electromagnetic impulse arrived. Moreover, an analogous identity such that of Eq. (\[Sham identity\]) is valid in the present context, namely $$\begin{aligned} \label{Sham id per ops} && \frac{d}{dt} \Bigg( \hat{X}_{0,1 k} {X}_{0, 1 k_p} e^{-i (\omega_{1 k} + \omega_{1 k_p})(t-t')} \bigg) = \\ && \hspace{2.5cm} = \bigg( i \frac{V_k}{\hbar} \hat{Y}_{0,1 k} X_{0,1 k_p} + i \frac{V_{k_p}}{\hbar} Y_{0,1 k_p} \hat{X}_{0,1 k} \bigg)e^{-i (\omega_{1 k} + \omega_{1 k_p})(t-t')} \nonumber\, .\end{aligned}$$ With these tools at hand we are able to perform step by step the manipulations of the previous section for all the quantities at play. The final result reads $$\begin{aligned} \label{dt X completa1} &&\frac{d}{dt} \hat{B}_{\pm {\bf k}} = - \omega_{\bf k} \hat{B}_{\pm {\bf k}} + i \frac{V}{\hbar}\ \hat{a}_{\pm {\bf k}} - \nonumber \\ && \hspace{0.5cm} - \frac{i}{\hbar} \frac{V}{n_{\text{sat}}} \bigg( \hat{X}_{1 \pm {\bf k}_i,0} X_{0,1 \pm {\bf k}_p} Y_{0, 1 \pm {\bf k}_p} + X_{1 \pm {\bf k}_p,0} X_{0,1 \pm {\bf k}_p} \hat{Y}_{0, 1 \pm {\bf k}} + X_{1 \pm {\bf k}_p,0} \hat{X}_{0,1 \pm {\bf k}} Y_{0, 1 \pm {\bf k}_p} \bigg) - \nonumber \\ && \hspace{0.5cm} - \frac{i}{\hbar} \hat{X}_{1 \pm {\bf k}_i,0}(t) \Bigg\{ V_{\text{xx}} X_{0,1 \pm {\bf k}_p}(t) X_{0,1 \pm {\bf k}_p}(t) - i \int_{-\infty}^t dt' F^{\pm \pm}(t-t') X_{0,1 \pm {\bf k}_p}(t')X_{0,1 \pm {\bf k}_p}(t') \Bigg\} - \nonumber \\ && \hspace{0.0cm} - 2 \frac{i}{\hbar} X_{1 \sigma_{{\bf k}_p} {\bf k}_p,0}(t) \Bigg\{ V_{\text{xx}} \hat{X}_{0,1 \pm {\bf k}}(t) X_{0,1 \pm {\bf k}_p}(t) - i \int_{-\infty}^t dt' F^{\pm \pm}(t-t') \hat{X}_{0,1 \pm {\bf k}}(t') X_{0,1 \pm {\bf k}_p}(t') \Bigg\}\, ,\end{aligned}$$ where ${\bf k}_i = 2{\bf k}_p-{\bf k}$, and again $V_{\text{xx}}$ and $F^{\pm \pm}(t-t')$ have reabsorbed the $1/2$ originating from Eq. (\[Sham id per ops\]). In the specific case under analysis we are considering co-circularly polarized waves and the mean field term, $V_{\text{xx}}$ as well as the the kernel function $F(t)$ can be found in Refs. [@Takayama; @EPJ; @Kwong-Binder; @PRB; @2001]. Eventually, the lowest order ($\chi^{(3)}$) nonlinear optical response in SMCs is given by the following set of coupled equations where, in the same spirit of the final remark in the previous section, we account for multiple scattering simply by replacing the linear excitonic polarization and light fields with the total fields: $$\begin{aligned} \label{final sys} &&\frac{d}{dt} \hat{a}_{\pm {\bf k}} = -i \omega^c_{\bf k} \hat{a}_{\pm {\bf k}} + i \frac{V}{\hbar}\ \hat{B}_{\pm {\bf k}} + t_c \frac{E_{\pm {\bf k}}}{\hbar} \nonumber \\ &&\frac{d}{dt} \hat{B}_{\pm k} = -i\omega_{\bf k} \hat{B}_{\pm {\bf k}} + \hat{s}_{\pm {\bf k}} + i \frac{V}{\hbar}\ \hat{a}_{\pm {\bf k}} - \frac{i}{\hbar}{R}^{NL}_{\pm {\bf k}}\, ,\end{aligned}$$ where ${R}^{NL}_{\pm {\bf k}}=(R^{sat}_{\pm {\bf k}}+{R}^{\text{xx}}_{\pm {\bf k}})$ $$\begin{aligned} \label{NN terms} && R^{sat}_{\pm {\bf k}} = \frac{V}{n_{\text{sat}}} B_{\pm {\bf k}_p} a_{\pm {\bf k}_p} \hat{B}^\dag_{\pm {\bf k}_i} \nonumber \\ && R^{\text{xx}}_{\pm {\bf k}} = \hat{B}^\dag_{\pm {\bf k}_i}(t) \bigg( V_{\text{xx}} B_{\pm {\bf k}_p}(t) B_{\pm {\bf k}_p}(t) - \nonumber \\ && - i \int_{-\infty}^t dt' F^{\pm \pm}(t-t') B_{\pm {\bf k}_p}(t') B_{\pm {\bf k}_p}(t') \bigg)\, .\end{aligned}$$ The pump induced renormalization of the exciton dispersion gives a frequency shift $$\begin{aligned} && \hat{s}_{\pm {\bf k}} = -i \bigg( \frac{V}{n_{\text{sat}}} \big(B^*_{\pm {\bf k}_p} a_{\pm {\bf k}_p} \hat{B}_{\pm {\bf k}} + B^*_{\pm {\bf k}_p} B_{\pm {\bf k}_p} \hat{a}_{\pm {\bf k}} \big) + \nonumber \\ && \hspace{2.5cm} 2 \frac{V_{\text{xx}}}{\hbar} B^*_{\pm {\bf k}_p} B_{\pm {\bf k}_p} \hat{B}_{\pm {\bf k}} - \nonumber \\ && \hspace{2.5cm} -2 \frac{i}{\hbar} B^*_{\pm {\bf k}_p}(t) \int_{-\infty}^t dt' F^{\pm \pm}(t-t') \hat{B}_{\pm {\bf k}}(t') B_{\pm {\bf k}_p}(t') \bigg)\, .\end{aligned}$$ Equations, (\[final sys\]) are the main result of this paper. They can be considered the starting point for the microscopic description of quantum optical effects in SMCs. These equations extend the usual semiclassical description of Coulomb interaction effects, in terms of a mean-field term plus a genuine non-instantaneous four-particle correlation, to quantum optical effects. Analogous equations can be obtained starting from an effective Hamiltonian describing excitons as interacting bosons [@CiutiBE]. The resulting equations (usually developed in a polariton basis) do not include correlation effects beyond Hartree-Fock. Moreover the interaction terms due to phase space filling differs from those obtaind within the present approach not based on an effective Hamiltonian. Only the many-body electronic Hamiltonian, the intracavity-photon Hamiltonian and the Hamiltonian describing their mutual interaction have been taken into account. Losses through mirrors, decoherence and noise due to environment interactions as well as applications of this theoretical framework will be addressed in another paper of ours [@nostro; @PRB]. Conclusion ========== In this paper we set a dynamics controlled truncation scheme approach to nonlinear optical processes in cavity embedded semiconductor QWs without any assumption on the quantum statistics of the excitons involved. This approach represents the starting point for the microscopic analysis to quantum optics experiments in the strong coupling regime. We presented a systematic theory of Coulomb-induced correlation effects in the nonlinear optical processes in SMCs. We end up with dynamical equations for exciton and photon operators which extend the usual semiclassical description of Coulomb interaction effects, in terms of a mean-field term plus a genuine non-instantaneous four-particle correlation, to quantum optical effects. The proper inclusion of the detrimental environment interactions as well as applications of the present theoretical scheme will be presented in another paper of ours [@nostro; @PRB]. The Equation of Motion At Any Order {#gen eq} =================================== The equation of motion for the operators in (\[Hubbard\]), under the Hamiltonian $\hat{H} = \hat{H}_{e} + \hat{H}_{c} + \hat{H}_{I} + \hat{H}_p$ reads: $$\begin{aligned} \frac{d}{dt} \left(\right. \hspace{-0.4cm} && \hat{X}_{N\alpha;M\beta} \hat{Y}_{n \lambda;m\mu} \left.\right) = -i(\omega_{M \beta} -\omega_{N \alpha} + \sum_{i=1}^m \omega^c_{k_i} - \sum_{j=1}^n \omega^c_{k_j} ) \left(\right. \hat{X}_{N\alpha;M\beta} \hat{Y}_{n \lambda;m\mu} \left.\right) + \nonumber \\ && + \hat{X}_{N\alpha;M\beta} \big( \delta_{m,1} t_c \frac{E_{\mu}}{\hbar} \hat{Y}_{n \lambda;0} + \delta_{n,1} t_c \frac{E^{*}_{\lambda}}{\hbar} \hat{Y}_{0;m\mu} \big) - \hat{X}_{N\alpha;M\beta} \sum_{\bar{k}} t_c \big( \delta_{m,0} \frac{E^{*}_{\bar{k}}}{\hbar} \hat{Y}_{n \lambda;1 \bar{k}} + \delta_{n,0} \frac{E_{\bar{k}}}{\hbar} \hat{Y}_{1 \bar{k};m\mu} \big) + \nonumber \\ && + \sum_{\bar{k} \nu} t_c \hat{X}_{N\alpha;M\beta} \Bigg[ \Theta(m-2) {\langlem \mu\mid}\hat{a}^\dag_{\bar{k}} {\mid(m-1) \nu\rangle}\frac{E_{\bar{k}}}{\hbar} \hat{Y}_{n \lambda;(m-1)\nu} + \nonumber \\ &&\hspace{5.0cm} \Theta(n-2) {\langle(n-1) \nu\mid}\hat{a}_{\bar{k}} {\midn \lambda\rangle} \frac{E^{*}_{\bar{k}}}{\hbar} \hat{Y}_{(n-1) \nu;m \mu} - \nonumber \\ &&\hspace{3.0cm} - \Theta(m-1) {\langlem \mu\mid}\hat{a}_{\bar{k}} {\mid(m+1) \nu\rangle} \frac{E^{*}_{\bar{k}}}{\hbar} \hat{Y}_{n \lambda;(m+1)\nu} - \nonumber \\ && \hspace{5.0cm} \Theta(n-1) {\langle(n+1) \nu\mid}\hat{a}^\dag_{\bar{k}} {\midn \lambda\rangle} \frac{E_{\bar{k}}}{\hbar} \hat{Y}_{(n+1) \nu;m \mu} \Bigg] + \nonumber \\ && +\frac{i}{\hbar} \delta_{M,0} \delta_{\beta,0} \delta_{m,1} \sum_{\bar{n}}V^*_{\bar{n}\mu} \hat{X}_{N\alpha;1 \bar{n} \mu} \hat{Y}_{n \lambda;0} - \frac{i}{\hbar} \delta_{N,0} \delta_{\alpha,0} \delta_{n,1} \sum_{\bar{n}} V_{\bar{n}\lambda} \hat{X}_{1 \bar{n} \lambda;M \beta} \hat{Y}_{0;m \mu} - \nonumber \\ && -\frac{i}{\hbar} \delta_{N,1} \delta_{n,0}\delta_{\lambda,0} V^*_{\alpha} \hat{X}_{0;M\beta} \hat{Y}_{1 k_{\alpha};m \mu} +\frac{i}{\hbar} \delta_{M,1} \delta_{m,0}\delta_{\mu,0} V_{\beta} \hat{X}_{N \alpha;0} \hat{Y}_{n \lambda;1 k_{\beta}} + \nonumber \\ && +\frac{i}{\hbar} \delta_{m,1} \Theta(M-1) \sum_{\bar{n} \delta} V^*_{\bar{n} \mu} {\langleM \beta\mid}\hat{B}_{\bar{n}\mu} {\mid(M+1) \delta\rangle} \hat{X}_{N \alpha;(M+1) \delta} \hat{Y}_{n \lambda,0} - \nonumber\\ && -\frac{i}{\hbar}\delta_{n,1} \Theta(N-1) \sum_{\bar{n} \eta} V_{\bar{n} \lambda} {\langle(N+1) \eta\mid}\hat{B}^\dag_{\bar{n}\lambda} {\midN \alpha\rangle} \hat{X}_{(N+1) \eta;M \beta} \hat{Y}_{0;m \mu} - \nonumber\\ && -\frac{i}{\hbar} \delta_{n,0} \delta_{\lambda,0} \Theta(N-2) \sum_{\bar{n} \bar{k} \eta} V^*_{\bar{n} \bar{k}} {\langle(N-1) \eta\mid}\hat{B}_{\bar{n}\bar{k}} {\midN \alpha\rangle} \hat{X}_{(N-1) \eta;M \beta} \hat{Y}_{1 \bar{k};m \mu} + \nonumber\\ && +\frac{i}{\hbar} \delta_{m,0} \delta_{\mu,0} \Theta(M-2) \sum_{\bar{n} \bar{k} \delta} V_{\bar{n} \bar{k}} {\langleM \beta\mid}\hat{B}^\dag_{\bar{n}\bar{k}} {\mid(M-1) \delta\rangle} \hat{X}_{N \alpha;(M-1) \delta} \hat{Y}_{n \lambda;1 \bar{k}} + \nonumber\\ && +\frac{i}{\hbar} \delta_{M,0} \delta_{\beta,0} \Theta(m-2) \sum_{\bar{n} \bar{k} \nu} V^*_{\bar{n} \bar{k}} {\langlem \mu\mid}\hat{a}^\dag_{\bar{k}} {\mid(m-1) \nu\rangle} \hat{X}_{N \alpha;1 \bar{n} \bar{k}} \hat{Y}_{n \lambda;(m-1) \nu} - \nonumber\\ && -\frac{i}{\hbar} \delta_{N,0} \delta_{\alpha,0} \Theta(n-2) \sum_{\bar{n} \bar{k} \gamma} V_{\bar{n} \bar{k}} {\langle(n-1) \gamma\mid}\hat{a}_{\bar{k}} {\midn \lambda\rangle} \hat{X}_{1 \bar{n} \bar{k};M \beta} \hat{Y}_{(n-1) \gamma;m \mu} - \nonumber\\ && -\frac{i}{\hbar} \delta_{N,1} \Theta(n-1) V_{\alpha} \sum_{\gamma} {\langle(n+1) \gamma\mid}\hat{a}^\dag_{k_{\alpha}} {\midn \lambda\rangle} \hat{X}_{0;M \beta} \hat{Y}_{(n+1) \gamma;m \mu} + \nonumber\\ && +\frac{i}{\hbar} \delta_{M,1} \Theta(m-1) V_{\beta} \sum_{\nu} {\langlem \mu\mid}\hat{a}_{k_{\beta}} {\mid(m+1) \nu\rangle} \hat{X}_{N \alpha;0} \hat{Y}_{n \lambda;(m+1) \nu} + \nonumber\end{aligned}$$ $$\begin{aligned} \label{dt gen op} && + \frac{i}{\hbar} \sum_{\bar{n} \bar{k}} \sum_{\nu \delta} {\Bigg [} V^*_{\bar{n} \bar{k}} {\Big (} \Theta(M-1) \Theta(m-2) {\langleM \beta\mid}\hat{B}_{\bar{n} \bar{k}} {\mid(M+1) \delta\rangle} \nonumber \\ && \hspace{3.0cm} {\langlem \mu\mid}\hat{a}^\dag_{\bar{k}} {\mid(m-1) \nu\rangle} \hat{X}_{N \alpha;(M+1)\delta} \hat{Y}_{n \lambda;(m-1) \nu} - \nonumber \\ && \hspace{2.5cm} - \Theta(N-2) \Theta(n-1) {\langle(N-1) \delta\mid}\hat{B}_{\bar{n} \bar{k}} {\midN \alpha\rangle} \nonumber \\ && \hspace{3.0cm} {\langle(n+1)\nu\mid}\hat{a}^\dag_{\bar{k}} {\midn \lambda\rangle} \hat{X}_{(N-1) \delta;M\beta} \hat{Y}_{(n+1) \nu;m \mu} {\Big )} + \nonumber \\ && \hspace{2.0 cm} - \frac{i}{\hbar}V_{\bar{n} \bar{k}} {\Big (} \Theta(N-1) \Theta(n-2) {\langle(N+1) \delta\mid}\hat{B}^\dag_{\bar{n} \bar{k}} {\midN \alpha\rangle} \nonumber \\ && \hspace{3.0cm} {\langle(n-1) \nu\mid}\hat{a}_{\bar{k}} {\midn \lambda\rangle} \hat{X}_{(N+1) \delta;M \beta} \hat{Y}_{(n-1) \nu;m \mu} - \nonumber \\ && \hspace{2.5cm} - \Theta(M-2) \Theta(m-1) {\langleM \beta\mid}\hat{B}^\dag_{\bar{n} \bar{k}} {\mid(M-1) \delta\rangle} \nonumber \\ && \hspace{3.0cm} {\langlem\mu\mid}\hat{a}_{\bar{k}} {\mid(m+1) \nu\rangle} \hat{X}_{N \alpha;(M-1)\delta} \hat{Y}_{n \lambda;(m+1) \nu} {\Big )} {\Bigg ]}\, . \end{aligned}$$ Here $\Theta(x)$ is the Heaviside function equal to 1 for positive argument and zero otherwise. N [*eh*]{} pair aggregates {#Npair states} ========================== We start from the usual model for the electronic Hamiltonian of a direct two-band semiconductor [@Haugh; @AxtKuhn]. It is obtained from the many-body Hamiltonian of the interacting electron system in a lattice, keeping explicitly only those terms in the Coulomb interaction preserving the number of electrons in a given band and can be expressed as $$\label{Ham electron1} \hat{H}_e = \hat{H}_0 + \hat{V}_{\text{Coul}}\, .$$ It comprises the single-particle Hamiltonian terms for electrons in conduction band and holes in valence band (here $k\equiv ({\bf k}, \sigma)$ and $\hat{c}_{\sigma,{\bf k}}$ ($\hat{d}_{\sigma,{\bf k}}$) annihilates an electron (a hole)) : $$\label{H zero} \hat{H}_0 = \sum_{k} E_{c,{k}} \hat{c}^\dag_{k} \hat{c}_{k} + \sum_{k} E_{h,{k}}\hat{d}^\dag_{k} \hat{d}_{k}\, ,$$ and the Coulomb interaction term of three contributions, the two repulsive electron-electron (e-e) and hole-hole (h-h) terms and the attractive (e-h) one: $$\begin{aligned} \label{V coul} \hat{V}_{\text{Coul}} && = \frac{1}{2} \sum_{{\bf q} \neq 0} V_q \sum_{\sigma,{\bf k},\sigma',{\bf k}'} \hat{c}^\dag_{\sigma,{\bf k}+{\bf q}} \hat{c}^\dag_{\sigma',{\bf k}'-{\bf q}} \hat{c}_{\sigma',{\bf k}'} \hat{c}_{\sigma,{\bf k}} + \frac{1}{2} \sum_{{\bf q} \neq 0} V_q \sum_{\sigma,{\bf k},\sigma',{\bf k}'} \hat{d}^\dag_{\sigma,{\bf k}+{\bf q}} \hat{d}^\dag_{\sigma',{\bf k}'-{\bf q}} \hat{d}_{\sigma',{\bf k}'} \hat{d}_{\sigma,{\bf k}} - \nonumber \\ && - \sum_{{\bf q} \neq 0} V_q \sum_{\sigma,{\bf k},\sigma',{\bf k}'} \hat{c}^\dag_{\sigma,{\bf k}+{\bf q}} \hat{d}^\dag_{\sigma',{\bf k}'-{\bf q}} \hat{d}_{\sigma',{\bf k}'}c_{\sigma,{\bf k}}\, .\end{aligned}$$ A many-body interacting state is usually very different from a product state, however a common way to express the former is by a superposition of uncorrelated product states. The physical picture that arises out of it expresses the *dressing* the interaction performs over a set of noninteracting particles. The general many-body Schrödinger equation for this Coulomb-correlated system is $$\label{Schr eq} \hat{H}_e {\mid\Psi\rangle} = (\hat{H}_0 + \hat{V}_{\text{Coul}}) {\mid\Psi\rangle} = E {\mid\Psi\rangle}\, ,$$ with ${\mid\Psi\rangle}$ the global interacting many-body state of the whole Fock space and $E$ its corresponding energy. The system Hamiltonian commutes with the total-number operators for electron and holes, i.e. $\hat{N}_e = \sum_{k} \hat{c}^\dag_{k}\hat{c}_{k}$ and $\hat{N}_h = \sum_{k} \hat{d}^\dag_{k}\hat{d}_{k}$. Therefore the state ${\mid\Psi\rangle}$ may be build up corresponding on a given number of electrons and of holes. Moreover, because we shall consider the case of intrinsic semiconductors materials where $N_e = N_h \doteq N $, the good quantum number for the Schrödinger equation (\[Schr eq\]) is the total number of electron-hole pairs $N$, explicitly $$\label{Schr in N} \hat{H}_e {\midN \alpha\rangle} = E_{N \alpha} {\midN \alpha\rangle}\, ,$$ where $\alpha$ is the whole set of proper quantum numbers needed to specify univocally the many-body state. For any given number $N$ of electron-hole pairs, the product-state set, built up from the single-particle states $\{ {\midN a\rangle} \}$ eigenstates of the noninteracting carrier Hamiltonian $\hat{H}_0$, is a natural complete basis of the N-pair sector of the global Fock space: $$\label{nonint H} \hat{H}_0 {\midN a\rangle} = \epsilon_{N a} {\midN a\rangle}\, ,$$ where N identifies the N-pair subspace and $a$ is a compact form for all the single particle indexes, i.e. $a \equiv {j}_{e1},{j}_{e2}, ...,{j}_{eN};{j}_{h1},{j}_{h2},...,{j}_{hN}$. Indeed $$\label{prod} {\midN,a\rangle} = \otimes_{n=1}^N \hat{c}^\dag_{j_{en}} \hat{d}^\dag_{j_{hn}} {\mid0,0\rangle}\ \ \text{and} \ \ \epsilon_{N a} = \sum_{n=1}^N (\epsilon_{{j}_{en}} + \epsilon_{{j}_{hn}} )\, .$$ Being a complete orthonormal basis for the N-pair subspace we may expand the many-body state ${\midN, \alpha\rangle}$ over it, it yields $$\label{U} {\midN \alpha\rangle} = \sum_{a} U^{N \alpha}_{a} {\midN a\rangle}\, .$$ It is only a matter of calculation to show that $U^{N \alpha}_a$ is nothing but the envelope function of the N-pair aggregate, solution of the corresponding secular equation. Indeed the eigenvalue problem (\[Schr in N\]) is transformed into: $$\label{dressing} \sum_{a'} ( {\langleN a\mid}\hat{H}_e {\midN a'\rangle} - E_{N \alpha} \delta_{a,a'} ) U^{N \alpha}_{a'} = 0\, .$$ Namely N=1 leads to the exciton secular equation, whereas N=2 represents the biexciton (two pairs) Coulomb problem. In order to be clearer we shall propose in details the N=1 exciton calculation. We shall work in the direct lattice ${\bf r} \leftrightarrow {\bf r}_i$ (the former is a continuous variable whereas the latter is a point in the $3D$ lattice). Using the general mapping [@Cohen-Tannoudji; @QED] $\sum_{r_i} \leftrightarrow (1/v_0) \int d^3r$, $\delta({\bf r} - {\bf r}') = (\delta_{{\bf r}_i,{\bf r_j}}/v_0)$, and $ \hat{a}^\dag_{{\bf r}_i} = (\hat{a}^\dag({{\bf r}_i})/{\sqrt{v_0}}) $, here $v_0$ is the unit cell volume and for simplicity the spin selection rules for the optically active states has been already taken into account, (\[dressing\]) reads $$\label{sec eq dir l} \sum_{{\bf r}'_e,{\bf r}'_h} {\Bigg (} {\langle{\bf r}_e,{\bf r}_h\mid}\hat{H}_e {\mid{\bf r}'_e,{\bf r}'_h\rangle} - E_{n \sigma {\bf k}} \delta_{{\bf r}_e {\bf r}_h,{\bf r}'_e {\bf r}'_h} {\Bigg )} U^{\alpha}_a({\bf r}'_e,{\bf r}'_h) = 0\, ,$$ with $$\label{Ham in r} {\langle{\bf r}_e,{\bf r}_h\mid}\hat{H}_e {\mid{\bf r}'_e,{\bf r}'_h\rangle} = {\Bigg (} - \frac{\hbar^2}{2m_e} \nabla^2_{r_e} - \frac{\hbar^2}{2m_h} \nabla^2_{r_h} - \frac{e^2}{\varepsilon_r |{\bf r}_e - {\bf r}_h|}+ V({\bf r}_e,{\bf r}_h) {\Bigg )} \delta_{{\bf r}_e {\bf r}_h,{\bf r}'_e {\bf r}'_h}\, ,$$ here $ V({\bf r}_e,{\bf r}_h)$ represents all the additional potential, e.g. those of the heterostructures or those of disorder effects, $ V({\bf r}_e,{\bf r}_h) = V^e(z_e) + V^h(z_h)$. Typically, the energy difference between the lowest QW subband level and the first excited one (at least a few meV) is much larger than the Coulomb interaction between particles (a few meV). As a consequence, at least at low temperatures, particles are confined at the lowest quantization level and the (possible) distorsion of the wave function due to the Coulomb-activated admixture of different subbands can be safely neglected. In some extent, then, the particle wave function dependence along the growth (say $z$) direction can be factorized out and the dynamics becomes essentially two-dimensional. However, a purely $2D$ approximation for excitons would miss important effects of the geometrical QWs parameters on the binding energy and would not be able to account for the interaction with a $3D$ continuum environment of surrounding modes (e.g. acoustic phonon modes in heterostructures with alloy lattice constant in close proximity [@Takagahara]). In addition in QWs, light and heavy holes in valence band are split off in energy. Assuming that this splitting is much larger than kinetic energies of all the involved particles and, as well, much larger than the interaction between them, we shall consider only heavy hole states as occupied. In Eq. (\[sec eq dir l\]) the $3D$ Coulomb interaction prevents form factorizing into (free) in-plane and confined directions. Nevertheless if we assume that the quantization energy along $z$ is much larger than the Coulomb energy, at leading order we can factorize out the $z$-dependence $$\begin{aligned} \label{in z} {\Bigg (} - \frac{\hbar^2}{2m_e} \frac{d^2}{dz^2_e} + V^e(z_e) - \frac{\hbar^2}{2m_h} \frac{d^2}{dz^2_h} &+& V^h(z_h) {\Bigg )} U^{\alpha}({\bf r}_e,{\bf r}_h) = \nonumber \\ && E^{z} U^{\alpha}({\bf r}_e,{\bf r}_h)\, .\end{aligned}$$ It means we are solving our secular equation with solutions built up as linear combination of $F^{\alpha}_{n_c,n_v,a}({\bf r}^\|_e,{\bf r}^\|_h)c_{n_c}(z_e)v_{n_v}(z_h)$, with ${\bf r} = ({\bf r}^\|,z)$. Equation (\[in z\]) expresses the lack of translational symmetry along the growth $z$ direction, thus single particle states experience confinement and two additional QW subband quantum numbers $n_v,n_c$ (for valence and conduction states respectively) appear. We still leave $a$ as a reminder that new possible indexes could still arise in due course. Projecting Eq. (\[sec eq dir l\]) on these confined states we end up with an effective Schrödinger equation in the plane $$\begin{aligned} \label{Wannier prima} {\Bigg (} - \frac{\hbar^2}{2m_e} \nabla^2_{\bf r^\|_e} - \frac{\hbar^2}{2m_h} \nabla^2_{\bf r^\|_h} - U_{n_c,n_v;n'_c,n'_v}(|{\bf r}^\|_e - {\bf r}^\|_h|) {\Bigg )} F^{\alpha}_{n_c,n_v,a}({\bf r}^\|_e,{\bf r}^\|_h) = \nonumber \\ && \hspace{-5.0cm} = (E_{\alpha} - E^z_{n_c} - E^z_{n_v}) F^{\alpha}_{n_c,n_v,a}({\bf r}^\|_e,{\bf r}^\|_h)\, ,\end{aligned}$$ with $$U_{n_c,n_v;n'_c,n'_v}(|{\bf r^\|_e} - {\bf r^\|_h|}) = \int dz_e \int dz_h \frac{e^2}{\varepsilon_r \sqrt{|{\bf r}^\|_e - {\bf r}^\|_h|^2 +(z_e - z_h)^2}} c_{n_c}(z_e) c_{n'_c}(z_e) v_{n_v}(z_h)v_{n'_v}(z_h)\, .$$ For what already stated, we shall consider only the lowest confined subband levels, then the resulting effective in-plane secular equation becomes $$\begin{aligned} \label{Wannier GS} {\Bigg (} - \frac{\hbar^2}{2m_e} \nabla^2_{\bf r^\|_e} - \frac{\hbar^2}{2m_h} \nabla^2_{\bf r^\|_h} - \int dz_e \int dz_h \frac{e^2}{\varepsilon_r \sqrt{|{\bf r}^\|_e - {\bf r}^\|_h|^2 +(z_e - z_h)^2}} |c_{n_c}(z_e)|^2 |v_{n_v}(z_h)|^2 {\Bigg )} F^{\alpha}_{a}({\bf r}^\|_e,{\bf r}^\|_h) = \nonumber \\ && \hspace{-8.0cm} = (E_{\alpha} - E^z_c - E^z_v) F^{\alpha}_{a}({\bf r}^\|_e,{\bf r}^\|_h)\, ,\end{aligned}$$with the product exciton envelope function $U^{\alpha}({\bf r}_e,{\bf r}_h)=F^{\alpha}_{a}({\bf r}^\|_e,{\bf r}^\|_h)c(z_e)v(z_h)$. Equation (\[Wannier GS\]) is solvable by separation of variables once we employ a coordinate transformation into center of mass (CM) ${\bf R}=(m_e{\bf r}^\|_e+m_h{\bf r}^\|_h)/(m_e+m_h)$ and relative ${\bf \rho}=({\bf r}^\|_e - {\bf r}^\|_h)$ exciton coordinates. It reads $$\label{Wannier R rho} {\Bigg (} - \frac{\hbar^2}{2M} \nabla^2_{\bf R} - \frac{\hbar^2}{2\mu} \nabla^2_{\bf \rho} - U({\bf \rho}) {\Bigg )} F^{\alpha}_{a}({\bf R}, {\bf \rho}) = E F^{\alpha}_{a}({\bf R}, {\bf \rho})\, ,$$ with a solution we can arrange as $F^{\alpha}_{a}({\bf R}, {\bf \rho}) = \frac{e^{i\, {\bf K} \cdot {\bf R}}}{\sqrt{A}} W^{\alpha}_a({\bf \rho})$ the latter solution of the relative hydrogen-like 2D problem. Eventually, in real-space representation, we have our exciton wave function with total in-plane CM wave vector ${\bf K}$ ($A$ is the in-plane quantization surface in the free directions) which reads $$\label{exciton} {\midn \sigma {\bf K}\rangle}= \frac{v_0}{\sqrt{A}} \sum_{\bf r_e,r_h} e^{i \bf k \cdot R} W_{n \sigma}(\rho) c(z_e) v(z_h) a^\dag _{c,{\bf r}_e} a^\dag_{h,{\bf r}_h } {\mid0\rangle}\, ,$$ being $\hat{a}^\dag _{c/v,{\bf r}} (\hat{a}_{c/v,{\bf r }})$ creation (annihilation) operator of the conduction- or valence- band electron in the Wannier representation and ${\bf r}_{e/h} = ({\bf r}^{\|}_{e/h},z_{e/h})$ are to be considered coordinates of the direct lattice, ${\mid\!0\rangle}$ is the crystal ground state. When e.g. exploring the exciton-phonon interaction, it is useful to express exciton states in reciprocal space. With the usual transformation to Bloch representation, ($N = v_0 A\,L$ is the number of unit cells and $L$ is the quantization dimension along the confined direction, $\nu = c,v$), $$\label{BlochWannier} \hat{a}_{\nu ({\bf r},z)} = \frac{1}{\sqrt{N}} \sum_{{\bf k},k_z} e^{i {\bf k} \cdot {\bf r}_n} \hat{a}_{\nu ({\bf k},k_z)}\, ,$$ one obtains: $$\begin{aligned} \label{exc in Bloch rep 1} && {\midn \sigma {\bf K}\rangle} = \sum_{\substack{{\bf k}, {\bf k}'\\{k_z,k'_z}}} \delta_{{\bf K},{\bf k}-{\bf k}'} {\Bigg (} \frac{1}{\sqrt{A}\, L} \int d{\rho} \int dz_e \int dz_h W_{n \sigma}(\rho) c(z_e) v(z_h) e^{-i\, {\bf \rho} \cdot (\eta_h {\bf k} + \eta_e {\bf k}')} e^{-i\, k_z z_e} e^{-i\, k'_z z_h} {\Bigg )} \nonumber \\ && \hspace{3.0cm} a^\dag _{c,({\bf k},k_z)}a_{v,({\bf k}',k'_z)} \bigl| 0\bigr>\, .\end{aligned}$$ In order to end up with a form as much as possible in analogy with its bulk counterpart we shall define CM and relative coordinates even in the reciprocal lattice: $$\begin{aligned} \label{rules} \left\{ \begin{array}{l} {\bf K} = {\bf k} - {\bf k}' \\ {\bf k}_r = \eta_h {\bf k} + \eta_e {\bf k}' \end{array} \right. \Longrightarrow \left\{ \begin{array}{l} {\bf k} = {\bf k}_r + \eta_e {\bf K} \\ {\bf k}' = {\bf k}_r - \eta_h {\bf K} \end{array} \right.\, .\end{aligned}$$ It becomes $$\begin{aligned} \label{exc in Bloch rep 2} && {\midn \sigma {\bf K}\rangle} = \sum_{{\bf K}, {\bf k}_r} \delta_{{\bf K},{\bf k}-{\bf k}'} \sum_{k_z,k'_z} {\Bigg (} \frac{1}{\sqrt{A}} \int d{\rho} W_{n \sigma}(\rho) e^{-i\, {\bf \rho} \cdot {\bf k}_r} {\Bigg )} {\Bigg (} \frac{1}{\sqrt{L}} \int dz_e c(z_e) e^{-i\, k_z z_e} {\Bigg )} \nonumber \\ && \hspace{3.0cm} {\Bigg (} \frac{1}{\sqrt{L}}\int dz_h v(z_h) e^{-i\, k'_z z_h} {\Bigg )} a^\dag _{c,({\bf k}_r + \eta_e {\bf K},k_z)}a_{v,({\bf k}_r - \eta_h {\bf K},k'_z)} \bigl| 0\bigr>\, .\end{aligned}$$ Thus $$\begin{aligned} \label{exc in Bloch rep final} && {\midn \sigma {\bf K}\rangle} = \sum_{{\bf k}_r} \sum_{k_z,k'_z} \Phi^{\bf K}_{n \sigma,{\bf k}_r} u^c_{k_z} u^{v *}_{k'_z} a^\dag _{c,({\bf k}_r + \eta_e {\bf K},k_z)}a_{v,({\bf k}_r - \eta_h {\bf K},k'_z)} \bigl| 0\bigr>\, ,\end{aligned}$$ or in the electron-hole picture ($\hat{a}_{v,{\bf k}} = \hat{d}^\dag_{-{\bf k}}$ and $-{\bf k}\left.\right|_{el} = {\bf k}\left.\right|_{hole}$) $$\begin{aligned} \label{exc in Bloch rep final hole picture} && {\midn \sigma {\bf K}\rangle} = \sum_{{\bf k}_r} \sum_{k_z,k'_z} \Phi^{\bf K}_{n \sigma,{\bf k}_r} u^e_{k_z} u^{h}_{k'_z} c^\dag _{({\bf k}_r + \eta_e {\bf K},k_z)}d^\dag_{(-{\bf k}_r + \eta_h {\bf K},-k'_z)} \bigl| 0\bigr>\, ,\end{aligned}$$ with the relations in Eq. (\[rules\]) changed accordingly. [999]{} Axt V M and Stahl A, Z. Phys. B [**93**]{}, 195 (1994). Th. Österich, K. Schönhammer, and L. J. Sham, Phys. Rev. Lett. 74 4698 (1995), Phys. Rev. B [**58**]{}, 19920 (1998). L. Esaki and R. Tsu, IBM J. Dev. [**14**]{}, 61 (1970); L. Esaki and L. L. Chang, Phys. Rev. Lett. [**33**]{}, 495 (1974); R. Dingle, W. Wiegmann and C. H. Henry, Phys. Rev. Lett. [**33**]{}, 872 (1974). M. A. Nielsen and I. L. Chuang, [*Quantum Computation and Quantum Information*]{} (Cambridge University Press, Cambridge, 2000). C. Weisbuch [*et al.*]{}, Phys. Rev. Lett. [**69**]{}, 3314 (1992); R. Houdré [*et al.*]{}, Phys. Rev. Lett. [**85**]{}, 2793 (2000). R. M. Stevenson [*et al.*]{}, Phys. Rev. Lett. [**85**]{}, 3680 (2000). J. Erland [*et al.*]{}, Phys. Rev. Lett. [**86**]{}, 5791 (2001). W. Langbein, Phys. Rev. B [**70**]{}, 205301 (2004). P. Schwendimann, C. Ciuti, and A. Quattropani, Phys. Rev. B [**68**]{}, 165324 (2003). C. Ciuti, Phys. Rev. B [**69**]{}, 245304 (2004). S. Savasta, O. D. Stefano, V. Savona, and W. Langbein, Phys. Rev. Lett. [**94**]{}, 246401 (2005). S. Kundermann, M. Saba, C. Ciuti, T. Guillet, U. Oesterle, J. L. Staehli, and B. Deveaud, Phys. Rev. Lett. [**91**]{}, 107402 (2003). S. Savasta, G. Martino, and R. Girlanda, Solid State Communication, [**111**]{}, 495 (1999). J. Ph. Karr, A. Baas, R. Houdré, and E. Giacobino, Phys. Rev. A [**69**]{}, 031802(R) (2004). V. M. Axt and T. Kuhn, Rep. Progr. Phys. [**67**]{}, 433 (2004). K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, Nature [**431**]{}, 167 (2004). B. Krummheuer, V. M. Axt, T. Kuhn, I. D’Amico, and F. Rossi, Phys. Rev. B [**71**]{}, 235329 (2005). S. Savasta and R. Girlanda, Phys. Rev. Lett. [**77**]{}, 4736 (1996). S. Savasta, O. Di Stefano, and R. Girlanda, Phys. Rev. Lett. [**90**]{}, 096403 (2003). See e.g. D. F. Walls and G. J. Milburn *Quantum Optics* Springer-Verlag (Berlin-Heidelberg) 1994. S. Savasta, O. Di Stefano, and R. Girlanda, Phys. Rev. B [**64**]{}, 073306 (2001). M. Buck, L. Wischmeier, S. Schumacher, G. Czycholl, F. Jahnke, T. Voss, I. Rückmann, and J. Gutowski, Eur. Phys. J. B [**42**]{}, 175 (2004). S. Portolan, S. Savasta, O. Di Stefano, F. Rossi, and R. Girlanda, arXiv:cond-mat/0611381v2 \[cond-mat.mes-hall\], submitted. H. Haugh and S. W. Koch, [*Quantum Theory of the Optical and Electronic Properties of Semiconductors*]{} (World Scientific, Singapore, 1993), 2nd ed. K. V. Kavokin, I. A. Shelykh, A. V. Kavokin, G. Malpuech, and P. Bigenwald, Phys. Rev. Lett. [**92**]{}, 017401 (2004). S. Savasta and R. Girlanda, Phys. Rev. B [**59**]{}, 15409 (1999). V. Savona, L. C. Andreani, P. Schwendimann, A. Quattropani, Solid State Communication, [**93**]{}, 733 (1995). Inserting two photon identities at both sides $\hat{a}_{k} = \sum_{n,\gamma} Y_{n,\gamma;n,\gamma}\hat{a}_{k}\sum_{m,\lambda} Y_{m,\lambda;m,\lambda}$, where $n= \sum_{i= modes} n_i$ and $m=\sum_{i =modes} m_i$, we have non-zero matrix elements only if $n_i = m_i \forall i \neq k$ and $n_k + 1= m_k$. Thus it becomes $\hat{a}_{k} = \sum_{(n_k,n'),\gamma} (\sqrt{{n_k}+1} Y_{n_k k;(n_k+1) k}) \otimes Y_{n' \gamma; n' \gamma}$, where $n'$ stands for the string $(n_1,n_2, ...)$ without the $k$-th entry. When we make the one photon addendum explicit, we end up with $Y_{0;1 k}$. R. Takayama, N.H. Kwong, I. Rumyantsev, M. Kuwata-Gonokami, and R. Binder, Eur. Phys. J. B [**25**]{}, 445 (2002). N. H. Kwong, R. Takayama, I. Rumyantsev, M. Kuwata-Gonokami, and R. Binder, Phys. Rev. B [**64**]{}, 045316 (2001). ${\bf q}_0 = {\bf q}+\eta_h({\bf k'}-{\bf k})$, ${\bf q}_1 = {\bf q}-{\bf k''}+\eta_h{\bf \tilde k}+\eta_e{\bf k})$, ${\bf q}_2 = {\bf q}-{\bf k''}+\eta_e{\bf k}+\eta_h{\bf k''})$, ${\bf q}_3 = {\bf q}-\eta_e({\bf k'}-{\bf k})$, ${\bf q}_4 = {\bf q}+{\bf k''}-\eta_e{\bf \tilde k}-\eta_h{\bf k})$ and ${\bf q}_5 = {\bf q}+{\bf k''}-\eta_h{\bf k}-\eta_e{\bf k''})$ C. Ciuti, P. Schwendimann, A. Quattropani, Semicond. Sci. Technol. [**18**]{}, S279 (2003). C. C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, *Photons and Atoms: Introduction to Quantum Electrodynamics* (Wyley Interscience publication). T. Takagahara, Phys. Rev. B, [**31**]{}, 6552 (1985). [^1]: Electronic address: stefano.portolan@epfl.ch

--- abstract: 'We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen’s theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. Along the way, we introduce an appropriate generalisation of LOCC operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von Neumann algebras. As an application of our result in the setting of ${\ensuremath{\mathrm{II}_1}}$-factors, we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of Vidal, thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we show that trace vectors play the role of maximally entangled states for general ${\ensuremath{\mathrm{II}_1}}$-factors. Examples are drawn from infinite spin chains, quasi-free representations of the CAR, and discretised versions of the CCR.' address: - 'School of Mathematics & Statistics, Carleton University, Ottawa, ON, Canada H1S 5B6' - 'Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1 & Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada N2L 3G1' - 'School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland' - 'Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom, and School of Mathematical Sciences, Nankai University, 300071 Tianjin, China' author: - Jason Crann - 'David W. Kribs' - 'Rupert H. Levene' - 'Ivan G. Todorov' date: 18 April 2019 title: State Convertibility in the von Neumann Algebra Framework --- Introduction {#s_bc} ============ Quantum entanglement is a central notion in quantum information theory and a key resource in the applications that are driving efforts to develop quantum technologies. While there is a growing depth of understanding of the concept and its many potential uses, the theory of quantum entanglement remains an active, challenging, and fundamental area of investigation in quantum information theory with, of particular note here, relatively little progress having been made in the general infinite-dimensional and von Neumann algebra settings. The mathematical theory that provides the foundation for these efforts rests in many ways on an understanding of how entanglement between quantum states can be transformed through various tasks and processes. A very natural and central question is to ask whether a given state, entangled between multiple parties, can be transformed by certain restricted classes of quantum operations to other types of entangled states, with restrictions determined by theoretical or physical limitations. This is a basic question that is relevant, for instance, in the development of any quantum communication scheme, realisations of quantum algorithms, physical implementations of quantum networks, etc. As the simplest starting point in this subject, consider the scenario in which two parties $A$ and $B$ each have the ability to implement all local quantum operations, described mathematically by completely positive and trace-preserving maps on the algebras of bounded linear operators on their respective system Hilbert spaces $H_A$, $H_B$, but such that the parties are limited in that they can only communicate with each other using classical communication. An initial core problem then is to start with an entangled state $\psi \in H_A \otimes H_B$ shared by the parties, and to determine what are the possible entangled states that $\psi$ can be converted to through local operations and classical communication (LOCC) between them. This question has a neat matrix theoretic solution in the finite-dimensional case, known as Nielsen’s Theorem [@nielsen]. For every pure state $\psi \in H_A \otimes H_B$, let $\rho_\psi = \operatorname{tr}_B (\psi\psi^*)$ be the (in general, mixed) state on $H_A$ found by applying the partial trace map over $H_B$ to the projection $\psi\psi^*$ with range the one-dimensional space of scalar multiples of $\psi$. The eigenvalues of $\rho_\psi$ (including multiplicities) form a probability distribution and can be arranged in non-increasing order, thus giving rise to a real vector $\lambda_\psi$. Nielsen’s Theorem states that $\psi$ can be converted into another state $\phi$ by LOCC between $A$ and $B$ if and only if $\lambda_\psi$ is majorised by $\lambda_\phi$, that is, all partial sums of values from $\lambda_\psi$ (respecting the non-increasing indexing) are bounded above by the corresponding partial sums of values from $\lambda_\phi$. Recall that $\rho_\psi$ is a pure (rank one) state if and only if ${\psi}$ is separable. Obviously if we start with a separable state ${\psi}$, then we can only transform it to another separable state via LOCC, and this case is easily captured by the theorem with $\lambda_\psi=(1,0,\ldots ,0)$. On the other hand, given any separable state ${\phi}$, any arbitrary state ${\psi}$ can be transformed to it via LOCC, in particular by making use of local depolarising maps on the individual systems. The power of Nielsen’s theorem lies in the fact that it gives a matrix and spectral theoretic description of which entangled states are attainable through LOCC when we start with a given entangled state. The theorem has far-reaching implications and applications throughout finite-dimensional quantum information theory; indeed, it is one of the most important and widely used results in the entire field. While the primary focus of research in quantum information has been on challenges and applications in the finite-dimensional and qubit setting for more than two decades now, ultimately general quantum mechanics is an infinite-dimensional theory that is rooted in the theory of von Neumann algebras [@vN]. Thus, one can reasonably expect that continued long-term progress in quantum information theory and its connections within theoretical physics will depend at least partly on the successful extension of central results in the field to the infinite-dimensional and general von Neumann algebra settings, with new peculiarities and connections uncovered along the way. This is clearly a desirable goal, and there has been a recent reemergence of activity in this direction, including quantum error correction and privacy (e.g. [@bkk2; @cklt]), entropy theory (e.g. [@bfs; @hiaif1; @hiaif2; @longo; @lx]), Bell inequalities (e.g. [@jungep]), the Connes embedding conjecture (e.g. [@musath; @slofstra; @dpp]), and entanglement in quantum field theory (e.g. [@hs] and the references therein). In this paper, we make new progress in this direction, establishing, as our main result, a generalisation of Nielsen’s Theorem to the context of von Neumann algebras, and more specifically for bipartite quantum systems modelled by commuting semi-finite von Neumann algebras, say ${\mathcal{A}}$ and ${\mathcal{B}}$ [@haagerup2; @t1; @t2]. We note that the case where ${\mathcal{A}}$ and ${\mathcal{B}}$ are separably acting factors of type I was considered in [@obnm]. While some parts of the theory extend in a somewhat straightforward way, there are, as one would expect, significant technical challenges to overcome, well beyond the generalisation provided in [@obnm]. En route, we introduce an appropriate generalisation of LOCC operations [@clmow] to our context. The setting for our version of Nielson’s Theorem is provided by the theory of singular numbers [@fk] and majorisation [@h; @h2] in von Neumann algebras. We build our analysis on key aspects of operator algebra theory, such as the standard form of a von Neumann algebra, the theory of completely bounded maps, the Haagerup tensor product, and dilation theory [@fk; @t2; @blecher_smith; @haagerup; @haagerup2; @h; @ksw]. We include a number of examples and applications of our results. In particular, we show that the entropy of the singular value distribution relative to the unique tracial state of a type ${\ensuremath{\mathrm{II}_1}}$-factor is an entanglement monotone in the sense of Vidal [@v; @op], thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we also show that trace vectors play the role of maximally entangled states for general ${\ensuremath{\mathrm{II}_1}}$-factors. Examples are drawn from quantum measurement theory [@vN], infinite spin chains [@keylsw; @kmsw], quasi-free representations of the CAR [@bcs; @dg; @dmm], and discretised versions of the CCR [@arv; @bkk2; @Faddeev]. This paper is organised as follows. The next section includes requisite preliminaries, focussed mainly on von Neumann algebra theory. In Section \[s\_locc\], we introduce the general notion of LOCC maps and derive some basic properties. In Section \[s:conv\], we investigate approximate convertibility of states by LOCC maps, showing that we can restrict attention to one-way convertibility. This is used in Section \[s:main\] to establish our main result on state convertibility and majorisation; this section also includes several supporting technical results. Section \[s\_ttf\] includes the aforementioned applications and examples. Other illustrative examples are presented throughout the paper. We finish with a brief outlook discussion on our results and the subject. Preliminaries ============= Let ${\mathcal{A}}$ be a von Neumann algebra. We denote by ${\mathcal{A}}_*$ its predual; thus, the elements of ${\mathcal{A}}_*$ are normal (that is, weak\* continuous) linear functionals on ${\mathcal{A}}$. If ${\mathcal{B}}$ is another von Neumann algebra with ${\mathcal{A}}\subseteq {\mathcal{B}}$, a map $\Phi : {\mathcal{B}}\to {\mathcal{B}}$ is called an ${\mathcal{A}}$-bimodule map if $\Phi(axb) = a\Phi(x)b$, for all $a,b\in {\mathcal{A}}$ and all $x\in {\mathcal{B}}$. We denote by $\operatorname{CP}_{{\mathcal{A}}}^\sigma({\mathcal{B}})$ the cone of all normal completely positive ${\mathcal{A}}$-bimodule maps on ${\mathcal{B}}$. We let $\operatorname{UCP}_{{\mathcal{A}}}^\sigma({\mathcal{B}})$ stand for the convex subset of all unital maps in $\operatorname{CP}^\sigma_{{\mathcal{A}}}({\mathcal{B}})$. We set $\operatorname{CP}^\sigma({\mathcal{B}}) = \operatorname{CP}_{{\mathbb{C}}I}^\sigma({\mathcal{B}})$ and $\operatorname{UCP}^\sigma({\mathcal{B}}) = \operatorname{UCP}_{{\mathbb{C}}I}^\sigma({\mathcal{B}})$. We call the elements of $\operatorname{UCP}^\sigma({\mathcal{B}})$ *quantum channels* on ${\mathcal{B}}$. We refer the reader to [@paulsen_book] for basics of the theory of completely positive and completely bounded maps and some standard notation. We denote by $S({\mathcal{A}})$ the convex set of all normal states of ${\mathcal{A}}$, and by $S_{{\rm ext}}({\mathcal{A}})$ the set of all pure normal states of ${\mathcal{A}}$. If $\omega\in {\mathcal{A}}_*$ and $a\in {\mathcal{M}}$, define $a\omega, \omega a\in {\mathcal{A}}_*$ by $(a\omega)(x) = \omega(xa)$ and $(\omega a)(x) = \omega(ax)$. We sometimes use the duality pairing notation ${{{ \left\langlex,\omega\right\rangle}}}:=\omega(x)$ with angled brackets (in contrast, we use rounded parentheses for inner products in Hilbert spaces); in this notation, the bimodule action just described may be written as ${{{ \left\langlex,a\omega b\right\rangle}}}={{{ \left\langlebxa,\omega\right\rangle}}}$, $a,b\in {\mathcal{A}}$. As usual, we let ${\mathcal{B}}(H)$ be the $C^*$-algebra of all bounded operators on a Hilbert space $H$, and set $M_n={{\mathcal{B}}}({{\mathbb{C}}}^n)$, the algebra of $n\times n$ matrices with complex entries. We assume throughout this paper that all Hilbert spaces under consideration are separable, sometimes mentioning this explicitly for emphasis. For an element $a\in {\mathcal{B}}(H)$, we write $\operatorname{Ad}(a)$ for the map given by $\operatorname{Ad}(a)(x) = axa^*$, where $a^*$ denotes the adjoint of $a$; clearly, $\operatorname{Ad}(a) \in \operatorname{CP}^\sigma({\mathcal{B}}(H))$. If $\nph,\psi\in H$, we let $\nph\psi^*$ be the rank one operator on $H$ given by $(\nph\psi^*)(\xi) = (\xi,\psi)\nph$ (note that our inner products are linear in the first argument). Let ${\mathcal{T}}(H)$ be the space of all trace class operators in ${\mathcal{B}}(H)$ and ${\rm tr} : {\mathcal{T}}(H)\to {\mathbb{C}}$ be the trace. We have a canonical identification ${\mathcal{T}}(H) \equiv {\mathcal{B}}(H)_*$, given by letting ${{{ \left\langleT,S\right\rangle}}} = {\rm tr}(TS)$, $T\in {\mathcal{B}}(H)$, $S\in {\mathcal{T}}(H)$. We denote by $H_1$ the set of unit vectors in a Hilbert space $H$. If $\phi\in H$, we write $\omega_\phi\in {\mathcal{B}(H)}_*$ for the (positive, normal) functional given by $\omega_\phi(x) = {{ \left(x\phi,\phi\right)}}$. Throughout the paper, we will let ${\mathcal{A}}\subseteq {\mathcal{B}(H)}$ be a von Neumann algebra, for some (separable) Hilbert space $H$, with unit $1$, projection lattice $\mathcal P({\mathcal{A}})$ and positive cone ${\mathcal{A}}^+$. We will mainly be interested in the case when ${\mathcal{A}}$ is semi-finite, equipped with a normal semi-finite faithful trace $\tau$. Let $\tilde{{\mathcal{A}}}$ be the \*-algebra of all $\tau$-measurable operators [@fk], that is, the set of all densely defined closed operators $T : {\mathcal{D}}(T)\to H$, (where ${\mathcal{D}}(T) \subseteq H$ is the domain of $T$), affiliated with ${\mathcal{A}}$, with the property that for every $\epsilon > 0$ there exists a projection $e\in {\mathcal{A}}$ such that $\tau(1-e) \leq \epsilon$ and $eH \subseteq {\mathcal{D}}(T)$. For $p\geq 1$, let $${\mathcal{A}}_p = \{a\in {\mathcal{A}} : \tau(|a|^p) < \infty\},$$ and let $L^p({\mathcal{A}},\tau)$ be the completion of ${\mathcal{A}}_p$ with respect to the norm $\|\cdot\|_p$, given by $\|a\|_p = \tau(|a|^p)^{1/p}$, $a\in {\mathcal{A}}_p$. Set $L^{\infty}({\mathcal{A}},\tau) = {\mathcal{A}}$. We will extensively use the fact that the elements of the space $L^p({\mathcal{A}},\tau)$ can be canonically identified with operators in $\tilde{{\mathcal{A}}}$ (see [@fk]). Note that $L^2({\mathcal{A}},\tau)$ is a Hilbert space, which is separable since ${\mathcal{A}}$ is separably acting, with inner product given by $${{ \left(a,b\right)}} = \tau(b^*a), \ \ \ a,b\in {\mathcal{A}}_2.$$ In fact, $L^2({\mathcal{A}},\tau)$ is the Hilbert space arising from the GNS construction applied to $\tau$. The associated (normal, faithful) \*-representation $\pi_\tau : {\mathcal{A}} \to {\mathcal{B}}(H)$ is given by $$\pi_{\tau}(a)b = ab, \ \ \ b\in {\mathcal{A}}_2, \ a\in {\mathcal{A}}.$$ We will suppress the use of the notation $\pi_{\tau}$; in this way, we will consider ${\mathcal{A}}$ as a von Neumann subalgebra of ${\mathcal{B}}(L^2({\mathcal{A}},\tau))$. We then have that ${\mathcal{A}}$ is in its standard form [@haagerup2]; we also say that ${{\mathcal{A}}}$ is standardly represented on $L^2({\mathcal{A}},\tau)$. Working in this standard representation, let ${\mathcal{A}}'$ be the commutant of ${\mathcal{A}}$, and let $J : L^2({\mathcal{A}},\tau) \to L^2({\mathcal{A}},\tau)$ be the associated conjugate linear isometry with the property that ${\mathcal{A}}' = J{\mathcal{A}}J$. Note that $J$ is the (unique) extension of the adjoint map $a\mapsto a^*$ on ${\mathcal{A}}_2$, and for $\psi\in L^2({\mathcal{A}},\tau)$, we have that $\psi^* = J\psi$, where the left hand side in the latter identity is the adjoint of the linear operator $\psi$ (by which we mean the element of $\tilde{{\mathcal{A}}}$ canonically identified with $\psi$). For $\xi,\eta\in L^2({\mathcal{A}},\tau)$, we have $$\label{eq_jaj} (J\xi,J\eta) = (\xi^*,\eta^*) = \tau(\eta\xi^*) = \tau(\xi^*\eta) = (\eta,\xi).$$ The map $\pi'_\tau : {\mathcal{A}}\rightarrow {\mathcal{B}}(L^2({\mathcal{A}},\tau))$, given by $$\pi'_\tau(a)b = ba, \ \ \ b\in {\mathcal{A}}_2, \ a\in {\mathcal{A}},$$ is a faithful anti-\*-homomorphism, satisfying $\pi'_\tau(a) = Ja^*J$, $a\in {\mathcal{A}}$ (see [@t1 Theorem V.2.22]). Let $R : {\mathcal{A}}'\rightarrow {\mathcal{A}}$ be the anti-\*-isomorphism given by $$R(a') = Ja'^*J, \ \ \ a'\in{\mathcal{A}}'.$$ We note that $L^1({\mathcal{A}},\tau)$ can be identified in a canonical way with the predual of ${\mathcal{A}}$. In fact, if $\omega\in {\mathcal{A}}_*$ then there exists a unique $\rho_{\omega}\in L^1({\mathcal{A}},\tau)$ such that $$\omega(a) = \tau(\rho_{\omega}a), \ \ \ a\in {\mathcal{A}}.$$ Here, and in the sequel, we use the fact that $L^1({\mathcal{A}},\tau)$ is an ${\mathcal{A}}$-bimodule, that is, given $\rho\in L^1({\mathcal{A}},\tau)$ and $a,b\in {\mathcal{A}}$, we have that $a\rho b$ is a well-defined $\tau$-measurable operator and belongs to $L^1({\mathcal{A}},\tau)$. Note that if $\omega\in {\mathcal{A}}_*^+$, then $\rho_{\omega}$ is a positive (in general unbounded) operator, which we call the *density operator* of $\omega$. If $x\in \tilde{{\mathcal{A}}}$ and $t > 0$, the *$t$-th singular value* $\mu_t(x)$ of $x$ is defined by letting $$\mu_t(x) = \mu_t(x;{{\mathcal{A}}},\tau)= \inf\{\|xp\| : p\in\mathcal P({\mathcal{A}}),\, \tau(1-p)\leq t\}.$$ The *singular value function of $x$*, namely $\mu(x)\colon (0,\infty)\to (0,\infty)$, $t\mapsto \mu_t(x)$, is decreasing and continuous from the right [@fk Lemma 2.5]. If $x,y\in \tilde{{\mathcal{A}}}^+$, we say that $x$ is *majorised* by $y$ if $$\int_0^s \mu_t(x)\, dt \leq \int_0^s \mu_t(y)\, dt, \ \ \ 0< s \leq \infty;$$ we write $x\prec y$ to designate the fact that $x$ is majorised by $y$ and $\tau(x) = \tau(y)$. We refer to [@h] for extensive details on majorisation of elements of $\tilde{{\mathcal{A}}}^+$ and to [@fk] for background on the theory of singular values. Let $\tau' : {\mathcal{A}}'\to {\mathbb{C}}$ be the functional given by $\tau'(a') = \tau(R(a'))$, $a'\in {\mathcal{A}}'$. Then $\tau'$ is a normal faithful semi-finite trace on ${\mathcal{A}}'$. Since ${\mathcal{A}}'\subseteq {\mathcal{B}}(L^2({\mathcal{A}},\tau))$, the elements of $L^1({\mathcal{A}}',\tau')$ can be identified with linear densely defined operators on $L^2({\mathcal{A}},\tau)$. Given a normal functional $\omega'$ on ${\mathcal{A}}'$, there exists, by the preceding discussion, a (unique) element $\rho'_{\omega'}\in L^1({\mathcal{A}}',\tau')$ such that $\omega'(b) = \tau'(\rho'_{\omega'}b)$, $b\in {\mathcal{A}}'$. The constructions described above can be performed relative to the pair $({\mathcal{A}}',\tau')$; in particular, one may define the corresponding singular values $\mu'_t(x'):=\mu_t(x';{{\mathcal{A}}}',\tau')$ associated with any $\tau'$-measurable operator $x'$, relative to $({\mathcal{A}}',\tau')$. We finish this section with two important examples of the previous notions. ${\mathcal{A}}=L^\infty(X,m)$, for a $\sigma$-finite measure space $(X,m)$. In this case, $\tau$ is integration by the measure $m$ and, for any $p\geq 1$, $L^p({\mathcal{A}},\tau)=L^p(X,m)$. In particular, the standard representation is given by the pointwise action of $L^\infty(X,m)$ on $L^2(X,m)$. For a non-negative element $f\in L^1(X,m)$, its singular value function $\mu_t(f)$ satisfies $$\mu_t(f)=\inf\left\{\strut s\geq 0\mid m(\{x\in X\mid f(x)>s\})\leq t\right\};$$ in other words, $t\mapsto \mu_t(f)$ is the non-increasing rearrangement of $f$. ${\mathcal{A}}={\mathcal{B}}(H)$ for a Hilbert space $H$. Here, the (essentially unique) normal semi-finite faithful trace $\tau$ is the canonical trace $\operatorname{tr}$. In this case, for $p\geq 1$, the space $L^p({\mathcal{A}},\tau)$ coincides with the Schatten $p$-class ${\mathcal{S}}_p(H)$. In particular, the standard representation of ${\mathcal{B}}(H)$ is given by the left multiplication action on the Hilbert–Schmidt operators ${\mathcal{S}}_2(H)$. Equivalently, fixing a unitary equivalence ${\mathcal{S}}_2(H)\cong H{\otimes}\overline{H}$ (where $\overline{H}$ is the conjugate Hilbert space of $H$), the standard representation is the canonical action of ${\mathcal{B}}(H){\otimes}1_{\overline{H}}$ on $H{\otimes}\overline{H}$. Given a positive element $\rho\in {\mathcal{T}}(H)={\mathcal{S}}_1(H)$, its singular value function $\mu_t(\rho)$ satisfies $$\mu_t(\rho)=\sum_{n=1}^\infty\lambda_n \chi_{[n-1,n)}(t),$$ where $\lambda_n$ is the $n^{th}$ largest eigenvalue of $\rho$ (including multiplicity). Local Operations and Classical Communication {#s_locc} ============================================ In this section, we introduce the class of maps that realise local operations and classical communication (LOCC) in our general setting and establish some of their properties needed in the sequel. The origins of our approach lie within the development of algebraic quantum field theory where the framework is typically encoded in two commuting \*-subalgebras of a larger $C^*$-algebra. In this paper, a bipartite quantum system is given by a Hilbert space $H$, together with a von Neumann algebra ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$, and its commutant ${\mathcal{B}}:={\mathcal{A}}'$. In the language of [@kmsw §4], this forms a simple, bipartite system satisfying Haag duality. The standard intuition comes from viewing ${\mathcal{A}}$ and ${\mathcal{B}}$ as the observable algebras of parties Alice and Bob, respectively, which have joint access to a quantum system modelled on the Hilbert space $H$. For example, when $H=H_A{\otimes}H_B$ for Hilbert spaces $H_A$ and $H_B$, then ${\mathcal{A}}={\mathcal{B}}(H_A){\otimes}1_{H_B}$ and ${\mathcal{B}} = 1_{H_A}{\otimes}{\mathcal{B}}(H_B)$ define the canonical bipartite system structure in the tensor product framework. We now examine a suitable generalisation of local operations and classical communication (LOCC) in this general bipartite setting, inspired by the approach in [@clmow] and related to, but slightly different from, the proposed notion in [@vw §5]. \[d\_locc\] Let ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$ be a von Neumann algebra on a Hilbert space $H$, and let ${{\mathcal{B}}}={{\mathcal{A}}}'$. (i) A *one-way right local map relative to ${\mathcal{A}}$* is a normal completely positive map $\Theta : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $\Theta = \Phi \circ\Psi$, where $\Phi\in \operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ and $\Psi\in \operatorname{CP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$. Similarly, a *one-way left local map relative to ${\mathcal{A}}$* is a normal completely positive map $\Theta : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $\Theta = \Phi \circ\Psi$, where $\Phi\in \operatorname{CP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ and $\Psi\in \operatorname{UCP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$. (ii) An *instrument* is a collection ${\mathcal{I}} = (\Theta_k)_{k\in {\mathbb{N}}}$ of normal completely positive maps on ${\mathcal{B}}(H)$ such that, for every $x\in {\mathcal{B}}(H)$, the series $\sum_{k=1}^{\infty} \Theta_k(x)$ converges in the weak\* topology to a limit, say $\Theta_{{\mathcal{I}}}(x)$, and the map $x\to \Theta_{{\mathcal{I}}}(x)$ is a normal unital completely positive map. In this case, we sometimes write $\sum_{k=1}^{\infty}\Theta_k$ to denote the map $\Theta_{{\mathcal{I}}}$. We will identify two instruments if they differ only by a bijective relabelling of the index set. (iii) A *one-way right instrument relative to ${\mathcal{A}}$* is an instrument ${\mathcal{I}} = (\Theta_k)_{k\in {\mathbb{N}}}$, where each of the maps $\Theta_k$ is a one-way right local map relative to ${\mathcal{A}}$. A *one-way right LOCC map relative to ${\mathcal{A}}$* is a normal completely positive map $\Theta : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $\Theta = \Theta_{{\mathcal{I}}}$, where ${\mathcal{I}}$ is a one-way right instrument relative to ${\mathcal{A}}$. Similarly, a *one-way left instrument relative to ${\mathcal{A}}$* is an instrument ${\mathcal{I}} = (\Theta_k)_{k\in {\mathbb{N}}}$, where each of the maps $\Theta_k$ is a one-way left local map relative to ${\mathcal{A}}$, and a *one-way left LOCC map relative to ${\mathcal{A}}$* is a normal completely positive map $\Theta : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $\Theta = \Theta_{{\mathcal{I}}}$, where ${\mathcal{I}}$ is a one-way left instrument relative to ${\mathcal{A}}$. (iv) An instrument $(\Gamma_j)_{j\in {\mathbb{N}}}$ is a *coarse-graining* of an instrument $(\Theta_k)_{k\in {\mathbb{N}}}$ if there is a partition ${{\mathbb{N}}}=\bigcup_{j\in {{\mathbb{N}}}}S_j$ so that $\Gamma_j=\sum_{k\in S_j}{\Theta_k}$ for $j\in {\mathbb{N}}$, where each series converges point-weak\*. (v) An instrument ${\mathcal{I}}$ is called *one-way local relative to ${\mathcal{A}}$* if ${\mathcal{I}}$ is either a one-way right instrument relative to ${\mathcal{A}}$ or a one-way left instrument relative to ${\mathcal{A}}$. We say that an instrument ${\mathcal{J}}$ is *linked* to an instrument ${\mathcal{I}} = (\Theta_k)_{k\in {\mathbb{N}}}$ if there exist one-way instruments $(\Theta_{ki})_{i\in {\mathbb{N}}}$, $k\in {\mathbb{N}}$, such that ${\mathcal{J}}$ is a coarse-graining of the instrument $(\Theta_k\circ \Theta_{ki})_{i,k\in {\mathbb{N}}}$. (vi) A map $\Theta\in \operatorname{UCP}^\sigma({\mathcal{B}}(H))$ is an *LOCC map relative to ${\mathcal{A}}$* if there exists a sequence $({\mathcal{I}}_0, \dots,{\mathcal{I}}_n)$ of instruments such that ${\mathcal{I}}_0$ is a one-way local instrument relative to ${\mathcal{A}}$, ${\mathcal{I}}_{l+1}$ is linked to ${\mathcal{I}}_l$, $l = 0,\dots,n-1$, and $\Theta = \Theta_{{\mathcal{I}}_n}$. We denote by $\operatorname{LOCC}({\mathcal{A}})$ the set of all LOCC maps relative to ${\mathcal{A}}$. We also write $\operatorname{LOCC}^r({{\mathcal{A}}})$ and $\operatorname{LOCC}^l({{\mathcal{A}}})$ for the subsets of $\operatorname{LOCC}({{\mathcal{A}}})$ consisting of the one-way right and left LOCC maps relative to ${{\mathcal{A}}}$, respectively. Thus, any $\Theta$ in $\operatorname{LOCC}^r({{\mathcal{A}}})$ is given by a point weak-\* convergent series $$\label{eq:locc-r} \Theta=\sum_{k\in {\mathbb{N}}} \Phi_k\circ \Psi_k$$ where $\Phi_k\in \operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ and $\Psi_k\in \operatorname{CP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$. \[ex\_pol\] Let $H$ be a Hilbert space, ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$ be a von Neumann algebra, and $\{a_k\mid k\in{\mathbb{N}}\} \subseteq {\mathcal{A}}$ be a countably infinite measurement system, that is, a sequence in ${\mathcal{A}}$ for which $\sum_{k=1}^\infty a_k^*a_k = 1$ in the weak\* topology. For each $k\in{\mathbb{N}}$, let $\Psi_k=\operatorname{Ad}(a_k^*)$ on ${\mathcal{B}}(H)$ and let $\Phi_k\in \operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ be a channel. Then the series $\Theta = \sum_{k=1}^\infty\Phi_k\circ\Psi_k$ defines a one-way right LOCC map relative to ${\mathcal{A}}$. Indeed, $$\sum_{k=1}^N \Phi_k\circ\Psi_k(1) = \sum_{k=1}^N \Psi_k\circ\Phi_k(1) = \sum_{k=1}^N a_k^* a_k \to_{N\to\infty} 1$$ in the weak\* topology. If $x\in {\mathcal{B}}(H)^+$ then $x\leq \|x\|1$ and hence the partial sums $\sum_{k=1}^N \Phi_k\circ\Psi_k(x)$ are dominated by $\|x\|1$. Since they form an increasing sequence, the series $\sum_{k=1}^{\infty} \Phi_k\circ\Psi_k(x)$ converges in the weak\* topology. If $x\in {\mathcal{B}}(H)$ is arbitrary then, using polarisation, we can write $x = \sum_{l=1}^4 \lambda_l x_l$, where $x_l\in {\mathcal{B}}(H)^+$ and $\lambda_l\in {\mathbb{C}}$, $l = 1,2,3,4$. It follows that the partial sums $$\sum_{k=1}^N \Phi_k\circ\Psi_k(x) = \sum_{k=1}^N \sum_{l=1}^4 \lambda_l \Phi_k\circ\Psi_k(x_l) = \sum_{l=1}^4 \sum_{k=1}^N \lambda_l \Phi_k\circ\Psi_k(x_l)$$ form a weak\* convergent sequence. The predual $\Theta_* : {\mathcal{T}}(H)\to {\mathcal{T}}(H)$ of the map $\Theta$ is given by $$\Theta_*(\rho) = \sum_{k=1}^\infty(\Phi_k)_*(a_k\rho a_k^*), \ \ \ \rho\in {\mathcal{T}}(H),$$ so one can think of $\Theta_*$ as a protocol where Alice makes a measurement corresponding to the system $\{a_k\mid k\in{\mathbb{N}}\}$, sends the result $k$ to Bob, who then applies $(\Phi_k)_*$. Below we show that any one-way right LOCC map relative to ${\mathcal{A}}$ is of this form, similar to the finite-dimensional setting. The use of measurements with countably many outcomes is natural in an infinite-dimensional context. Indeed, even the measurement of an observable modelled by a (possibly unbounded) self-adjoint operator with continuous spectrum can, within an arbitrarily small amount of error, be modelled by an observable with countably many disjoint outcomes [@vN §III.3]. \[r\_locc\] Let $\Theta\in \LOCC^r({{\mathcal{A}}})$ and let $\Phi_k$, $\Psi_k$ be as in . Then $\Phi_k(a) = a \Phi_k(1) = a$, $a\in {\mathcal{A}}$, $k\in{\mathbb{N}}$, and, in the weak\* topology we have $$\label{eq_psi1} 1 = \Theta(1) = \sum_{k=1}^\infty \Psi_k\circ \Phi_k(1) = \sum_{k=1}^{\infty} \Psi_k(1).$$ It follows that, for every positive element $x\in {\mathcal{B}}(H)$, the partial sums $\sum_{i=1}^m \Psi_k(x)$ are norm bounded; since they form an increasing sequence, they converge in the weak\* topology. Using polarisation, we conclude, as in Example \[ex\_pol\], that the sequence $\left(\sum_{i=1}^m \Psi_k(x)\right)_{m\in {\mathbb{N}}}$ converges in the weak\* topology for every $x\in {\mathcal{B}}(H)$. The locality of an LOCC operation is reflected through the bimodule structure of its implementing maps. For example, if $\Theta$ is a map of the form in , then Alice’s local operations are modelled by the maps $\Psi_k$, and Bob’s local operations by the channels $\Phi_k$. Since the $\Phi_k$ are ${\mathcal{A}}$-bimodule maps, they do not affect any of Alice’s observables and, as shown below, they admit Kraus decompositions with operators belonging to Bob’s observable algebra ${\mathcal{B}}$. A similar intuition is applied for Alice’s local operations $\Psi_k$. In the case where $H_A$ and $H_B$ are finite dimensional Hilbert spaces, $H = H_A\otimes H_B$, ${\mathcal{A}} = {\mathcal{B}}(H_A)\otimes 1$ and ${\mathcal{B}} = 1\otimes {\mathcal{B}}(H_B)$, Definition \[d\_locc\] reduces to the usual notions as described, for example, in [@clmow]. A notion of LOCC operation for general bipartite systems was proposed by Verch–Werner in [@vw §5]. There, a bipartite system is modelled by commuting unital $C^*$-subalgebras ${\mathcal{A}}$ and ${\mathcal{B}}$ of an ambient unital $C^*$-algebra ${\mathcal{C}}$. In this setting, they defined a one-way right LOCC map between bipartite systems $({\mathcal{A}}_1,{\mathcal{B}}_1, {\mathcal{C}}_1)$ and $({\mathcal{A}}_2,{\mathcal{B}}_2, {\mathcal{C}}_2)$, where ${\mathcal{A}}_i$ and ${\mathcal{B}}_i$ are commuting C\*-subalgebras of ${\mathcal{C}}_i$, $i = 1,2$, by a UCP map $\Theta:{\mathcal{C}}_1\rightarrow{\mathcal{C}}_2$, for which there exist finitely many completely positive maps $\Psi_k:{\mathcal{A}}_1\rightarrow{\mathcal{A}}_2$ and UCP maps $\Phi_k:{\mathcal{B}}_1\rightarrow{\mathcal{B}}_2$ satisfying $$\label{e:vw}\Theta(ab)=\sum_{k}\Psi_k(a)\Phi_k(b), \ \ \ a\in{\mathcal{A}}_1, \ b\in{\mathcal{B}}_1.$$ In the special case when ${\mathcal{C}}_1={\mathcal{C}}_2={\mathcal{B}}(H)$, ${\mathcal{A}}_1={\mathcal{A}}_2 =: {\mathcal{A}}$ is a von Neumann algebra, and ${\mathcal{B}}_1={\mathcal{B}}_2={\mathcal{A}}'$, our definition of a one-way right LOCC map relative to ${\mathcal{A}}$ satisfies this condition, albeit, allowing a countable summation over a classical” index $k$. This follows from the fact that any normal completely positive ${\mathcal{A}}'$-bimodule map on ${\mathcal{B}}(H)$ admits a Kraus decomposition with operators from ${\mathcal{A}}$ (see e.g. [@blecher_smith; @haagerup]), and similarly for ${\mathcal{B}}$. If, in addition, one assumes that ${\mathcal{A}}$ and ${\mathcal{B}}$ are injective factors, then by [@cs Theorem 4.2], any completely positive map $\Psi:{\mathcal{A}}\rightarrow {\mathcal{A}}$ admits a net of completely positive elementary operators $\Psi_i:{\mathcal{A}}\rightarrow{\mathcal{A}}$ (i.e., operators admitting finitely many Kraus operators from ${\mathcal{A}}$) satisfying $\norm{\Psi_i}_{cb} \leq \norm{\Psi}_{cb}$ and $\Psi_i\rightarrow\Psi$ in the point weak\* topology of $\operatorname{CB}({\mathcal{A}})$. The maps $\Psi_i$ admit canonical extensions to maps in $\operatorname{CP}^\sigma_{{\mathcal{A}}'}({\mathcal{B}}(H))$ (through their finite Kraus decompositions), so that we may approximate the (potentially non-normal) completely positive maps $\Psi_k$ occurring in by normal maps satisfying our bimodule requirements. Similar considerations hold for the maps $\Phi_k$. Hence, in the case of injective factors, one may view the proposed definition of Verch–Werner as a limit case” of ours. Note that by [@cs Remark 4.3], when ${\mathcal{A}}$ is an injective factor of type II or III, it is *not* true that every *normal* completely positive map $\Psi : {\mathcal{A}}\rightarrow{\mathcal{A}}$ extends to a *normal* completely positive map $\widetilde{\Psi}\in\operatorname{CP}^\sigma_{{\mathcal{A}}'}({\mathcal{B}}(H))$. \[p\_comp\] Let ${{\mathcal{A}}}\subseteq {\mathcal{B}(H)}$ be a von Neumann algebra on a separable Hilbert space $H$. (i) The class $\LOCC^r({{\mathcal{A}}})$ of one-way right LOCC maps is closed under finite compositions. (ii) The maps $\Psi_k$ in  can be taken of the form $\Psi_k = \operatorname{Ad}(a_k^*)$, for some $a_k\in {\mathcal{A}}$, $k\in {\mathbb{N}}$, with $\sum_{k=1}^{\infty} a_k^*a_k = 1$ in the weak\* topology. (iii) The maps $\Phi_k$ in  can be taken of the form $\Phi_k=\sum_{i=1}^\infty \operatorname{Ad}(c_{ki}^*)$, a point weak\*-convergent series, for some $c_{ki}\in {{\mathcal{A}}}'$, $k,i\in {{\mathbb{N}}}$, with $\sum_{i=1}^\infty c_{ki}^*c_{ki}=1$ in the weak\* topology for every $k\in {{\mathbb{N}}}$. Let, as before, ${\mathcal{B}} = {\mathcal{A}}'$. We first claim that if $\Phi \in \operatorname{CP}_{{\mathcal{A}}}^\sigma({\mathcal{B}}(H))$ and $\Psi \in \operatorname{CP}_{{\mathcal{B}}}^\sigma({\mathcal{B}}(H))$, then $$\label{eq_comm} \Phi \circ\Psi = \Psi \circ\Phi.$$ Indeed, since $H$ is separable, by [@haagerup] (see also [@blecher_smith]), there exists a bounded column operator $(a_{i})_{i\in {\mathbb{N}}}$ with entries ${\mathcal{A}}$, such that $$\Psi(x) = \sum_{i=1}^{\infty} a_{i}^* x a_{i}, \ \ \ x\in {\mathcal{B}}(H),$$ where the series converges in the weak\* topology. For every $x\in {\mathcal{B}}(H)$ we now have $$\Phi\circ\Psi(x) = \Phi \left(\sum_{i=1}^{\infty} a_{i}^* x a_{i}\right) = \sum_{i=1}^{\infty} \Phi(a_{i}^* x a_{i}) = \sum_{i=1}^{\infty} a_{i}^* \Phi(x) a_{i}=\Psi\circ \Phi(x),$$ showing . Let $\Theta \in \LOCC^r({{\mathcal{A}}})$, with corresponding maps $\Phi_k\in \UCP^\sigma_{{{\mathcal{A}}}}({\mathcal{B}(H)})$ and $\Psi_k\in \CP^\sigma_{{{\mathcal{B}}}}({\mathcal{B}(H)})$ for $k\in {{\mathbb{N}}}$, as in . Write $$\Psi_k(x) = \sum_{i=1}^{\infty} a_{ki}^* x a_{ki}, \ \ \ x\in {\mathcal{B}}(H),$$ for some bounded column operator $(a_{ki})_{i\in {\mathbb{N}}}$ with entries in ${\mathcal{A}}$ [@haagerup]. By , $\Psi_k(1) \leq 1$, and hence the column operator $(a_{ki})_{i\in {\mathbb{N}}}$ is contractive. Thus, $a_{ki}$ is a contraction for all $k,i\in {\mathbb{N}}$. Set $\Phi_{ki} = \Phi_k$ for all $k,i\in {\mathbb{N}}$ and note that, in the weak\* topology, we have $$\Theta(x) = \lim_{p\to\infty} \lim_{q\to\infty} \sum_{k=1}^p \sum_{i=1}^q \Phi_{ki}\circ \operatorname{Ad}(a_{ki}^*)(x), \ \ \ x\in {\mathcal{B}}(H).$$ Let $x\in {\mathcal{B}}(H)^+$ and $\rho\in {\mathcal{T}}(H)^+$. Then the double limit $$\label{eq_doub} \lim_{p\to\infty} \lim_{q\to\infty} \sum_{k=1}^p \sum_{i=1}^q {{ \left\langle\Phi_{ki}\circ \operatorname{Ad}(a_{ki}^*)(x),\rho\right\rangle}}$$ exists. Since the terms of the sequence in are positive, the limit $$\lim_{L\to\infty} \sum_{k,i=1}^L {{ \left\langle\Phi_{ki}\circ \operatorname{Ad}(a_{ki}^*)(x),\rho\right\rangle}}$$ exists. Thus, the partial sums $\sum_{k,i=1}^L \Phi_{ki}\circ \operatorname{Ad}(a_{ki}^*)(x)$ converge in the weak\* topology for every $x\in {\mathcal{B}}(H)^+$ and hence, by the polarisation identity, for every $x\in {\mathcal{B}}(H)$. It is now straightforward to see that the limit coincides with $\Theta(x)$. Note, moreover, that identity shows that $\sum_{k,i=1}^{\infty} a_{ki}^* a_{ki} = 1$ in the weak\* topology. This establishes (ii). The assertion in (iii) follows by considering the Kraus decomposition of the maps $\Phi_k$. To show (i), assume that $\Theta_i\in \LOCC^r({{\mathcal{A}}})$ and let $\Phi_k^{(i)}$, $\Psi_k^{(i)}$, $k\in {\mathbb{N}}$, be the maps as in , associated with $\Theta_i$, $i = 1,2$. By , for every $x\in {\mathcal{B}}(H)$, we have that, in the weak\* topology, $$\begin{aligned} (\Theta_1\circ \Theta_2)(x) & = & \lim_{p\to \infty} \lim_{q\to \infty} \sum_{k=1}^p \sum_{l=1}^q \left(\Phi_k^{(1)}\circ \Psi_k^{(1)} \circ \Phi_l^{(2)} \circ \Psi_l^{(2)}\right)(x)\\ & = & \lim_{p\to \infty} \lim_{q\to \infty} \sum_{k=1}^p \sum_{l=1}^q \left(\Phi_k^{(1)}\circ \Phi_l^{(2)} \circ \Psi_k^{(1)}\circ \Psi_l^{(2)}\right)(x).\end{aligned}$$ Set $\Phi_{kj} = \Phi_k^{(1)}\circ \Phi_l^{(2)}$ and $\Psi_{kj} = \Psi_k^{(1)}\circ \Psi_l^{(2)}$. An argument similar to the one in the previous paragraph now implies that $$(\Theta_1\circ \Theta_2)(x) = \sum_{k,l=1}^{\infty} (\Phi_{kj} \circ \Psi_{kj})(x), \ \ \ x\in {\mathcal{B}}(H),$$ in the weak\* topology, so $\Theta_1\circ \Theta_2\in \LOCC^r({{\mathcal{A}}})$. \[r\_mirror\] (i) The expression of one-way right LOCC maps given in Proposition \[p\_comp\](ii) reflects the notion of fine-graining of LOCC channels described in [@clmow]. \(ii) By symmetry, observations analogous to those above for one-way right LOCC maps also hold for the one-way left LOCC maps. State Convertibility via $\operatorname{LOCC}({{\mathcal{A}}})$ {#s:conv} =============================================================== Having established an appropriate generalisation of LOCC operations in the preceding section, we now define the corresponding notions of convertibility. \[d\_conv\] Let ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$ be a von Neumann algebra on a Hilbert space $H$, and let $\psi, {\varphi}\in H_1$. (i) We say that $\psi$ *is convertible to* ${\varphi}$ *via* $\operatorname{LOCC}({\mathcal{A}})$ if there exists $\Theta\in \operatorname{LOCC}({\mathcal{A}})$ such that $\Theta_*({\omega}_\psi) = {\omega}_{\varphi}$. (ii) We say that $\psi$ *is approximately convertible to* ${\varphi}$ *via* $\operatorname{LOCC}({\mathcal{A}})$ if for every $\varepsilon > 0$ there exists $\Theta\in \operatorname{LOCC}({\mathcal{A}})$ such that ${\left\Vert \Theta_*({\omega}_\psi)- {\omega}_{\varphi}\right\Vert}<{\varepsilon}$. We also make analogous definitions with $\operatorname{LOCC}^l({{\mathcal{A}}})$ and $\operatorname{LOCC}^r({{\mathcal{A}}})$ in place of $\operatorname{LOCC}({{\mathcal{A}}})$. The goal of the next few results is to show that approximate convertibility can be realised by using only one-way LOCC maps (see Corollary \[c\_oneway\]). This generalises to the commuting operator framework a result of Lo-Popescu [@lp] for finite-dimensional bipartite systems. We note that another generalisation of this theorem was established in [@obnm], which we recover from our results by taking the special case ${{\mathcal{A}}}={\mathcal{B}(H)}$ in its standard representation, for $H$ a separable Hilbert space. The essential feature of our argument, similar to the finite-dimensional case, is the symmetry induced from the standard form of a von Neumann algebra [@haagerup2], as highlighted in Section \[s\_bc\]. Let $H$ be a Hilbert space, ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$ be a (semi-finite) von Neumann algebra equipped with a normal semi-finite faithful trace $\tau$, and $\psi\in H$. To avoid double subscripts, we will write $\rho_\psi\in L^1({{\mathcal{A}}},\tau)$ for the density operator $\rho_{\omega_\psi}$ arising from the restriction of the vector state ${\omega}_\psi$ to ${\mathcal{A}}$. Recall that, in the case where $H=L^2({{\mathcal{A}}},\tau)$ and ${\mathcal{A}}$ is in standard form, we write $\rho_\psi'\in L^1({{\mathcal{A}}}',\tau')$ for the density operator affiliated to ${\mathcal{A}}'$ satisfying $${{ \left(a'\psi,\psi\right)}}={\omega}_{\psi}|_{{\mathcal{A}}'}(a')=\tau'(\rho'_\psi a'), \ \ \ a'\in {\mathcal{A}}',$$ where $\tau'=\tau\circ R$ is the canonical trace on ${\mathcal{A}}'$. Recall that for $t > 0$, we write $\mu'_t(\rho'_\psi)=\mu_t(\rho'_\psi;{{\mathcal{A}}}',\tau')$ for the $t$-th singular value of $\rho'_\psi$ relative to $({{\mathcal{A}}}',\tau')$. \[l\_Schmidt\] Let $({\mathcal{A}},\tau)$ be a semi-finite von Neumann algebra, represented in its standard form on $L^2({{\mathcal{A}}},\tau)$, and let $\psi\in L^2({\mathcal{A}},\tau)$. Then $$J\rho_\psi'J=\rho_{\psi^*}{{\quad\text{and}\quad}}\mu'_t(\rho_\psi')=\mu_t(\rho_{\psi^*})=\mu_t(\rho_\psi),\quad t>0.$$ First we show that $\rho_{\psi^*}=J\rho'_\psi J$ and $\mu_t'(\rho_\psi')=\mu_t(\rho_{\psi^*})$. Recall that $\psi^*=J\psi$. For $a\in {{\mathcal{A}}}$, we therefore have $$\begin{aligned} \tau(\rho_{\psi^*}a)&={{ \left(aJ\psi,J\psi\right)}}={{ \left(Ja^*J\psi,\psi\right)}}=\tau'(\rho'_\psi Ja^*J)\\&=\tau\left(J(JaJ\rho_\psi')J\right)=\tau(J\rho_\psi'Ja).\end{aligned}$$ Hence, $$\label{eq_psiJ} \rho_{\psi^*} = J\rho'_\psi J.$$ Recall that $J$ is isometric, with $J^2=1$, and note that $R\colon {{\mathcal{A}}}'\to {{\mathcal{A}}}$ induces a trace-preserving bijection ${{\mathcal{P}}}({{\mathcal{A}}}')\to {{\mathcal{P}}}({{\mathcal{A}}})$, $p'\mapsto Jp'J$. For $t > 0$, using we obtain $$\begin{aligned} \mu'_t(\rho_\psi')&=\inf\{{\left\Vert \rho_\psi' p'\right\Vert}\colon {p'\in {{\mathcal{P}}}({{\mathcal{A}}}'),\,\tau'(1-p')\le t}\}\\ &=\inf\{{\left\Vert \rho_\psi'JpJ\right\Vert}\colon {p\in {{\mathcal{P}}}({{\mathcal{A}}}),\,\tau(1-p)\le t}\}\\ &=\inf\{{\left\Vert J\rho_\psi'Jp\right\Vert}\colon {p\in {{\mathcal{P}}}({{\mathcal{A}}}),\,\tau(1-p)\le t}\}\\ &=\inf\{{\left\Vert \rho_{\psi^*}p\right\Vert}\colon {p\in {{\mathcal{P}}}({{\mathcal{A}}}),\,\tau(1-p)\le t}\} = \mu_t(\rho_{\psi^*}).\end{aligned}$$ Now, by [@t2 Exercises IX.1.2–3], there exists a partial isometry $u\in{\mathcal{A}}$ such that $|\psi^*|=\psi u^*$, and $\psi^*=u^*|\psi^*|$. It follows that $$\psi^* = u^*\psi u^* = u^* |\psi^*| = u^* J |\psi^*| = u^* J u \psi^* = u^*JuJ\psi.$$ Since $u\in{\mathcal{A}}$, we have $\rho_{\psi*}=\rho_{u^*JuJ\psi}=u^*\cdot\rho_{JuJ\psi}\cdot u$. By [@fk Lemma 2.5(vi)], $$\label{eq_mut} \mu_t(\rho_{\psi*})\leq{\left\Vert u^*\right\Vert}{\left\Vert u\right\Vert}\mu_t(\rho_{JuJ\psi})\leq\mu_t(\rho_{JuJ\psi}),$$ for $t > 0$. But $JuJ$ is a contraction in ${\mathcal{A}}'$, so ${\omega}_{JuJ\psi}|_{{\mathcal{A}}}\leq {\omega}_{\psi}|_{{\mathcal{A}}}$, that is, $\rho_{JuJ\psi}\leq \rho_{\psi}$. By [@fk Lemma 2.5(iii)], $\mu_t(\rho_{JuJ\psi})\leq\mu_t(\rho_\psi)$, $t>0$. Thus, by , $\mu_t(\rho_{\psi^*})\leq\mu_t(\rho_{\psi})$, for $t>0$. By symmetry, we obtain equality. \[p\_lopopescu\] Let $({\mathcal{A}},\tau)$ be a semi-finite factor in its standard form on $L^2({{\mathcal{A}}},\tau)$, let ${\mathcal{B}} = {\mathcal{A}}'$ and let $\epsilon > 0$. For any $(\psi,b)\in H\times {{\mathcal{B}}}$, there exist a unitary $u\in {\mathcal{B}}$ and partial isometries $v,w\in {{\mathcal{A}}}$ so that $\psi=v^*v\psi$ and, if $z=JbJv$, then $${\left\Vert b\psi\right\Vert}={\left\Vert z\psi\right\Vert}{{\quad\text{and}\quad}}{\left\Vert b\psi-uwz\psi\right\Vert}<\epsilon.$$ Moreover, the partial isometry $v$ can be chosen independently of $b$. Let $\psi^*=v|\psi^*|$ be the polar decomposition of $\psi^*$ [@t2 Exercises IX.1.2–3]; thus, $v\in {\mathcal{A}}$ is a partial isometry with $\psi=v^*|\psi|$, and the projections $v^*v$ and $vv^*$ have ranges $\overline{{\mathcal{B}}\psi}$ and $\overline{{\mathcal{B}}|\psi|}$, respectively, so in particular, $\psi=v^*v\psi$. It follows that $$\label{eq_vpsi} v\psi=vv^*|\psi|=|\psi|=J|\psi|=Jv\psi.$$ Note that $z \in {{\mathcal{A}}}$. We have $v^*vb\psi = b v^*v\psi = b\psi$, and hence, using , $${\left\Vert b\psi\right\Vert}={\left\Vert vb\psi\right\Vert}={\left\Vert bv\psi\right\Vert}={\left\Vert bJv\psi\right\Vert}={\left\Vert JbJv\psi\right\Vert}={\left\Vert z\psi\right\Vert},$$ as desired. Note that $$\label{eq_taudash} \tau'(\rho') = \tau(J\rho'\mbox{}^{*} J), \ \ \ \rho'\in L^1({\mathcal{B}}, \tau');$$ indeed, the formula holds by the definition of $\tau'$ in the case $\rho' \in {\mathcal{B}}$, and the general case follow by approximating $\rho'$ by a sequence in ${\mathcal{B}}$ in the norm $\|\cdot\|_1$. Let $\alpha=z\psi$ and $\beta=b\psi$. For $c\in{\mathcal{B}}$, by , and , we have $$\begin{aligned} \tau(\rho_\alpha R(c)) &= (R(c)z\psi,z\psi) ={{ \left(Jc^*JJbJv\psi,JbJv\psi\right)}}={{ \left(Jc^*bv\psi,Jbv\psi\right)}}\\ &={{ \left(bv\psi,c^*bv\psi\right)}} = {{ \left(cv^*vb\psi,b\psi\right)}} ={{ \left(c \beta,\beta\right)}} =\tau'(\rho'_\beta c)\\ &=\tau(J\rho_\beta'J R(c)).\end{aligned}$$ Thus, by Lemma \[l\_Schmidt\], $\rho_\alpha=J\rho_\beta'J=\rho_{\beta^*}$ and $$\mu'_t(\rho'_{\alpha})=\mu_t(\rho_{\alpha})=\mu_t(\rho_{\beta^*})=\mu_t'(\rho'_{\beta}), \ \ \ t>0.$$ Since ${{\mathcal{A}}}$ is a factor, by [@h Theorem 3.4(1)] for every ${\varepsilon}>0$ there exists a unitary $u\in{\mathcal{B}}$ such that $${\left\Vert \rho'_{\beta}-\rho'_{u\alpha}\right\Vert}_1 = {\left\Vert \rho'_{\beta}-u \rho'_{\alpha} u^*\right\Vert}_1 < {\varepsilon}^2.$$ By the continuity of Stinespring’s representation [@ksw Theorem 1], there exist a Hilbert space $K$, a \*-homomorphism $\pi:{\mathcal{B}}\rightarrow {\mathcal{B}}(K)$ and vectors $\xi,\eta\in K$ such that ${\omega}_{\beta}|_{{\mathcal{B}}}={\omega}_\xi\circ\pi$ and ${\omega}_{u\alpha}|_{{\mathcal{B}}}={\omega}_\eta\circ\pi$, and $${\left\Vert \xi-\eta\right\Vert}\leq{\left\Vert \omega_\beta|_{{{\mathcal{B}}}}-\omega_{u\alpha}|_{{{\mathcal{B}}}}\right\Vert}^{1/2}={\left\Vert \rho'_{\beta}-\rho'_{u\alpha}\right\Vert}_1^{1/2}<{\varepsilon}.$$ By the uniqueness of Stinespring representations, there exist partial isometries $w_1 : L^2({\mathcal{A}},\tau)\rightarrow K$ and $w_2 : K\rightarrow L^2({\mathcal{A}},\tau)$ such that $w_1c=\pi(c)w_1$ and $w_2\pi(c)=cw_2$ for all $c\in {\mathcal{B}}$, $w_1u\alpha=\eta$ and $w_2\xi=\beta$. Then $w:=w_2w_1$ is a contraction in ${\mathcal{A}}$, and $${\left\Vert b\psi-uwz\psi\right\Vert}={\left\Vert \beta-wu\alpha\right\Vert} = {\left\Vert w_2\xi-w_2\eta\right\Vert} < {\varepsilon}.\qedhere$$ The following estimate is straightforward and we will make use of it multiple times. \[l\_piso\] Let $\psi\in H$, ${\left\Vert \psi\right\Vert}\leq 1$, and let $v\in{\mathcal{B}(H)}$ be a contraction. If ${\left\Vert \psi\right\Vert}-{\left\Vert v\psi\right\Vert}<{\varepsilon}$, then ${\left\Vert (1-v^*v)^{1/2}\psi\right\Vert}<\sqrt{2{\varepsilon}}$. The assumption implies $$\begin{aligned} {\left\Vert (1-v^*v)^{1/2}\psi\right\Vert}^2&={{ \left(\strut(1-v^*v)\psi,\psi\right)}}={\left\Vert \psi\right\Vert}^2-{\left\Vert v\psi\right\Vert}^2\\ &=({\left\Vert \psi\right\Vert}+{\left\Vert v\psi\right\Vert})({\left\Vert \psi\right\Vert}-{\left\Vert v\psi\right\Vert}) \leq 2({\left\Vert \psi\right\Vert}-{\left\Vert v\psi\right\Vert}) <2{\varepsilon}.\qedhere\end{aligned}$$ \[l\_dini\] Let ${\mathcal{A}}$ be a von Neumann algebra equipped with a normal faithful semi-finite trace $\tau$, and let $(\rho_k)_{k\in {\mathbb{N}}}$ be an increasing sequence of hermitian elements of $L^1({\mathcal{A}},\tau)$. If $\rho_k\to_{k\to \infty}\rho$ in the weak topology, for some $\rho\in L^1({\mathcal{A}},\tau)$, then $\rho_k\to_{k\to \infty}\rho$ in norm. We have that $\rho_k\leq \rho$; thus, $\rho - \rho_k \geq 0$ for every $k$. Thus, $$\|\rho - \rho_k\| = (\rho - \rho_k)(1) \to_{k\to\infty} 0.\qedhere$$ \[l\_ineq\] Let $H$ be a Hilbert space and $\Phi:{\mathcal{B}(H)}\rightarrow{\mathcal{B}(H)}$ be positive. Then for any self-adjoint $T\in{\mathcal{B}(H)}$ and any self-adjoint $\rho\in{\mathcal{T}}(H)$, $$|{\langle}\Phi(T),\rho{\rangle}|\leq\norm{T}{\langle}\Phi(1),|\rho|{\rangle}.$$ Let $(e_n)_{n=1}^\infty$ be orthonormal eigenvectors of $\rho$ with corresponding eigenvalues $(\lambda_n)_{n=1}^\infty\subseteq{\mathbb{R}}$. Since $-\norm{T}1\leq T\leq\norm{T}1$, by positivity of ${\omega}_{e_n}\circ\Phi$ we have $$-\norm{T}(\Phi(1)e_n,e_n)\leq(\Phi(T)e_n,e_n)\leq\norm{T}(\Phi(1)e_n,e_n),$$ so that $|(\Phi(T)e_n,e_n)|\leq\norm{T}(\Phi(1)e_n,e_n)$ for all $n\in{\mathbb{N}}$. Hence $$\begin{aligned} |{\langle}\Phi(T),\rho{\rangle}|&=\bigg|\sum_{n=1}^\infty\lambda_n(\Phi(T)e_n,e_n)\bigg|\leq\sum_{n=1}^\infty|\lambda_n||(\Phi(T)e_n,e_n)|\\ &\leq\sum_{n=1}^\infty|\lambda_n|\norm{T}(\Phi(1)e_n,e_n)=\norm{T}{\langle}\Phi(1),|\rho|{\rangle}.\qedhere\end{aligned}$$ \[th\_lopopescu\_std\] Let $({\mathcal{A}},\tau)$ be a semi-finite factor in its standard form on $H=L^2({{\mathcal{A}}},\tau)$. For any $\Theta\in\operatorname{LOCC}({\mathcal{A}})$, $\psi\in H$ and ${\varepsilon}> 0$, there exists $\Theta_{\varepsilon}\in \LOCC^r({{\mathcal{A}}})$ such that ${\left\Vert \Theta_*({\omega}_\psi)-\Theta_{{\varepsilon}*}({\omega}_\psi)\right\Vert} < {\varepsilon}$. We may assume that ${\left\Vert \psi\right\Vert}\le1$. Any LOCC map $\Theta\in\operatorname{LOCC}({\mathcal{A}})$ is of the form $\Theta=\Theta_{{\mathcal{I}}_n}$ where $({\mathcal{I}}_0,\dots,{\mathcal{I}}_n)$ is a sequence of instruments such that ${\mathcal{I}}_0$ is one-way local relative to ${\mathcal{A}}$, and ${\mathcal{I}}_{l+1}$ is linked to ${\mathcal{I}}_{l}$ for each $l=0,\dots,n-1$. Since the trivial instrument $(\operatorname{id},0,0,\dots)$ is one-way right local and any one-way left local instrument is linked to it, without loss of generality, we may suppose that ${\mathcal{I}}_0$ is a one-way right instrument. Writing ${\mathcal{I}}_n=(\Theta_k^{(n)})_{k}$, consider the following proposition $P(n)$: for every ${\varepsilon}>0$ there exists an instrument ${\mathcal{I}}_{{\varepsilon}}=(\Gamma_{k})_{k}$ which is a coarse-graining of some one-way right instrument relative to ${\mathcal{A}}$, such that $$\sum_{k=1}^\infty\norm{\Theta_{k*}^{(n)}({\omega}_\psi)-\Gamma_{k*}({\omega}_\psi)}<{\varepsilon}.$$ If $P(n)$ were true for every $n\in{\mathbb{N}}$ then the Theorem follows with $\Theta_{{\varepsilon}}=\Theta_{{\mathcal{I}}_{{\varepsilon}}}$. We therefore proceed by induction on $n$, starting with the base case $n=1$ (as the claim for $n=0$ is vacuous). Let ${\varepsilon}> 0$ and write ${\mathcal{I}}_0=(\Theta_k)_{k\in {\mathbb{N}}}$. Each $\Theta_k$ is one-way right local, so by Proposition \[p\_comp\](ii) we may write $\Theta_k=\sum_{j,l=1}^\infty \operatorname{Ad}(a_{kj}^*)\circ\operatorname{Ad}(c_{kl}^*)$, where $a_{kj}\in{\mathcal{A}}$ satisfy $\sum_{k=1}^\infty\sum_{j=1}^{\infty}a_{kj}^*a_{kj}=1$ and $c_{kl}\in{\mathcal{B}}$ satisfy $\sum_{l=1}^\infty c_{kl}^*c_{kl}=1$ for each $k\in{\mathbb{N}}$. We define $\psi_{kj}:=a_{kj}\psi\in H$, for $k,j\in {\mathbb{N}}$. Since ${\mathcal{I}}_1$ is linked to ${\mathcal{I}}_0$, the instrument ${\mathcal{I}}_1$ is a coarse-graining of an instrument of the form $(\Theta_k\circ \Theta_{ki})_{k,i}$, for a collection of one-way instruments ${\mathcal{J}}_k=(\Theta_{ki})_i$ indexed by $k\in{\mathbb{N}}$. Write $L=\{k\in{\mathbb{N}}\mid {\mathcal{J}}_k \ \textnormal{is one-way left}\}$ and $R=\{k\in{\mathbb{N}}\mid {\mathcal{J}}_k \ \textnormal{is one-way right}\}$. Suppose first that $k\in L$. By Proposition \[p\_comp\](ii) and Remark \[r\_mirror\](ii), we may assume that $\Theta_{ki}=\Psi^L_{ki}\circ\operatorname{Ad}(b_{ki}^*)$, where $\Psi^L_{ki}\in\operatorname{UCP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$, $b_{ki}\in{\mathcal{B}}$ and $\sum_{i=1}^\infty b_{ki}^*b_{ki}=1$. Hence, in the point weak\*-topology, we have $$\label{e_rl} \Theta_k\circ \Theta_{ki}={\sum_{j=1}^\infty} {\sum_{l=1}^\infty} \operatorname{Ad}((b_{ki}c_{kl}a_{kj})^*)\circ \Psi^L_{ki}.$$ Since ${{\mathcal{A}}}$ is a factor, we can apply Proposition \[p\_lopopescu\] to the pairs $(\psi_{kj},b_{ki}c_{kl})\in H\times {{\mathcal{B}}}$. We obtain unitaries $u_{kijl}\in {{\mathcal{B}}}$ and partial isometries $v_{kj},w_{kijl}\in {{\mathcal{A}}}$ so that $\psi_{kj}=v_{kj}^*v_{kj}\psi_{kj}$ and $z_{kijl}=Jb_{ki}c_{kl}Jv_{kj}\in {{\mathcal{A}}}$ satisfy ${\left\Vert b_{ki}c_{kl}\psi_{kj}\right\Vert}={\left\Vert z_{kijl}\psi_{kj}\right\Vert}$ and $$\label{e_bound1} \|b_{ki}c_{kl}\psi_{kj}-u_{kijl}w_{kijl}z_{kijl}\psi_{kj}\|<\frac{\epsilon}{2^{i+j+k+l+2}}$$ for $i,j,k,l\in {{\mathbb{N}}}$. Let $a_{kijl}=w_{kijl}z_{kijl}\in {{\mathcal{A}}}$, and observe that $${\left\Vert z_{kijl}\psi_{kj}\right\Vert}-\norm{a_{kijl}\psi_{kj}}={\left\Vert b_{ki}c_{kl}\psi_{kj}\right\Vert}-\norm{u_{kijl}w_{kijl}z_{kijl}\psi_{kj}}<\frac{{\varepsilon}}{2^{i + j + k+l+2}}.$$ Let $\widetilde{a_{kijl}} := (1-w_{kijl}^*w_{kijl})z_{kijl}\in {{\mathcal{A}}}$. By Lemma \[l\_piso\], the preceding inequality implies $$\label{eq_kij1} {\left\Vert \widetilde{a_{kijl}}\psi_{kj}\right\Vert}={\left\Vert (1-w_{kijl}^*w_{kijl}) z_{kijl}\psi_{kj}\right\Vert}<\sqrt{\frac{{\varepsilon}}{2^{i + j+k+l+1}}}.$$ Now suppose that $k\in R$. The map $\Theta_{{\mathcal{J}}_k}=\sum_{i=1}^\infty \Theta_{ki}$ is then a one-way right LOCC map relative to ${\mathcal{A}}$, with $\Theta_{ki}=\Psi^R_{ki}\circ\Phi_{ki}$, where $\Psi^R_{ki}\in\operatorname{CP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$ and $\Phi_{ki}\in\operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$. Then $$\label{eq_kR}\Theta_k\circ \Theta_{ki}=\sum_{j=1}^\infty\sum_{l=1}^\infty \operatorname{Ad}(a_{kj}^*)\circ\Psi^R_{ki}\circ(\operatorname{Ad}(c_{kl}^*)\circ\Phi_{ki})$$ in the point weak\*-topology. Recall that $v_{kj}$ was defined above for $k\in L$ and $j\in {\mathbb{N}}$; we define $v_{kj}:=1$ for $k\in R$ and $j\in{\mathbb{N}}$. Then $$A:=\sum_{k=1}^\infty\sum_{j=1}^\infty a_{kj}^*v_{kj}^*v_{kj}a_{kj}\leq\sum_{k=1}^\infty\sum_{j=1}^\infty a_{kj}^*a_{kj}=1;$$ thus, the operator $(1-A)^{1/2}$ is a well-defined element of ${\mathcal{A}}$. Moreover, since $a_{kj}\psi=\psi_{kj}=v_{kj}^*v_{kj}\psi_{kj}=v_{kj}^*v_{kj}a_{kj}\psi$ for all $j,k\in{\mathbb{N}}$, we have $\psi=\sum_{k=1}^\infty\sum_{j=1}^\infty a_{kj}^*a_{kj}\psi=A\psi$, showing that $\psi\in\operatorname{Ker}(1-A)$ and thus that $\psi \in \operatorname{Ker}((1-A)^{1/2})$. In particular, $$\label{eq_1-A} \left(\operatorname{Ad}((1-A)^{1/2})\right)_*(\omega_\psi) = 0.$$ Fix a probability distribution $(p_n)$ over ${\mathbb{N}}$ with $p_n > 0$ for each $n$. For $k\in L$ and $i,j,l\in {\mathbb{N}}$, define $\Gamma^L_{kijl}\in \operatorname{CP}_{{\mathcal{B}}}^\sigma({\mathcal{B}}(H))$ for $i\in{{\mathbb{N}}}$ by $$\Gamma^L_{kijl}=\left(\operatorname{Ad}(a_{kj}^*)\circ (\operatorname{Ad}(a_{kijl}^*)+\operatorname{Ad}(\widetilde{a_{kijl}}^*)) + p_kp_ip_j p_l\operatorname{Ad}((1-A)^{1/2})\right)\circ \Psi^L_{ki}.$$ Since $$a_{kijl}^*a_{kijl}+\widetilde{a_{kijl}}^*\widetilde{a_{kijl}}=z_{kijl}^*z_{kijl}=v_{kj}^*Jc_{kl}^*b_{ki}^*b_{ki}c_{kl}Jv_{kj}$$ and ${\sum_{i=1}^\infty} {\sum_{l=1}^\infty} c_{kl}^*b_{ki}^*b_{ki}c_{kl}=1$ for every $k\in L$, in the weak\* topology we have $$\begin{aligned} \sum_{k\in L}\sum_{i=1}^\infty{\sum_{j=1}^\infty}{\sum_{l=1}^\infty}\Gamma^L_{kijl}(1) &=\sum_{k\in L}\sum_{i=1}^\infty\sum_{j=1}^\infty\sum_{l=1}^\infty a_{kj}^*z_{kijl}^*z_{kijl} a_{kj}+p_kp_ip_jp_l(1-A)\\ &=\sum_{k\in L}\sum_{j=1}^\infty a_{kj}^*v_{kj}^*v_{kj}a_{kj} +p_kp_j(1-A).\end{aligned}$$ For $k\in R$, define one-way right local maps $\Gamma^R_{ki}$ by $$\Gamma^R_{ki}=\sum_{j=1}^\infty\sum_{l=1}^\infty(\operatorname{Ad}(a_{kj}^*)\circ\Psi^R_{ki}+p_kp_ip_j\operatorname{Ad}((1-A)^{1/2}))\circ(\operatorname{Ad}(c_{kl}^*)\circ\Phi_{ki}).$$ Then $$\begin{aligned} \sum_{k\in R}\sum_{i=1}^\infty\Gamma^R_{ki}(1)&=\sum_{k\in R}\sum_{i=1}^\infty\sum_{j=1}^\infty a_{kj}^*\Psi^R_{ki}(1)a_{kj}+p_kp_ip_j(1-A)\\ &=\sum_{k\in R}\sum_{j=1}^\infty a_{kj}^*a_{kj}+p_kp_j(1-A).\\\end{aligned}$$ Putting things together, we get $$\begin{aligned} &\sum_{k\in L}\sum_{i=1}^\infty{\sum_{j=1}^\infty}{\sum_{l=1}^\infty}\Gamma^L_{kijl}(1)+ \sum_{k\in R}\sum_{i=1}^\infty\Gamma^R_{ki}(1)\\ &=\sum_{k\in L}\sum_{j=1}^\infty a_{kj}^*v_{kj}^*v_{kj}a_{kj} +p_kp_j(1-A)+\sum_{k\in R}\sum_{j=1}^\infty a_{kj}^*a_{kj}+p_kp_j(1-A)\\ &=\sum_{k=1}^\infty\sum_{j=1}^\infty a_{kj}^*v_{kj}^*v_{kj}a_{kj} +p_kp_j(1-A)\\ &=A+(1-A)\\ &=1.\end{aligned}$$ It follows that the series $$\sum_{k\in L}\sum_{i=1}^\infty\sum_{j=1}^\infty \sum_{l=1}^\infty \operatorname{Ad}(u_{kijl}^*)\circ \Gamma^L_{kijl}(x)+\sum_{k\in R}\sum_{i=1}^\infty\Gamma^R_{ki}(x)$$ is convergent in the weak\* topology for every positive $x\in{\mathcal{B}}(H)$. By polarisation, it is convergent in the weak\* topology for every $x\in{\mathcal{B}}(H)$. With $$\Gamma^L_{ki}:=\sum_{j=1}^\infty \sum_{l=1}^\infty \operatorname{Ad}(u_{kijl}^*)\circ \Gamma^L_{kijl},$$ and $\Gamma_{ki}:=\Gamma^L_{ki}$ for $k\in L$, $i\in{\mathbb{N}}$, and $\Gamma_{ki}:=\Gamma^R_{ki}$ for $k\in R$, $i\in{\mathbb{N}}$, it follows that $(\Gamma_{ki})_{k,i}$ is a coarse-graining of a one-way right instrument relative to ${\mathcal{A}}$. By and , for $k\in R$ we have $\Theta_{ki*}(\Theta_{k*}({\omega}_\psi))=\Gamma_{ki*}({\omega}_\psi)$, $i\in{\mathbb{N}}$. Also, using , , and with the identity $\operatorname{Ad}(a_{kj})\omega_\psi=\omega_{\psi_{kj}}$ and Lemma \[l\_dini\] consecutively, we have $$\begin{aligned} &\sum_{k\in L}{\sum_{i=1}^\infty}{\left\Vert \Gamma_{ki*}({\omega}_\psi)-\Theta_{ki*}(\Theta_{k*}({\omega}_\psi))\right\Vert}\\ &\leq\sum_{k\in L}\sum_{i,j,l=1}^\infty{\left\Vert \Psi^L_{ki*} \circ\left(\operatorname{Ad}(u_{kijl})\circ(\operatorname{Ad}(a_{kijl})+\operatorname{Ad}(\widetilde{a_{kijl}})) -\operatorname{Ad}(b_{ki}c_{kl}) \right)\omega_{\psi_{kj}}\right\Vert}\\ &\leq\sum_{k\in L}\sum_{i,j,l=1}^\infty{\left\Vert u_{kijl}(a_{kijl}{\omega}_{\psi_k}a_{kijl}^*+\widetilde{a_{kijl}}{\omega}_{\psi_{kj}}\widetilde{a_{kijl}}^*)u_{kijl}^*-b_{ki}c_{kl}{\omega}_{\psi_{kj}}c_{kl}^*b_{ki}\right\Vert}\\ &\leq\sum_{k\in L}\sum_{i,j,l=1}^\infty{\left\Vert u_{kijl}a_{kijl}{\omega}_{\psi_{kj}}a_{kijl}^*u_{kijl}^*-b_{ki}c_{kl}{\omega}_{\psi_{kj}}c_{kl}^*b_{ki}\right\Vert}+{\left\Vert \widetilde{a_{kijl}}{\omega}_{\psi_{kj}}\widetilde{a_{kijl}}^*\right\Vert}\\ &\leq\sum_{k\in L}\sum_{i,j,l=1}^\infty{\left\Vert u_{kijl}a_{kijl}\psi_{kj}-b_{ki}c_{kl}\psi_{kj}\right\Vert}({\left\Vert a_{kijl}\psi_{kj}\right\Vert}+{\left\Vert b_{ki}c_{kl}\psi_{kj}\right\Vert})+{\left\Vert \widetilde{a_{kijl}}\psi_{kj}\right\Vert}^2\\ &\leq2\sum_{k\in L}\sum_{i,j,l=1}^\infty\frac{{\varepsilon}}{2^{k + i + j+l+1}} \leq {\varepsilon}.\end{aligned}$$Since ${\mathcal{I}}_1$ is a coarse-graining of $(\Theta_k\circ \Theta_{ki})_{k,i}$, applying the same coarse-graining to $(\Gamma_{ki})_{k,i}$ produces an instrument ${\mathcal{I}}_{{\varepsilon}}$ satisfying $P(1)$. Now, assuming $P(n)$ is true, let $\Theta$ be an LOCC map of the form $\Theta=\Theta_{{\mathcal{I}}_{n+1}}$ where $({\mathcal{I}}_0,\dots,{\mathcal{I}}_{n+1})$ is a sequence of instruments such that ${\mathcal{I}}_0$ is a one-way local right instrument relative to ${\mathcal{A}}$, and ${\mathcal{I}}_{l+1}$ is linked to ${\mathcal{I}}_{l}$ for each $l=0,\dots,n$. Then ${\mathcal{I}}_{n+1}$ is a coarse-graining of $(\Theta_k\circ \Theta_{ki})_{k,i}$, where ${\mathcal{I}}_{n}=(\Theta_k)_k$ and ${\mathcal{J}}_k=(\Theta_{ki})_i$ is a one-way instrument for each $k\in{\mathbb{N}}$. Given ${\varepsilon}>0$, by $P(n)$ there exists a coarse-graining ${\mathcal{I}}_{{\varepsilon}}=(\Gamma_{k})_{k}$ of some one-way right instrument such that $$\sum_{k=1}^\infty\norm{\Theta_{k*}({\omega}_\psi)-\Gamma_{k*}({\omega}_\psi)}<\frac{{\varepsilon}}{2}.$$ We have $\Gamma_k=\sum_{j\in S_k} \tilde \Gamma_j$ for some one-way right instrument ${\mathcal{I}}_0'=(\tilde\Gamma_j)_j$ and some partition $(S_k)_k$ of ${\mathbb{N}}$. Let $\tilde \Theta_{ji}=\Theta_{ki}$ for $j\in S_k$, $k\in {\mathbb{N}}$. Then $(\tilde \Theta_{ji})_i$ is a one-way instrument for each $j\in {\mathbb{N}}$, and $(\Gamma_k\circ \Theta_{ki})_{k,i}$ is a coarse-graining of ${\mathcal{I}}_1'=(\tilde \Gamma_j\circ \tilde \Theta_{ji})_{j,i}$, and ${\mathcal{I}}_1'$ is linked to ${\mathcal{I}}_0'$. Applying $P(1)$ to the pair $({\mathcal{I}}_0', {\mathcal{I}}_1')$, we obtain an instrument $(\hat \Gamma_{ji})_{j,i}$ which is a coarse-graining of some one-way right instrument, and satisfies $$\sum_{j,i=1}^\infty{\left\Vert \tilde \Theta_{ji*}( \tilde \Gamma_{j*}(\omega_\psi))-\hat\Gamma_{ji*}(\omega_\psi)\right\Vert}<\frac\epsilon2.$$ Setting $\Gamma_{ki}:=\sum_{j\in S_k}\hat \Gamma_{ji}$, we obtain an instrument $(\Gamma_{ki})_{k,i}$ which is a coarse-graining of a one-way right instrument, and satisfies $$\begin{aligned} {\sum_{k=1}^\infty}{\sum_{i=1}^\infty}\norm{\Theta_{ki*}(\Gamma_{k*}({\omega}_\psi))-\Gamma_{ki*}({\omega}_\psi)}&\le {\sum_{k=1}^\infty}{\sum_{i=1}^\infty}\sum_{j\in S_k}\norm{\tilde\Theta_{ji*}(\tilde \Gamma_{j*}({\omega}_\psi))-\hat\Gamma_{ji*}({\omega}_\psi)}\\&={\sum_{i=1}^\infty}{\sum_{j=1}^\infty} \norm{\tilde\Theta_{ji*}(\tilde \Gamma_{j*}({\omega}_\psi))-\hat\Gamma_{ji*}({\omega}_\psi)}<\frac{{\varepsilon}}{2}.\end{aligned}$$ For each $k,i\in{\mathbb{N}}$, $\Theta_{ki*}((\Theta_{k*}-\Gamma_{k*})({\omega}_\psi))\in{\mathcal{T}}(H)$ is self-adjoint, and attains its norm on self-adjoint operators in ${\mathcal{B}(H)}$. Hence, there exists $T_{ki}\in{\mathcal{B}(H)}$ with $T_{ki}=T_{ki}^*$ and $\norm{T_{ki}}\leq 1$ satisfying $$\norm{\Theta_{ki*}((\Theta_{k*}-\Gamma_{k*})({\omega}_\psi))}=|{\langle}\Theta_{ki}(T_{ki}),(\Theta_{k*}-\Gamma_{k*})({\omega}_\psi){\rangle}|.$$ By Lemma \[l\_ineq\], $$\begin{aligned} \norm{\Theta_{ki*}((\Theta_{k*}-\Gamma_{k*})({\omega}_\psi))}&\leq\norm{T_{ki}}{\langle}\Theta_{ki}(1),|(\Theta_{k*}-\Gamma_{k*})({\omega}_\psi)|{\rangle}\\ &\leq{\langle}\Theta_{ki}(1),|(\Theta_{k*}-\Gamma_{k*})({\omega}_\psi)|{\rangle}.\end{aligned}$$ Hence, $$\begin{aligned} \sum_{k,i=1}^\infty\norm{\Theta_{ki*}(\Theta_{k*}({\omega}_\psi))-\Theta_{ki*}(\Gamma_{k*}({\omega}_\psi))}&\leq\sum_{k=1}^\infty\sum_{i=1}^\infty{\langle}\Theta_{ki}(1),|(\Theta_{k*}-\Gamma_{k*})({\omega}_\psi)|{\rangle}\\ &=\sum_{k=1}^\infty\norm{\Theta_{k*}({\omega}_\psi)-\Gamma_{k*}({\omega}_\psi)}\\ &<\frac{{\varepsilon}}{2}.\end{aligned}$$ By the triangle inequality we obtain $${\sum_{k=1}^\infty}{\sum_{i=1}^\infty}\norm{\Theta_{ki*}(\Theta_{k*}({\omega}_\psi))-\Gamma_{ki*}({\omega}_\psi)}<{\varepsilon}.$$ Recall that ${\mathcal{I}}_{n+1}$ is a coarse-graining of $(\Theta_k\circ \Theta_{k,i})_{k,i}$. Letting ${\mathcal{I}}_\epsilon$ be the result of applying the same coarse-graining to $(\Gamma_{ki})_{k,i}$, the preceding inequality and the triangle inequality then show that ${\mathcal{I}}_\epsilon$ satisfies $P(n+1)$. The proof of Theorem \[th\_lopopescu\_std\] may seem complicated when compared to Lo and Popescu’s intuitive argument for the special case ${{\mathcal{A}}}=M_n\otimes 1$. This may be explained by our approximate version of convertibility together with the additional approximation provided by Proposition \[p\_lopopescu\], the latter not being required in the type I case. Using the representation theory of properly infinite von Neumann algebras, we now remove the standardness assumption in the previous theorem. Recall that a von Neumann algebra is $\sigma$-finite if every set of mutually orthogonal projections is at most countable, and that every von Neumann algebra ${{\mathcal{A}}}\subseteq{\mathcal{B}(H)}$ on a separable Hilbert space $H$ enjoys this property. \[t:lopopescu\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a semi-finite factor on a separable Hilbert space $H$. Given $\Theta\in\operatorname{LOCC}({\mathcal{A}})$ and $\psi\in H$, for every ${\varepsilon}>0$ there exists $\Xi \in \LOCC^r({{\mathcal{A}}})$ such that $${\left\Vert \Theta_*({\omega}_\psi)-\Xi_*({\omega}_\psi)\right\Vert} < {\varepsilon}.$$ Clearly, we may assume that $\psi\in H_1$. Let $K$ be a separable infinite-dimensional Hilbert space and fix a unit vector $\xi\in K$. Consider the factors $\widetilde{{\mathcal{A}}} :={\mathcal{A}}{\otimes}{\mathcal{B}}(K){\otimes}1_K$ and $\widetilde{{\mathcal{B}}}:=\widetilde{{\mathcal{A}}}'={\mathcal{B}}{\otimes}1_K{\otimes}{\mathcal{B}}(K)$, acting on $H{\otimes}K{\otimes}K$, and equip $\widetilde{{\mathcal{A}}}$ with the trace $\widetilde{\tau} = \tau\otimes {\rm tr} \otimes {\rm tr}$. Letting $L^{\infty}(\widetilde{{\mathcal{A}}})\subseteq{\mathcal{B}}(L^2(\widetilde{{\mathcal{A}}}))$ denote the standard representation of $\widetilde{{\mathcal{A}}}$, one sees that both $\widetilde{{\mathcal{B}}}=\widetilde{{\mathcal{A}}}'$ and $L^{\infty}(\widetilde{{\mathcal{A}}})'\cong L^{\infty}(\widetilde{{\mathcal{A}}})$ are $\sigma$-finite and properly infinite factors. Hence, by [@t1 Proposition V.3.1], the representations $(\widetilde{{\mathcal{A}}},H{\otimes}K{\otimes}K)$ and $(\widetilde{{\mathcal{A}}},L^2(\widetilde{{\mathcal{A}}}))$ are unitarily equivalent, implemented by the unitary operator $U : H{\otimes}K {\otimes}K\rightarrow L^2(\widetilde{{\mathcal{A}}})$, say. Clearly, $\operatorname{Ad}(U)\circ (\Theta{\otimes}\operatorname{id}_{{\mathcal{B}}(K{\otimes}K)})\circ \operatorname{Ad}(U^*)\in\operatorname{LOCC}(L^\infty(\widetilde{{\mathcal{A}}}))$. Since $L^\infty(\widetilde{{{\mathcal{A}}}})$ is a factor, by Theorem \[th\_lopopescu\_std\], for every ${\varepsilon}>0$ there exists a one-way right LOCC map $\widetilde{\Xi}:{\mathcal{B}}(L^2(\widetilde{{\mathcal{A}}}))\rightarrow{\mathcal{B}}(L^2(\widetilde{{\mathcal{A}}}))$ relative to $L^\infty(\widetilde{{\mathcal{A}}})$ such that $$\norm{(\operatorname{Ad}(U)\circ (\Theta{\otimes}\operatorname{id}_{{\mathcal{B}}(K{\otimes}K)})\circ \operatorname{Ad}(U^*))_*({\omega}_{U(\psi{\otimes}\xi{\otimes}\xi)})-\widetilde{\Xi}_*({\omega}_{U(\psi{\otimes}\xi{\otimes}\xi)})}<{\varepsilon}.$$ Then $\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U):{\mathcal{B}}(H{\otimes}K{\otimes}K)\rightarrow{\mathcal{B}}(H{\otimes}K{\otimes}K)$ is a one-way right LOCC map relative to $\widetilde{{\mathcal{A}}}$ satisfying $$\norm{(\Theta{\otimes}\operatorname{id}_{{\mathcal{B}}(K{\otimes}K)})_*({\omega}_{\psi{\otimes}\xi{\otimes}\xi})-(\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U))_*({\omega}_{\psi{\otimes}\xi{\otimes}\xi})}<{\varepsilon}.$$ Let $\widetilde{\Psi}_k\in \CP^\sigma_{\widetilde{{\mathcal{B}}}}({\mathcal{B}}(H{\otimes}K{\otimes}K))$ and $\widetilde{\Phi}_k\in \UCP^\sigma_{\widetilde{{\mathcal{A}}}}({\mathcal{B}}(H{\otimes}K{\otimes}K))$ satisfy $\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U)= \sum_{k=1}^\infty \widetilde{\Phi}_k\circ\widetilde{\Psi}_k$, and let ${\mathcal{E}}:{\mathcal{B}}(H{\otimes}K {\otimes}K)\rightarrow {\mathcal{B}}(H)$ denote the normal unital completely positive map given by ${\mathcal{E}}(X)=(\operatorname{id}{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)(X)$. Define $\Psi_k\in \CP^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$ and $\Phi_k\in \UCP^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ by $$\Psi_k={\mathcal{E}}\circ \widetilde{\Psi}_k|_{{\mathcal{B}(H)}{\otimes}1_K{\otimes}1_K}\circ \iota{{\quad\text{and}\quad}}\Phi_k={\mathcal{E}}\circ \widetilde{\Phi}_k|_{{\mathcal{B}(H)}{\otimes}1_K{\otimes}1_K}\circ \iota,$$ where $\iota\colon {\mathcal{B}(H)}\to {\mathcal{B}(H)}{\otimes}1_K{\otimes}1_K$ is the embedding $\iota(T)= T{\otimes}1{\otimes}1$. Then $\Xi:=\sum_{k=1}^\infty \Phi_k\circ \Psi_k$ is a one-way right LOCC map on ${\mathcal{B}(H)}$ relative to ${\mathcal{A}}$. Moreover, letting $\sigma_{r,s}$, for $r,s\in \{2,3,4,5\}$ be the flip between terms $r$ and $s$ acting on the tensor product $H\otimes K \otimes K \otimes K \otimes K$, for every $T\in{\mathcal{B}(H)}$, we have $$\begin{aligned} &{{{ \left\langleT,\Xi_*({\omega}_\psi)\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle{\mathcal{E}}(\widetilde{\Phi}_k(T{\otimes}1{\otimes}1)){\otimes}1{\otimes}1,(\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1,({\mathcal{E}}_*{\otimes}\operatorname{id}{\otimes}\operatorname{id})((\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi))\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1,\sigma_{24}\sigma_{35}((\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi){\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1,\sigma_{35}((\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi){\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1,(\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi){\otimes}{\omega}_\xi{\otimes}{\omega}_\xi\right\rangle}}}\\ &={{{ \left\langleT{\otimes}1{\otimes}1{\otimes}1{\otimes}1,(\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U))_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi){\otimes}{\omega}_\xi{\otimes}{\omega}_\xi\right\rangle}}}\\ &={{{ \left\langleT{\otimes}1{\otimes}1,(\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U))_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)\right\rangle}}},\end{aligned}$$ where in the fourth equality we used the fact that $\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1\in {\mathcal{B}}(H){\otimes}1{\otimes}B(K){\otimes}1{\otimes}1$ is symmetric under $\sigma_{24}$ and in the fifth equality we used the fact that $(\widetilde{\Psi}_k)_*$ acts trivially on the third leg. (These facts follow from Proposition \[p\_comp\], for example.) It follows that $$\begin{aligned} &\norm{\Theta_*({\omega}_\psi)-\Xi_*({\omega}_\psi)}\\ &\le \norm{(\Theta{\otimes}\operatorname{id}_{{\mathcal{B}}(K{\otimes}K)})_*({\omega}_{\psi{\otimes}\xi{\otimes}\xi})-(\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U))_*({\omega}_{\psi{\otimes}\xi{\otimes}\xi})}<{\varepsilon}.\;\qedhere\end{aligned}$$ By left-right symmetry, Theorem \[t:lopopescu\] immediately yields the following corollary: \[c\_oneway\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a semi-finite factor on a separable Hilbert space. For unit vectors $\psi,{\varphi}\in H$, the following are equivalent: (i) $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$; (ii) $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}^r({\mathcal{A}})$; (iii) $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}^l({\mathcal{A}})$. The Main Theorem {#s:main} ================ In this section we establish Theorem \[th\_maj\], a version of Nielsen’s theorem for bipartite systems modelled by semi-finite, $\sigma$-finite von Neumann algebras (or by standardly represented von Neumann algebras). The next group of lemmas will help justify certain technical arguments in its proof. \[l\_M\] Let $({\mathcal{A}},\tau)$ be a semi-finite von Neumann algebra. For any $\rho_1,\dots,\rho_n\in {{\mathcal{A}}}_*^+$, there exist $M_1,\dots,M_{n+1}\in {\mathcal{A}}$ with $\sum_{i=1}^{n+1}M_i^*M_i=1$ such that, for $\rho=\sum_{i=1}^n \rho_i$, we have $$M_i \rho M_i^*=\rho_i,\quad 1\le i\le n,{{\quad\text{and}\quad}}M_{n+1}\rho M_{n+1}=0.$$ We may assume that ${{\mathcal{A}}}$ is standardly represented on $H=L^2({{\mathcal{A}}},\tau)$, and identify ${{\mathcal{A}}}_*$ with $L^1({{\mathcal{A}}},\tau)$. Let ${{\mathcal{B}}}={{\mathcal{A}}}'$. Since $\rho$ is a positive element of $L^1({{\mathcal{A}}},\tau)$, the (positive, densely defined) operator $\xi:=\rho^{1/2}$ is an element of $H=L^2({{\mathcal{A}}},\tau)$. Let $p$ and $p'$ be the orthogonal projections onto $\overline{{{\mathcal{B}}}\xi}$ and $\overline{{{\mathcal{A}}}\xi}$, respectively, and note that $$p\in {{\mathcal{A}}},\quad p'\in {{\mathcal{B}}},\quad Jp'J=p{{\quad\text{and}\quad}}JpJ=p'$$ (see [@t2 Section IX.1]; for the last two equalities, $J\xi=\xi^*=\xi$ so $JpJa\xi=Jp(JaJ\xi)=J(JaJ\xi)=a\xi$, so $p'\le JpJ$ and similarly $p\le Jp'J$.) For $1\le i\le n$, we have $\rho_i\leq\rho$, that is, ${\omega}_{\rho_i^{1/2}}\leq {\omega}_\xi$. By the Radon–Nikodym theorem (see [@kr2 Proposition 7.3.5] and its proof) there exists $b_i\in {\mathcal{B}}^+$ such that $${\omega}_{\rho_i^{1/2}}(a) = {{ \left(a b_i\xi,\xi\right)}} = {{ \left(a b_i^{1/2}\xi,b_i^{1/2}\xi\right)}} = {\omega}_{b_i^{1/2} \xi}(a),\quad a\in {\mathcal{A}}.$$ By the uniqueness of the GNS representation, there exists a partial isometry $v_i\in {\mathcal{B}}$ such that for $c_i:=v_ib_i^{1/2}\in {{\mathcal{B}}}$, we have $$\label{eq_cixi} c_i \xi=\rho_i^{1/2},\quad 1\le i\le n.$$ Consider $M_1,\dots,M_{n+1}\in {{\mathcal{A}}}$, given by $$M_i:=Jc_iJp,\quad 1\le i\le n,\quad M_{n+1}:=1-p.$$ For $1\le i\le n$, using we have $$M_i\xi=Jc_iJ\xi=\xi c_i^*=(c_i\xi)^*=(\rho_i^{1/2})^*=\rho_i^{1/2},$$ so $$M_i \rho M_i^*=(M_i\xi)(M_i\xi)^*=\rho_i,\quad 1\le i\le n.$$ Similarly, $M_{n+1}\xi=(1-p)\xi=0$, so $M_{n+1}\rho M_{n+1}^*=0$. For $a,b\in {{\mathcal{A}}}$, we have $p'a\xi=a\xi$ and $p'b\xi=b\xi$; hence, for $1\le i\le n$, $$\begin{aligned} {{ \left(JM_i^*M_iJa\xi,b\xi\right)}}&={{ \left(JpJc_i^*c_iJpJa\xi,b\xi\right)}}= {{ \left(c_ip'a\xi,c_ip'b\xi\right)}} \\&={{ \left(c_ia\xi,c_ib\xi\right)}}={{ \left(ac_i\xi,bc_i\xi\right)}}={{ \left(a\rho_i^{1/2},b\rho_i^{1/2}\right)}} \\&=\tau(\rho_i^{1/2}b^*a\rho_i^{1/2})=\tau(b^*a\rho_i). \end{aligned}$$ Thus, the operator $S:=\sum_{i=1}^n M_i^*M_i$ satisfies $ {{ \left(JSJ a\xi,b\xi\right)}}=\tau(b^*a\rho)={{ \left(a\xi,b\xi\right)}}$. Hence, $p'JSJp'=p'$, so $$S = pSp = Jp'JSJp'J = Jp'J = p,$$ and $$\sum_{i=1}^{n+1}M_i^*M_i=S+M_{n+1}^*M_{n+1}=p+(1-p)=1.\qedhere$$ The version of the following lemma for the case where ${\varepsilon}= 0$ is well-known. We require the following approximate extension. \[l\_pure\] Let $H$ be a Hilbert space, ${\varphi}\in H_1$ and $(\omega_k)_{k=1}^\infty$ be a sequence in ${\mathcal{T}}(H)^+$ such that $\sum_{k=1}^\infty{\omega}_k$ converges weakly to an element in the closed unit ball of ${\mathcal{T}}(H)$. Set $\alpha_k=\langle \phi\phi^*,{\omega}_k\rangle$, $k\in {\mathbb{N}}$. If $\epsilon > 0$ and $$\label{l_1}{\left\Vert {\omega}_{\varphi}- \sum_{k=1}^\infty{\omega}_k\right\Vert}<{\varepsilon},$$ then $\sum_{k=1}^\infty{\left\Vert {\omega}_k-\alpha_k{\omega}_{\varphi}\right\Vert} < 2\sqrt{{\varepsilon}}+{\varepsilon}$. Let $\operatorname{tr}$ denote the canonical trace on ${\mathcal{T}}(H)$ and, for simplicity, write $p = {\varphi}{\varphi}^*$ and $p^{\perp} = 1 - p$. Observe that $p^\perp {\omega}_{\varphi}p^\perp =0$. Thus, by , $$\begin{aligned} \label{eq_tlines} \sum_{k=1}^\infty{\left\Vert p^\perp{\omega}_k p^\perp\right\Vert}&=\sum_{k=1}^\infty\operatorname{tr}(p^{\perp}{\omega}_kp^{\perp})=\operatorname{tr}\bigg(p^{\perp}\bigg(\sum_{k=1}^\infty{\omega}_k\bigg)p^{\perp}\bigg)\\ &={\left\Vert p^{\perp}\bigg(\sum_{k=1}^\infty{\omega}_k\bigg)p^{\perp}\right\Vert} < {\varepsilon}. \nonumber\end{aligned}$$ By the Cauchy–Schwarz inequality, for any $T\in{\mathcal{B}(H)}$, we have $$\begin{aligned} \left|p\omega_k p^{\perp}(T)\right| = \left|{\omega}_k(p^{\perp}Tp)\right| & \leq {\omega}_k\left(p^{\perp}TT^*p^{\perp}\right)^{1/2}{\omega}_k(p)^{1/2}\\ &\leq {\left\Vert T\right\Vert}{\left\Vert p^{\perp}{\omega}_kp^{\perp}\right\Vert}^{1/2}{\omega}_k(p)^{1/2}.\end{aligned}$$ Hence, ${\left\Vert p{\omega}_kp^\perp\right\Vert}\leq{\left\Vert p^{\perp}{\omega}_kp^{\perp}\right\Vert}^{1/2}{\omega}_k(p)^{1/2}$. Applying the Cauchy–Schwarz inequality once again and using , we obtain $$\begin{aligned} \sum_{k=1}^\infty{\left\Vert p{\omega}_kp^\perp\right\Vert}&\leq\sum_{k=1}^\infty {\omega}_k(p)^{1/2} {\left\Vert p^{\perp}{\omega}_kp^{\perp}\right\Vert}^{1/2}\\ &\leq \bigg(\sum_{k=1}^\infty{\omega}_k(p)\bigg)^{1/2} \bigg(\sum_{k=1}^\infty{\left\Vert p^{\perp}{\omega}_kp^{\perp}\right\Vert}\bigg)^{1/2} < \sqrt{{\varepsilon}}.\end{aligned}$$ Since ${\left\Vert p^\perp {\omega}_k p \right\Vert}={\left\Vert (p{\omega}_k p^\perp)^*\right\Vert}={\left\Vert p{\omega}_k p^\perp\right\Vert}$, we have $\sum_{k=1}^\infty {\left\Vert p^\perp {\omega}_k p \right\Vert}< \sqrt{{\varepsilon}}$. Decomposing ${\omega}_k=p{\omega}_kp+p{\omega}_kp^{\perp}+p^\perp{\omega}_k p+p^\perp{\omega}_k p^\perp$, and noting $p{\omega}_k p={\omega}_k(p){\omega}_{\varphi}=\alpha_k {\omega}_{\varphi}$, we see that $$\begin{aligned} \sum_{k=1}^\infty{\left\Vert {\omega}_k-\alpha_k{\omega}_{\varphi}\right\Vert}&\leq\sum_{k=1}^\infty \Big( {\left\Vert p{\omega}_kp^\perp\right\Vert}+{\left\Vert p^\perp {\omega}_k p \right\Vert}+{\left\Vert p^\perp{\omega}_k p^\perp\right\Vert}\Big) < 2\sqrt{{\varepsilon}}+{\varepsilon}.\qedhere\end{aligned}$$ We are now in a position to prove the main result of the paper. It provides a version of Nielsen’s theorem for bipartite systems without any explicit (spatial) tensor product structure. \[th\_maj\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a semi-finite factor on a separable Hilbert space $H$. For unit vectors $\psi,{\varphi}\in H$, the following are equivalent: \(i)  $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$; \(ii) $\rho_{\psi}\prec \rho_{{\varphi}}$. $(i)\Rightarrow(ii)$ Let ${\varepsilon}> 0$ and $\delta > 0$ be such that $2(\delta+\sqrt{\delta}) < \tfrac12{\varepsilon}$. By Corollary \[c\_oneway\], there exists a one-way right LOCC map $\Theta$ relative to ${\mathcal{A}}$ such that $$\label{eq_nor}{\left\Vert {\omega}_{\varphi}-\Theta_*({\omega}_\psi)\right\Vert}<\delta.$$ By Proposition \[p\_comp\](ii), we may write $$\Theta(x) = \sum_{k=1}^\infty \Phi_k(a_k^*xa_k), \ \ x\in{\mathcal{B}(H)},$$ for some $\Phi_k\in \operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ and $a_k\in {\mathcal{A}}$, $k\in{\mathbb{N}}$, with $$\label{eq_akak} \sum_{k=1}^{\infty} a_k^* a_k = 1,$$ where the series converge in the weak\* topology. Let $$\omega_k=\Phi_{k*}(a_k\omega_\psi a_k^*) = a_k\Phi_{k^*}(\omega_\psi) a_k^*\in {\mathcal{T}}(H)^+;$$ then the series $\sum_{k=1}^\infty \omega_k$ is weakly convergent to $\Theta_*(\omega_\psi)$. Let us write $\alpha_k:={{ \left\langle\phi\phi^*,\omega_k\right\rangle}}$, $k\in {\mathbb{N}}$. Since ${{ \left\langle\phi\phi^*,\omega_\phi\right\rangle}}=1$, the bound  implies $$\label{eq_alphak} \sum_{k=1}^\infty \alpha_k={{ \left\langle\phi\phi^*,\Theta_*(\omega_\psi)\right\rangle}}\in (1-\delta,1].$$ By Lemma \[l\_pure\], $\sum_{k=1}^\infty{\left\Vert \omega_k -\alpha_k{\omega}_{\varphi}\right\Vert}<2\sqrt{\delta}+\delta$. Taking restrictions to ${\mathcal{A}}$, and using the fact that $\Phi_k|_{{\mathcal{A}}}$ coincides with the identity map, we obtain $$\label{eq_no1} \sum_{k=1}^\infty{\left\Vert a_k\rho_\psi a_k^*-\alpha_k\rho_{\varphi}\right\Vert}_1 < 2 \sqrt{\delta} + \delta.$$ For each $a\in{\mathcal{A}}$, by we have $$\begin{aligned} \sum_{k=1}^l &{{{ \left\langlea,\rho_\psi^{1/2}a_k^*a_k\rho_\psi^{1/2}\right\rangle}}} = \sum_{k=1}^l \tau(a\rho_\psi^{1/2}a_k^*a_k\rho_\psi^{1/2}) = \sum_{k=1}^l \tau(a_k^*a_k\rho_\psi^{1/2}a\rho_\psi^{1/2})\\ &= \sum_{k=1}^l {{{ \left\langlea_k^*a_k,\rho_\psi^{1/2}a\rho_\psi^{1/2}\right\rangle}}} \to_{l\to\infty} {{{ \left\langle1,\rho_\psi^{1/2}a\rho_\psi^{1/2}\right\rangle}}} ={{{ \left\langlea,\rho_\psi\right\rangle}}}.\end{aligned}$$ Hence, the series $\sum_{k=1}^\infty\rho_\psi^{1/2}a_k^*a_k\rho_\psi^{1/2}$ converges weakly to $\rho_\psi$. By Lemma \[l\_dini\], the convergence is in norm. Choose $L\in {\mathbb{N}}$ so that $$\label{eq_norrho} {\left\Vert \rho_\psi-\sum_{k=1}^L \rho_\psi^{1/2} a_k^*a_k\rho_\psi^{1/2}\right\Vert}_1<\tfrac12\epsilon.$$ By the right polar decomposition, there exists a partial isometry $v_k\in{\mathcal{A}}$ such that $a_k\rho_\psi^{1/2} = (a_k\rho_\psi a_k^*)^{1/2}v_k^*$, $k\in {\mathbb{N}}$. Writing $\alpha_0 = 1 - \sum_{k=1}^\infty\alpha_k$, we have by that $\alpha_0 <\delta$. Setting $v_0 = 1$ and using and , we see that $$\begin{aligned} {\left\Vert \rho_\psi-\sum_{k=0}^L\alpha_kv_k\rho_{\varphi}v_k^*\right\Vert}_1 &< \delta+{\left\Vert \sum_{k=1}^L \rho_\psi^{1/2}a_k^*a_k \rho_\psi^{1/2}-\alpha_kv_k\rho_\phi v_k^*\right\Vert}_1+\tfrac12\epsilon\\ &= \delta+{\left\Vert \sum_{k=1}^L v_k(a_k \rho_\psi a_k^*-\alpha_k\rho_\phi) v_k^*\right\Vert}_1+\tfrac12\epsilon\\ &\le \delta+\sum_{k=1}^L {\left\Vert a_k\rho_\psi a_k^*-\alpha_k\rho_\phi\right\Vert}_1+\tfrac12\epsilon \\ & < 2(\delta+\sqrt \delta)+\tfrac12\epsilon<{\varepsilon}.\end{aligned}$$ Since ${\varepsilon}>0$ was arbitrary, it follows from [@h Theorem 2.5(3)] that $\rho_\psi\prec\rho_{\varphi}$. $(ii)\Rightarrow(i)$ Suppose $\rho_\psi \prec \rho_{\varphi}$, and fix ${\varepsilon}>0$. Pick $\delta>0$ such that $4\sqrt{\delta}<{\varepsilon}$. Since ${{\mathcal{A}}}$ is a factor, by [@h Theorem 2.5], there exist a family $(u_i)_{i=1}^n$ of unitary operators in ${\mathcal{A}}$ and a probability distribution $(p_i)_{i=1}^n$, such that, if $\widetilde{\rho_\psi}=\sum_{i=1}^n p_i u_i\rho_{\varphi}u_i^*$, then $\norm{\rho_\psi-\widetilde{\rho_\psi}}_1 < \delta$. Set $m=n+1$. By Lemma \[l\_M\], there exist $M_1,\dots,M_m\in {\mathcal{A}}$ with $\sum_{i=1}^m M_i^*M_i=1$, such that $$\label{e_M} M_i\widetilde{\rho_\psi}M_i^*= p_i\rho_{\varphi}\text{ for $1\le i\le n$,}{{\quad\text{and}\quad}}M_m\widetilde{\rho_\psi}M_m^*= 0.$$ Let $e_1,\dots,e_m$ be the standard basis of ${\mathbb{C}}^m$, and consider the UCP maps $\Psi,\Phi:{\mathcal{B}}(H{\otimes}{\mathbb{C}}^m)\rightarrow{\mathcal{B}}(H)$ given by $$\Psi(T) = \sum_{i=1}^mM_i^*(\operatorname{id}{\otimes}{\omega}_{e_i})(T)M_i{{\quad\text{and}\quad}}\Phi(T)=\sum_{i=1}^n p_i(\operatorname{id}{\otimes}{\omega}_{e_i})(T)$$ for $T\in {\mathcal{B}}(H{\otimes}{\mathbb{C}}^m)$. We have that $$\label{e_pre} \Psi_*(\rho)=\sum_{i=1}^m M_i\rho M_i^*{\otimes}e_ie_i^*{{\quad\text{and}\quad}}\Phi_*(\rho)=\sum_{i=1}^n p_i\rho{\otimes}e_ie_i^*,$$ for $\rho\in {\mathcal{T}}(H)$. Letting $V,W: H\rightarrow H{\otimes}{\mathbb{C}}^m{\otimes}{\mathbb{C}}^m$ be the isometries given by $$V\eta = \sum_{i=1}^mM_i\eta{\otimes}e_i{\otimes}e_i{{\quad\text{and}\quad}}W\eta=\sum_{i=1}^n \sqrt p_i\eta{\otimes}e_i{\otimes}e_i,\quad\eta\in H,$$ we have Stinespring representations $$\label{st_VW}\Psi(T)=V^*(T{\otimes}1)V {{\quad\text{and}\quad}}\Phi(T) = W^*(T{\otimes}1)W,\quad T\in {\mathcal{B}}(H{\otimes}{\mathbb{C}}^m).$$ Consider the states $\nu_\psi,\nu_\phi\colon {\mathcal{A}}{\otimes}M_m\to {\mathbb{C}}$ given by $$\nu_\psi=\omega_\psi\circ \Psi|_{{\mathcal{A}}{\otimes}M_m}{{\quad\text{and}\quad}}\nu_\phi=\omega_\phi\circ \Phi|_{{\mathcal{A}}{\otimes}M_m}.$$ By  and , we have $$\begin{aligned} {\left\Vert \nu_\psi-\nu_\phi\right\Vert}_{cb}&={\left\Vert \nu_\psi-\nu_\phi\right\Vert}\\ &={\left\Vert (\Psi|_{{\mathcal{A}}{\otimes}M_m})_*(\rho_\psi)-(\Phi|_{{\mathcal{A}}{\otimes}M_m})_*(\rho_{\varphi})\right\Vert}\\ &={\left\Vert \sum_{i=1}^mM_i\rho_\psi M_i^* {\otimes}e_ie_i^*- M_i\widetilde{\rho_\psi} M_i^*{\otimes}e_ie_i^*\right\Vert}\\ &={\left\Vert (\Psi|_{{\mathcal{A}}{\otimes}M_m})_*(\rho_\psi-\widetilde{\rho_\psi})\right\Vert} < \delta.\end{aligned}$$ Let $\theta\colon {\mathcal{A}}\otimes M_m\to {\mathcal{B}} (H\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^m)$ be the $*$-homomorphism given by $\theta(X)=X\otimes 1$, $X\in {\mathcal{A}}\otimes M_m$. By , the maps $\nu_\psi$ and $\nu_\phi$ have Stinespring representations $$\nu_\psi=\omega_{V\psi}\circ \theta{{\quad\text{and}\quad}}\nu_\phi=\omega_{W\phi}\circ \theta.$$ By the continuity of the Stinespring representation [@ksw Theorem 1] there exist a Hilbert space $K$, a $*$-homomorphism $\pi:{\mathcal{A}}{\otimes}M_m\rightarrow {\mathcal{B}}(K)$, and vectors $\eta_1,\eta_2\in K$ yielding Stinespring representations $$\nu_\psi={\omega}_{\eta_1}\circ \pi{{\quad\text{and}\quad}}\nu_\phi={\omega}_{\eta_2}\circ\pi$$ with $$\label{eq_sqrd} {\left\Vert \eta_1-\eta_2\right\Vert} < \sqrt{\delta}.$$ By the uniqueness of Stinespring representations, there exist partial isometries $U_1:H{\otimes}{\mathbb{C}}^m{\otimes}{\mathbb{C}}^m\rightarrow K$ and $U_2:K\rightarrow H{\otimes}{\mathbb{C}}^m{\otimes}{\mathbb{C}}^m$ satisfying $U_1V\psi=\eta_1$, $U_2\eta_2=W{\varphi}$, $$U_1(X{\otimes}1)=\pi(X)U_1 \quad \mbox{ and } \quad U_2\pi(X) = (X{\otimes}1)U_2,\quad X\in {\mathcal{A}}{\otimes}M_m.$$ Let ${\mathcal{B}}={\mathcal{A}}'$. The preceding relations imply that the contraction $U:=U_2U_1$ satisfies $$U\in ({\mathcal{A}}{\otimes}M_m{\otimes}1)'={\mathcal{A}}'{\otimes}1{\otimes}M_m={\mathcal{B}}{\otimes}1{\otimes}M_m;$$ moreover, by , $$\label{eq_uvw} {\left\Vert UV\psi-W{\varphi}\right\Vert} < \sqrt{\delta}.$$ Since $U \in {\mathcal{B}}{\otimes}1{\otimes}M_m$, we have $U = \sum_{k,l=1}^mb_{kl}\otimes 1\otimes e_ke_l^*$ for some $b_{kl}\in{\mathcal{B}}$. Set $b_i = b_{ii}$, $1\leq i\leq m$. Then $b_i$ is a contraction in ${\mathcal{B}}$. Let $\Phi_i\in\operatorname{UCP}_{{\mathcal{A}}}^\sigma({\mathcal{B}(H)})$ be the channel given by $$\Phi_i(x) = b_i^*xb_i + (1 - b_i^*b_i)^{1/2}x(1 - b_i^*b_i)^{1/2}, \ \ \ x\in{\mathcal{B}(H)},$$ and define $\Theta\in\operatorname{LOCC}({\mathcal{A}})$ by $$\Theta(x) = \sum_{i=1}^m \Phi_i(M_i^*x M_i), \ \ \ x\in{\mathcal{B}(H)}.$$ We claim that ${\left\Vert \Theta_*({\omega}_\psi)-{\omega}_{\varphi}\right\Vert} < {\varepsilon}$, which will complete the proof. To see this, let $P : {\mathbb{C}}^m{\otimes}{\mathbb{C}}^m\rightarrow{\mathbb{C}}^m{\otimes}{\mathbb{C}}^m$ denote the orthogonal projection onto $\operatorname{span}\{e_i{\otimes}e_i\mid 1\leq i\leq m\}$, and consider the contraction $$\tilde{U} = (1{\otimes}P)U \in {\mathcal{B}}(H\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^m).$$ A calculation shows that $$\label{tUV} \tilde U V\psi = \sum_{i=1}^m b_i M_i\psi\otimes e_i\otimes e_i.$$ Since $W\phi$ lies in the range of $1{\otimes}P$, the bound  implies $$\label{eq_wdelta} {\left\Vert \tilde UV\psi-W{\varphi}\right\Vert} = {\left\Vert (1{\otimes}P)\left(UV\psi-W{\varphi}\right)\right\Vert} < \sqrt{\delta},$$ and so $$\begin{aligned} \label{eq_anewo} {\left\Vert \omega_{\tilde U V\psi}-\omega_{W\phi}\right\Vert}\le 2{\left\Vert \tilde U V\psi-W\phi\right\Vert}< 2 \sqrt{\delta}.\end{aligned}$$ Observe that for $x\in {\mathcal{B}(H)}$, equation  yields $$\begin{aligned} {{{ \left\langlex,\left(\sum_{i=1}^m\omega_{b_iM_i\psi}\right)-\omega_\phi\right\rangle}}}={{{ \left\langlex{\otimes}1{\otimes}1,\omega_{\tilde U V\psi}-\omega_{W\phi}\right\rangle}}}\end{aligned}$$ so, in particular, using , we have $$\begin{aligned} \label{eq_mipsi} {\left\Vert \sum_{i=1}^m b_iM_i\omega_\psi M_i^*b_i^*-\omega_\phi\right\Vert} &= {\left\Vert \left(\sum_{i=1}^m \omega_{b_iM_i\psi}\right)-\omega_{\phi}\right\Vert} \\&\leq {\left\Vert \omega_{\tilde UV\psi}-\omega_{W\phi}\right\Vert}<2\sqrt\delta. \nonumber\end{aligned}$$ Since $V$ and $W$ are isometries, we have $\|V\psi\|=1=\|W\phi\|$. Using , we thus have $$\|V\psi\| - \|\tilde{U}V\psi\| = \|W{\varphi}\| - \|\tilde{U}V\psi\| < \sqrt{\delta}.$$ By Lemma \[l\_piso\], $$\begin{aligned} \sum_{i=1}^m{\left\Vert (1-b_i^*b_i)^{1/2}M_i\psi\right\Vert}^2 &= \sum_{i=1}^m \|M_i\psi\|^2-\|b_iM_i\psi\|^2 =\|V\psi\|^2-\|\tilde UV\psi\|^2\\ &={\left\Vert (1-\tilde{U}^*\tilde{U})^{1/2}V\psi\right\Vert}^2 < 2\sqrt{\delta}.\end{aligned}$$ Thus, using , we have $$\begin{aligned} & {\left\Vert \Theta_*({\omega}_\psi)-{\omega}_{\varphi}\right\Vert}\\ &= {\left\Vert \sum_{i=1}^mb_iM_i{\omega}_\psi M_i^*b_i^*+(1-b_i^*b_i)^{1/2}M_i{\omega}_\psi M_i^*(1-b_i^*b_i)^{1/2}-{\omega}_{\varphi}\right\Vert}\\ &\leq {\left\Vert \sum_{i=1}^mb_iM_i{\omega}_\psi M_i^*b_i^*-{\omega}_{\varphi}\right\Vert} +\sum_{i=1}^m{\left\Vert (1-b_i^*b_i)^{1/2}M_i{\omega}_\psi M_i^*(1-b_i^*b_i)^{1/2}\right\Vert}\\ &<2\sqrt\delta+\sum_{i=1}^m{\left\Vert (1-b_i^*b_i)^{1/2}M_i\psi\right\Vert}^2 < 4\sqrt{\delta} < {\varepsilon}.\qedhere\end{aligned}$$ The question of convertibility for purely infinite factors becomes trivial. For instance, if ${\mathcal{A}}$ is a factor of type $\mathrm{III}_1$ with separable predual then, by [@t2 Theorem XII.5.12], for all normal states ${\omega}_1,{\omega}_2$ on ${\mathcal{A}}$, we have $$\inf\{\norm{u^*{\omega}_1 u - {\omega}_2}\mid u\in {\mathcal{U}}({\mathcal{A}})\}=0.$$ Hence, given states $\psi,{\varphi}$ in the representation space $H$ of ${\mathcal{A}}$, for every ${\varepsilon}>0$ there exists a unitary $u\in{\mathcal{A}}$ such that $\norm{u^*{\omega}_\psi|_{{\mathcal{A}}} u - {\omega}_{\varphi}|_{{\mathcal{A}}}}<{\varepsilon}$. Appealing to continuity and uniqueness of Stinespring representations (as in Proposition \[p\_lopopescu\]) one can build a channel $\Theta\in\operatorname{LOCC}({\mathcal{A}})$ for which $$\norm{\Theta_*({\omega}_\psi)-{\omega}_{\varphi}} < 2\sqrt{{\varepsilon}}.$$ Hence, $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$ and vice-versa. It is natural to ask if the statement of Theorem \[th\_maj\] holds in the case of general semi-finite von Neumann algebras. Such an extension would require a treatment of integral decompositions of normal completely positive maps, and is left for a further study. Here we only include an illustration involving a typical non-factor case. Let ${\mathcal{D}}$ be a maximal abelian selfadjoint algebra with separable predual, acting on a Hilbert space $H$. We may assume, without loss of generality, that $(X,\mu)$ is a probability measure space such that $H = L^2(X,\mu)$ and ${\mathcal{D}} = \{M_a : a\in L^{\infty}(X,\mu)\}$, where, for $a\in L^{\infty}(X,\mu)$, we have let $M_a\in {\mathcal{B}}(H)$ be the operator of multiplication by $a$. We equip ${\mathcal{D}}$ with the trace $\tau$ given by $\tau(M_a) = \int_{X} a\, d\mu$. Note that, since ${\mathcal{D}} = {\mathcal{D}}'$, we have $\operatorname{LOCC}^r({\mathcal{D}})=\operatorname{LOCC}^l({\mathcal{D}})=\operatorname{LOCC}({\mathcal{D}})$, and these sets consist of all unital positive Schur multipliers relative to $(X,\mu)$, that is, the maps $\Phi : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $$\label{eq_schur} \Phi(T) = \sum_{i=1}^{\infty} M_{a_i}^* T M_{a_i}, \ \ \ T\in {\mathcal{B}}(H),$$ where $a_i\in L^\infty(X,\mu)$, $i\in {\mathbb{N}}$ and $$\sum_{i=1}^{\infty} |a_i(s)|^2 = 1 \ \ \mbox{ for almost all } s\in X.$$ Let $\psi,\nph\in H$. We claim that the following are equivalent: - there exists $\Phi\in \operatorname{LOCC}({\mathcal{D}})$ such that $\Phi_*({\omega}_\psi) = {\omega}_\nph$; - $\psi$ is approximately convertible to $\nph$ via $\LOCC({{\mathcal{D}}})$; - $|\psi| = |\nph|$ almost everywhere. Indeed, the implication (i)$\Rightarrow$(ii) is trivial. Assuming (ii), fix $\epsilon > 0$ and let $\Phi\in \operatorname{LOCC}({\mathcal{D}})$ be such that $\|\omega_{\nph} - \omega_{\psi} \circ \Phi\| < \epsilon$. Writing $\Phi$ in the form , we have $$\begin{aligned} \sup_{\|c\|_{\infty} \leq 1} \left| \int_X c (|\nph|^2 - |\psi|^2) \,d\mu \right| & = & \sup_{\|c\|_{\infty} \leq 1} \left| \int_X c (|\nph|^2 - \left(\sum_{i=1}^{\infty}|a_i|^2\right) |\psi|^2) \,d\mu \right|\\ & = & \sup_{\|c\|_{\infty} \leq 1} |{\omega}_\nph(M_c) - {\omega}_\psi(\Phi(M_c))|\\ & \leq & \|{\omega}_\nph - {\omega}_\psi \circ \Phi\| < \epsilon.\end{aligned}$$ Thus, $\||\nph|^2 - |\psi|^2\|_1 < \epsilon$. Hence $|\nph|^2 = |\psi|^2$ in $L^1(X,\mu)$, and (iii) follows. Finally, assuming (iii), let $\theta : X\to{\mathbb{C}}$ be a unimodular function such that $\nph = \theta \psi$, and let $\Phi : {\mathcal{B}}(H) \to {\mathcal{B}}(H)$ be the map given by $\Phi(T) = M_{\theta}^* T M_{\theta}$. Then $\Phi\in \operatorname{LOCC}({\mathcal{D}})$ and $\Phi_*({\omega}_\psi) = {\omega}_\nph$. Trace Vectors and Entanglement in ${\ensuremath{\mathrm{II}_1}}$-factors {#s_ttf} ======================================================================== This section is dedicated to some examples and applications of our convertibility result from Section \[s:main\]. In its first part, we consider a generalisation of maximally entangled vectors to the commuting von Neumann algebra setting, while in its second part we show that entropy of states, relative to the trace, is an entanglement monotone in the sense of [@v]. Trace vectors ------------- Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a finite factor on a Hilbert space. A unit vector $\psi\in H$ is said to be a *trace vector* for ${\mathcal{A}}$ if ${\omega}_{\psi}|_{{\mathcal{A}}}=\tau$, the unique (normal) tracial state on ${\mathcal{A}}$. \[r\_1A\] Since $\omega_{\psi}|_{{{\mathcal{A}}}}(a)=\tau(\rho_\psi a)$ for $a\in {{\mathcal{A}}}$, we see that $\psi$ is a trace vector if and only if $\rho_\psi=1_{{\mathcal{A}}}$. It follows from Nielsen’s theorem [@nielsen] that the maximally entangled state $\psi=\frac{1}{\sqrt{n}}\sum_{i=1}^ne_n{\otimes}e_n\in{\mathbb{C}}^n{\otimes}{\mathbb{C}}^n$ is LOCC-convertible (that is, convertible via $\operatorname{LOCC}(M_n\otimes 1)$) to any other state ${\varphi}\in{\mathbb{C}}^n{\otimes}{\mathbb{C}}^n$. Notice that ${\omega}_\psi|_{M_n{\otimes}1}=\frac{1}{n}\operatorname{tr}$, the normalised trace on $M_n$. Hence, $\psi$ is a trace vector for $M_n{\otimes}1_n\subseteq {\mathcal{B}}({\mathbb{C}}^n{\otimes}{\mathbb{C}}^n)$. The next proposition shows that trace vectors play the role of maximally entangled states relative to ${\ensuremath{\mathrm{II}_1}}$-factors, and provides additional evidence for viewing maximal entanglement through the lens of tracial states [@keylsw §V.A]. \[II1\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a ${\ensuremath{\mathrm{II}_1}}$-factor on a separable Hilbert space $H$. If $\psi\in H$ is a trace vector for ${\mathcal{A}}$, then $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$ for any ${\varphi}\in H_1$. Conversely, if there exists a trace vector $\psi_0\in H$ for ${\mathcal{A}}$, and $\psi\in H_1$ is approximately convertible to any ${\varphi}\in H_1$ via $\operatorname{LOCC}({\mathcal{A}})$, then $\psi$ is a trace vector for ${\mathcal{A}}$. Suppose that $\psi$ is a trace vector for ${\mathcal{A}}$. By Remark \[r\_1A\], $\rho_\psi = 1_{{\mathcal{A}}}$. The map on ${{\mathcal{A}}}$, given by $a\mapsto \tau(a)1_{{\mathcal{A}}}$, is doubly stochastic (i.e., it is positive, normal, unital and trace-preserving) and its extension to $L^1({{\mathcal{A}}},\tau)$ maps $\rho_\phi$ to $1_{{\mathcal{A}}}=\rho_\psi$, since $\tau(\rho_\phi)=\omega_\phi(1_{{\mathcal{A}}})=(\phi,\phi)=1$. It follows from [@h Theorem 4.5] that $\rho_\psi\prec\rho_{\varphi}$, hence, $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$ by Theorem \[th\_maj\]. For the converse statement, suppose $\psi_0\in H_1$ is a trace vector for ${\mathcal{A}}$, and that $\psi\in H_1$ is approximately convertible to every ${\varphi}\in H_1$ via $\operatorname{LOCC}({\mathcal{A}})$. Then $\psi$ is approximately convertible to $\psi_0$ and vice-versa. By Theorem \[th\_maj\] and Remark \[r\_1A\], $\rho_\psi\prec 1_{{\mathcal{A}}}$ and $1_{{\mathcal{A}}} \prec\rho_\psi$, that is, $\rho_\psi$ and $1_{{\mathcal{A}}}$ are spectrally equivalent in the sense of [@h §3]. By [@h Theorem 3.4(2)], for every ${\varepsilon}>0$, there exists a unitary $u\in{\mathcal{A}}$ such that $$\norm{\rho_\psi - 1_{{\mathcal{A}}}}_1 = \norm{\rho_\psi-u\cdot\rho_{\psi_0}\cdot u^*}_1 < {\varepsilon}.$$ Since ${\varepsilon}> 0$ was arbitrary we have $\rho_\psi = 1_{{\mathcal{A}}}$; by Remark \[r\_1A\], $\psi$ is a trace vector for ${\mathcal{A}}$. Amongst ${\ensuremath{\mathrm{II}_1}}$-factors, the hyperfinite (i.e., approximately finite dimensional) ${\ensuremath{\mathrm{II}_1}}$-factor is best suited for applications in mathematical physics. In that context, it typically appears through an infinite tensor product construction, an algebra of canonical commutation/anti-commutation relations, or an irrational rotation algebra. We now present examples of maximally entangled states relative to the hyperfinite ${\ensuremath{\mathrm{II}_1}}$-factor in each of the three aforementioned manifestations. This example is based on [@kmsw §4.2]. Consider an infinite spin chain consisting of infinitely many qubits arranged on a one-dimensional lattice, say ${\mathbb{Z}}$. The underlying $C^*$-algebra of the system is given by the infinite tensor product $A=\bigotimes_{{\mathbb{Z}}} M_2$, that is, the inductive limit of the system $A_F=\bigotimes_{n\in F} M_2$, with canonical inclusion maps, where $F$ ranges through the finite subsets of ${\mathbb{Z}}$. For $n\in{\mathbb{Z}}$, let $\psi_n$ be the maximally entangled state on $A_{\{-n,n+1\}}$. Then ${\omega}=\bigotimes_{n\in{\mathbb{Z}}}\psi_n \psi_n^*$ defines a state on $A$. Let ${\mathcal{A}}=\pi_{\omega}(A_{(-\infty,0)})''\subseteq{\mathcal{B}}(H_{\omega})$ be the von Neumann algebra generated by the left half-chain in the cyclic GNS-representation $(H_{\omega},\pi_{\omega},\psi_{\omega})$ of ${\omega}$. Then ${\mathcal{B}}:={\mathcal{A}}'=\pi_{\omega}(A_{[0,\infty)})''$ is the von Neumann algebra generated by the right half-chain. Both ${\mathcal{A}}$ and ${\mathcal{B}}$ are ${\ensuremath{\mathrm{II}_1}}$-factors, and by construction it follows that $\psi_{\omega}$ is a trace vector for ${\mathcal{A}}$. Thus, by Proposition \[II1\], $\psi_{\omega}$ is approximately convertible to any state ${\varphi}\in H_{\omega}$ via $\operatorname{LOCC}({\mathcal{A}})$. Naturally, one may view $\psi_{\omega}$ as a state representing infinitely many pairs of entangled qubits. \[eg\_fock\] Let $K$ be a real Hilbert space and $H=K\oplus iK$ its complexification. Let ${\mathcal{F}}_a(H)$ denote the anti-symmetric Fock space over $H$, given by $${\mathcal{F}}_a(H)=\bigoplus_{n\geq 0}\wedge^n H,$$ where $\wedge^n H$ is the anti-symmetric subspace of $H^n:=\bigotimes_{k=1}^n H$ for $n\ge1$ and $\wedge^0:={\mathbb{C}}$. For $\psi\in H$, let $a(\psi)^*$ and $a(\psi)$ denote the Fock creation and annihilation operators, namely the bounded [@de] linear maps ${\mathcal{F}}_a(H)\to {\mathcal{F}}_a(H)$ given by $$a(\psi)^*{\varphi}=\sqrt{n+1}P_a^{n+1}(\psi{\otimes}{\varphi}), \ \ a(\psi){\varphi}=\sqrt{n}P_a^n(\psi^*{\otimes}\operatorname{id}){\varphi},$$ where $n\geq 1$, ${\varphi}\in\wedge^n H$ and $P_a^n:H^n\rightarrow\wedge^n H$ is the canonical projection. Let $S\in {{\mathcal{B}}}({\mathcal{F}}_a(H))$ denote the parity operator defined by $S=\bigoplus_{n\geq0}(-1)^{{\otimes}n}$. Letting $B(\psi):=a(\psi)^*+a(\psi)$ represent the corresponding (self-adjoint) Fermionic field operators, it follows that ${\mathcal{A}}:=\{B(\psi)\mid\psi\in K\}''$ is a ${\ensuremath{\mathrm{II}_1}}$-factor associated to a real-wave representation of the canonical anti-commutation relations [@dg §13] whose commutant satisfies ${\mathcal{B}}:={\mathcal{A}}'=\{S B(i\psi)\mid \psi\in K\}''$. It is known that the vacuum vector ${\Omega}=(1,0,0,\ldots)\in{\mathcal{F}}_a(H)$ is a quasi-free trace vector for ${\mathcal{A}}$, with $$\label{e:vac}(B(\psi)B({\varphi}){\Omega},{\Omega})=(\psi,{\varphi}), \ \ \ \psi,{\varphi}\in K.$$ More generally, given an anti-symmetric tensor $c\in H\wedge H$ (seen as a Hilbert-Schmidt operator from $\overline{H}$ to $H$), the Fermionic Gaussian vector associated with $c$ is given by $${\Omega}_c=\det(1+c^*c)^{-\frac{1}{4}}e^{-\frac{1}{2}a^*(c)}{\Omega},$$ where $\det(\cdot)$ is the Fredholm determinant and $a^*(c)$ is the two particle creation operator defined by $$a^*(c)\psi=\sqrt{(n+2)(n+1)}P_a^{n+2}(c{\otimes}\psi), \ \ \ \psi\in\wedge^n H, \ n\geq 1.$$ Such vectors occur in the Hartree–Fock–Bogoliubov method for approximating Fermionic systems (see e.g. [@dmm §4]), which is related to the Bardeen–Cooper–Schrieffer theory of superconductivity [@bcs]. For every $c$ there exists an orthogonal transformation $O_c$ on $K$, and a unitary $U_c$ on ${\mathcal{F}}_a(H)$ satisfying ${\Omega}_c=U_c^*{\Omega}$ and $U_cB(\psi)U_c^*=B(O_c\psi)$, $\psi\in K$ [@dg]. Thus, ${\omega}_{{\Omega}_c}|_{{\mathcal{A}}}={\omega}_{{\Omega}}\circ\operatorname{Ad}(U_c)$, and it follows from that ${\Omega}_c$ is also a trace vector for ${\mathcal{A}}$. Thus, by Proposition \[II1\], any of the Fermionic Gaussian vectors ${\Omega}_c$ may be converted into any Fock state ${\varphi}\in{\mathcal{F}}_a(H)$ by means of local operations and classical communication relative to the real-wave representation ${\mathcal{A}}$ of the CAR. In particular, the vectors $\Omega_c$ display properties of maximal entanglement relative to ${\mathcal{A}}$ and its commutant. We present one more instance of Proposition \[II1\], based on the example from [@bkk2 §7], which in turn was partly motivated by [@Faddeev]. This example is a particular realization of the irrational rotation algebra and is related to discretised CCR relations, whose relevance to numerical analysis of quantum systems was advocated by Arveson [@arv]. Suppose Alice and Bob have access to a quantum system represented by the Hilbert space $L^2({\mathbb{R}})$. Let $q$ and $p$ denote the self-adjoint operators corresponding to position and momentum: $$q\psi(x)=x\psi(x), \ \ \ p\psi(x)=i\frac{d}{dx}\psi(x),$$ where $\psi$ belongs to a common dense domain for $q$ and $p$. Suppose that Alice can measure periodic functions of position and momentum, with periods $t_q$ and $t_p$, respectively. Such functions are given (respectively) by integer powers of the unitary operators $$U:=e^{i\omega_q q} {{\quad\text{and}\quad}}V:=e^{i\omega_p p},$$ where, following [@bkk2], we let ${\omega}_q:=\frac{2\pi}{t_q}$ and ${\omega}_p:=\frac{2\pi}{t_p}$. The operators $U$ and $V$ satisfy $$UV=e^{2\pi i\theta}VU,$$ where $\theta:=\frac{{\omega}_q{\omega}_p}{2\pi}$. In what follows, we assume that ${\omega}_q{\omega}_p>4\pi$ and that $\theta$ is irrational. The algebra describing Alice’s measurement statistics is the von Neumann subalgebra ${\mathcal{A}}$ of ${\mathcal{B}}(L^2({\mathbb{R}}))$ generated by $U$ and $V$, and is known to be a type ${\ensuremath{\mathrm{II}_1}}$-factor. The $C^*$-algebra generated by $U$ and $V$ is known as the irrational rotation algebra corresponding to $\theta$. The von Neumann algebra describing Bob’s measurement statistics, ${\mathcal{B}} = {\mathcal{A}}'$, is generated by $$U':=e^{i\frac{{\omega}_q}{\theta}q}, {{\quad\text{and}\quad}}V':=e^{i\frac{{\omega}_p}{\theta}p},$$ and is also a type ${\ensuremath{\mathrm{II}_1}}$-factor. Let $\psi=\frac{1}{\sqrt{2t_q}}\chi_{[-t_q,t_q]}\in L^2({\mathbb{R}})$. If $x\in[-t_q,t_q]$ and $m\in{\mathbb{Z}}$ then, since ${\omega}_p>\frac{4\pi}{{\omega}_q}=2t_q$, it follows that $x+m{\omega}_p\in[-t_q,t_q]$ if and only if $m=0$. Hence, for all $n,m\in{\mathbb{Z}}$ we have $$\begin{aligned} (U^nV^m\psi,\psi)&=\frac{1}{2t_q}\int_{{\mathbb{R}}}e^{in\frac{2\pi}{t_q} x}\chi_{[-t_q,t_q]}(x+m{\omega}_p)\chi_{[-t_q,t_q]}(x) \ dx\\ &=\delta_{m,0}\frac{1}{2t_q}\int_{-t_q}^{t_q}e^{in\frac{2\pi}{t_q}x} \ dx\\ &=\delta_{m,0}\delta_{n,0}1.\end{aligned}$$ By [@d Corollary VI.1.2], ${\omega}_\psi|_{{\mathcal{A}}}$ is the unique normal tracial state $\tau$ on ${\mathcal{A}}$. By Proposition \[II1\], we have that $\psi$ is approximately convertible to any unit vector ${\varphi}\in L^2({\mathbb{R}})$ via $\operatorname{LOCC}({\mathcal{A}})$. One can think of $\psi$ as representing the state of a particle whose position is uniformly distributed over the interval $[-t_q,t_q]$. This uniformity is playing the role of maximal entanglement relative to the bipartitite system $({\mathcal{A}}, {\mathcal{B}})$. Entanglement monotones ---------------------- The practical importance of quantifying the degree of entanglement present in a given state cannot be overestimated. In the standard finite-dimensional tensor product framework, this quantification is studied through notions of entanglement measures. Since entanglement at its very core is a form of non-local quantum correlation, any reasonable entanglement measure ought to be monotonic with respect to local operations and classical communication. The term entanglement monotone has since emerged for such a measure, and it was argued by Vidal [@v] that monotonicity under LOCC is the *only* natural requirement for measures of entanglement. As an application of our main result, we show that the entropy of the singular value distribution satisfies this requirement for pure states relative to ${\ensuremath{\mathrm{II}_1}}$-factors, thus yielding an entanglement monotone. Let ${\mathcal{A}}$ be a ${\ensuremath{\mathrm{II}_1}}$-factor with unique tracial state $\tau$. Given a normal state $\rho\in{\mathcal{S}}({\mathcal{A}})$, we define the *entropy of $\rho$ relative to $\tau$* by $$H_\tau(\rho):=H(\mu(\rho))=-\int_0^1\mu_t(\rho)\log(\mu_t(\rho)) \ dt.$$ By splitting the entropy function $\eta(x)=-x\log(x)$ into $\chi_{[0,1]}\eta+\chi_{(1,\infty)}\eta$, and applying [@fk Remark 3.3] to the non-negative Borel functions $\chi_{[0,1]}\eta$ and $-\chi_{(1,\infty)}\eta$, it follows that $$H_\tau(\rho)=\tau(\eta(\rho))=-\tau(\rho\log(\rho))=-S(\rho,\tau),$$ whenever $|H_\tau(\rho)|<\infty$, where $S(\cdot,\cdot)$ is the relative entropy between normal states of ${\mathcal{A}}$ (see e.g. [@op §5]). As such, we see that $H_\tau(\rho)\leq 0$ and $H_\tau(\rho)=0$ if and only if $\rho=\tau$. In particular, if ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ and $\psi\in H_1$, then $H_\tau(\rho_\psi)=0$ if and only if $\psi$ is a trace vector for ${\mathcal{A}}$. \[p:monotone\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a ${\ensuremath{\mathrm{II}_1}}$-factor with tracial state $\tau$. The function $$H_\tau:{\mathcal{S}}({\mathcal{A}})\ni \rho\mapsto H_\tau(\rho)\in[-\infty,0]$$ is non-increasing under approximate convertibility by $\operatorname{LOCC}({\mathcal{A}})$, when restricted to states of the form $\rho_\psi$, $\psi\in H_1$. First note that for any state $\rho\in{\mathcal{S}}({\mathcal{A}})$, $$H_\tau(\rho)=-S(\rho,\tau)=-S(\mu(\rho),\chi_{[0,1]}),$$ where $S(\mu(\rho),\chi_{[0,1]})$ is the relative entropy of the density $\mu(\rho)\colon t\mapsto \mu_t(\rho)$ on $[0,1]$ with respect to the uniform distribution. Given $\psi,{\varphi}\in H_1$, if $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$ then by Theorem \[th\_maj\] we have $\rho_{\psi}\prec \rho_{{\varphi}}$, meaning that $\mu_t(\rho_\psi)\prec\mu_t(\rho_{\varphi})$ as probability densities on $[0,1]$. By [@veh Theorem 10], it follows that $$S(\mu_t(\rho_\psi),\chi_{[0,1]})\leq S(\mu_t(\rho_{\varphi}),\chi_{[0,1]}).$$ Hence, $H_\tau(\rho_\psi)\geq H_\tau(\rho_{\varphi})$. In the proof of Proposition \[p:monotone\] we could instead appeal to [@h Theorem 4.7(1)] for the connection between majorisation and double stochasticity together with monotonicity of the relative entropy [@op Theorem 5.3]. Recall that any density matrix $\rho\in M_n$ satisfies $$S(\rho)=-S(\rho,\tau)+\log(n),$$ where $S(\cdot)$ is the von Neumann entropy and $\tau=\frac{1}{n}\operatorname{tr}$ is the maximally mixed state (the $\log(n)$ factor would disappear if we used the unnormalised trace $\operatorname{tr}$). It is known that the restriction of any entanglement monotone to pure states $\psi\in {\mathbb{C}}^n{\otimes}{\mathbb{C}}^n$ is a concave function of the reduced density $\rho_\psi=(\operatorname{id}{\otimes}\operatorname{tr})(|\psi{\rangle}{\langle}\psi|)$ [@v Theorem 3]. The entanglement monotone $-S(\rho_\psi,\tau)$ is equivalent (up to the translational factor $\log(n)$) to the common choice of $S(\rho_\psi)$, and is the finite-dimensional analogue of our proposed monotone above. Note that $-S(\rho_\psi,\tau)\in[-\log(n),0]$, with the largest value of 0 occurring for maximally entangled $\psi$, and the lowest value of $-\log(n)$ occurring for separable $\psi$. Outlook ======= Several natural lines of investigation arise from this work. First, we intend to study the generalisation of our main result to the non-factor setting in connection with [@h2], as mentioned at the end of Section \[s:main\]. This could be useful for the study of entanglement in hybrid systems [@kuper; @DevShor; @bkk0; @bkk1]. Second, a rectangular version of our main theorem, describing convertibility between states $\psi\in H$ and ${\varphi}\in K$, with respect to distinct bipartite systems $({\mathcal{A}}_1,{\mathcal{B}}_1)$ in ${\mathcal{B}(H)}$ and $({\mathcal{A}}_2,{\mathcal{B}}_2)$ in ${\mathcal{B}}(K)$, would be desirable. Among other things, this could have applications to the structure of quantum correlation matrices and values of certain non-local games [@psstw]. We also plan to explore notions of distillability/dilution of entanglement for general bipartite systems in connection with this and previous work [@vw; @kmsw]. In that direction it would be interesting to explore uniqueness of entanglement monotones in the asymptotic regime for pure states, analogous to the finite-dimensional setting [@pr]. Finally, we observe that while many of the von Neumann algebras of interest in algebraic quantum field theory are of type III [@haag], and hence not semi-finite, it is conceivable that semi-finite von Neumann algebras may be relevant in the discretisation of space-time via tensor networks—a current area of research—analogous to those arising from discretisations of the canonical commutation relations [@slawny]. [*Acknowledgements.*]{} J.C. was partially supported by the NSERC Discovery Grant RGPIN-2017-06275. D.W.K. was partly supported by NSERC Discovery Grant 400160 and by a Royal Society grant allowing research visits to Queen’s University Belfast. R.H.L. was partly supported by UCD Seed Funding. [99]{} , [*Discretized CCR algebras*]{}, [J. Operator Thy. 26 (1991), no. 2, 225–239]{}. , [*Theory of superconductivity*]{}, [Phys. Rev. 108 (1957), 1175]{}. , [*Generalization of quantum error correction via the Heisenberg picture*]{}, [Phys. Rev. Lett. 98 (2007), 100502]{}. , [*Quantum error correction of observables*]{}, [Phys. Rev. A 76 (2007), 042303, 22 pp]{}. , [*Quantum error correction on infinite-dimensional Hilbert spaces*]{}, [J. Math. Phys. 50 (2009), no. 6, 062108, 24 pp]{}. , [*The smooth entropy formalism for von Neumann algebras*]{}, [J. Math. Phys. 57 (2016), no. 1, 015213, 25 pp]{}. , [*The dual of the Haagerup tensor product*]{}, . , [*The central Haagerup tensor product and maps between von Neumann algebras*]{}, [J. Funct. Anal. 112 (1993), no. 1, 97–120]{}. , [*Everything you always wanted to know about LOCC (but were afraid to ask)*]{}, . , [*Private algebras in quantum information and infinite-dimensional complementarity,*]{} . , [*$C^*$-algebras by Example*]{}, [Fields Institute Monographs, 1996]{}. , [*Introduction to representations of the canonical commutation and anticommutation relations*]{}, in [*Large Coulomb Systems: Lecture Notes on Mathematical Aspects of QED*]{}, [Springer Berlin Heidelberg, 2006. pp 63–143]{}. , [*Mathematics of quantization and quantum fields*]{}, [Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2013]{}. , [*On the energy-momentum spectrum of a homogeneous Fermi gas*]{}, [Ann. Henri Poincaré 14 (2013), no. 1, 1–36]{}. , [*The capacity of a quantum channel for simultaneous transmission of classical and quantum information*]{}, [Comm. Math. Phys. 256 (2005), no. 2, 287–303]{}. , [*Non-closure of the set of quantum correlations via graphs*]{}, [Comm. Math. Phys., to appear]{}. , [*Generalised $s$-numbers of $\tau$-measurable operators*]{}, [Pacific J. Math. 123 (1986), no. 2, 269–300]{}. , [*Discrete Heisenberg-Weyl group and modular group*]{}, [Lett. Math. Phys. 34 (1995), 249]{}. , [*Local quantum physics. Fields, particles, algebras*]{}, [Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992]{}. , [*Decomposition of completely bounded maps on operator algebras*]{}, [unpublished manuscript]{}. , [*The standard form of von Neumann algebras*]{}. [Math. Scand. 37 (1975), 271–283]{}. , [*As asymptotic property of factorizable completely positive maps and the Connes embedding problem*]{}, [Comm. Math. Phys. 338 (2015), 721–752]{}. , [*Majorization and stochastic maps in von Neumann algebras*]{}, [J. Math. Anal. Appl. 127 (1987), 18–48]{}. , [*Spectral relations and unitary mixing in semifinite von Neumann algebras*]{}, [Hokkaido Math. J. 17 (1988), no. 1, 117–137]{}. , [*Quantum $f$-divergences in von Neumann algebras I. Standard $f$-divergences*]{}, [arXiv:1805.02050]{}. , [*Quantum $f$-divergences in von Neumann algebras II. Maximal $f$-divergences*]{}, [arXiv:1807.03118]{}. , [*Entanglement measures and their properties in quantum field theory*]{}, [arXiv:1702.04924]{} , [*Large violation of Bell inequalities with low entanglement*]{}, [Comm. Math. Phys. 306 (2011), 695–746]{}. , [*Infinitely entangled states*]{}, [Quantum Inf. Comput. 3 (2003), no. 4, 281–306]{}. , [*Entanglement Haag-duality and type properties of infinite quantum spin chains*]{}, [Rev. Math. Phys. 18 (2006), no. 9, 935–970]{}. , [*Fundamentals of the theory of operator algebras, Volume II: Advanced theory*]{}, [AMS Grad. Studies in Math. vol. 16, 1997]{}. , [*A continuity theorem for Stinespring’s dilation*]{}, [J. Funct. Anal. 255 (2008), 1889–1904]{}. , [*The capacity of hybrid quantum memory*]{}, [IEEE Trans. Inf. Thy. 49 (2003), no. 6, 1465–1473]{}. , [*Concentrating entanglement by local actions: Beyond mean values*]{}, [Phys. Rev. A 63 (2001), no. 2, 022301]{}. , [*On Landauer’s principle and bound for infinite systems*]{}, [Comm. Math. Phys. 363 (2018), no. 2, 531–560]{}. , [*Relative entropy in CFT*]{}, [Adv. Math. 337 (2018), 139–170]{}. , [*Conditions for a class of entanglement transformations*]{}, [Phys. Rev. Lett. 83 (1999), 436]{}. , [*Quantum entropy and its use*]{}, [Texts and Monographs in Physics, Springer-Verlag, Berlin-Heidelberg 1993]{}. , [*${\varepsilon}$-convertibility of entangled states and extension of Schmidt rank in infinite-dimensional systems*]{}, [Quantum Inf. & Comp. 8 (2008), no. 1 & 2, 30–52]{}. , [*Completely bounded maps and operator algebras*]{}, [Cambridge University Press, 2002]{}. , [*Estimating quantum chromatic numbers*]{}, [J. Funct. Anal. 270 (2016), no. 6, 2188–2222]{}. , [*Thermodynamics and the measure of entanglement*]{}, [Phys. Rev. A (3) 56 (1997), no. 5, R3319–R3321]{}. , [*On factor representations and the $C^*$-algebra of canonical commutation relations*]{}, [Comm. Math. Phys. 24 (1972), 151–170]{}. , [*The set of quantum correlations is not closed*]{}, [preprint (2017), arXiv:1703.08618]{}. , [*Theory of operator algebras I*]{}, [Encyclopedia of Mathematical Sciences 124, Springer-Verlag Berlin-Heidelberg-New York, 2002]{}. , [*Theory of operator algebras II*]{}, [Encyclopedia of Mathematical Sciences 125, Springer-Verlag Berlin-Heidelberg-New York, 2003]{}. , [*Rényi Divergence and majorization*]{}, [2010 IEEE International Symposium on Information Theory, Conference Proceedings DOI:10.1109/ISIT.2010.5513784]{}. , [*Distillability and positivity of partial transposes in general quantum field systems*]{}, [Rev. Math. Phys. 17 (2005), no. 5, 545–576]{}. , [*Entanglement monotones*]{}, [J. Modern Opt. 47 (2000), no. 2–3, 355–376]{}. , [*Mathematical foundations of quantum mechanics*]{}, [Princeton University Press, 1955]{}.

--- address: - 'L. Ornea, Universitatea din Bucureşti, Facultatea de Matematică, Str. Academiei nr. 14, 70109, Bucureşti, România, *also,* Institutul de Matematică “Simion Stoilow” al Academiei Române, C.P. 1-764, 014700, Bucureşti, România' - 'R. Pantilie, Institutul de Matematică “Simion Stoilow” al Academiei Române, C.P. 1-764, 014700, Bucureşti, România' author: - Liviu Ornea and Radu Pantilie title: | holomorphic maps between generalized\ complex manifolds --- [^1] \[section\] \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Definition]{} \[thm\][Remark]{} \[thm\][Example]{} Abstract {#abstract .unnumbered} ======== > [We introduce a natural notion of *holomorphic map* between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kähler manifolds.]{} Introduction {#introduction .unnumbered} ============ The *generalized complex structures* [@Gua-thesis],[@Hit-gc_QJM] contain, as particular cases, the complex and symplectic structures. Although for the latter structures there exist well known definitions which give the corresponding morphisms (holomorphic maps and Poisson morphisms, respectively), it still lacks a suitable notion of *holomorphic map* with respect to which the class of generalized complex manifolds to become a category.\ In this paper we introduce such a notion (Definition \[defn:ogc\], below) based on the following considerations. Firstly, holomorphic maps between generalized complex manifolds should be invariant under *$B$-field transformations*. This is imposed by the fact that the group of (orthogonal) automorphisms of the Courant bracket (which defines the integrability in Generalized Complex Geometry) on a manifold is the semidirect product of the group of diffeomorphisms and the additive group of closed two-forms on the manifold [@Gua-thesis]. Secondly, by [@Gua-thesis], underlying any linear generalized complex structure there are:\ $\bullet$ a linear Poisson structure (that is, a constant Poisson structure on the vector space; see Section \[section:lDs\]), and\ $\bullet$ a linear co-CR structure (that is, a linear CR structure on the dual vector space; see Section \[section:gclm\]),\ both of which are preserved under linear $B$-field transformations. Moreover, these two structures determine, up to a (non-unique) linear $B$-field transformation, the given generalized linear complex structure; furthermore, if we choose a compatible linear $f$-structure (Definition \[defn:compatible\_f\_Poisson\]) then there exists a distinguished linear $B$-field transformation with this property (see ). It follows that a linear map is *generalized complex* (Definition \[defn:gcl\]) if and only if, up to linear $B$-fields transformations, it is an $f$-linear Poisson morphism between linear generalized complex structures in normal form (Proposition \[prop:gcl\]).\ A *holomorphic map* between generalized (almost) complex manifolds is a map whose differential is generalized complex (Definition \[defn:ogc\]). Then, essentially, all of the above mentioned (linear) facts hold, locally, in the setting of generalized complex manifolds (Theorem \[thm:local\_gcs\] and Proposition \[prop:ogc\_basic\]).\ The first examples are the classical holomorphic maps, the Poisson morphisms between symplectic manifolds and their products (Example \[exm:ogc\_first\]).\ Other large classes of natural examples can be obtained by working with compact Lie groups (Examples \[exm:ogc\_second\] and \[exm:ogc\_third\]).\ Further motivation for our notion of holomorphicity comes from generalized Kähler geometry. For example, if $(g,b,J_+,J_-)$ is the bi-Hermitian structure corresponding to a generalized Kähler manifold $(M,L_1,L_2)$ then the holomorphic functions of $(M,L_1)$ and $(M,L_2)$ are the bi-holomorphic functions of $(M,J_+,J_-)$ and $(M,J_+,-J_-)$, respectively (Remark \[rem:assocF\_holo-functions\]). Other natural properties of the holomorphic maps between generalized Kähler manifolds are obtained in Sections \[section:gKm\] and \[section:gK\_tamed\] (Remark \[rem:Riem\_subm\_descend\](2) and Corollaries \[cor:holo\_diffeo\], \[cor:Poisson\_holo\_map\]).\ Along the way, we obtain results on generalized Kähler manifolds, such as the factorisation result Theorem \[thm:H+-int\]; see, also, Corollaries \[cor:H+-\], \[cor:VH+\] and \[cor:L2\_normal\], the first of which is a significant improvement of [@ApoGua Theorem A].\ The paper is organized as follows. In Section \[section:lDs\], after recalling [@Courant] some basic facts on linear Dirac structures, we give explicit descriptions (Proposition \[prop:pfpb\]) for the pull-back and push-forward of a linear Dirac structure, which we then use to show that any linear Dirac structure is, in a natural way, the pull-back of a linear Poisson structure (Corollary \[cor:pbP\]; cf. [@BurRad],[@BurWei-2005]), which we call the *canonical (linear) Poisson quotient* (cf. [@BurWei-2005]), of the given linear Dirac structure. The smooth version (Theorem \[thm:pbP\]; cf. [@Courant],[@BurRad],[@BurWei-2005]) of this result is proved in Section \[section:Ds\] together with some other results on Dirac structures. For example, there we show (Corollary \[cor:blank\_up\_to\_B\]) that, locally, any regular Dirac structure is, up to a $B$-field transformation, of the form ${\mathscr{V}}\oplus{\rm Ann}({\mathscr{V}})$, where ${\mathscr{V}}$ is (the tangent bundle of) a foliation.\ In Section \[section:gclm\], we introduce the notion of generalized complex linear map, along the above mentioned lines. It follows that two generalized linear complex structures $L_1$ and $L_2$, on a vector space $V$, can be identified if and only if $L_2$ is the linear $B$-field transform of the push-forward of $L_1$, through a linear isomorphism of $V$ (Corollary \[cor:gcl\]). Also, we explain (Remark \[rem:nonBinv\]) why another definition of the notion of generalized complex linear map is, in our opinion, inadequate.\ In Section \[section:h\_gc\], we review some basic facts on generalized complex manifolds and we introduce the corresponding notion of holomorphic map. It follows that if a real analytic map $\phi$, between real analytic regular generalized complex manifolds, is holomorphic then, locally, up to the complexification of a real analytic $B$-field tranformation, the complexification of $\phi$ descends to a complex analytic Poisson morphism between canonical Poisson quotients (Proposition \[prop:holo\_Dirac\]). Also, we show that the pseudo-horizontally conformal submersions with minimal two-dimensional fibres, from Riemannian manifolds, provide natural constructions of generalized complex structures (Example \[exm:gc\_harmorphs\]).\ In Section \[section:gKm\], we prove (Theorem \[thm:H+geod\]) that if $(g,b,J_+,J_-)$ is the bi-Hermitian structure corresponding to a generalized Kähler structure and we denote ${\mathscr{H}}^{\pm}={\rm ker}(J_+\mp J_-)$ then the following conditions are equivalent:\ $\bullet$ ${\mathscr{H}}^{\pm}$ integrable;\ $\bullet$ ${\mathscr{H}}^{\pm}$ geodesic;\ $\bullet$ ${\mathscr{H}}^{\pm}$ holomorphic, with respect to $J_+$ or $J_-$.\ It follows that, under natural conditions, the holomorphic maps between generalized Kähler manifolds descend to holomorphic maps between Kähler manifolds (Remark \[rem:Riem\_subm\_descend\]). Also, we classify the generalized Kähler manifolds $M$ for which $TM={\mathscr{H}}^+\oplus{\mathscr{H}}^-$ (Corollary \[cor:H+-\]).\ In Section \[section:gK\_tamed\], we describe, in terms of *tamed symplectic manifolds* (see Definition \[defn:tamed\_symp\]) the generalized Kähler manifolds for which either ${\mathscr{H}}_+$ or ${\mathscr{H}}_-$ is zero; the obtained result (Theorem \[thm:gK\_tamed\]) also appears, in a different form, in [@Gua-Pbranes]. Also, in Corollary \[cor:VH+\], we prove a factorisation result for generalized Kähler manifolds with ${\mathscr{H}}^+$ an integrable distribution and ${\mathscr{H}}^-=0$ (or ${\mathscr{H}}^+=0$ and ${\mathscr{H}}^-$ an integrable distribution); see, also, Corollary \[cor:L2\_normal\] for a similar result and Theorem \[thm:H+-int\] for a generalization.\ Furthermore, we explain how the associated holomorphic Poisson structures of [@Hit-gc_CMP] fit into our approach (Theorem \[thm:holo\_Poisson\_tamed\], Remark \[rem:holo\_Poisson\_tamed\]), we deduce some consequences for holomorphic diffeomorphisms (Corollary \[cor:holo\_diffeo\]), and we show that, under natural conditions, the holomorphic maps between generalized Kähler manifolds are holomorphic Poisson morphisms (Corollary \[cor:Poisson\_holo\_map\]). Linear Dirac structures {#section:lDs} ======================= In this section we recall ([@Courant]; see [@BurRad],[@BurWei-2005],[@Gua-thesis]) some basic facts on linear Dirac structures.\ Let $V$ be a (real or complex, finite dimensional) vector space. The symmetric bilinear form $<\cdot,\cdot>\,$ on $V\oplus V^*$ defined by $$<u+\a,v+\b>=\tfrac12\bigl(\a(v)+\b(u)\bigr)\;,$$ for any $u+\a\,,\,v+\b\in V\oplus V^*$, corresponds, up to the factor $\tfrac12$, to the canonical isomorphism $V\oplus V^*{\overset{\sim}{\setbox0=\hbox{$\longrightarrow$}\ht0=0.2\ht0\box0}}\bigl(V\oplus V^*\bigr)^*$ defined by $u+\a\longmapsto\a+u$, for any $u+\a\in V\oplus V^*$. In particular, $<\cdot,\cdot>$ is nondegenerate and, if $V$ is real, its index is $\dim V$. Thus, the dimension of the maximal isotropic subspaces of $V\oplus V^*$ (endowed with $<\cdot,\cdot>$) is equal to $\dim V$. A *linear Dirac structure* on $V$ is a maximal isotropic subspace of $V\oplus V^*$. If $b$ is a bilinear form on $V$ then we shall denote by the same letter the corresponding linear map from $V$ to $V^*$; thus, $b(u)(v)=b(u,v)$, for any $u\,,v\in V$.\ Let $E\subseteq V$ be a vector subspace and let $\ep\in\Lambda^2E^*$; denote $$L(E,\ep)=\bigl\{\,u+\a\,\big|\,u\in E\,,\,\a|_E=\ep(u)\,\bigr\}\;.$$ From the fact that $\ep$ is skew-symmetric it follows easily that $L(E,\ep)$ is isotropic. Also, $L(E,0)=E\oplus{\rm Ann}(E)$, where ${\rm Ann}(E)=\bigl\{\a\in V^*\big|\,\a|_E=0\,\bigr\}$.\ \ We shall denote by $\p$ and $^{*\!}\p$ the projections from $V\oplus V^*$ onto $V$ and $V^*$, respectively. Also, if $L\subseteq V\oplus V^*$ then $L^{\perp}$ denotes the ‘orthogonal complement’ of $L$ with respect to $<\cdot,\cdot>$. \[prop:L\] Let $L$ be an isotropic subspace of $V\oplus V^*$ and let $E=\p(L)$.\ Then there exists a unique $\ep\in\Lambda^2E^*$ such that $L\subseteq L(E,\ep)$. In particular, if $L$ is a linear Dirac structure then $L=L(E,\ep)$. Furthermore, $V\cap L={\rm ker}\,\ep$ and $^{*\!}\p(L)={\rm Ann}(V\cap L)$. Let $L$ be a linear Dirac structure on $V$. If $^{*\!}\p(L)=V^*$ then $L$ is called a *linear Poisson structure* (see [@Courant]). By Proposition \[prop:L\], if $L$ is a linear Poisson structure then $L=L(V^*,\eta)$ for some *bivector* $\eta\in\Lambda^2V$ (cf. [@Wei-local_P]).\ Let $V$ and $W$ be vector spaces endowed with linear Dirac structures $L_V$ and $L_W$, respectively, and let $f:V\to W$ be a linear map. Denote $$\begin{split} f_*(L_V)&=\bigl\{f(X)+\e\,|\,X+f^*(\e)\in L_V\bigr\}\;,\\ f^*(L_W)&=\bigl\{X+f^*(\e)\,|\,f(X)+\e\in L_W\bigr\}\;. \end{split}$$ \[prop:pfpb\] Let $f:V\to W$ be a linear map. Let $L(E,\ep)$ and $L(F,\e)$ be linear Dirac structures on $V$ and $W$, respectively. Then $$\begin{split} f_*\bigl(L(E,\ep)\bigr)&=L\bigl(f\bigl((E\cap\ker\!f)^{\perp_{\ep}}\bigr),\check{\ep}\,\bigr)\;,\\ f^*\bigl(L(F,\e)\bigr)&=L\bigl(f^{-1}(F),f^*(\e)\bigr)\;, \end{split}$$ where $\check{\ep}$ is characterised by $f^*(\check{\ep})=\ep$ on $(E\cap\ker\!f )^{\perp_{\ep}}$. It is easy to prove that $f_*(L_V)$ and $f^*(L_W)$ are isotropic subspaces of $W\oplus W^*$ and $V\oplus V^*$, respectively.\ Next, we show that there exists a unique two-form $\check{\ep}$ on $f\bigl((E\cap\ker\!f )^{\perp_{\ep}}\bigr)$ such that $f^*(\check{\ep})=\ep$ on $(E\cap\ker\!f )^{\perp_{\ep}}$. For this, it is sufficient to prove that if $X_1,X_2\in (E\cap\ker\!f )^{\perp_{\ep}}$ are such that $f(X_1)=f(X_2)$ then $\ep(X_1,Y)=\ep(X_2,Y)$, for any $Y\in(E\cap\ker\!f )^{\perp_{\ep}}$. Now, if $X_1,X_2\in (E\cap\ker\!f )^{\perp_{\ep}}$, then $X_1,X_2\in E$ and, as $X_1-X_2\in\ker\!f $, we have $\ep(X_1-X_2,Y)=0$, for any $Y\in(E\cap\ker\!f )^{\perp_{\ep}}$.\ Thus, to complete the proof it is sufficient to show that $$\label{e:pfpb1} \begin{split} f_*\bigl(L(E,\ep)\bigr)&\supseteq L\bigl(f\bigl((E\cap\ker\!f )^{\perp_{\ep}}\bigr),\check{\ep}\,\bigr)\;,\\ f^*\bigl(L(F,\e)\bigr)&\supseteq L\bigl(f^{-1}(F),f^*(\e)\bigr)\;. \end{split}$$ Let $Y+\xi\in L\bigl(f\bigl((E\cap\ker\!f )^{\perp_{\ep}}\bigr),\check{\ep}\,\bigr)$; equivalently, there exists $X\in(E\cap\ker\!f )^{\perp_{\ep}}$ such that $f(X)=Y$ and $\xi(f(X'))=\ep(f(X),f(X'))$, for any $X'\in(E\cap\ker\!f )^{\perp_{\ep}}$.\ We claim that $Y+\xi\in f_*\bigl(L(E,\ep)\bigr)$; equivalently, there exists $X\in(E\cap\ker\!f )^{\perp_{\ep}}$ such that $f(X)=Y$ and $\xi(f(X'))=\ep(X,X')$, for any $X'\in E$.\ It is easy to prove that, if $X\in(E\cap\ker\!f )^{\perp_{\ep}}$ is such that $f(X)=Y$, then $\xi(f(X'))=\ep(X,X')$, for any $X'\in(E\cap\ker\!f )\cup(E\cap\ker\!f )^{\perp_{\ep}}$.\ It follows that, for any $X\in(E\cap\ker\!f )^{\perp_{\ep}}$ with $f(X)=Y$, there exists $X_1\in\ker(\ep|_{E\cap\ker\!f })$ such that $\xi(f(X'))=\ep(X+X_1,X')$, for any $X'\in E$; as, then, we also have $X_1\in(E\cap\ker\!f )^{\perp_{\ep}}$ and $f(X_1)=0$, this shows that $Y+\xi=f(X+X_1)+\xi\in f_*(L_V)$.\ To prove the second relation of , let $X+\xi\in L\bigl(f^{-1}(F),f^*(\e)\bigr)$; equivalently, $f(X)\in F$ and $\xi(X')=\e(f(X),f(X'))$ for any $X'\in f^{-1}(F)$. As $f^{-1}(F)\supseteq\ker\!f $, there exists $\check{\xi}$ in the dual of $f(V)$ such that $\xi=f^*(\check{\xi})$. Obviously, we can extend $\check{\xi}$ to an one-form on $W$, which we shall denote by the same symbol $\check{\xi}$, such that $\check{\xi}(Y)=\e(f(X),Y)$, for any $Y\in F$; equivalently, $f(X)+\check{\xi}\in L(F,\e)$. Therefore $X+\xi=X+f^*(\check{\xi})\in f^*\bigl(L(F,\e)\bigr)$.\ The proof is complete. Let $V$ and $W$ be vector spaces endowed with linear Dirac structures $L_V$ and $L_W$, respectively, and let $f:V\to W$ be a linear map.\ Then $f_*(L_V)$ and $f^*(L_W)$ are called the *push forward *and* pull back, by $f$, of $L_V$ *and* $L_W$*, respectively. Note that, if $f:(V,L_V)\to(W,L_W)$ is a linear map between vector spaces endowed with linear Poisson structures then the following assertions are equivalent (see [@BurRad],[@BurWei-2005]):\ (i) $f$ is a *linear Poisson morphism* (that is, $f(\eta_{\,V})=\eta_{\,W}$, where $\eta_{\,V}$ and $\eta_{\,W}$ are the bivectors defining $L_V$ and $L_W$, respectively; see [@Vai-Poisson_book]).\ (ii) $f_*(L_V)=L_W$.\ From Proposition \[prop:pfpb\], we easily obtain the following result. \[cor:pbP\] Let $V$ be a vector space endowed with a linear Dirac structure $L=L(E,\ep)$. Let $W=\ker\ep$ and denote by $\phi:V\to V/W$ the projection.\ Then $L=\phi^*(\phi_*(L))$ and $\phi_*(L)$ is a linear Poisson structure on $V/W$. Dirac structures {#section:Ds} ================ In this section, we shall work in the smooth and (real or complex) analytic categories. All the notations of Section \[section:lDs\] will be applied to tangent bundles of manifolds and to (differentials of) maps between manifolds. An *almost Dirac structure* on a manifold $M$ is a maximal isotropic subbundle of $TM\oplus T^*M$.\ An almost Dirac structure is *integrable* if it’s space of sections is closed under the *Courant bracket* defined by $$[X+\a,Y+\b]=[X,Y]+\tfrac12\operatorname{d}\bigl(\iota_X\b-\iota_Y\a\bigr)+\iota_{X\!}\operatorname{d}\!\b-\iota_{Y\!}\operatorname{d}\!\a\,,$$ for any sections $X+\a$ and $Y+\b$ of $TM\oplus\Lambda(T^*M)$, where $\iota$ denotes the interior product.\ A *Dirac structure* is an integrable almost Dirac structure. Let $L$ be a Dirac structure on $M$. If $\p(L)=TM$ then $L$ is a *presymplectic structure* whilst if $^{*\!}\p(L)=T^*M$ then $L$ is a *Poisson structure* [@Courant] (cf. [@Wei-local_P]).\ Recall [@Courant §4] that a point of a manifold endowed with an almost Dirac structure $L$ is called *regular* if, in some open neighbourhood of it, $\p(L)$ and $^{*\!}\p(L)$ are bundles.\ The following result follows from the fact that it is sufficient to be proved for maps of constant rank between manifolds endowed with regular almost Dirac structures. \[prop:funct\_integr\] Let $M$ and $N$ be manifolds endowed with the almost Dirac structures $L_M$ and $L_N$, respectively. Let $\phi:M\to N$ be a map which maps regular points of $L_M$ to regular points of $L_N$.\ If $L_M$ is integrable, $\phi_*(L_M)=L_N$ and $\phi$ is surjective then $L_N$ is integrable.\ If $L_N$ is integrable and $\phi^*(L_N)=L_M$ then $L_M$ is integrable. Next, we prove the following result. \[thm:pbP\] Let $L$ be a Dirac structure on $M$ such that $^{*\!}\p(L)$ is a subbundle of $T^*M$. Then, locally, there exist submersions $\phi$ on $M$ such that $\phi_*(L)$ is a Poisson structure and $L=\phi^*(\phi_*(L))$; moreover, these submersions are (germ) unique, up to Poisson diffeomorphisms of their codomains. By hypothesis, $TM\cap L$ is a subbundle of $TM$. Furthermore, as $L$ is integrable, $TM\cap L$ is (the tangent bundle to) a foliation.\ Let $F={}^{*\!}\p(L)$ and let $\eta$ be the section of $\Lambda^2F^*$ such that $L=L(F,\eta)$. Note that, $F\bigl(={\rm Ann}(TM\cap L)\bigr)$ is locally spanned by the differentials of functions which are basic with respect to $TM\cap L$.\ Let $f$ and $g$ be functions, locally defined on $M$, such that $\operatorname{d}\!f$ and $\operatorname{d}\!g$ are sections of $F$. Then there exists vector fields $X$ and $Y$, locally defined on $M$, such that $X+\operatorname{d}\!f$ and $Y+\operatorname{d}\!g$ are local sections of $L$; in particular, we have $\eta(\operatorname{d}\!f,\operatorname{d}\!g)=X(g)=-Y(f)$. Hence $[X+\operatorname{d}\!f,Y+\operatorname{d}\!g]=[X,Y]+\operatorname{d}\bigl(\eta(\operatorname{d}\!f,\operatorname{d}\!g)\bigr)$ and we deduce that $\eta(\operatorname{d}\!f,\operatorname{d}\!g)$ is basic with respect to $TM\cap L$.\ The proof follows quickly from Corollary \[cor:pbP\] and Proposition \[prop:funct\_integr\]. Under the same hypotheses, as in Theorem \[thm:pbP\], we call $\phi_*(L)$ the *canonical (local) Poisson quotient* of $L$.\ Next, we prove the following (cf. [@Courant Proposition 4.1.2]). \[prop:pf\_to\_presym\] Let $L=L(E,\ep)$ be a Dirac structure on $M$ and let $x\in M$ be a regular point of $L$; denote by $P$ the leaf of $E$ through $x$.\ Then for any submanifold $Q$ of $M$ transversal to $E$, such that $x\in Q$ and $\dim Q=\dim M-\dim P$, there exists a submersion $\r$ from some open neighbourhood $U$ of $x$ in $M$ onto some open neighbourhood $V$ of $x$ in $P$ such that $\r_*(L|_U)=L(TV,\ep|_V)$ and the fibre of $\r$ through $x$ is an open set of $Q$. From Theorem \[thm:pbP\] it follows that we may assume $L$ a Poisson structure.\ If we ignore the fact that the fibre of $\r$ through $x$ is fixed then the proposition is a consequence of [@Wei-local_P Corollary 2.3] and Proposition \[prop:pfpb\]. To complete the proof just note that in the proof of [@Wei-local_P Theorem 2.1] (and, consequently, of [@Wei-local_P Corollary 2.3], as well), at each step, the two functions involved may be assumed constant along $Q$. Recall (see [@Gua-thesis],[@BurRad]) that any closed two-form $B$ on $M$ corresponds to a *$B$-field transformation* which is the automorphism of $TM\oplus T^*M$, preserving the Courant bracket, defined by $${\rm exp}(B)(X+\a)=X+B(X)+\a\;$$ for any $X+\a\in TM\oplus T^*M$, where, as before, we have identified $B$ with the corresponding section of ${\rm Hom}(TM,T^*M)$. It is easy to prove that if $L=L(E,\ep)$ is an almost Dirac structure on $M$ then ${\rm exp}(B)(L)=L(E,\ep+B|_E)$. \[cor:blank\_up\_to\_B\] Let $L$ be a regular Dirac structure on $M$; denote $E=\p(L)$. Then, locally, there exist two-forms $B$ on $M$ such that ${\rm exp}(B)(L)=E\oplus{\rm Ann}E$. By Proposition \[prop:pf\_to\_presym\], locally, there exist submersions $\r:M\to P$ onto presymplectic manifolds $\bigl(P,L(TP,\o)\bigr)$ such that $\r_*(L)=L(TP,\o)$.\ Then $B=-\r^*(\o)$ is as required. We end this section with the following result which will be used later on. \[prop:Poisson\_OK\] Let $\phi:(M,L_M)\to(N,L_N)$ be a Poisson morphism, of constant rank, between regular Poisson manifolds such that $\operatorname{d}\!\phi(E_M)\subseteq E_N$, where $E_M$ and $E_N$ are the (symplectic) foliations determined by $L_M$ and $L_N$, respectively.\ Then, locally, there exist submersions $\r:M\to(P,\o)$ and $\s:N\to(Q,\e)$ onto symplectic manifolds, and a Poisson morphism $\psi:(P,\o)\to(Q,\e)$ such that:\ $TM=E_M\oplus{\rm ker}\operatorname{d}\!\r$ and $\r_*(L_M)=L(TP,\o)$;\ $TN=E_N\oplus{\rm ker}\operatorname{d}\!\s$ and $\s_*(L_N)=L(TQ,\e)$;\ $\s\circ\phi=\psi\circ\r$. From Proposition \[prop:pfpb\] we obtain that $\operatorname{d}\!\phi(E_M)=E_N$. As, locally, $\phi$ is the composition of a submersion followed by an immersion, it follows that we may assume that $\phi$ is a surjective submersion.\ By Proposition \[prop:pf\_to\_presym\], locally, there exists a submersion $\s:M\to(Q,\e)$ onto a symplectic manifold such that assertion (ii) is satisfied.\ Let ${\mathscr{V}}$ be the distribution on $M$ generated by all of the Hamiltonian vector fields determined by $u\circ\s\circ\phi$, with $u$ a function on $Q$; obviously, ${\mathscr{V}}\subseteq E_M$. Then arguments similar to the inductive step of the proof of [@Wei-local_P Theorem 2.1] show that:\ (a) ${\mathscr{V}}$ is a foliation mapped by $\s\circ\phi$ onto $TQ$;\ (b) ${\mathscr{V}}$ and $E_M\cap{\rm ker}\operatorname{d}\!\phi$ are nondegenerate and complementary orthogonal with respect to the symplectic structure $\o_M$ of $E_M$;\ (c) $\o_M$ restricted to ${\mathscr{V}}$ is projectable (onto $\e$) with respect to $\s\circ\phi$;\ (d) $\o_M$ restricted to $E_M\cap{\rm ker}\operatorname{d}\!\phi$ is projectable with respect to ${\mathscr{V}}$.\ Consequently, $(E_M,\o_M)$ induces on any fibre $M'$ of $\s\circ\phi$ a Poisson structure $L'$ such that, locally, $(M,L_M)$ is the product of $(M',L')$ and $\bigl(Q,L(TQ,\e)\bigr)$.\ By Proposition \[prop:pf\_to\_presym\], locally, there exists a submersion $\r':M'\to(P',\o')$ such that ${\rm ker}\operatorname{d}\!\r'\oplus(E_M\cap TM')=TM'$ and $\r'_*(L')=L(TP',\o')$.\ If we define $(P,\o)=(P',\o')\times(Q,\e)$, $\r=\r'\times\s$ and $\psi:P\to Q$ the projection then it is easy to see that $\r$, $\s$ and $\psi$ are as required. Generalized complex linear maps {#section:gclm} =============================== Let $V$ be a (real) vector space. A *linear CR structure* on $V$ is a complex vector subspace $C$ of $V^{\C}$ such that $C\cap\overline{C}=\{0\}$.\ Dually, a *linear co-CR structure* on $V$ is a complex vector subspace $C$ of $V^{\C}$ such that $C+\overline{C}=V^{\C}$.\ A complex vector subspace of $V^{\C}$ is a linear co-CR structure if and only if its annihilator is a linear CR structure.\ Note that, the eigenspaces of a linear complex structure are both linear CR and co-CR structures.\ A *linear $f$-structure* on $M$ is an endomorphism $F$ of $V$ such that $F^3+F=0$. Any linear $f$-structure corresponds to a pair formed of a linear CR structure $C$ and a linear co-CR structure $D$, which are compatible [@fq]; these are given by $C=V^{1,0}$ and $D=V^0\oplus V^{1,0}$, where $V^0$ and $V^{1,0}$ are the eigenspaces of $F$ corresponding to $0$ and ${\rm i}$, respectively.\ Note that, a linear map $t:(V,F_V)\to(W,F_W)$ between vector spaces endowed with linear $f$-structures satisfies $t\circ F_V=F_W\circ t$ if and only if $t(C_V)\subseteq C_W$ and $t(D_V)\subseteq D_W$, where $C_V$ and $D_V$ ($C_W$ and $D_W$) are the linear CR and co-CR structures, respectively, corresponding to $F_V$ ($F_W$); equivalently, $t$ is *$f$-linear* if and only if it is *CR linear* and *co-CR linear* [@fq].\ A *linear generalized complex structure* on $V$ is a maximal isotropic subspace $L=L(E,\ep)$ of $V^{\C}\oplus\bigl(V^{\C}\bigr)^*$ such that $L\cap\overline{L}=\{0\}$ [@Gua-thesis],[@Hit-gc_QJM]; equivalently, $E$ is a linear co-CR structure and ${\rm Im}\bigl(\ep|_{E\cap\overline{E}}\bigr)$ is nondegenerate [@Gua-thesis].\ If $L=L(E,\ep)$ is a linear generalized complex structure then we call $E$ and $L\bigl(E\cap\overline{E},{\rm Im}\bigl(\ep|_{E\cap\overline{E}}\bigr)\bigr)$ *the associated linear co-CR* and *Poisson structures*, respectively. \[defn:compatible\_f\_Poisson\] A linear $f$-structure $F$ and a two-form $\o$ on $V$ are *compatible* if $\o|_{V^0}$ is nondegenerate and ${\rm ker}\,\o=V^{1,0}\oplus V^{0,1}$.\ We say that a linear generalized complex structure $L$ on $V$ is *in normal form* if there exist a linear $f$-structure on $V$ and a compatible two-form $\o$ with respect to which $L=L\bigl(V^0\oplus V^{1,0},{\rm i}\,\o\bigr)$. \[rem:LJ\] 1) Let $L$ be a linear generalized complex structure on $V$. Denote by ${\mathcal{J}}$ the linear complex structure on $V\oplus V^*$ whose eigenspace corresponding to ${\rm i}$ is $L$. Then the bivector corresponding to the linear Poisson structure associated to $L$ is $\p\circ\bigl({\mathcal{J}}|_{V^*}\bigr)$ [@AboBoy].\ 2) Let $L=L\bigl(V^0\oplus V^{1,0},{\rm i}\,\o\bigr)$ be a linear generalized complex structure in normal form, determined by the compatible linear $f$-structure $F$ and two-form $\o$. Then the linear Poisson structure associated to $L$ is $L\bigl(V^0,\o\bigr)$; denote by $\e$ the corresponding bivector. Furthermore, if ${\mathcal{J}}$ is the linear complex structure on $V\oplus V^*$ whose eigenspace corresponding to ${\rm i}$ is $L$, then (cf. [@Gua-thesis]) $${\mathcal{J}}= \begin{pmatrix} F & \e \\ -\o & -F^* \end{pmatrix} \;,$$ where $(\cdot)^*$ denotes the transposition.\ In the terminology of [@Gua-thesis], we have that $L$ is the *product* of a complex vector space and a symplectic vector space. However, in the smooth category, the corresponding two notions are no longer equivalent. The next result (which reformulates [@Gua-thesis Theorem 4.13]) shows that any linear generalized complex structure is determined, up to a linear $B$-field transformation, by its associated linear co-CR and Poisson structures. \[prop:linear\_normal\_form\] Let $L$ be a linear generalized complex structure on $V$ and let $F$ be a linear $f$-structure on $V$ such that $\p(L)$ is the linear co-CR structure associated to $F$.\ Then there exists a unique $B\in\Lambda^2V^*$ such that $({\rm exp}B)(L)$ is in normal form, with $F$ the corresponding linear $f$-structure. If $L=L\bigl(V^0\oplus V^{1,0},\ep\bigr)$ then $B$ is characterised by the relations $B=-{\rm Re}\,\ep$, on $V^0$, and $B=-\ep$, on $V^{1,0}$ and $V^0\otimes V^{1,0}$. Next, we make the following: \[defn:gcl\] A linear map $t:V\to W$, between vector spaces endowed with linear generalized complex structures $L_V$ and $L_W$, respectively, is *generalized complex linear* if it is a co-CR linear Poisson morphism, with respect to the associated linear co-CR and Poisson structures. Note that, Definition \[defn:gcl\] is invariant under linear $B$-field transformations. \[prop:gcl\] Let $t:V\to W$ be a linear map between vector spaces endowed with linear generalized complex structures $L_V$ and $L_W$, respectively.\ Then the following assertions are equivalent:\ $t$ is generalized complex linear.\ Up to linear $B$-field transformations, $L_V$ and $L_W$ are in normal form and $t$ is an $f$-linear Poisson morphism, with respect to the corresponding linear $f$-structures and Poisson structures, on $\bigl(V,L_V\bigr)$ and $\bigl(W,L_W\bigr)$.\ Up to linear $B$-field transformations, $t$ is the direct sum of a linear Poisson morphism, between symplectic vector spaces, and a complex linear map, between complex vector spaces. From Proposition \[prop:pfpb\] it follows that it is sufficient to prove (i)$\Longrightarrow$(ii). Furthermore, if (i) holds then $t\bigl(E_V\cap\overline{E_V}\bigr)=E_W\cap\overline{E_W}$. Hence, $$t^{-1}\bigl(E_W\cap\overline{E_W}\bigr)={\rm ker}\,t+\bigl(E_V\cap\overline{E_V}\bigr)$$ and, consequently, there exist complementary vector spaces $V'$ and $W'$ of $E_V\cap\overline{E_V}$ and $E_W\cap\overline{E_W}$ in $V$ and $W$, respectively, such that $t\bigl(V'\bigr)\subseteq W'$.\ Then (i)$\Longrightarrow$(ii) and (ii)$\Longrightarrow$(iii) follow from Propositions \[prop:linear\_normal\_form\] and \[prop:pfpb\], respectively, whilst (iii)$\Longrightarrow$(i) is trivial. Note that, by using Remark \[rem:LJ\](2), assertion (ii) of Proposition \[prop:gcl\] can be formulated in terms of the corresponding linear complex structures of $V\oplus V^*$ and $W\oplus W^*$.\ The next result is an immediate consequence of Proposition \[prop:gcl\]. \[cor:gcl\] Let $t:V\to W$ be a linear isomorphism between vector spaces endowed with linear generalized complex structures $L_V$ and $L_W$, respectively.\ Then the following assertions are equivalent:\ $t$ is generalized complex linear.\ $t_*\bigl(L_V\bigr)=L_W$, up to linear $B$-field transformations. We end this section with the following: \[rem:nonBinv\] It has been proposed another definition for the notion of generalized complex linear map by imposing that the product of the graphs of the map and of its transpose be invariant under the product of the (endomorphisms corresponding to the) generalized linear complex structures, of the domain and codomain [@Cra] (see [@Vai-red_gc]).\ However this notion is not invariant under linear $B$-field transformations as we shall now explain.\ Let $(V,J)$ be a complex vector space and let $b$ be a two-form on $V$; denote by $L_J$ the linear generalized complex structure corresponding to $J$. Then the map ${\rm Id}_V:\bigl(V,L_J\bigr)\to\bigl(V,L_{({\rm exp}\,b)(L_J)}\bigr)$ satisfies the above mentioned condition if and only if $b$ is of type $(1,1)$, with respect to $J$.\ Certainly, this inconvenience would be removed if we take this definition up to linear $B$-field transformations. However, a straightforward calculation shows that there are no such maps between symplectic vector spaces $U$ and $V$ with $\dim U-\dim V=2$, a rather unnatural restriction. Holomorphic maps between generalized complex manifolds {#section:h_gc} ====================================================== From now on, unless otherwise stated, all the manifolds are assumed connected and smooth and all the maps are assumed smooth.\ An *almost $($co-$)$CR structure* on a manifold $M$ is a complex vector subbundle ${\mathcal{C}}$ of $T^{\C}\!M$ such that ${\mathcal{C}}_x$ is a linear (co-)CR structure on $T_xM$, for any $x\in M$. An *integrable* almost (co-)CR structure is an almost (co-)CR structure whose space of sections is closed under the (Lie) bracket. A *$($co-$)$CR structure* is an integrable almost (co-)CR structure.\ Note that, the eigenbundles of a complex structure are both CR and co-CR structures.\ Let $\phi:M\to N$ be a submersion onto a complex manifold $(N,J)$; denote by $T^{1,0}N$ the eigenbundle of $J$ corresponding to ${\rm i}$. Then $\operatorname{d}\!\phi^{-1}\bigl(T^{1,0}N\bigr)$ is a co-CR structure on $M$. Conversely, any co-CR structure is, locally, obtained this way.\ An *almost $f$-structure* is a $(1,1)$-tensor field $F$ such that $F^3+F=0$. Any almost $f$-structure on $M$ corresponds to a pair formed of an almost CR structure ${\mathcal{C}}$ and an almost co-CR structure ${\mathcal{D}}$, which are compatible [@fq]; these are given by ${\mathcal{C}}=T^{1,0}M$ and ${\mathcal{D}}=T^0M\oplus T^{1,0}M$, where $T^0M$ and $T^{1,0}M$ are the eigenbundles of $F$ corresponding to $0$ and ${\rm i}$, respectively.\ An almost $f$-structure is *(co-)CR integrable* if the associated almost (co-)CR structure is integrable. An *(integrable almost) $f$-structure* is an almost $f$-structure which is both CR and co-CR integrable [@fq].\ A map between manifolds endowed with almost (co-)CR structures ($f$-structures) is *$($co-$)$CR holomorphic* (*$f$-holomorphic*) if, at each point, its differential is linear ($f$-linear).\ A *generalized almost complex structure* on $M$ is a complex vector subbundle $L$ of $T^{\C\!}M\oplus\bigl(T^{\C\!}M\bigr)^*$ such that $L_x$ is a linear generalized complex structure on $T_xM$, for any $x\in M$. An *integrable* generalized complex structure is a generalized almost complex structure whose space of sections is closed under the (complexification of the) Courant bracket; a *generalized (almost) complex manifold* is a manifold endowed with a generalized (almost) complex structure [@Gua-thesis],[@Hit-gc_QJM].\ A point $x$ of a generalized almost complex manifold $(M,L)$ is *regular* if it is regular for the associated almost Poisson structure; equivalently, in some open neighbourhood of $x$, $\p(L)$ is a complex vector subbundle of $T^{\C\!}M$ (note that, then $\p(L)$ is an almost co-CR structure on $M$).\ An almost $f$-structure $F$ and a two form $\o$ on $M$ are *compatible* if $\o$ is nondegenerate on $T^0M$ and $\iota_X\o=0$, for any $X\in T^{1,0}M\oplus T^{0,1}M$.\ A generalized (almost) complex structure $L$ on $M$ is in *normal form* if $L=L\bigl(T^0M\oplus T^{1,0}M,{\rm i}\,\o\bigr)$ for some compatible almost $f$-structure and two-form $\o$ on $M$. Note that, a generalized almost complex structure in normal form is regular. \[prop:int\_normal\_form\] Let $L=L\bigl(T^0M\oplus T^{1,0}M,{\rm i}\,\o\bigr)$ be the generalized almost complex structure in normal form, corresponding to the compatible almost $f$-structure $F$ and two-form $\o$ on $M$.\ Then the following assertions are equivalent:\ $L$ is integrable.\ $F$ is integrable, $L(T^0M,\o)$ is a Poisson structure and $\o$ is invariant under the parallel displacement of $T^{1,0}M\oplus T^{0,1}M$. From [@Gua-thesis Proposition 4.19] it follows quickly that assertion (i) is equivalent to the fact that $F$ is co-CR integrable and $(\operatorname{d}\!\o)|_{T^0M\oplus T^{1,0}M}=0$. Assuming $F$ co-CR integrable, the latter condition is equivalent to the fact that $L(T^0M,\o)$ is a Poisson structure, $F$ is CR integrable and $(\mathcal{L}_X\o)|_{T^0M}=0$ for any vector field $X$ tangent to $T^{1,0}M\oplus T^{0,1}M$, where $\mathcal{L}$ denotes the Lie derivative. A generalized complex structure in normal form $L\bigl(T^0M\oplus T^{1,0}M,{\rm i}\,\o\bigr)$ is *special* if $T^{1,0}M\oplus T^{0,1}M$ is integrable (note that, an $f$-structure $F$ has this property if and only if $[F,F]=0$, where $[\cdot,\cdot]$ is the Nijenhuis bracket; see [@KoNo page 38]). All of the examples of generalized complex structures of [@CavGua-nil] are in normal form. Similarly, we have the following example, due to [@AleDav-GC]. \[exm:normal\_GC\_on\_Lie\_groups\] Let $G$ be a compact Lie group of even rank assumed, for simplicity, semisimple. Let $\mathfrak{g}$ be the Lie algebra of $G$ and let $\mathfrak{k}$ be the Lie algebra of a maximal torus in $G$.\ Let $\mathfrak{c}$ be a Borel subalgebra of $\mathfrak{g}^{\C}$ containing $\mathfrak{k}^{\C}$. Any such Borel subalgebra is obtained by choosing a base for the root system of $\mathfrak{g}^{\C}$ corresponding to $\mathfrak{k}^{\C}$ (see [@Hum]): $\mathfrak{c}=\mathfrak{k}^{\C}\oplus\bigoplus_{\a\succ0}\mathfrak{g}^{\a}$, where $\mathfrak{g}^{\a}$ is the root space of $\mathfrak{g}^{\C}$ corresponding to the root $\a$.\ As $\overline{\mathfrak{g}^{\a}}=\mathfrak{g}^{-\a}$ (see [@BurRaw]), we have $\mathfrak{c}+\overline{\mathfrak{c}}=\mathfrak{g}^{\C}$ and $\mathfrak{c}\cap\overline{\mathfrak{c}}=\mathfrak{k}^{\C}$. Consequently, $\mathfrak{c}$ corresponds to a left invariant co-CR structure ${\mathcal{C}}$ on $G$ (for any $a\in G$, we have that ${\mathcal{C}}_a$ is the left translation of $\mathfrak{c}$, at $a$).\ Let $\o$ be a linear symplectic form on $\mathfrak{k}$ ($\dim\mathfrak{k}=\operatorname{rank}G$ is even), extended to $\mathfrak{g}$ such that $\iota_X\o=0$ for any $X\in\bigoplus_{\a\succ0}\mathfrak{g}^{\a}$. We shall denote by the same letter $\o$ the left invariant two-form on $G$, determined by $\o$.\ Then $L\bigl({\mathcal{C}},{\rm i}\,\o\bigr)$ is a generalized complex structure on $G$ in normal form. The next result follows from the proof of [@Gua-thesis Theorem 4.35]. \[thm:local\_gcs\] Let $L$ be a regular generalized almost complex structure on $M$ and let $L'$ be the associated almost Poisson structure.\ Then the following assertions are equivalent:\ $L$ is integrable.\ $\p(L)$ and $L'$ are integrable and, locally, for any submersion $\r:M\to P$, with $\dim P=\operatorname{rank}\bigl(\p(L')\bigr)$ and $\r_*(L')$ a symplectic structure on $P$, we have that, up to a $B$-field transformation, $L$ is in special normal form with respect to the $f$-structure on $M$ determined by $\p(L)$ and $\p(L)\cap{\rm ker}\operatorname{d}\!\r$. Next, we formulate the notion of holomorphic map between generalized complex manifolds. \[defn:ogc\] A map between generalized almost complex manifolds is *holomorphic* if, at each point, its differential is generalized complex linear. The following result is the smooth version of Proposition \[prop:gcl\]. \[prop:ogc\_basic\] Let $\phi:(M,L_M)\to(N,L_N)$ be a map between generalized complex manifolds.\ Then the following assertions are equivalent:\ $\phi$ is holomorphic.\ On an open neighbourhood of each regular point of $L_M$ on which $\phi$ has constant rank, up to $B$-field transformations, $\phi$ is an $f$-holomorphic Poisson morphism between generalized complex manifolds in special normal form.\ On an open neighbourhood of each regular point of $L_M$ on which $\phi$ has constant rank, up to $B$-field transformations, $\phi$ is the product of a Poisson morphism between symplectic manifolds and a holomorphic map between complex manifolds. This is an immediate consequence of Proposition \[prop:Poisson\_OK\], [@Gua-thesis Theorem 4.35] and Theorem \[thm:local\_gcs\]. Next, we give examples of holomorphic maps between generalized complex manifolds. \[exm:ogc\_first\] The classical holomorphic maps, the Poisson morphisms between symplectic manifolds, and their products are, obviously, holomorphic maps between generalized complex manifolds.\ Moreover, by Proposition \[prop:ogc\_basic\], any holomorphic map $\phi:(M,L_M)\to(N,L_N)$ between generalized complex manifolds is, locally, of this form on an open neighbourhood of each regular point of $L_M$ on which $\phi$ has constant rank. \[exm:ogc\_second\] Let $G$ be a compact Lie group endowed with the generalized complex structures $L=L\bigl({\mathcal{C}},{\rm i}\,\o\bigr)$ of Example \[exm:normal\_GC\_on\_Lie\_groups\].\ Then $(G\times G,L\times L)\to\bigl(G,L\bigl({\mathcal{C}},\tfrac{\rm i}{2}\,\o\bigr)\bigr)$, $(a,b)\mapsto ab^{-1}$, is a holomorphic map.\ Furthermore, let $K$ be the maximal torus of $G$ whose Lie algebra is used to define ${\mathcal{C}}$. Obviously, $\operatorname{d}\!\phi({\mathcal{C}})$ defines a left invariant complex structure on $G/K$, where $\phi:G\to G/K$ is the projection.\ Then $\phi:(G,L)\to\bigl(G/K,\operatorname{d}\!\phi({\mathcal{C}})\bigr)$ is a holomorphic map. \[exm:ogc\_third\] Let $G/H$ be a compact inner symmetric space (see [@BurRaw page 23] for the definition and [@BurRaw page 38] for a table of examples) with $\operatorname{rank}G(=\operatorname{rank}H)$ even; denote by $\mathfrak{g}$ and $\mathfrak{h}$ the Lie algebras of $G$ and $H$, respectively.\ Endow $G$ with the generalized complex structures $L\bigl({\mathcal{C}},{\rm i}\,\o\bigr)$ of Example \[exm:normal\_GC\_on\_Lie\_groups\], determined by a Borel subalgebra $\mathfrak{c}$ of $\mathfrak{g}^{\C}$ containing the Lie algebra of a maximal torus of $H$ (also a maximal torus of $G$, as $G/H$ is inner).\ It follows that $\mathfrak{d}=\mathfrak{c}\cap\mathfrak{h}^{\C}$ is a Borel subalgebra of $\mathfrak{h}^{\C}$. Let ${\mathcal{D}}$ be the left invariant co-CR structure induced by $\mathfrak{d}$, on $H$, and let $\e=\o|_H$.\ Then the inclusion map from $\bigl(H,L\bigl({\mathcal{D}},{\rm i}\,\e\bigr)\bigr)$ to $\bigl(G,L\bigl({\mathcal{C}},{\rm i}\,\o\bigr)\bigr)$ is holomorphic.\ Fairly similar examples can be obtained by working with nilpotent Lie groups endowed with the generalized complex structures of [@CavGua-nil]. The following facts are immediate consequences of the definitions. 1\) A map between regular generalized almost complex manifolds is holomorphic if and only if it is a co-CR Poisson morphism, with respect to the associated almost co-CR and Poisson structures.\ 2) Let $\phi:(M,L_M)\to(N,L_N)$ be a diffeomorphism between generalized complex manifolds. Then $\phi$ is holomorphic if and only if, in an open neighbourhood of each regular point of $M$, we have $\phi_*\bigl(L_M\bigr)=L_N$, up to $B$-field tranformations.\ 3) The composition of two holomorphic maps, between generalized (almost) complex manifolds is holomorphic.\ 4) Let $L=L(E,\ep)$ be a regular generalized complex structure on $M$. Then the holomorphic (local) functions on $(M,L)$ are just the co-CR holomorphic functions on $(M,E)$. Equivalently, if $E$ is locally defined by the submersion $\phi:M\to(N,J)$ onto the complex manifold $(N,J)$ (that is, $E=\operatorname{d}\!\phi^{-1}\bigl(T^{1,0}N\bigr)$) then, locally, any holomorphic function on $(M,L)$ is the composition of $\phi$ followed by a holomorphic function on $(N,J)$. From Theorem \[thm:pbP\] we obtain the following result. \[prop:holo\_Dirac\] Let $(M,L_M)$ and $(N,L_N)$ be regular real analytic generalized complex manifolds and let $\phi:M\to N$ be a real analytic map.\ If $\phi$ is holomorphic then, locally, up to the complexification of a real analytic $B$-field tranformation, the complexification of $\phi$ descends to a complex analytic Poisson morphism between canonical Poisson quotients. Let $L(E,{\rm i}\,\ep)$ be a generalized complex structure in normal form on a Riemannian manifold $(M,g)$.\ Then $E$ is coisotropic (that is, $E^{\perp}$ is isotropic), with respect to $g$, if and only if $E\cap\overline{E}$ is locally defined by pseudo-horizontally conformal submersions onto complex manifolds (a map from a Riemannian manifold to an almost complex manifold is *pseudo-horizontally conformal* if it pulls back $(1,0)$-forms to isotropic forms).\ Also, if $\ep^k$ has constant norm, with respect to $g$, where $\dim(E\cap\overline{E})=2k$, then the leaves of $E\cap\overline{E}$ are minimal submanifolds of $(M,g)$.\ Conversely, we have the following: \[exm:gc\_harmorphs\] Let $\phi:(M,g)\to(N,J)$ be a pseudo-horizontally conformal submersion from a Riemannian manifold onto an almost complex manifold, with $\dim M=\dim N+2$.\ Denote ${\mathscr{V}}={\rm ker}\operatorname{d}\!\phi$, ${\mathscr{H}}={\mathscr{V}}^{\perp}$ and let $\o$ be the volume form of ${\mathscr{V}}$. Also, let $F$ be the unique skew-adjoint almost $f$-structure on $M$ such that ${\rm ker}F={\mathscr{V}}$ and, with respect to which, $\phi$ is co-CR holomorphic. Obviously, $F$ and $\o$ are compatible; denote by $L$ the corresponding generalized almost complex structure in normal form.\ From Proposition \[prop:int\_normal\_form\] it follows that $L$ is integrable if and only if $J$ is integrable, the fibres of $\phi$ are minimal and the integrability tensor of ${\mathscr{H}}$ is of type $(1,1)$; note that, if $\dim M=4$ then this is equivalent to the condition that $\phi$ is a harmonic morphism (see [@BaiWoo2]), where $N$ is endowed with the conformal structure with respect to which $J$ is a Hermitian structure.\ Moreover, any generalized complex structure, in normal form, on a Riemannian manifold such that the corresponding $f$-structure is skew-adjoint, the associated Poisson structure has rank two and its symplectic form has norm $1$ is, locally, obtained this way.\ The pseudo-horizontally conformal submersions with totally-geodesic fibres onto complex manifolds, for which the integrability tensor of the horizontal distribution is of type $(1,1)$, admit a twistorial description from which it follows that they abound on Riemannian manifolds of constant curvature [@Pan-tm] (cf. [@BaiWoo2]).\ Also, see [@AprAprBri-IJM] for a study of the harmonic pseudo-horizontally conformal submersions with minimal fibres and [@BaiWoo2] for twistorial constructions of harmonic morphisms with two-dimensional fibres on four-dimensional Riemannian manifolds. Generalized Kähler manifolds {#section:gKm} ============================ We start this section by recalling from [@Gua-thesis] a few facts on generalized Kähler manifolds.\ A *generalized (almost) Kähler manifold* is a manifold $M$ endowed with two generalized (almost) complex structures such that the corresponding sections ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$ of ${\rm End}(TM\oplus T^*M)$ commute and ${\mathcal{J}}_1{\mathcal{J}}_2$ is negative definite.\ Any generalized almost Kähler structure $(L_1,L_2)$ on a manifold $M$ corresponds to a quadruple $(g,b,J_+,J_-)$ where $g$ is a Riemannian metric, $b$ is a two-form and $J_{\pm}$ are almost Hermitian structures on $(M,g)$. The (bijective) correspondence is given by $L_1=L^+\oplus L^-$, $L_2=L^+\oplus\overline{L^-}$, where $$L^{\pm}=\bigl\{X+(b\pm g)(X)\,|\,X\in V^{\pm}\bigr\}$$ with $V^{\pm}$ the eigenbundles of $J_{\pm}$ corresponding to ${\rm i}$.\ According to [@Gua-thesis Theorem 6.28], the following assertions are equivalent:\ (i) $L_1$ and $L_2$ are integrable.\ (ii) $L_+$ and $L_-$ are integrable.\ (iii) $J_{\pm}$ are integrable and parallel with respect to $\nabla^{\pm}=\nabla^g\pm\tfrac12g^{-1}h$, where $\nabla^g$ is the Levi-Civita connection of $g$ and $h=\operatorname{d}\!b$ (equivalently, $J_{\pm}$ are integrable and $\operatorname{d}^c_{\pm}\!\o_{\pm}=\mp h$, where $\o_{\pm}$ are the Kähler forms of $J_{\pm}$).\ Now, if we (pointwisely) denote $E_j=\p(L_j)$, $(j=1,2)$, then $E_1=V^++V^-$ and $E_2=V^++\overline{V^-}$. Hence, $E_1^{\perp}=V^+\cap V^-$, $E_2^{\perp}=V^+\cap\overline{V^-}$ and, therefore, $E_1$ and $E_2$ are coisotropic. \[rem:assocF\_holo-functions\] Let $(M,L_1,L_2)$ be a generalized Kähler manifold.\ 1) The (skew-adjoint) almost $f$-structures $F_j$ determined by $E_j$ and $E_j^{\perp}$ are integrable; we call $F_j$ *the $f$-structures of $L_j$*, $(j=1,2)$.\ 2) The holomorphic functions of $(M,L_1)$ and $(M,L_2)$ are the bi-holomorphic functions of $(M,J_+,J_-)$ and $(M,J_+,-J_-)$, respectively. Let ${\mathscr{H}}^{\pm}={\rm ker}(J_+\mp J_-)$. Then ${\mathscr{H}}^+$ and ${\mathscr{H}}^-$ are orthogonal; this follows from ${\mathscr{H}}^+=\bigl(V^+\cap V^-\bigr)\oplus\overline{\bigl(V^+\cap V^-\bigr)}$ and ${\mathscr{H}}^-=\bigl(V^+\cap\overline{V^-}\bigr)\oplus\overline{\bigl(V^+\cap\overline{V^-}\bigr)}$. Denote ${\mathscr{V}}=\bigl({\mathscr{H}}^+\oplus{\mathscr{H}}^-\bigr)^{\perp}$.\ Note that, ${\mathscr{H}}^+$, ${\mathscr{H}}^-$ and ${\mathscr{V}}$ are invariant under $J_+$ and $J_-$. Also, $J_+-J_-$ and $J_++J_-$ are invertible on ${\mathscr{V}}$. \[prop:regular\] The following assertions are equivalent:\ $L_1$ and $L_2$ are regular.\ ${\mathscr{H}}^+$ and ${\mathscr{H}}^-$ are distributions on $M$.\ ${\mathscr{V}}$ is a distribution on $M$. The obvious relations $$\begin{split} E_1&=\bigl(V^+\cap V^-\bigr)^{\perp}=\bigl(V^+\cap V^-\bigr)\oplus{\mathscr{H}}^-\oplus{\mathscr{V}}\;,\\ E_2&=\bigl(V^+\cap\overline{V^-}\bigr)^{\perp}=\bigl(V^+\cap\overline{V^-}\bigr)\oplus{\mathscr{H}}^+\oplus{\mathscr{V}}\\ \end{split}$$ imply $$\begin{split} E_1\cap\overline{E_1}&={\mathscr{H}}^-\oplus{\mathscr{V}}=\bigl({\mathscr{H}}^+\bigr)^{\perp}\;,\\ E_2\cap\overline{E_2}&={\mathscr{H}}^+\oplus{\mathscr{V}}=\bigl({\mathscr{H}}^-\bigr)^{\perp} \end{split}$$ which show that (i)$\Longleftrightarrow$(ii).\ Also, as the dimensions of ${\mathscr{H}}^+$ and ${\mathscr{H}}^-$ are upper semicontinuous functions on $M$, assertion (ii) holds if and only if ${\mathscr{H}}^+\oplus{\mathscr{H}}^-\bigl(={\mathscr{V}}^{\perp}\bigr)$ is a distribution on $M$. Next, we prove the following result. \[thm:H+geod\] Let $(M,L_1,L_2)$ be a generalized Kähler manifold with $L_1$ regular.\ Then the following assertions are equivalent:\ ${\mathscr{H}}^+$ is integrable.\ ${\mathscr{H}}^+$ is geodesic.\ ${\mathscr{H}}^+$ is a holomorphic distribution on $(M,J_+)$.\ ${\mathscr{H}}^+$ is a holomorphic distribution on $(M,J_-)$.\ Furthermore, if [(i)]{}, [(ii)]{}, [(iii)]{} or [(iv)]{} holds then ${\mathscr{H}}^+$ is a holomorphic foliation on $(M,J_{\pm})$ and its leaves, endowed with $(g,J_{\pm})$, are Kähler manifolds. To prove Theorem \[thm:H+geod\] we need some preparations.\ Let ${\mathscr{H}}$ be a distribution on a Riemannian manifold $(M,g)$ endowed with a linear connection $\nabla$; denote ${\mathscr{V}}={\mathscr{H}}^{\perp}$.\ The *second fundamental form* of ${\mathscr{H}}$, with respect to $\nabla$, is the ${\mathscr{V}}$-valued symmetric two-form $\operatorname{{\it B}^{{\mathscr{H}}}}$ on ${\mathscr{H}}$ defined by $\operatorname{{\it B}^{{\mathscr{H}}}}(X,Y)=\tfrac12\,{\mathscr{V}}\bigl(\nabla_XY+\nabla_YX\bigr)$; then ${\mathscr{H}}$ is geodesic, with respect to $\nabla$, if and only if $\operatorname{{\it B}^{{\mathscr{H}}}}=0$ (cf. [@BaiWoo2]).\ The next result follows from a straightforward calculation. \[lem:Watson\] Let $(M,g,J)$ be a Hermitian manifold endowed with a distribution ${\mathscr{H}}$ and a conformal connection $\nabla$ such that $\nabla J=0$.\ If ${\mathscr{V}}$ is integrable and $J$-invariant then the following relation holds: $$2\,g\bigl(\operatorname{{\it B}^{{\mathscr{H}}}}(JX,Y),V\bigr)+g\bigl(I^{{\mathscr{H}}}(X,Y),JV\bigr)=g\bigl(T(V,JX),Y\bigr)+g\bigl(T(V,X),JY\bigr)\;,$$ for any $X,Y\in{\mathscr{H}}$ and $V\in{\mathscr{V}}$, where $T$ is the torsion of $\nabla$ and $I^{{\mathscr{H}}}$ is the integrability tensor of ${\mathscr{H}}$, defined by $I^{{\mathscr{H}}}(X,Y)=-{\mathscr{V}}[X,Y]$, for any sections $X$ and $Y$of ${\mathscr{H}}$. To prove Theorem \[thm:H+geod\] we also need the following lemma. \[lem:holoH\] Let $(M,J)$ be a complex manifold and let ${\mathscr{H}}$ be a complex vector subbundle of $(TM,J)$. The following assertions are equivalent:\ ${\mathscr{H}}$ is integrable.\ ${\mathscr{H}}^{1,0}$ is a CR structure and ${\mathscr{H}}$ is a holomorphic distribution on $(M,J)$. This follows from the fact that assertion (ii) holds if and only if for any any sections $X,Y$ of ${\mathscr{H}}^{1,0}$ and $Z$ of $T^{0,1}M$ we have that $[X,Y]$ is a section of ${\mathscr{H}}^{1,0}$ and $[X,Z]$ is a section of ${\mathscr{H}}^{1,0}\oplus T^{0,1}M$. We may assume that, also, $L_2$ is regular.\ Obviously, the second fundamental form of ${\mathscr{H}}^+$, with respect to $\nabla^g$, is equal to the second fundamental forms of ${\mathscr{H}}^+$, with respect to $\nabla^{\pm}$.\ As $L_1$ and $L_2$ are integrable we have that $E_1$ and $E_2$ are integrable and, consequently, ${\mathscr{H}}^+\oplus{\mathscr{V}}$ and ${\mathscr{H}}^-\oplus{\mathscr{V}}$ are integrable; in particular, the integrability tensor of ${\mathscr{H}}^+$ takes values in ${\mathscr{V}}$. Furthermore, ${\mathscr{H}}^+\oplus{\mathscr{V}}$ and ${\mathscr{H}}^-\oplus{\mathscr{V}}$ are holomorphic with respect to both $J_+$ and $J_-$.\ Now, by applying Lemma \[lem:Watson\] to ${\mathscr{H}}={\mathscr{H}}^+$ twice, with respect to $\nabla^+$ and $\nabla^-$, we quickly obtain $$4\,g\bigl(B^{{\mathscr{H}}^+}\!(J_{\pm}X,Y),V\bigr)=-g\bigl(I^{{\mathscr{H}}^+}\!(X,Y),(J_++J_-)(V)\bigr)\;,$$ for any $X,Y\in{\mathscr{H}}^+$ and $V\in{\mathscr{H}}_-\oplus{\mathscr{V}}$. As $J_++J_-$ is invertible on ${\mathscr{V}}$, we obtain that (i)$\Longleftrightarrow$(ii).\ The equivalences (iii)$\Longleftrightarrow$(i)$\Longleftrightarrow$(iv) follow quickly from Lemma \[lem:holoH\] and the fact that the eigenbundles of $J_{\pm}|_{{\mathscr{H}}^+}$ corresponding to ${\rm i}$ are equal to $V^+\cap V^-$ which is integrable.\ To complete the proof just note that if ${\mathscr{H}}^+$ is integrable then $(g,b,J_+,J_-)$ induces, by restriction, a generalized Kähler structure on each leaf $L$ of ${\mathscr{H}}^+$ and $J_+=J_-$ on $L$. \[rem:Riem\_subm\_descend\] 1) Let $(M,L_1,L_2)$ be a generalized Kähler manifold with $L_1$ regular. If ${\mathscr{H}}^+$ is integrable then, by Theorem \[thm:H+geod\], the co-CR structure associated to $L_1$ (that is, $E_1$) is, locally, given by holomorphic Riemannian submersions from $(M,g,J_{\pm})$ onto Kähler manifolds $(P,h,J)$; in particular, the leaves of ${\mathscr{H}}^+$, endowed with $(g,J_{\pm})$ can be, locally, identified with $(P,h,J)$.\ 2) If $(M,L_1^M,L_2^M)$ and $(N,L_1^N,L_2^N)$ are generalized Kähler manifolds with ${\mathscr{H}}^+_M$ and ${\mathscr{H}}^+_N$ integrable distributions then any holomorphic map $\phi:(M,L_1^M)\to(N,L_1^N)$ descends, locally (with respect to the Riemannian submersions of Remark \[rem:Riem\_subm\_descend\](1)), to a holomorphic map between Kähler manifolds. Let $(M_j,g_j,J_j)$ be Kähler manifolds, $(j=1,2)$. Then on $M_1\times M_2$ there are two nonequivalent natural generalized Kähler structures: the *first product* is just the Kähler product structure whilst the *second product* is given by $L_1=L\bigl(T^{1,0}M_1\times TM_2,{\rm i}\,\o_2\bigr)$ and $L_2=L\bigl(T^{1,0}M_2\times TM_1,{\rm i}\,\o_1\bigr)$, where $\o_j$ are the Kähler forms of $J_j$, $(j=1,2)$; see Section \[section:gK\_tamed\], below, for the corresponding definitions in a more general setting. Note that, both $L_1$ and $L_2$ are in normal form; moreover, the corresponding almost $f$-structures are skew-adjoint (and, thus, unique with this property).\ We end this section with the following consequence of Theorem \[thm:H+geod\] (cf. [@ApoGua Theorem A]). \[cor:H+-\] Any generalized Kähler manifold with ${\mathscr{V}}=0$ is, up to a unique $B$-field transformation, locally given by the second product of two Kähler manifolds. Let $(M,L_1,L_2)$ be a generalized Kähler manifold with ${\mathscr{V}}=0$. Then, Proposition \[prop:regular\] implies that ${\mathscr{H}}^{\pm}$ are complementary orthogonal distributions on $M$.\ As $L_1$ and $L_2$ are integrable, we have ${\mathscr{H}}^{\pm}$ integrable. Furthermore, by Theorem \[thm:H+geod\], we have that ${\mathscr{H}}^{\pm}$ are geodesic foliations which are holomorphic with respect to both $J_{\pm}$; moreover, $(g,J_{\pm})$ induce, by restriction, Kähler structures on their leaves.\ If $L_2=L(E_2,\ep_2)$ then, from the definitions it follows that $\ep_2=(b-{\rm i}\,\e)|_{E_2}$, where $\e$ is the two-form on $M$ characterised by $\iota_X\e=0$ if $X\in{\mathscr{H}}_-$ and $\e|_{{\mathscr{H}}^+}$ is the Kähler form of $J_+|_{{\mathscr{H}}^+}$. As $(\operatorname{\mathcal{L}}_X\!\e)(Y,Z)=0$ for any sections $X$ of ${\mathscr{H}}^-$ and $Y$, $Z$ of ${\mathscr{H}}^+$, and $(\operatorname{d}\!\ep_2)(X,Y,Z)=0$ for any $X,Y,Z\in E_2$, we obtain that $(\operatorname{d}\!b)(X,Y,\overline{Z})=0$ for any $X\in V^+\cap{\mathscr{H}}^-(=E_2\cap{\mathscr{H}}^-)$ and $Y,Z\in V^+\cap{\mathscr{H}}^+$. Furthermore, from Lemma \[lem:Watson\], applied to ${\mathscr{H}}={\mathscr{H}}^+$ with $J=J_+$ and $\nabla=\nabla^+$, we obtain $(\operatorname{d}\!b)(X,Y,Z)=0$ for any $X\in{\mathscr{H}}^-$ and $Y,Z\in V^+\cap{\mathscr{H}}^+$.\ It follows that $\operatorname{d}\!b=0$ and the proof is complete. Tamed symplectic and generalized Kähler manifolds {#section:gK_tamed} ================================================= The following definition is fairly standard. \[defn:tamed\_symp\] A *tamed almost symplectic manifold* is a manifold $M$ endowed with a nondegenerate two-form $\ep$ and an almost complex structure $J$ such that $\ep(JX,X)>0$ for any nonzero $X\in TM$.\ A *tamed symplectic manifold* is a tamed almost symplectic manifold $(M,\ep,J)$ such that $J$ and $\ep^{-1}J^*\ep$ are integrable and $\operatorname{d}\!\ep=0$. Obviously, $(M,\ep,J)$ is a tamed symplectic manifold if and only if $\ep$ is a symplectic form, $T^{1,0}M$ and $\bigl(T^{1,0}M\bigr)^{\perp_{\ep}}$ are integrable, and $\ep(JX,X)>0$, for any nonzero $X\in TM$.\ The next result also appears, in a different form, in [@Gua-Pbranes]. \[thm:gK\_tamed\] Let $M$ be a manifold endowed with a nondegenerate two-form $\ep$ and an almost complex structure $J$; denote $J_+=J$ and $J_-=-\ep^{-1}J^*\ep$. Let $g$ and $b$ be the symmetric and skew-symmetric parts, respectively, of $\ep J$.\ Then the following assertions are equivalent:\ $(M,\ep,J)$ is a tamed symplectic manifold.\ $(g,b,J_+,J_-)$ defines a generalized Kähler structure such that $J_++J_-$ is invertible.\ Moreover, up to a unique $B$-field transformation, any generalized Kähler structure, on $M$, with $J_++J_-$ invertible is obtained this way from a tamed symplectic structure. Firstly, note that $\ep(J_+X,Y)=-\ep(X,J_-Y)$, for any $X,Y\in TM$. This implies that $$\label{e:gb_ep} \begin{split} g(X,Y)=\,&\tfrac12\,\ep\bigl((J_++J_-)(X),Y\bigr)\;,\\ b(X,Y)=\,&\tfrac12\,\ep\bigl((J_+-J_-)(X),Y\bigr)\;, \end{split}$$ for any $X,Y\in TM$.\ Therefore $(M,\ep,J)$ is a tamed almost symplectic manifold if and only if the quadruple $(g,b,J_+,J_-)$ defines a generalized almost Kähler manifold with $J_++J_-$ invertible.\ Now, with respect to $J_{\pm}$, we have $\o_{\pm}=-\ep^{1,1}$, $b^{1,1}=0$ and $b^{2,0}=\pm{\rm i}\,\ep^{2,0}$. It quickly follows that if $J_{\pm}$ are integrable then $\operatorname{d}\!\ep=0$ if and only if $\operatorname{d}^c_{\pm}\!\o_{\pm}=\mp\operatorname{d}\!b$.\ We have thus proved that (i)$\Longleftrightarrow$(ii).\ Suppose that $(g,b,J_+,J_-)$ corresponds to the generalized Kähler structure $(L_1,L_2)$ on $M$. Then $J_++J_-$ is invertible if and only if $\p(L_2)=TM$. Hence, if $J_++J_-$ is invertible then, up to a unique $B$-field transformation, we have $L_2=L(TM,{\rm i}\,\ep)$ for some symplectic form $\ep$ on $M$ and, consequently, $$\label{e:ep_gb} \begin{split} {\rm i}\,\ep(X-{\rm i}J_+X,Y)=\,&(b+g)(X-{\rm i}J_+X,Y)\;,\\ {\rm i}\,\ep(X+{\rm i}J_-X,Y)=\,&(b-g)(X+{\rm i}J_-X,Y)\;, \end{split}$$ for any $X,Y\in TM$. By using the fact that $J_++J_-$ is invertible, from we quickly obtain that $g$ and $b$ satisfy . Together with the fact that $g$ and $b$ are symmetric and skew-symmetric, respectively, this shows that $J_-=-\ep^{-1}J_+^*\ep$ and the proof follows. It is easy to rephrase Theorem \[thm:gK\_tamed\] so that to obtain the description of generalized Kähler manifolds with $J_+-J_-$ invertible.\ Let $(M,L^M_1,L^M_2)$ and $(N,L^N_1,L^N_2)$ be generalized Kähler manifolds corresponding to the tamed symplectic manifolds $(M,\ep_M,J_M)$ and $(N,\ep_N,J_N)$, respectively. Then $(M\times N,L^M_1\times L^N_1,L^M_2\times L^N_2)$ and $(M\times N,L^M_1\times L^N_2,L^M_2\times L^N_1)$ are called the *first* and *second product* of $(M,L^M_1,L^M_2)$ and $(N,L^N_1,L^N_2)$, respectively; note that, the first product is the generalized Kähler manifold corresponding to $(M\times N,\ep_M+\ep_N,J_M\times J_N)$. \[cor:VH+\] Any generalized Kähler manifold with ${\mathscr{H}}^+$ an integrable distribution and ${\mathscr{H}}^-=0$ is, up to a unique $B$-field transformation, locally given by the first product of a Kähler manifold and a generalized Kähler manifold for which both $J_++J_-$ and $J_+-J_-$ are invertible. Let $(M,L_1,L_2)$ be a generalized Kähler manifold with ${\mathscr{H}}^+$ a distribution and ${\mathscr{H}}_-=0$. Then, by Theorem \[thm:gK\_tamed\], up to a unique $B$-field transformation, we have that $(M,L_1,L_2)$ corresponds to the tamed symplectic manifold $(M,\ep,J)$.\ Thus, by , we have $\iota_Xb=0$ for any $X\in{\mathscr{H}}^+$ and $\ep=\e+\ep'$ where $\e$ and $\ep'$ are the two-forms on $M$ characterised by $\iota_X\e=0$, $(X\in{\mathscr{V}})$, $\iota_X\ep'=0$, $(X\in{\mathscr{H}}^+)$, $\e=\o_+$ on ${\mathscr{H}}^+$, and $\ep'=\ep$ on ${\mathscr{V}}$.\ If, further, ${\mathscr{H}}^+$ is integrable then, by Theorem \[thm:H+geod\], it is also geodesic and its leaves endowed with $(g,J)$ are Kähler manifolds; in particular, $\operatorname{d}\!\e=0$ on ${\mathscr{H}}^+$. As, also, ${\mathscr{H}}^+$ and ${\mathscr{V}}$ are holomorphic foliations, it quickly follows that $(\operatorname{\mathcal{L}}_X\!\e)(Y,Z)=0$ for any sections $X$ of ${\mathscr{V}}$ and $Y$, $Z$ of ${\mathscr{H}}^+$; consequently, $\operatorname{d}\!\e=0$.\ We have thus obtained $\operatorname{d}\!\ep'=0$ which implies $(\operatorname{\mathcal{L}}_X\!\ep')(Y,Z)=0$ for any sections $X$ of ${\mathscr{H}}^+$ and $Y$, $Z$ of ${\mathscr{V}}$. Together with , this gives $(\operatorname{\mathcal{L}}_X\!b)(Y,Z)=0$ and $(\operatorname{\mathcal{L}}_X\!g)(Y,Z)=0$ for any sections $X$ of ${\mathscr{H}}^+$ and $Y$, $Z$ of ${\mathscr{V}}$; in particular, this shows that ${\mathscr{V}}$ is geodesic. The proof follows. Obviously, a result similar to Corollary \[cor:VH+\] holds for any generalized Kähler manifold with ${\mathscr{H}}^+=0$ and ${\mathscr{H}}^-$ an integrable distribution. \[cor:L2\_normal\] Let $(M,L_1,L_2)$ be a generalized Kähler manifold such that $L_2$ is in normal form with respect to its $f$-structure and the two-form $\ep$ on $M$.\ Then, in a neighbourhood of each regular point of $L_1$, we have that $(M,L_1,L_2)$ is the second product of a Kähler manifold and a generalized Kähler manifold determined by a tamed symplectic manifold. Assume $L_1$ regular. Define $\ep_{\pm}$ to be the (complex linear) two-forms on $T^{1,0}_+M+T^{1,0}_-M$ such that $\ep_{\pm}=\ep$ on $T^{1,0}_{\pm}M$ and $\iota_X\ep_{\pm}=0$ if $X\in T^{1,0}_{\mp}M$.\ Obviously, $\operatorname{d}\!\ep_{\pm}=0$ on $T^{1,0}_{\pm}M$. Also, from the fact that $\iota_X\ep_{\pm}=0$ if $X\in T^{1,0}_{\mp}M$ it quickly follows that if $X_{\pm},\,Y_{\pm},\,Z_{\pm}\in T^{1,0}_{\pm}M$ then $\operatorname{d}\!\ep_{\pm}(X_{\mp},Y_{\mp},Z_{\pm})=0$; together with the fact that $\ep=\ep_++\ep_-$ on $T^{1,0}_+M+T^{1,0}_-M$, this implies that $\operatorname{d}\!\ep_{\pm}(X_{\pm},Y_{\pm},Z_{\mp})=0$. Thus, we have proved that $\operatorname{d}\!\ep_{\pm}=0$ on $T^{1,0}_+M+T^{1,0}_-M$.\ Therefore ${\rm ker}\,\ep_{\pm}=T^{1,0}_{\mp}\oplus\bigl(T^{0,1}_{\mp}\cap{\mathscr{H}}^-\bigr)$ is integrable which implies that ${\mathscr{H}}^-$ is an antiholomorphic distribution on $(M,J_{\mp})$. Hence, by Lemma \[lem:holoH\], we have that ${\mathscr{H}}^-$ is integrable and the proof follows from Theorem \[thm:H+geod\] and the fact that ${\rm ker}\,\ep={\mathscr{H}}^-$. Let $(M,\ep,J)$ be a tamed almost symplectic manifold. With the same notations as in Corollary \[cor:L2\_normal\], if $(M,L_1,L_2)$ is the generalized Kähler manifold determined by $(M,\ep,J)$ then, from , it follows that $L_1=L\bigl(T^{1,0}_+M+T^{1,0}_-M,\,{\rm i}\,\ep_+-{\rm i}\,\ep_-\bigr)$. \[thm:holo\_Poisson\_tamed\] Let $(M,\ep,J)$ be a tamed almost symplectic manifold and let $(M,L_1,L_2)$ be the corresponding generalized almost Kähler manifold; denote by $\r^{\pm}:T^{\C\!}M\to T^{1,0}_{\pm}M$ the projections.\ If $(M,L_1,L_2)$ is generalized Kähler then $J_{\pm}$ are integrable and $\r^{\pm}_*(L_2)$ are holomorphic Poisson structures on $(M,J_{\pm})$, respectively. Furthermore, the converse holds if also $J_+-J_-$ is invertible; moreover, in this case, if $(M,L_1,L_2)$ is generalized Kähler then $\r^{\pm}_*(L_2)$ are holomorphic symplectic structures on $(M,J_{\pm})$, respectively. Assume, for simplicity, that $(M,\ep,J)$ is real analytic. Also, we may assume $L_1$ regular. If $(M,L_1,L_2)$ is generalized Kähler then, by passing to the complexification of $(M,\ep,J)$, from Proposition \[prop:pfpb\] and the proof of Corollary \[cor:L2\_normal\] we obtain that $\r^{\pm}_*(L_2)$ are the canonical Poisson quotients of $L\bigl(T^{0,1}_+M+T^{0,1}_-M,{\rm i}\,\overline{\ep_{\mp}}\bigr)$.\ If $J_+\pm J_-$ are invertible and $J_{\pm}$ are integrable then $\r^{\pm}_*(L_2)$ are holomorphic Poisson structures on $(M,J_{\pm})$ if and only if $\operatorname{d}\!\ep_{\pm}=0$. We call the $\r^{\pm}_*(L_2)$ of Theorem \[thm:holo\_Poisson\_tamed\] the *holomorphic Poisson structures associated to $(M,L_1,L_2)$*. \[rem:holo\_Poisson\_tamed\] 1) Let $(M,L_1,L_2)$ be a generalized Kähler manifold with $J_++J_-$ invertible. Denote by $\e_{\pm}$ the (real) bivectors on $M$ which determine the holomorphic Poisson structures on $(M,J_{\pm})$, respectively, associated to $(M,L_1,L_2)$; that is, with respect to $J_{\pm}$, we have $\e_{\pm}^{1,1}=0$ and the holomorphic bivectors corresponding to $\r^{\pm}_*(L_2)$ are $\e_{\pm}^{2,0}$, respectively.\ It quickly follows that $$\e_-=-\e_+=\tfrac12\bigl(J\ep^{-1}+\ep^{-1}J^*\bigr)=\tfrac12\bigl(J_+-J_-\bigr)\ep^{-1}=\tfrac14[J_+,J_-]g^{-1}\;,$$ where $(M,\ep,J)$ is the tamed symplectic manifold associated to $(M,L_1,L_2)$.\ Hence, the symplectic foliation associated to $\e_+$ is given by ${\mathscr{V}}(={\rm im}(J_+-J_-)\,)$.\ 2) If the generalized almost Kähler structure $(L_1,L_2)$ on $M$ corresponds to the quadruple $(g,b,J_+,J_-)$ then $(L_2,L_1)$ corresponds to $(g,b,J_+,-J_-)$. Assume that $(M,L_1,L_2)$ is a generalized Kähler manifold with $J_++J_-$ and $J_+-J_-$ invertible and let $\e_+$ and $\e_+'$ be the bivectors which determine, as in (1), the holomorphic symplectic structures associated to $(M,L_1,L_2)$ and $(M,L_2,L_1)$, respectively. Then implies that $\e_+'=-\e_+$. Next, we prove some results on holomorphic maps between generalized Kähler manifolds. \[cor:holo\_diffeo\] Let $(M,L_1,L_2)$ be a generalized almost Kähler manifold with $J_++J_-$ and $J_+-J_-$ invertible.\ If $\phi:M\to M$ is a diffeomorphism then any two of the following assertions imply the third:\ $\phi:(M,L_1)\to(M,L_1)$ is holomorphic.\ $\phi:(M,L_2)\to(M,L_2)$ is holomorphic.\ $\bigl[\operatorname{d}\!\phi\,,J_+J_-\bigr]=0$. Let $L=L\bigl(T^{1,0}_+M+T^{1,0}_-M,\ep_1\bigr)$. By using the first relation of , we obtain $$\label{e:Im-ep1_ep} ({\rm Im}\,\ep_1)(J_+-J_-)=\ep(J_++J_-)\;,$$ which, firstly, shows that if (iii) holds then (i)$\Longleftrightarrow$(ii).\ Furthermore, implies that $\ep^{-1}({\rm Im}\,\ep_1)$ is skew-adjoint, with respect to $g$, and, consequently, $\ep-{\rm Im}\,\ep_1$ is invertible. This fact together with proves that (i),(ii)$\Longrightarrow$(iii). \[cor:Poisson\_holo\_map\] Let $(M,L^M_1,L^M_2)$ and $(N,L^N_1,L^N_2)$ be generalized Kähler manifolds, with $J^M_++J^M_-$ and $J^N_++J^N_-$ invertible, and let $\phi:M\to N$ be a map.\ If $\phi:(M,L^M_1)\to(N,L^N_1)$ and $\phi:(M,J^M_{\pm})\to(N,J^N_{\pm})$ are holomorphic then $\phi$ is a holomorphic Poisson morphism between the corresponding associated holomorphic Poisson manifolds; moreover, the converse holds if $\phi$ is an immersion.\ If $\phi:(M,L^M_2)\to(N,L^N_2)$ and, either, $\phi:(M,J^M_+)\to(N,J^N_+)$ or $\phi:(M,J^M_-)\to(N,J^N_-)$ are holomorphic maps then $\phi$ is a holomorphic Poisson morphism between the associated holomorphic Poisson structures. Assertion (i) follows from Proposition \[prop:pfpb\] and the proof of Theorem \[thm:holo\_Poisson\_tamed\].\ To prove (ii), note that if $\phi:(M,L^M_2)\to(N,L^N_2)$ is holomorphic then $\phi:(M,J^M_+)\to(N,J^N_+)$ is holomorphic if and only if $\phi:(M,J^M_-)\to(N,J^N_-)$ is holomorphic. The proof quickly follows from Remark \[rem:holo\_Poisson\_tamed\](1). If $(g,J_{\pm})$ are Kähler structures on $M$ then $(g,0,J_+,J_-)$ corresponds to a generalized Kähler structure $(L_1,L_2)$ on $M$; furthermore, if $b$ is a closed two-form on $M$ then $(g,b,J_+,J_-)$ corresponds to $\bigl((\exp b)(L_1),(\exp b)(L_2)\bigr)$. \[exm:hK\_gK\] Let $(M,g,I,J,K)$ be a hyper-Kähler manifold. Denote by $\o_I$, $\o_J$, $\o_K$ the Kähler forms of $I$, $J$, $K$, respectively, and let $\ep=-(\o_J+\o_K)$.\ Then $(M,\ep,J)$ is a tamed symplectic manifold. The corresponding generalized Kähler structure $(L_1,L_2)$ is given by $(g,b,J_+,J_-)$, where $b=\o_I$, $J_+=J$ and $J_-=K$. Also, $L_1=L\bigl(T^{\C\!}M,2\,\o_I-{\rm i}(\o_J-\o_K)\bigr)$, $L_2=L\bigl(T^{\C\!}M,-{\rm i}(\o_J+\o_K)\bigr)$ and $\ep_+=-{\rm i}(\o_I-{\rm i}\,\o_J)$, $\ep_-=-(\o_K-{\rm i}\,\o_I)$. We end with a generalization of Corollaries \[cor:H+-\] and \[cor:VH+\]. \[thm:H+-int\] Let $(M,L_1,L_2)$ be a generalized Kähler manifold. Then the following assertions are equivalent:\ ${\mathscr{H}}^+\oplus{\mathscr{H}}^-$ is an integrable distribution.\ Locally, up to a $B$-field transformation, $(M,L_1,L_2)$ is the first product of a generalized Kähler manifold for which $J_+\pm J_-$ are invertible and the second product of two Kähler manifolds. By applying Lemma \[lem:Watson\] to ${\mathscr{H}}={\mathscr{H}}^+\oplus{\mathscr{H}}^-$ twice, with respect to $\nabla^+$ and $\nabla^-$, we obtain $$\label{e:H+-int} \begin{split} 2g\bigl(B^{{\mathscr{H}}^+\oplus{\mathscr{H}}^-\!}(J_+X_+,X_-)&,V\bigr)+g\bigl(I^{{\mathscr{H}}^+\oplus{\mathscr{H}}^-\!}(X_+,X_-),J_+V\bigr)\\ &=(\operatorname{d}\!b)(V,J_+X_+,X_-)+(\operatorname{d}\!b)(V,X_+,J_+X_-)\;,\\ 2g\bigl(B^{{\mathscr{H}}^+\oplus{\mathscr{H}}^-\!}(J_+X_+,X_-)&,V\bigr)+g\bigl(I^{{\mathscr{H}}^+\oplus{\mathscr{H}}^-\!}(X_+,X_-),J_-V\bigr)\\ &=-(\operatorname{d}\!b)(V,J_+X_+,X_-)+(\operatorname{d}\!b)(V,X_+,J_+X_-)\;, \end{split}$$ for any $X_{\pm}\in{\mathscr{H}}^{\pm}$ and $V\in{\mathscr{V}}$. Consequently, we, also, have $$\label{e:H+-int_} g\bigr(I^{{\mathscr{H}}^+\oplus{\mathscr{H}}^-\!}(X_+,X_-),(J_+-J_-)(V)\bigl)=2\operatorname{d}\!b(V,J_+X_+,X_-)\;,$$ for any $X_{\pm}\in{\mathscr{H}}^{\pm}$ and $V\in{\mathscr{V}}$.\ Suppose that (i) holds. Then, by , we have $\operatorname{d}\!b(V,X_+,X_-)=0$, for any $X_{\pm}\in{\mathscr{H}}^{\pm}$ and $V\in{\mathscr{V}}$. Moreover, from Corollaries \[cor:H+-\] and \[cor:VH+\] it follows that $\operatorname{d}\!b(X,Y,Z)=0$ if $X,Y,Z\in{\mathscr{H}}^+\oplus{\mathscr{H}}^-$ or $X\in{\mathscr{H}}^{\pm}$ and $Y,Z\in{\mathscr{V}}\oplus{\mathscr{H}}^{\pm}$.\ As $\operatorname{d}(\operatorname{d}\!b)=0$, this shows that $\operatorname{d}\!b$ is basic with respect to ${\mathscr{H}}^+\oplus{\mathscr{H}}^-$. Hence, locally, there exists a two-form $b'$, basic with respect to ${\mathscr{H}}^+\oplus{\mathscr{H}}^-$, such that $\operatorname{d}\!b=\operatorname{d}\!b'$.\ Furthermore, from and we obtain $B^{{\mathscr{H}}^+\oplus{\mathscr{H}}^-\!}(X_+,X_-)=0$, for any Together with Theorem \[thm:H+geod\] and Corollary \[cor:VH+\], this shows that ${\mathscr{V}}$ and ${\mathscr{H}}^+\oplus{\mathscr{H}}^-$ are geodesic foliations on $(M,g)$.\ Thus, we have proved that $(M,L_1,L_2)$ is the first product of a generalized Kähler manifold with ${\mathscr{H}}^+=0={\mathscr{H}}^-$ and a generalized Kähler manifold with ${\mathscr{V}}=0$. Hence, by Corollary \[cor:H+-\], assertion (ii) holds.\ The implication (ii)$\Longrightarrow$(i) is trivial. [**Acknowledgements.**]{} We are grateful to Henrique Bursztyn for bringing to our attention [@BurRad] and [@BurWei-2005], and to Marco Gualtieri for informing us about [@Gua-Pbranes]. [10]{} M. Abouzaid, M. Boyarchenko, Local structure of generalized complex manifolds, *J. Symplectic Geom.*, [**4**]{} (2006) 43–62. D. Alekseevsky, L.David, Invariant generalized complex and Kahler structures on Lie groups, Preprint, I.M.A.R., Bucharest, 2008. V. Apostolov, M. Gualtieri, Generalized Kähler manifolds, commuting complex structures, and split tangent bundles, *Comm. Math. Phys.*, [**271**]{} (2007) 561–575. M. A. Aprodu, M. Aprodu, V. Brînzănescu, A class of harmonic submersions and minimal submanifolds, *Internat. J. Math.*, [**11**]{} (2000) 1177–1191. P. Baird, J. C. Wood, *Harmonic morphisms between Riemannian manifolds*, London Math. Soc. Monogr. (N.S.), no. 29, Oxford Univ. Press, Oxford, 2003. F. E. Burstall, J. H. Rawnsley, *Twistor theory for Riemannian symmetric spaces. With applications to harmonic maps of Riemann surfaces*, Lecture Notes in Mathematics, 1424, Springer-Verlag, Berlin, 1990. H. Bursztyn, O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, *Ann. Inst. Fourier (Grenoble)*, [**53**]{} (2003) 309–337. H. Bursztyn, A. Weinstein, Poisson geometry and Morita equivalence, *Poisson geometry, deformation quantisation and group representations*, 1–78, London Math. Soc. Lecture Note Ser., 323, Cambridge Univ. Press, Cambridge, 2005. G. R. Cavalcanti, M. Gualtieri, Generalized complex structures on nilmanifolds, *J. Symplectic Geom.*, [**2**]{} (2004) 393–410. T. J. Courant, Dirac manifolds, *Trans. Amer. Math. Soc.*, [**319**]{} (1990) 631–661. M. Crainic, Generalized complex structures and Lie brackets, Preprint, ([arXiv:math/0412097](http://arxiv.org/abs/math/0412097)). M. Gualtieri, Generalized complex geometry, D. Phil. Thesis, University of Oxford, 2003. M. Gualtieri, Branes on Poisson varieties, Preprint, ([arXiv:0710.2719](http://arxiv.org/abs/0710.2719)). N. J. Hitchin, Generalized Calabi-Yau manifolds, *Q. J. Math.*, [**54**]{} (2003) 281–308. N. J. Hitchin, Instantons, Poisson structures and Generalized Kähler Geometry, *Comm. Math. Phys.*, [**265**]{} (2006) 131-–164. J. E. Humphreys, *Introduction to Lie algebras and representation theory*, third printing, revised, Graduate Texts in Mathematics, 9, Springer-Verlag, New York-Berlin, 1980. S. Kobayashi, K. Nomizu, *Foundations of differential geometry*, I, Wiley Classics Library (reprint of the 1963 original), Wiley-Interscience Publ., Wiley, New-York, 1996. S. Marchiafava, L. Ornea, R. Pantilie, Twistor Theory for CR-quaternionic manifolds and related structures, Preprint, I.M.A.R., Bucharest, 2009, ([arXiv:09051455](http://arxiv.org/abs/0905.1455)). R. Pantilie, On a class of twistorial maps, *Differential Geom. Appl.*, [**26**]{} (2008) 366–376. R. Pantilie, J. C. Wood, Harmonic morphisms with one-dimensional fibres on Einstein manifolds, *Trans. Amer. Math. Soc.*, [**354**]{} (2002) 4229–4243. I. Vaisman, *Lectures on the geometry of Poisson manifolds*, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994. I. Vaisman, Reduction and submanifolds of generalized complex structures, *Differential Geom. Appl.*, [**25**]{} (2008) 147–166. B. Watson, Almost Hermitian submersions, *J. Differential Geometry*, [**11**]{} (1976) 147–165. A. Weinstein, The local structure of Poisson manifolds, *J. Differential Geometry*, [**18**]{} (1983) 523–557. [^1]: The authors gratefully acknowledges that this work was partially supported by a CNCSIS Grant code 1193

--- abstract: 'A fast algorithm for Antoine and Vandergheynst’s (1998) directional Continuous Spherical Wavelet Transform () is presented. Computational requirements are reduced by a factor of $\complexity(\sqrt{\num_{\rm pix}})$, when $\num_{\rm pix}$ is the number of pixels on the sphere. The spherical  wavelet Gaussianity analysis of the  1-year data performed by Vielva  (2003) is reproduced and confirmed using the fast . The proposed extension to directional analysis is inherently afforded by the fast  algorithm.' address: 'Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, U.K.' author: - 'J.D. MCEWEN, M.P. HOBSON, A.N. LASENBY and D.J. MORTLOCK' title: | A FAST DIRECTIONAL CONTINUOUS SPHERICAL\ WAVELET TRANSFORM --- Introduction ============ A range of primordial processes imprint signatures on the temperature fluctuations of the  (). For instance, various cosmic defect and non-standard inflationary models predict non-Gaussian anisotropies. By studying the Gaussianity of the  anisotropies evidence may be provided for competing scenarios of the early Universe. In addition, a number of astrophysical processes introduce secondary sources of non-Gaussianity. Measurement systematics or contamination may also be highlighted by Gaussianity analysis. Wavelets are a powerful tool for probing the Gaussianity of anisotropies. Previous wavelet analysis of the , however, has been restricted to simple spherical Haar and isotropic Mexican Hat wavelets. A directional analysis on the full sky has previously been prohibited by the computational infeasibility of any implementation. We rectify this problem by providing a fast algorithm for Antoine and Vandergheynst’s[@antoine:1998] Continuous Spherical Wavelet Transform (), based on the fast spherical convolution proposed by Wandelt and Górski[@wandelt:2001]. The remainder of this paper is organised as follows. The  is presented in and the fast implementation in . In the fast  is applied to reproduce the non-Gaussianity  analysis performed by Vielva [@vielva:2003]. Concluding remarks are made in . A directional Continuous Spherical Wavelet Transform {#sec:cswt} ==================================================== Antoine and Vandergheynst[@antoine:1998] extend Euclidean wavelet analysis to spherical geometry by constructing a wavelet basis on the sphere. The natural extension of Euclidean motions on the sphere are rotations, defined by where we parameterise $\rho$ by the Euler angles $(\eulers)$. Dilations on the sphere, denoted $(\dil_\scale f)(\sa) = f_\scale(\sa)$, are constructed by first lifting the sphere  to the plane by a norm preserving stereographic projection from the South pole, performing the usual Euclidean dilation in the plane, before re-projecting back onto . Mother spherical wavelets are constructed simply by projecting Euclidean planar wavelets onto the sphere by a norm preserving inverse stereographic projection. A wavelet basis on  may be constructed from rotations and dilations of an admissible mother spherical wavelet. The corresponding wavelet family $\{ \wav_{\scale,\rho} \equiv \rot_\rho \dil_\scale \wav, \, \rho \in SO(3), \, \scale \in \real_{\ast}^{+} \}$ provides an over-complete set of functions in $L^2(\sphere)$. The  is given by the projection onto each wavelet basis function $$\skywav(\scale, \eulers) = \int_{\sphere} (\rot_{\eulers} \wav_\scale)^\conj(\sa) \: \sky(\sa) \: d\mu(\sa) \spcend , \label{eqn:cswt}$$ where the  denotes complex conjugation and $d\mu(\sa)=\sin(\saa) \, d\saa \, d\sab$ is the usual rotation invariant measure on the sphere. Fast algorithm {#sec:fast} ============== A direct implementation of the  is simply not computationally feasible for a data set of any practical size; a fast algorithm is essential. At a particular scale the  is essentially a spherical convolution, hence it is possible to apply Wandelt and Górski’s[@wandelt:2001] fast spherical convolution algorithm to rapidly evaluate the transform. Fast implementation {#sec:harmonic} ------------------- There does not exist any finite point set on the sphere that is invariant under rotations, hence it is more natural, and in fact more accurate for numerical purposes, to recast the  in spherical harmonic space. The Wigner rotation matrices (defined by Brink and Satchler[@brink:1993], for example) introduced to characterise the rotation of a spherical harmonic may be decomposed as $ \dmatbig_{mm\p}^{l}(\eulers) = e^{-\img m\eulera} \: \dmatsmall_{mm\p}^l(\eulerb) \: e^{-\img m\p\eulerc} $. This decomposition is exploited by factoring the rotation into two separate rotations, both of which contain a constant $\pm \pi/2$ polar rotation: $ \rot_{\eulers} = \rot_{\eulera-\pi/2, \; -\pi/2, \; \eulerb} \:\: \rot_{0, \; \pi/2, \; \eulerc+\pi/2} $. By uniformly sampling and applying some algebra the  may be recast as $$\skywav [\ind_\eulera, \ind_\eulerb, \ind_\eulerc] = e^{-\img 2\pi [ \frac{\ind_\eulera \lmax}{\num_\eulera} + \frac{\ind_\eulerb \lmax}{\num_\eulerb} + \frac{\ind_\eulerc \mmax}{\num_\eulerc}]} \sum_{j=0}^{\num_\eulera-1} \: \sum_{j\p=0}^{\num_\eulerb-1} \: \sum_{j\pp=0}^{\num_\eulerc-1} \cswtfftterm_{j, j\p, j\pp} \: e^{ \img 2\pi [ \frac{j\ind_\eulera}{\num_\eulera} + \frac{j\p \ind_\eulerb}{\num_\eulerb} + \frac{j\pp \ind_\eulerc}{\num_\eulerc}]} \spcend , \label{eqn:cswt_fast}$$ where the summation is simply the unnormalised  inverse discrete Fourier transform of $$\cswtfftterm_{m+\lmax, m\p+\lmax, m\pp+\mmax} = e^{i(m\pp-m)\pi/2} \sum_{l=\max(\mid m \mid, \mid m\p \mid, \mid m\pp \mid)}^{\lmax} \dmatsmall_{m\p m}^l(\pi/2) \: \dmatsmall_{m\p m\pp}^l(\pi/2) \: \shcoeff{\wav}_{lm\pp}{}^\conj \: \shcoeff{\sky}_{lm} \spcend , \label{eqn:cswt_fast_term}$$ where $\shcoeff{\cdot}_{lm}$ denote spherical harmonic coefficients, $\lmax$ and $\mmax$ define the general and azimuthal band limits of the wavelet respectively and the shifted indices show the conversion between the harmonic and Fourier conventions. The  may be performed very rapidly in spherical harmonic space by using fast Fourier techniques to rapidly and simultaneously evaluate , once is precomputed.[^1] Comparison with other algorithms -------------------------------- Direct and semi-fast (where only one rotation is performed in Fourier space) implementations of the  are also possible. A comparison of the theoretical complexity and typical execution times of each algorithm is presented in . The fast  algorithm provides a saving of $\complexity(\sqrt{\num_{\rm pix}})$ for $\num_{\rm pix}$ pixels on the sphere.  non-Gaussianity analysis {#sec:cmb} ========================= We reproduce the Gaussianity analysis of Vielva [@vielva:2003], preprocessing the  data in the same manner. The resolution of the co-added map defined by Komatsu [@komatsu:2003] is down-sampled by a factor or 4, before the *[Kp0]{}* exclusion mask is applied to remove emissions due to the Galactic plane and known point sources. Spherical wavelet analysis -------------------------- The  is a linear operation, hence the wavelet coefficients of a Gaussian map will also obey a Gaussian distribution. To test for deviations from Gaussianity, skewness and kurtosis statistics are calculated for each wavelet coefficient map at each scale. Monte Carlo simulations are performed to construct confidence bounds on the test statistics. The application of the *Kp0* mask distorts coefficients corresponding to wavelets that overlap with the mask exclusion region. These wavelet coefficients must be removed from any subsequent analysis. Our construction of an extended coefficient exclusion mask differs to that of Vielva  and inherently accounts for the dominant distortion (either point-source or Galactic plane) at each scale. The only non-zero coefficients in a  of the original mask are those that are distorted (due to the zero-mean property of spherical wavelets). These may be easily detected and the coefficient exclusion mask extended accordingly. Results ------- We reproduce the results of Vielva [@vielva:2003] for the spherical  wavelet analysis of the co-added  data. The  wavelet scales $\{ \scale_i \}_{i=1}^{11} = \{ 14, 25, 50, 75, 100, 150, 200, \linebreak 250, 300, 400, 500\}_{i=1}^{11}$ acrmin are considered, corresponding to an effective size of the sky of $\effsize_i=4\tan^{-1}( a_i/\sqrt{2} ) \approx 2\sqrt{2} \, \scale_i $ (defined as the angular separation between opposite zero-crossings). shows the skewness and kurtosis of the coefficients at each scale. The wavelet analysis inherently allows one to localise signal components on the sky, as illustrated in . We make similar observations to Vielva [@vielva:2003], although the different coefficient exclusion masks [produce]{} slight discrepancies. These discrepancies do not alter the general findings of the analysis. Conclusions and future work {#sec:conc} =========================== A fast algorithm is presented and evaluated for performing a directional  on the sphere. The fast implementation reduces the complexity of the  by $\complexity(\sqrt{\num_{\rm pix}})$, where $\num_{\rm pix}$ is the number of pixels on the sphere. Furthermore, the numerical accuracy of the  is improved by elegantly representing rotations in harmonic space. The Gaussianity analysis of the  1-year data performed by Vielva [@vielva:2003] has been reproduced and confirmed using the fast . We consider the extension to a full directional analysis in an upcoming publication by McEwen [@mcewen:2004]; preliminary findings indicate deviations from Gaussianity outside of the 99% confidence level. References {#references .unnumbered} ========== [99]{} , [*J. Math. Phys.*, 39, 8, 3987–4008]{}. [(1998)]{}. , [*Angular Momentum* (3rd Ed.)]{}, [Clarendon Press, Oxford]{}, [(1993)]{}. , [*ApJ*, 148, 119]{}, [(2003)]{}. , [*Preprint* (/0406604)]{}, submitted to MNRAS, [(2004)]{}. , [*Preprint* (/0310273)]{}, submitted to ApJ, [(2003)]{}. , [*Phys. Rev.*, 63, 123002, 1–6]{}, [(2001)]{}. [^1]: Memory and computational requirements may be reduced by a further factor of two for real signals by exploiting the conjugate symmetry relationship $\cswtfftterm_{-m,-m\p,-m\pp}=\cswtfftterm_{m,m\p,m\pp}^\conj$.

--- abstract: 'Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. We prove that $p$ does not divide the conjugacy class size (index) of each $p$-regular element of prime power order $x\in A\cup B$ if and only if $G$ is $p$-decomposable, i.e. $G=O_p(G) \times O_{p''}(G)$.' author: - | María José Felipe[^1], Lev S. Kazarin[^2],\ Ana Martínez-Pastor, and Víctor Sotomayor date: '*Dedicated to the memory of Carlo Casolo*' title: On products of groups and indices not divisible by a given prime --- Introduction and statement of results ===================================== All groups considered in this paper are finite. In recent years a new research line its being developed in the confluence of two well-established areas of study in group theory. On the one hand the theory of products of groups, and on the other hand the study of the influence of conjugacy class sizes, also called *indices*, on the structure of finite groups. The present paper is a contribution to this current development. Regarding products of groups, the main objective is to obtain information about the structure of a factorised group from the one of the subgroups in the factorization (and vice versa). In this setting, the fact that a product of two normal supersoluble groups is not necessarily supersoluble, has led to the approach of assuming certain permutability relations between the factors involved (see [@BEA] for a detailed account). In particular, among others, *mutually permutable products* have been considered. These are factorised groups $G = AB$ such that $A $ permutes with every subgroup of $B$ and $B$ permutes with every subgroup of $A$. Besides, during the last decades, several authors have carried out in-depth investigations with the purpose of understanding how the structure of a finite group is affected by the indices of its elements. In particular, it has been examined whether the indices of some subsets of elements are enough in order to provide features of the group. The survey [@CC] gives a general overview about this subject until 2011. In the recent approach, which combines the above mentioned lines within the theory of groups, the main aim is to analyze how the indices of some elements in the factors of a factorised group influence the structure of the whole group. Most of the contributions in this framework consider additional hypotheses on the subgroups in the factorization. Some of them ([@LWW; @ZGS]) impose some (sub)normality conditions on either both factors. Other papers consider mutually permutable products (see [@BCL; @CL; @FMOsquare]). Recent work done by some of the authors ([@FMOvan; @FMOpi]) extends previous developements by considering some special type of factorisations, the so-called *core-factorisations*. Only [@FMOprime] treats prime power indices without considering any additional restriction on the factors. On the other hand, it is to be said that in most cases the conditions on the indices are imposed only on some subsets of elements of the factors, namely prime power order elements, $p$-regular elements, zeros of irreducible characters, etc. The notation and terminology are as follows. For an element $x$ in a group $G$, we call $i_G(x)$ the *index of $x$*, i.e $i_G(x)=|G:{{\operatorname}{C}_{G}(x)}|$. A *$p$-regular* element is an element whose order is not divisible by $p$, where $p$ will always be a prime number. If $n$ is a positive integer, $n_p$ denotes the highest power of $p$ dividing $n$. We represent by $\pi(G)$ the set of all prime divisors of $|G|$. The set of all Sylow $p$-subgroups of $G$ is ${{\operatorname}{Syl}_{p}\left(G\right)}$ and ${{\operatorname}{Hall}_{\pi}\left(G\right)}$ is the set of all Hall $\pi$-subgroups of $G$ for a set of primes $\pi$. A group such that $G={{\operatorname}{O}_{\pi}(G)}\times {{\operatorname}{O}_{\pi'}(G)}$ is said to be *$\pi$-decomposable*. If $H$ is a subgroup of $G$, we denote by $H^G$ the normal closure of $H$ in $G$. The remaining notation and terminology are standard within the theory of finite groups, and they mainly follow those of the book [@DH], apart from some terminology on simple groups which will be introduced later. It is well known that if $p$ does not divide $i_G(x) $ for every $p$-regular element in a group $G$, then the Sylow $p$-subgroup is a direct factor of $G$ (see for instance [@CC Lemma 2]). This result was improved in [@LWW Theorem 5] by proving that the same conclusion remains true if the conditions on the indices are only imposed on $p$-regular elements of prime power order. In this paper, we deal with the corresponding result for factorised groups, but avoiding the consideration of any additional conditions on the factors, as were considered in [@BCL; @FMOpi; @ZGS]. This means that, in contrast to some of the mentioned results whose proofs are elementary, the classification of finite simple groups (CFSG) has been used in our proof. In particular, we derive some results on the center of the prime graph of an almost simple group, which will be used as a tool. The aim of this paper is then to prove the following result: \[mainth\] Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. Then $p$ does not divide $i_G(x)$ for every $p$-regular element of prime power order $x\in A\cup B$ if and only if $G$ is $p$-decomposable, i.e. $G=O_p(G) \times O_{p'}(G)$. Note that if $G$ is $p$-decomposable, then clearly the conditions on the indices hold. For the converse, the following lemma shows that only the existence of a unique Sylow $p$-subgroup should be proved. \[pclos\] Let the group $G = AB$ be the product of the subgroups $A$ and $B$, and let $p$ be a prime. If $p$ does not divide $i_G(x)$ for every $p$-regular element of prime power order $x\in A\cup B$, then the following statements are equivalent: - $G$ is $p$-closed, i.e. $G$ has a normal Sylow $p$-subgroup. - $G$ is $p$-decomposable. Clearly, it is enough to prove that (i) implies (ii). Let $P\in{{\operatorname}{Syl}_{p}\left(G\right)}$ and assume that $P \unlhd G$. Since $p$ does not divide $i_G(x)=|G : {{\operatorname}{C}_{G}(x)}|$ it follows that $P\leq {{\operatorname}{C}_{G}(x)}$ for every $p$-regular element of prime power order $x\in A\cup B$. Since $G$ is $p$-separable, by Lemma \[1.3.2\], we may consider $H$ a Hall $p'$-subgroup of $G$ such that $H= (H \cap A)(H \cap B)$. Hence, for every element $x\in (H \cap A) \cup (H \cap B)$ of prime power order, it holds that $P\leq {{\operatorname}{C}_{G}(x)}$. Therefore $[P, H]=1$ and (ii) follows. As an inmediate consequence of the Main Theorem, we get: \[all\] Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. Then $p$ does not divide $i_G(x) $ for every element of prime power order $x\in A\cup B$ if and only if $G$ has a central Sylow $p$-subgroup, i.e. $G=O_p(G) \times O_{p'}(G)$ with $O_p(G)$ abelian. Our results provide an improvement of [@BCL Theorem 1.1] in the case of only two factors, since in that paper products of $n$ pairwise mutually permutable subgroups were considered. \[adolfo\] Let the group $G = AB$ be the mutually product of the subgroups $A$ and $B$, and let $p$ be a prime. Then: - No index $i_G(x) $, where $x$ is a $p$-regular element in $ A\cup B$, is divisible by $p$ if and only if $G=O_p(G) \times O_{p'}(G)$. - $i_G(x) $ is not divisible by $p$ for every element $ x \in A\cup B$ if and only if $G=O_p(G) \times O_{p'}(G)$ with $O_p(G)$ abelian. Finally, we also point out that [@FMOpi Theorem A] and [@ZGS Theorem 3.2] when $\pi=p'$ are direct consequences from our main result. Preliminary results =================== We will use without further reference the following elementary lemma: Let $N$ be a normal subgroup of a group $G$, and $H$ be a subgroup of $G$. We have: - $i_N(x)$ divides $i_G(x) $, for each $x\in N$. - $i_{G/N}(xN)$ divides $i_G(x) $, for each $x\in G$. - If $xN$ is a $\pi$-element of $HN/N$, for a set of primes $\pi$, then there exists a $\pi$-element $x_1\in H$ such that $xN = x_1N$. We will also need the following fact about Hall subgroups of factorised groups, which is a convenient reformulation of [@AFG 1.3.2]. We recall that a group is a D$_{\pi}$-group, for a set of primes $\pi$, if every $\pi$-subgroup is contained in a Hall $\pi$-subgroup, and any two Hall $\pi$-subgroups are conjugate. It is well known that any $\pi$-separable group is a D$_{\pi}$-group. Also, all finite groups are D$_{\pi}$-groups when $\pi$ consists of a single prime. \[1.3.2\] Let $G=AB$ be the product of the subgroups $A$ and $B$. Asume that $A, B$, and $G$ are D$_{\pi}$-groups for a set of primes $\pi$. Then there exists a Hall $\pi$-subgroup $H$ of $G$ such that $H= (H \cap A)(H \cap B)$, with $H \cap A$ a Hall $\pi$-subgroup of $A$ and $H \cap B$ a Hall $\pi$-subgroup of $B$. Next we record some arithmetical lemmas, that will be applied later. \[SylowSym\] Let $G$ be the symmetric group of degree $k$ and let $s$ be a prime. If $s^{ N}$ is the largest power of $s$ dividing $|G|=k!$, then $N \leq \frac{k-1}{s-1}$. \[cuentas\] Let $q,s,t$ be positive integers. Then: 1. $(q^s-1,q^t-1)=q^{(s,t)}-1$, 2. $(q^s+1,q^t+1)=\begin{cases} q^{(s,t)}+1 \quad \text{if both } s/(s,t) \text{ and } t/(s,t) \text{ are odd,}\\ (2,q+1) \quad \text{otherwise,}\end{cases}$ 3. $(q^s-1,q^t+1)=\begin{cases} q^{(s,t)}+1 \quad \text{if } s/(s,t) \text{ is even and } t/(s,t) \text{ is odd,}\\ (2,q+1) \quad \text{otherwise.}\end{cases}$ We introduce now some additional terminology. Let $n$ be a positive integer and $p$ be a prime number. A prime $r$ is said to be *primitive with respect to the pair $(p, n)$* (or a *primitive prime divisor of $p^n-1$*) if $r$ divides $p^n-1$ but $r$ does not divide $p^k-1$ for every integer $k$ such that $1\leq k< n$. \[Zsi\] Let $n$ be a positive integer and $p$ a prime. Then: - If $n \geq 2$, then there exists a prime $r$ primitive with respect to the pair $(p, n)$ unless $n=2$ and $p$ is a Mersenne prime or $(p, n)=(2, 6)$. - If the prime $r$ is primitive with respect to the pair $(p, n)$, then $r-1 \equiv 0\,(\mbox{mod }n)$. In particular, $r \geq n+1$. The following lemmas are used when dealing with prime power order elements. We remark that the proof of the first one uses CFSG. \[FKS\] Let $G$ be a group acting transitively on a set $\Omega$ with $|\Omega|>1$. Then there exists a prime power order element $x\in G$ which acts fixed-point-freely on $\Omega$. \[feinkantor\] Let $H$ be a subgroup of a group $G$. If every prime power order element of $G$ lies in $\bigcup_{g\in G} H^g$, then $G=H$. If $H$ is normal in $G$, then every prime power order element belongs to $H$, and since $G$ is generated by such elements, we get $G=H$. So we may assume that $H$ is not normal in $G$. Note that $G$ acts on $\Omega:=\{H^g\: : \: g\in G\}$ transitively. If $H<G$, then certainly $|\Omega|>1$ and, by Lemma \[FKS\], there exists a prime power order element $x\in G$ acting fixed-point-freely on $\Omega$. But the hypotheses imply that $x\in H^z$ for some $z\in G$, so $H^{zx}=H^z$ and this is a contradiction. Preliminaries on (almost) simple groups and their prime graphs ============================================================== We begin this section with a useful result on the centralisers of automorphisms of simple groups, which is a refinement of [@DPSS Lemma 2.6]. In fact, the own proof of that lemma provides this stronger result: \[aut\] Let $N$ be a simple group. Then there exists $r\in \pi(N) \setminus\pi({\operatorname}{Out}(N))$ such that $(r, {\ensuremath{\left| {{\operatorname}{C}_{N}(\alpha)} \right|}})=1$ for every non-trivial $\alpha \in {\operatorname}{Out}(N)$ of order coprime to ${\ensuremath{\left| N \right|}}$. Following the proof of [@DPSS Lemma 2.6], we can assume that $N=G(q)$ is a simple group of Lie type, with $q=p^{e}$, $p$ a prime and $e \geq 3$ a positive integer. In that proof it is shown that the prime $r$ is in fact a primitive prime divisor of $p^{em}-1$ for some integer $m \geq 2$, and that such $r$ always exist under the given assumptions. Now having in mind the orders of the outer automorphisms of the simple groups of Lie type (see for instance [@LPS Table 2.1]) and applying Lemma \[Zsi\] we can deduce that $r \not\in \pi({\operatorname}{Out}(N))$ (see also [@LPS 2.4. Proposition B]). We will denote the *prime graph* of a group $G$, also called, the *Grünberg-Kegel* graph, by $\Gamma(G)$. The set of vertices of such graph is the set $\pi(G)$ of prime divisors of $|G|$, and two vertices $r,s$ are adjacent in $\Gamma(G)$ if there exists an element of order $rs$ in $G$. The connected components of the prime graph of a simple group are known from [@Wil] and [@Kon]. We will denote by $\mathcal{Z}( \Gamma(G))$, the center of the graph, i.e. $\mathcal{Z}( \Gamma(G))=\{p \, | \, p \mbox{ is adjacent to } r, \forall r \in \pi(G)\}$. The following result on the center of the prime graph of alternating and symmetric groups will be used later: \[angraph\] Let $n \geq 5, n\neq 6$ be a positive integer. Let $k$ be the largest positive integer such that $\{n, n-1, \ldots, n-k+1\}$ are consecutive composite numbers. If $k=1$, then both $\Gamma(A_n)$ and $\Gamma(\Sigma_n)$ are non-connected. For $k \geq 2$, let $t$ be the largest prime number such that $t \leq k$. Then: - If $k=2$, then $\Gamma(A_n)$ is non-connected and ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{2\}$. - If $k=3$, then ${{\operatorname}{\mathcal{Z}}(\Gamma(A_n))}=\{3\}$ and ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{2,3\}$. - If $k\geq 4$, then ${{\operatorname}{\mathcal{Z}}(\Gamma(A_n))}={{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{s\in\pi(\Sigma_n)\, | \, 2\leq s \leq t\}$. It is well known that two odd primes $s, u$ are adjacent in $\Gamma(A_n)$, and so in $\Gamma(\Sigma_n)$, if and only if $s+u \leq n$. On the other hand, if $s$ is an odd prime, $s$ is adjacent to $2$ in $\Gamma(A_n)$ if and only if $s+4 \leq n$, and $2,s$ are adjacent in $\Gamma(\Sigma_n)$ only when $s+2 \leq n$. It is then clear that if $k=1$, then both $\Gamma(A_n)$ and $\Gamma(\Sigma_n)$ are non-connected. Consider the prime $r:=n-k$, which is the largest prime divisor of $n!$ by the choice of $k$. Clearly, $r>\frac{n}{2}>k=n-r \geq t$. Thus $r+t \leq n$, and we deduce that $t\in {\operatorname}{Z}(\Gamma(\Sigma_n))$. If $k=2$, then $r=n-2$, and so ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{2\}$. Since $r+4>n$, then $\Gamma(A_n)$ is non-connected. If $k=3$, then $r=n-3$. It follows that ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{2, 3\}$ and ${{\operatorname}{\mathcal{Z}}(\Gamma(A_n))}=\{3\}$. Finally, let us suppose that $k\geq 4$, so $n\geq 11$. Take a prime $s\in\pi(\Sigma_n)$. If $s\leq t$, then $r+s\leq r+t\leq r+k=n$, and so $s$ lies in ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}$. Assume now $s>t$, so $s>k$. It is known that there exist two primes $\frac{n}{2}<r_1<r_2\leq n$, and we may take $r_2=r$. If $s\neq r$, then $s+r=s+n-k>n$. If $s=r$, then $s+r_1>\frac{n}{2}+\frac{n}{2}=n$. Hence, in both cases $s\notin {{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}$. This proves that ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{s\in\pi(\Sigma_n)\, : \, 2\leq s \leq t\}$. Finally, since $k\geq 4$, then $r\leq n-4$, so $r+4\leq n$ and $2\in {\operatorname}{Z}(\Gamma(A_n))$. Therefore ${{\operatorname}{\mathcal{Z}}(\Gamma(A_n))}={{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}$. For the special case of the alternating group $A_6$ and its group of automorphisms we can derive the following result from [@Luc Lemma 2]: \[a6\] If $A_6=N\unlhd G \leq {\operatorname}{Aut}(N)$, then $\Gamma(G)$ is non-connected, except when $G={\operatorname}{Aut}(N)$. In this last case ${{\operatorname}{\mathcal{Z}}(\Gamma(G))}=\{2\}$. Also, for sporadic groups the following result is well known (see [@Atl] or [@Wil Theorem 2] and [@Luc Theorem 3]): \[sporgraph\] If $N$ is an sporadic simple group, then $\Gamma(N)$ is non-connected. Moreover, $\Gamma({{\operatorname}{\textup{Aut}}({N})})$ is also non-connected, except when $N= McL$ or $N=J_2$, and in both cases ${{\operatorname}{\mathcal{Z}}(\Gamma({{\operatorname}{\textup{Aut}}({N})}))}=\{2\}$. We will use the following facts on groups of Lie type in the proof of our Main Theorem. In the sequel, for $q=t^e$, $e\ge 1$, we will denote by $q_n$ *any* primitive prime divisor of $t^{en}-1$, i.e. primitive with respect to $(t, ne)$. \[class\] For $N=G(q)$ a classical simple group of Lie type of characteristic $t$ and $q=t^e$, there exist primes $r, \, s \in \pi(N) \setminus (\pi({{\operatorname}{\textup{Out}}({N})}) \cup \{t\})$ and maximal tori $T_1$ and $T_2$ of $N$, of orders divisible by $r$ and $s$, respectively, with $( |T_1|, |T_2|)=1$, as stated in Table 1. (In such table for the case $\star$, $l$ denotes a Mersenne prime.) Moreover, for the groups $N$ listed in Table 2, there exist a prime $s \in \pi(N) \setminus (\pi({{\operatorname}{\textup{Out}}({N})}) \cup \{t\})$ and a Sylow $s$-subgroup of order $s$ which is self-centralising in $N$. If $N=L_2(q)$, $C_N(x)$ is a $t$-group for each $t$-element $x \in N$. If $N=L_3(q)$, there exists a maximal torus $T$ of order $(1/d) (q^2+q+1)$, $d=(3, q-1)$, such that each prime $r \in \pi(T)$ is a primitive prime divisor of $q^3-1$ (for $q\neq 4$), and $(|T|, 2t)=1$. If $N=U_3(q)$, there exists a maximal torus $T$ of order $(1/d) (q^2-q+1)$, $d=(3, q+1)$, such that each prime $r \in \pi(T)$ is a primitive prime divisor of $q^6-1$, and $(|T|, 2t)=1$. $$\begin{array}{c|c|c|c|c|c} \hline & & & & & \\ N& r &s& |T_1|& |T_2| & Remarks \\ & & & & & \\ \hline & & & & & \\ L_n(q)& q_n &q_{n-1} & \frac{q^n-1}{(n, q-1)(q-1)}&\frac{q^{n-1}-1}{(n, q-1)}& (n, q) \neq (6, 2)\\ n \geq 4 & & & & & (n, q) \neq (4, 4), (7, 2)\\ && s=7&&&(n,q)=(4,4)\\ & & & & & \\ \hline & & & & & \\ U_{n}(q) & q_{n} & q_{2(n-1)} &\frac{q^{n}-1}{(n, q+1)(q+1)}& \frac{(q^{n-1}+1)}{(n, q+1)}& n \mbox{ even } \\ & & & & & (n, q) \neq (4, 2), (6, 2) \\ & & & & & \\ n\geq 4 & q_{2n} & q_{n-1} & \frac{q^{n}+1}{(n, q+1)(q+1)} & \frac{q^{n-1}-1}{(n, q+1)}& n \mbox{ odd} \\ & & & & & (n, q) \neq (7, 2) \\ & & & & & \\ \hline & & & & & \\ PSp_{4}(q) & q_{4} & q_{2} & \frac{q^{2}+1}{(2, q-1)}& \frac{(q^{2}-1)}{(2, q-1)}& q\neq 8, l \quad (\star) \\ && s=7&&& q=8\\ && s \neq 2&&& q=l \\ & & & & & \\ \hline & & & & & \\ PSp_{2n}(q) & q_{2n} & q_{2(n-1)} & \frac{q^{n}+1}{(2, q-1)}& \frac{(q^{n-1}+1)(q-1)}{(2, q-1)}& n \mbox{ even } \\ & && & & (n, q) \neq (4,2) \\ P\Omega_{2n+1}(q) & & & & & \\ & q_{2n} & q_{n} & \frac{q^{n}+1}{(2, q-1)} & \frac{(q^{n}-1)}{(2, q-1)}& n \mbox{ odd } \\ n\geq 3 & & & & & (n, q) \neq (3, 2) \\ & & & & & \\ \hline & & & & & \\ P\Omega_{2n}^{-}(q) & q_{2n} & q_{2(n-1)} & \frac{q^{n}+1}{(4, q^n+1)}& \frac{(q^{n-1}+1)(q-1)}{(4, q^n+1)}& (n, q) \neq (4, 2) \\ n \geq 4 &&&&&\\ & & & & & \\ \hline & & & & & \\ P\Omega_{2n}^{+}(q) & q_{2(n-1)} & q_{n-1} &\frac{(q^{n-1}+1)(q+1)}{(4, q^n-1)} &\frac{(q^{n-1}-1)(q-1)}{(4, q^n-1)} & n \mbox{ even } \\ & & &&&(n,q)\neq (4,2)\\ n \geq 4 & q_{2(n-1)} & q_{n} &\frac{(q^{n-1}+1)(q+1)}{(4, q^n-1)} & \frac{q^n-1}{(4, q^n-1)} & n \mbox{ odd}\\ & & & & & \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline \, N \, & \, s \, \\ \hline \, L_3(4) \, & \quad 7 \quad \\ \hline \, L_6(2) \, & \quad 31 \quad \\ \hline \, L_7(2) \, & \quad 127 \quad \\ \hline \, U_6(2) \, & \quad 11 \quad \\ \hline \, U_7(2) \, & \quad 43 \quad \\ \hline \, PSp_{4}(4) \, & \quad 17 \quad \\ \hline \, PSp_{6}(2) \, & \quad 7 \quad \\ \hline \, PSp_{8}(2) \, & \quad 17 \quad \\ \hline \, P\Omega_{8}^{-}(2) \, & \quad 17 \quad \\ \hline \, P\Omega_{8}^{+}(2) \, & \quad 7 \quad \\ \hline \end{array}$$ We recall that a torus is an abelian $t'$-group. The existence of the subgroups $T_1$ and $T_2$ appearing in Table 1 can be derived from the known facts about the maximal tori in these groups (see, [@Car Propositions 7-10] or [@VV Lemma 1.2]). The fact that the corresponding orders of the tori are coprime in each case can be deduced easily from Lemma \[cuentas\], while the assertion regarding the primitive prime divisors is deduced from Lemma \[Zsi\]. The information in Table 2 can be found either in [@Atl], or from the orders of maximal tori for the corresponding groups. Note that the case $PSp_{4}(2)\cong \Sigma_6$ has already been considered in Lemma \[a6\]. The assertion on $L_2(q)$ is well known (see for instance [@Car Proposition 7]). The existence of tori of the corresponding orders in $L_3(q)$ and $U_3(q)$ can be found in [@VV Lemma 1.2], and the claim on the prime divisors is easily deduced applying Lemma \[cuentas\]. \[excep\] For $N=G(q)$ an exceptional simple group of Lie type of characteristic $t$ and $q=t^e$, there exist primes $r, \, s \in \pi(N) \setminus (\pi({{\operatorname}{\textup{Out}}({N})}) \cup \{t\})$ and maximal tori $T_1$ and $T_2$ of $N$, of orders divisible by $r$ and $s$, respectively, with $( |T_1|, |T_2|)=1$, as stated in Table 3. In the cases denoted by $(\star)$, $r$ and $s$ denote the largest prime divisor of $|T_1|$ and $|T_2|$, respectively. The Tits group $N=F_4(2)'$ contains a Sylow 13-subgroup of order 13 which is self-centralising. $$\begin{array}{c|c|c|c|c|c} \hline & & & & & \\ N& r &s& |T_1|& |T_2| & Remarks \\ & & & & & \\ \hline & & & & & \\ G_2(q)& q_3 &q_{6} & q^2+q+1&q^2-q+1& q\neq 4\\ q > 2 & r=7 & & & & q=4\\ & & & & & \\ \hline & & & & & \\ F_{4}(q) & q_{8} & q_{12} &q^{4}+1 & q^4-q^2+1& \\ & & & & & \\ \hline & & & & & \\ E_6(q)& q_{9} & q_{12} & \frac{q^{6}+q^3+1}{(3, q-1)}& \frac{(q^4-q^2+1)(q^2+q+1)}{(3, q-1)}& \\ & & & & & \\ \hline & & & & & \\ E_7(q)& q_{9} & q_{14} & \frac{(q^6+q^3+1)(q-1)}{(2, q-1)} & \frac{q^{7}+1}{(2, q-1)}& \\ & & & & & \\ \hline & & & & & \\ E_8(q)& q_{20} & q_{24} & q^8-q^{6}+q^4-q^2+1& q^8-q^4+1& \\ & & & & & \\ \hline & & & & & \\ {}^3D_4(q)& q_{3} & q_{12} &(q^{3}-1)(q+1)& q^4-q^2+1& \\ & & & & & \\ \hline & & & & & \\ {}^2B_2(q) & r & s & q+\sqrt{2q}+1& q-\sqrt{2q}+1& (\star) \\ q=2^{2m+1} >2 &&&&&\\ & & & & & \\ \hline & & & & & \\ {}^2G_2(q) & r & s & q+\sqrt{3q}+1& q-\sqrt{3q}+1& (\star) \\ q=3^{2m+1} >3 &&&&&\\ & & & & & \\ \hline & & & & & \\ {}^2F_4(q) & r & s & \small{q^2+q \sqrt {2q}+q+\sqrt{2q}+1}& (q-\sqrt{2q}+1)(q-1)& (\star) \\ q=2^{2m+1} >2 &&&&&\\ & & & & & \\ \hline & & & & & \\ {}^{2}E_6(q)& q_{18} & q_{12} & \frac{q^{6}-q^3+1}{(3, q+1)}& \frac{(q^4-q^2+1)(q^2-q+1)}{(3, q+1)}& \\ & & & & & \\ \hline \end{array}$$ The existence of the subgroups $T_1$ and $T_2$ appearing in Table 3 can be derived from the information about the maximal tori in these groups (see [@VV Lemma 1.3] and [@VV2 Lemma 2.6]). The fact that they are coprime can be deduced from Lemma \[cuentas\] having in mind that $|T_i|$ divides $q^n-1$ when we state that $q_n \in \pi(T_i)$, $i=1, 2$, (for the case ${}^3D_4(q)$, $|T_1|$ divides $q^6-1$), for all groups except ${}^2B_2(q), {}^2G_2(q), {}^2F_4(q) $. In the latter cases, denoted by $(\star)$, the information can be obtained from [@VV2 Lemma 2.8] The previous lemmas provide the following result on the center of the prime graph of a simple group of Lie type, which can also be derived from [@center Proposition 2.9]. If $N$ is a simple group of Lie type, then ${{\operatorname}{\mathcal{Z}}(\Gamma(N))}=\emptyset$. Let $t$ be the characteristic of the group of Lie type $N$. It is well known that $t \not \in {{\operatorname}{\mathcal{Z}}(\Gamma(N))}$. If $p \in {{\operatorname}{\mathcal{Z}}(\Gamma(N))}$, then for each prime $r \neq t$, there exists an abelian $t'$-subgroup of $N$ whose order is divisible by $p$ and $r$. Since any abelian $t'$-subgroup is contained in a maximal torus, this means that $p \in \pi(T)$ for each maximal torus $T$ of $N$. Hence, the information given in Lemmas \[class\] and \[excep\] leads to a contradiction. The minimal counterexample: reduction to the almost simple case =============================================================== In this section we will give a description of the structure of a minimal counterexample to our Main Theorem. Hence, having in mind Lemma \[pclos\], we assume the following hypotheses: (H1) : $p$ is a prime number. (H2) : $G$ is a group satisfying the following conditions: 1. $G=AB$ is the product of the subgroups $A$ and $B$, and $p$ does not divide $i_G(x) $ for every $p$-regular element of prime power order $x \in A \cup B$. 2. $G$ does not have a normal Sylow $p$-subgroup. Among all such groups we choose $(G, A, B)$ such that $|G|+|A|+|B|$ is minimal.\ For such a group $G$ we have the following results. \[0\] $G$ has a unique minimal normal subgroup $N$ which is not a $p$-group. Moreover, $P \neq 1$, $PN \unlhd G$, $G/N=PN/N \times O_{p'}(G/N)$, and $G=NN_{G}(P)$, for each $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$. Since the hypotheses (H2)(i) are clearly inherited by quotients of $G$, and the class of all $p$-closed groups is a saturated formation, we deduce that ${{\operatorname}{\Phi}(G)}=1$ and that $G$ has a unique minimal normal subgroup, say $N$. Since $G/N$ has a normal Sylow $p$-subgroup, then ${{\operatorname}{O}_{p}(G)}=1$, and so $N$ is not a $p$-group. Also this implies that $PN \unlhd G$ for each $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$, and that $G/N$ is $p$-decomposable, by Lemma \[pclos\], as claimed. The last assertion follows from Frattini’s argument. From now on $N$ is the unique minimal normal subgroup of $G$. \[1\] $G=APN=BPN$, for each $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$ . Let $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$. Since $PN\unlhd G$, take for instance $T:=APN$. Let us suppose that $T<G$. Note that $T=A(T \cap B)$. If we take any $p$-regular element of prime power order $x\in A\cup (T\cap B)$, since $G=NN_{G}(P)$, then, by our hypotheses, there exists some $n\in N\leq T$ such that $P^n\leq {{\operatorname}{C}_{G}(x)}$, where $P\in{{\operatorname}{Syl}_{p}\left(T\right)}$. Whence $T$ satisfies the hypotheses and, by minimality, we deduce that $P \unlhd T$. But this means, by Lemma \[pclos\], that $T$, and so $N$, is $p$-decomposable. Since $N$ is not a $p$-group, we deduce that $N={{\operatorname}{O}_{p'}(N)}\leq {{\operatorname}{O}_{p'}(T)}\leq {{\operatorname}{C}_{G}(P)}$. But then $P$ is normal in $G=NN_{G}(P)$, a contradiction. Therefore, $G=APN$ and, analogously, $G=BPN$. \[2\] Either $p\in\pi(A)$ or $p\in\pi(B)$. Moreover, if $X\in\{A, B\}$ and $p\in\pi(XN)$, then $G=XN$. The first assertion is clear since $G=AB$ and $p \in \pi(G)$. Without loss of generality, let us assume that $p\in\pi(AN)$. Consider $1\neq P_0\in{{\operatorname}{Syl}_{p}\left(AN\right)}$ and take $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$ with $P_0\leq P$, that is, $P_0=P\cap AN$. Set $H:=AN$ and observe that $H=A(H \cap B)$. Note that for each $p$-regular element of prime power order $x\in A \cup(H\cap B)$, there exists some $n\in N$ with $P^n\leq {{\operatorname}{C}_{G}(x)}$. Hence $x\in {{\operatorname}{C}_{H}(P^n)}\leq {{\operatorname}{C}_{H}(P_0^{n})}$ with $n\in N$, so $i_H(x)$ is not divisible by $p$. If $H<G$, then, by minimality, we deduce that $1\neq {{\operatorname}{O}_{p}(H)}=P_0=P\cap AN\leq AN=H$. Thus ${{\operatorname}{O}_{p}(H)}^G={{\operatorname}{O}_{p}(H)}^P$ because $G=APN$, by Lemma \[1\]. It follows that $1\neq {{\operatorname}{O}_{p}(H)}^G\leq P$, and so ${{\operatorname}{O}_{p}(G)}\neq 1$, a contradiction. \[3\] If $p\in\pi(N)$, then $G=AN=BN=AB$ and $N$ is a non-abelian simple group. Hence $N \unlhd G \leq {{\operatorname}{\textup{Aut}}({N})}$, i.e. $G$ is an almost simple group. The first assertion follows from Lemma \[2\]. Since ${{\operatorname}{O}_{p}(G)}=1$ and $p\in\pi(N)$, certainly $N$ is non-abelian. Set $N=N_1\times N_2\times \cdots \times N_r$ with $N_i \cong N_1$ a non-abelian simple group, for $i=2, \ldots, r$, and assume that $r >1$. Since $G=AN=BN$, both $A$ and $B$ act transitively by conjugation on the set $\Omega=\{N_1, \ldots, N_r \}$. Suppose first that there exists some $p$-regular element of prime power order $1 \neq x \in A \cup B$ such that $N_1^x=N_i$, for some $i >1$. By the hypotheses, there exists some $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$ such that $P\leq {{\operatorname}{C}_{G}(x)}$. Moreover, $1 \neq P\cap N\in{{\operatorname}{Syl}_{p}\left(N\right)}$, and so $1\neq P\cap N_1\in{{\operatorname}{Syl}_{p}\left(N_1\right)}$. It follows that $P\cap N_1=(P\cap N_1)^x=P\cap N_1^x=P\cap N_i$, therefore $1 \neq N_1 \cap N_i$, a contradiction. Hence, we may assume that any $p$-regular element of prime power order in $A \cup B$ normalises $N_1$, and hence $N_i$ for $i=2, \ldots, r$, since $A$ and $B$ both act transitively on $\Omega$. But this means that if $R:= \cap_{i=1}^{r} N_G(N_i)$, then $G=A_pR=B_pR$, for any $A_p \in {{\operatorname}{Syl}_{p}\left(A\right)}$ and $B_p \in {{\operatorname}{Syl}_{p}\left(B\right)}$. Therefore $G=PR$ for any $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$, and so $P$ acts transitively on $\Omega$. But this contradicts the fact that $1 \neq Z(P) \cap N \leq C_N(P)={{\operatorname}{C}_{N_1}(P)}\times \cdots \times {{\operatorname}{C}_{N_r}(P)}$, unless $r=1$. Therefore $N$ is a simple group as claimed. In the next section the case when $N$ is a $p'$-group will be discarded. Hence, by Lemma \[3\], the minimal counterexample to our Main Theorem will be an almost simple group $G$, $N \unlhd G \leq {{\operatorname}{\textup{Aut}}({N})}$, with $p \in \pi(N)$ and $G=AB=AN=BN$. In Section 5 we will analyse such almost simple groups satisfying the hypotheses of our Main Theorem, and all possible cases for the simple group appearing as the socle of such a group will be ruled out. Case $N$ is a $p'$-group. ------------------------- Assume from now on in this section that $N$ is a $p'$-group. Note that, in particular, $G/N$ is $p$-decomposable, by Lemma \[pclos\], and so $G$ is $p$-separable. \[4\] $G={{\operatorname}{O}_{p'}(G)}\langle y\rangle=N{{\operatorname}{C}_{G}(y)}$, where $1 \neq y \in A$ and $ \langle y \rangle \in {{\operatorname}{Syl}_{p}\left(G\right)}$. Further, $B$ is a $p'$-group. Recall that $G/N=PN/N \times O_{p'}(G/N)$, for $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$. Now since $N$ is a $p'$-group, it follows that $H:={{\operatorname}{O}_{p'}(G)}\in{{\operatorname}{Hall}_{p'}\left(G\right)}$, so it is the unique Hall $p'$-subgroup of $G$ and $H= (H \cap A)(H \cap B)$, by Lemma \[1.3.2\]. We may consider $P=(P\cap A)(P\cap B)$. For some $p$-element $y\in (P\cap A)\cup (P\cap B)$, set $H_y:={{\operatorname}{O}_{p'}(G)}\langle y \rangle$. Note that $H_y= (H_y \cap A)(H_y \cap B)$. Now if $x\in (H_y\cap A)\cup (H_y \cap B)$ is a $p$-regular element of prime power order, then there exists some $n\in N$ with $\langle y \rangle^n\leq P^n\leq {{\operatorname}{C}_{G}(x)}$, so $x\in{{\operatorname}{C}_{H_y}(\langle y\rangle^n)}$. As $\langle y\rangle^n$ is a Sylow $p$-subgroup of $H_y$ because $n\in N\leq {{\operatorname}{O}_{p'}(G)}$, then $H_y$ satisfies the hypotheses (H2). If ${\ensuremath{\left| H_y \right|}}<{\ensuremath{\left| G \right|}}$, by minimality we obtain that $H_y$ has a normal Sylow $p$-subgroup, and so $[y, {{\operatorname}{O}_{p'}(G)}]=1$. If this holds for every $y\in (P\cap A)\cup (P\cap B)$, then $[P, {{\operatorname}{O}_{p'}(G)}]=1$, a contradiction. Hence we may suppose that, for instance, there exists $y\in P\cap A$ with $H_y={{\operatorname}{O}_{p'}(G)}\langle y \rangle=G$. Further, since we are assuming that ${\ensuremath{\left| A \right|}}+{\ensuremath{\left| B \right|}}$ is minimal, then we deduce that $A=({{\operatorname}{O}_{p'}(G)}\cap A)\langle y \rangle$ and $B={{\operatorname}{O}_{p'}(G)}\cap B$. Now, by coprime action and minimality, ${{\operatorname}{O}_{p'}(G)}=[{{\operatorname}{O}_{p'}(G)}, y]{{\operatorname}{C}_{{{\operatorname}{O}_{p'}(G)}}(y)}\leq N{{\operatorname}{C}_{G}(y)}.$ Thus $G={{\operatorname}{O}_{p'}(G)}\langle y\rangle = N{{\operatorname}{C}_{G}(y)}$. \[ncapa\] $N \cap A \neq 1$. Assume that $N\cap A=1$. By Lemma \[1\], we know that $G/N=\langle y \rangle N/N \times {{\operatorname}{O}_{p'}(G)}/N$. Hence $[\langle y \rangle, {{\operatorname}{O}_{p'}(G)}\cap A]\leq N\cap A=1$, so $\langle y\rangle$ is a Sylow $p$-subgroup of $G$ which is normal in $A$. Now, since $B$ is a $p'$-group, we have that for any $b \in B$ of prime power order, there exists $g \in G=AB$ such that $\langle y\rangle^g \leq C_G(b)$. This implies that $\langle y\rangle^{b_1} \leq C_G(b)$ for some $b_1 \in B$, since $\langle y\rangle^a=\langle y\rangle$ for any $a \in A$. It follows that each element of prime power order of $B$ lies in $\underset{x\in B}{\cup}{{\operatorname}{C}_{B}(\langle y\rangle)}^x$ and so, by Lemma \[feinkantor\], we deduce $[B, \langle y\rangle]=1$, a contradiction which proves our claim. \[5\] $N$ is a non-abelian group, so $N=N_1\times N_2\times \cdots \times N_r$, with $N_i \cong N_1$ a non-abelian simple group. Assume that $N$ is abelian. Therefore ${{\operatorname}{C}_{N}(y)}\leq N$ is a normal subgroup of $G=N{{\operatorname}{C}_{G}(y)}$. Since $N$ is a minimal normal subgroup of $G$, we deduce that either $N={{\operatorname}{C}_{N}(y)}$ or ${{\operatorname}{C}_{N}(y)}=1$. The first case yields to the contradiction $G={{\operatorname}{C}_{G}(y)}$. So we may assume ${{\operatorname}{C}_{N}(y)}=1$. If we take $1 \neq x\in N\cap A$ of prime power order (which is a $p$-regular element) then, by our hypotheses, there exists some $n\in N$ such that $\langle y\rangle^n\leq {{\operatorname}{C}_{G}(x)}$, and so $x \in {{\operatorname}{C}_{N}(\langle y\rangle^n)}=({{\operatorname}{C}_{N}(\langle y\rangle)}^n$, a contradiction. \[xi\] ${\ensuremath{\left| N \right|}}{\ensuremath{\left| \langle y\rangle \right|}}{\ensuremath{\left| A\cap B \right|}}={\ensuremath{\left| \frac{G}{N} \right|}}{\ensuremath{\left| N\cap A \right|}}{\ensuremath{\left| N\cap B \right|}}.$ Recall that $p\in\pi(AN)\smallsetminus\pi(BN)$ and $G=AN=BPN$ for any $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$. Hence $G=AN=B\langle y\rangle N$. Now it is enough to make some computations having in mind that ${\ensuremath{\left| B\cap \langle y\rangle N \right|}}={\ensuremath{\left| B\cap N \right|}}$ (recall that $G$ is $p$-separable). \[7\] The Sylow $p$-subgroups of $G$ are cyclic of order $p$, i.e. $\langle y\rangle \cong C_p$. Take $x\in \langle y\rangle$ of order $p$, and set $H:=BN\langle x\rangle$; this is a subgroup of $G$ since $BN={{\operatorname}{O}_{p'}(G)}\unlhd G$. Assume that $H=(H \cap A) B<G$. Now if $h\in (H\cap A)\cup B$ is $p$-regular of prime power order, by the hypotheses there exists $n\in N$ with $\langle x\rangle ^n \leq \langle y\rangle^n \leq {{\operatorname}{C}_{G}(h)}$, so $h\in {{\operatorname}{C}_{H}(\langle x\rangle^n)}$, where $\langle x\rangle^n\in{{\operatorname}{Syl}_{p}\left(H\right)}$. By minimality, $\langle x\rangle={{\operatorname}{O}_{p}(H)}$, so (recall that $G=BPN$, by Lemma \[1\], with $P=\langle y \rangle$) $1\neq {{\operatorname}{O}_{p}(H)}^G={{\operatorname}{O}_{p}(H)}^{BNP}={{\operatorname}{O}_{p}(H)}^P\leq P$. This contradicts the fact ${{\operatorname}{O}_{p}(G)}=1$ and it proves the claim. \[new\] The subgroup $\langle y \rangle$ does not normalise $N_i$, for each $i \in\{1, \ldots, r\}$. In particular, $r > 1$. Assume that $\langle y \rangle$ normalises some $N_i$ with $i \in\{1, \ldots, r\}$. Then $\langle y \rangle$ normalises $N_i$ for each $i \in\{1, \ldots, r\}$. We can view $\langle y \rangle$ as a subgroup of ${\operatorname}{Aut}(N_i)$, because ${{\operatorname}{C}_{\langle y\rangle}(N_i)}=1$ (recall that $y$ has order $p$). By Lemma \[aut\], there exists a prime $s\in\pi(N_i)\smallsetminus\pi({\operatorname}{Out}(N_i))$ such that $(s, |{{\operatorname}{C}_{N_i}(y)}|)=1$. Therefore $s$ cannot divide $|{{\operatorname}{C}_{N}(y)}|$ as ${{\operatorname}{C}_{N}(y)}={{\operatorname}{C}_{N_1}(y)}\times \cdots \times {{\operatorname}{C}_{N_r}(y)}$. Since each element of prime power order in $(N \cap A) \cup (N \cap B)$ centralises some Sylow $p$-subgroup (because our hypotheses), we deduce that $\pi(N\cap A)\cup\pi(N\cap B)\subseteq \pi({{\operatorname}{C}_{N}(y)})$. Thus, this last property and Lemma \[xi\] yield $s\in\pi(G/N)$. Note that $G/N\lessapprox {\operatorname}{Out}(N)$ and ${\operatorname}{Out}(N)\cong {\operatorname}{Out}(N_1) \text{ wr } \Sigma_{r}$, , the natural wreath product of ${\operatorname}{Out}(N_1)$ with $\Sigma_r$. As $s$ does not divide ${\ensuremath{\left| {\operatorname}{Out}(N_i) \right|}}$ for any $i$, it follows that $s\in\pi(\Sigma_r)$. Using Lemma \[xi\], we obtain that ${\ensuremath{\left| N \right|}}_s$ divides ${\ensuremath{\left| G/N \right|}}_s$, and so it divides ${\ensuremath{\left| \Sigma_r \right|}}_s$. Set ${\ensuremath{\left| N_1 \right|}}_s:=s^d$. Then ${\ensuremath{\left| N \right|}}_s=s^{dr}$ divides $s^{\frac{r-1}{s-1}}$, by Lemma \[SylowSym\], so $dr\leq \frac{r-1}{s-1}$ and necessarily $d=0$, which contradicts the fact that $s$ divides ${\ensuremath{\left| N_i \right|}}$. Set $C:={{\operatorname}{C}_{{{\operatorname}{O}_{p'}(G)}\cap A}(y)}={{\operatorname}{C}_{{{\operatorname}{O}_{p'}(A)}}(y)}$ and $A_0:=\langle y \rangle \times C$. Then $A=(N\cap A)A_{0}$ and $G=AN=A_{0}N$. By the minimal choice of $G$, we deduce that $G/N\cong A/A\cap N$ is $p$-decomposable. Hence $[{{\operatorname}{O}_{p'}(G)}\cap A, \langle y\rangle ]\leq N\cap A$. Thus, by coprime action, ${{\operatorname}{O}_{p'}(G)}\cap A=[{{\operatorname}{O}_{p'}(G)}\cap A, \langle y\rangle]{{\operatorname}{C}_{{{\operatorname}{O}_{p'}(G)}\cap A}(y)}\leq (N\cap A){{\operatorname}{C}_{{{\operatorname}{O}_{p'}(G)}\cap A}(y)}$. Hence $A=(N\cap A){{\operatorname}{C}_{{{\operatorname}{O}_{p'}(G)}\cap A}(y)}\langle y\rangle=(N \cap A)A_0$, and the assertion follows. Recall that $N=N_1\times N_2\times \cdots \times N_r$ with $N_i \cong N_1$ a non-abelian simple group, and set $\Omega:=\{N_1, N_2, \ldots, N_r\}$. By Lemma \[new\], $r > 1$. As $G=A_0N$, $A_0$ acts transitively on $\Omega$. We adapt here some arguments used in [@KMP3] and we claim some facts about this action: - The orbits of $B$ on $\Omega$ are the same as those of $C$. This is clear because ${{\operatorname}{O}_{p'}(G)}={{\operatorname}{O}_{p'}(A)}N=CN=BN$. - Let $\Delta$ be an orbit of $C$ on $\Omega$ of minimal lenght. If $c\in C$, then $\Delta^{yc}=\Delta^{cy}=\Delta^y$, so $\Delta^y$ and $\Delta\cap \Delta^y$ are also orbits of $C$. Therefore, by the choice of $\Delta$, either $\Delta=\Delta^y$ (and hence $\Delta=\Delta^{y^{i}}$ for $i\in\{1,\ldots, p\}$), or $\Delta\cap \Delta^y=\emptyset$ (and hence $\Delta^{y^{i}}\cap \Delta^{y^{j}}=\emptyset$ for $i\neq j$, $i, j \in \{1,\ldots, p\}$). It follows that there is a partition of $\Omega$ of the form $$\Omega=\Delta_1 \cup \Delta_2 \cup \cdots \cup \Delta_k,$$ where $\Delta_i:=\Delta^{y^{i-1}}$ for $i\in\{1,\ldots, k\}$, and $k \in \{1, p\}$. Note that all $\Delta_i$ have the same length, say $m$, and $\{\Delta_1, \ldots, \Delta_k\}$ are all the $C$-orbits (and $B$-orbits) on $\Omega$. Note also that $m$ is a $p'$-number, since $C$ is a $p'$-subgroup. Moreover, $\langle y \rangle$ acts transitively on $\{\Delta_1, \Delta_2, \ldots, \Delta_k\}$. - The length of an orbit $\nabla$ of $\langle y \rangle$ on $\Omega$ is $k=p$, and there are $m$ orbits $\nabla_1:=\nabla$, $\nabla_2\dots, \nabla_m$. Hence there is a partition $$\Omega=\nabla_1\cup \cdots \cup \nabla_m$$ and both ${{\operatorname}{O}_{p'}(A)}$ and $B$ act transitively on the set $\{\nabla_1, \nabla_2, \dots, \nabla_m\}$. In particular, for each $1\leq i\leq m$, there exists $a_i\in{{\operatorname}{O}_{p'}(A)}$ such that $\nabla_1^{a_i}=\nabla_i$; $a_1=1$. Since the lenght of an orbit of $\langle y \rangle$ on $\Omega$ divides $p$ and $\langle y \rangle$ does not normalise any $N_i$, by Lemma \[new\], the first assertion follows. Now, the fact that $G=\langle y \rangle {{\operatorname}{O}_{p'}(A)}N=\langle y\rangle BN$ gives the last assertion. - It follows from (ii) and (iii) that $r=pm$, with $1=(m,p)$. - Without loss of generality, we may consider $\Delta=\{N_1,\ldots, N_m\}$, and we set $M_{\Delta}:=N_1\times \cdots \times N_m$. Then $M_{\Delta}$ is a minimal normal subgroup of $NC$. Moreover, if $1\neq R\leq N$ and $R\unlhd NC$, then there exist $\{x_1, \ldots, x_d\}\subseteq \langle y \rangle$ such that $R=M_{\Delta}^{x_1} \times \cdots \times M_{\Delta}^{x_d}$. - Since $r > 1$, then $m>1$. Recall that $r >1$, by Lemma \[new\]. If $m=1$, then $\langle y \rangle$ has only one orbit on $\Omega$, i.e. $\langle y \rangle$ acts transitively on $\Omega=\{N_1, \ldots, N_k\}$. Suppose, for instance, that $N_i=N_1^y$, for $i > 1$. If there exists a non-trivial element $x\in{{\operatorname}{C}_{N_1}(y)}$, then $x=x^y\in N_1\cap N_1^y=N_1 \cap N_i$, a contradiction. Hence ${{\operatorname}{C}_{N_1}(y)}=1$, and it follows that ${{\operatorname}{C}_{N}(y)}=1$. But since $N\cap A\neq 1$, by Lemma \[ncapa\], we can choose an element of prime powe order $1 \neq x \in N \cap A$ such that $x \in C_N(y)$ because the hypotheses on the indices (recall that $N$ is a $p'$-group), which gives a contradiction. - Since $k=p>1$, then $A\cap M_{\Delta}=1={{\operatorname}{C}_{{{\operatorname}{O}_{p'}(A)}}(M_{\Delta})}$. Let $x\in A\cap M_{\Delta}$ of prime power order, which is a $p$-regular element. Then by hypotheses there exists $n\in N$ with $x\in{{\operatorname}{C}_{G}(\langle y\rangle^n)}$. Hence $x^{y^n}=x\in M_{\Delta}\cap M_{\Delta}^{y^n}=M_{\Delta}\cap M_{\Delta}^{y}=1$. We deduce that $A\cap M_{\Delta}=1$. Let $x\in {{\operatorname}{C}_{{{\operatorname}{O}_{p'}(A)}}(M_{\Delta})}$ of prime power order. Then there exists $n\in N$ such that $[\langle y \rangle^n, x]=1$. Therefore $[x, M_{\Delta}]=1=[x^{(y^j)^n}, M_{\Delta}^{(y^j)^n}]=[x, M_{\Delta}^{y^j}]$, for every $j\in\{1, \ldots, p-1\}$. So $x\in{{\operatorname}{C}_{G}(N)}=1$. Hence ${{\operatorname}{C}_{{{\operatorname}{O}_{p'}(A)}}(M_{\Delta})}=1$. - Without loss of generality, let $\nabla =\{N_1, \ldots, N_p\}$. Set $M_{\nabla}:=N_1\times \cdots\times N_p$. Then $M_{\nabla}$ is a minimal normal subgroup of $N\langle y \rangle$, and if $1\neq R\leq N$ with $R\unlhd N\langle y\rangle$, then there exist $\{d_1,\ldots, d_t\}\subseteq\{a_1,\ldots, a_m\} \subseteq {{\operatorname}{O}_{p'}(A)}$ such that $R=M_{\nabla}^{d_1}\times \cdots \times M_{\nabla}^{d_t}$. Moreover, if we set $F_1:=N_2\times \cdots \times N_p$, $F_i:=F_1^{a_i}$ for each $2\leq i\leq m$, and $F_{\nabla}:=F_1\times \cdots \times F_m$, then $F_{\nabla}\cap {{\operatorname}{O}_{p'}(A)}=1=F_{\nabla}\cap B$. The first assertion is clear. If $x\in F_{\nabla}\cap {{\operatorname}{O}_{p'}(A)}$ is of prime power order, then there exists $n\in N$ such that $\langle y \rangle^n$ centralises $x$, so for every $1\leq j \leq p$ we get $x=x^{(y^j)^n}\in F_{\nabla}\cap F_{\nabla}^{y^j}\leq E_{\nabla}:=\cap_{g\in\langle y \rangle} F_{\nabla}^g$. It follows that $F_{\nabla}\cap {{\operatorname}{O}_{p'}(A)}\leq E_{\nabla}$. Note that $E_{\nabla}\leq N$ and it is normal in $N\langle y \rangle$, hence we deduce from the above that $E_{\nabla}=1$, and so $F_{\nabla}\cap {{\operatorname}{O}_{p'}(A)}=1$. Analogously, $F_{\nabla}\cap B=1$. Now, we will use the above facts on the actions of $C$ (and so $B$) and $\langle y \rangle$ on the set $\Omega$ to see that the minimal normal subgroup $N$ in our minimal counterexample cannot be a $p'$-group. The proof of the next Lemma follows similar arguments as those in [@KMP3 Lemma 11], using (i)-(viii) above, with suitable changes. However, we include an outline of the proof for the convenience of the reader. \[ix\] Let $s\neq p$ be a prime, and assume that ${\ensuremath{\left| N_1 \right|}}_s=s^n$ and ${\ensuremath{\left| {\operatorname}{Out}(N_1) \right|}}_s=s^{\delta}$. Then $n(p-2)\leq \delta + \frac{m-1}{m(s-1)}$, where $r=pm$. In particular, $n(p-2)< \delta+1$. Recall that a $\langle y \rangle$-orbit $\nabla$ on $\Omega$ has length $k=p > 1$. Let $A_s\in{{\operatorname}{Syl}_{s}\left(A\right)}$ and $B_s\in{{\operatorname}{Syl}_{s}\left(B\right)}$. Note that $A_s\leq {{\operatorname}{O}_{p'}(A)}$. By (viii) above, $F_{\nabla}\unlhd N$ and $F_{\nabla}\cap N\cap A_s\leq F_{\nabla}\cap {{\operatorname}{O}_{p'}(A)}=1$. So it follows that ${\ensuremath{\left| A_s\cap N \right|}}\leq {\ensuremath{\left| N:F_{\nabla} \right|}}_s={\ensuremath{\left| N_1 \right|}}_s^m=s^{nm}$. Analogously ${\ensuremath{\left| B_s\cap N \right|}}\leq s^{nm}$. Set $M:=M_{\Delta}=N_1 \times \cdots \times N_m$. From (v) and (vii) above we have that $M\unlhd N{{\operatorname}{O}_{p'}(A)}=NB$ and $M\cap {{\operatorname}{O}_{p'}(A)}=1={{\operatorname}{C}_{{{\operatorname}{O}_{p'}(A)}}(M)}$. Hence ${{\operatorname}{O}_{p'}(A)}\cong {{\operatorname}{O}_{p'}(A)}{{\operatorname}{C}_{G}(M)}/{{{\operatorname}{C}_{G}(M)}} \lessapprox {\operatorname}{Aut}(M)$. Moreover ${\operatorname}{Aut}(M) \cong [{\operatorname}{Aut}(N_1) \times \cdots \times {\operatorname}{Aut}(N_m)]\Sigma_m \cong \, {\operatorname}{Aut}(N_1){\operatorname}{wr} \, \Sigma_m$, the natural wreath product of ${\operatorname}{Aut}(N_1)$ with $\Sigma_m$. Now applying Lemma \[SylowSym\] we deduce that ${\ensuremath{\left| A_s \right|}}$ divides ${\ensuremath{\left| {\operatorname}{Aut}(M) \right|}}$, and so $s^{(\delta+n)m}\cdot s^{\frac{m-1}{s-1}}$. On the other hand, if ${\ensuremath{\left| G/N \right|}}_s=s^{\gamma}$, then ${\ensuremath{\left| G \right|}}_s={\ensuremath{\left| G/N \right|}}_s {\ensuremath{\left| N \right|}}_s=s^{\gamma+nr}$. Further, ${\ensuremath{\left| B_s \right|}}={\ensuremath{\left| G/N \right|}}_s {\ensuremath{\left| B_s\cap N \right|}}$ divides $s^{\gamma+nm}$. Since ${\ensuremath{\left| G \right|}}_s$ divides ${\ensuremath{\left| A_s \right|}} {\ensuremath{\left| B_s \right|}}$, so $s^{\gamma+nr}$ divides $s^{\frac{m-1}{s-1}}\cdot s^{\gamma+nm} s^{(\delta + n)m}$. This fact, after some straightforward computations, leads to the desired conclusion, having in mind that $r=pm$. $N$ is not a $p'$-group. We take a prime $s\in \pi(N_1)\smallsetminus \pi({\operatorname}{Out}(N_1))$ (such prime always exists, see for instance [@KMP3 Lemma 5]). Note that $s \neq p$, since $N$ is a $p'$-group. Applying the previous Lemma for such prime we obtain that $n(p-2)<\delta +1$, but $\delta=0$ so necessarily $p=2$. This cannot happen, as it would imply that $N$ is a $2'$-group, so soluble, which is a contradiction by Lemma \[5\]. The almost simple case ====================== Let $N$ be a non-abelian simple group with $p \in \pi(N)$, and let $N \unlhd G \leq {{\operatorname}{\textup{Aut}}({N})}$ such that $G=AB=AN=BN$. Assume that $G$ satisfies the hypotheses of our main theorem, i.e. $p$ does not divide $i_G(x) $ for every $p$-regular element of prime power order $x \in A \cup B$. We will carry out a case-by-case analysis of the simple group $N$ occuring as the socle of $G$ to prove that there is no a counterexample to our Main Theorem. Our strategy will apply the following lemma and the results in Section 3. \[cent\] For any $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$ $$\pi(G)=\pi(C_G(P)) =\pi(G/N) \cup \pi(C_N(P)).$$ In particular, $p \in {{\operatorname}{\mathcal{Z}}(\Gamma(G))}$. Moreover, if $r \in \pi(G) \setminus \pi(G/N)$, then $r$ is adjacent to $p$ in $\Gamma(N)$. By our hypotheses, for any prime $r \in \pi(G) \setminus \{p\}$ there exists an element $x \in A \cup B$ of order $r$ such that $x \in C_G(P)$, for some $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$. Hence the first equality follows. Now, observe that $\pi(G)=\pi(G/N) \cup \pi(N)$ and, since $G=AN=BN=AB$, after some computations we also obtain $${\ensuremath{\left| N \right|}}{\ensuremath{\left| A\cap B \right|}}={\ensuremath{\left| \frac{G}{N} \right|}}{\ensuremath{\left| N\cap A \right|}}{\ensuremath{\left| N\cap B \right|}}.$$ But again our hypotheses lead to $\pi((N \cap A) \cup (N \cap B))\setminus \{p\} \subseteq \pi(C_{N}(P))$. Also, since $p \in \pi(N)$, $1 \neq Z(P) \cap N \leq C_{N}(P)$. Hence the second equality also holds. It is clear then that $p \in {{\operatorname}{\mathcal{Z}}(\Gamma(G))}$. Assume now that $r \in \pi(G) \setminus \pi(G/N)$, and so $r \in \pi(N)$. By the second equality, we deduce that $r \in \pi(C_N(P))$ and since $p \in \pi(N)$ the last assertion follows. \[notan\] $N$ is not an alternating group $A_n$. Let $N=A_n$ and assume first $n\neq 6$, so $G=N=A_{n}$ or $G=\Sigma_n$. Because our hypotheses, we may assume that $\Gamma(G)$ is connected. As in Lemma \[angraph\], let $k\geq 2$ be the largest positive integer such that $\{n, n-1, \ldots, n-k+1\}$ are consecutive composite numbers, $r:=n-k$ the largest prime divisor of $n!$, and $t$ the largest prime with $t \leq k$. Since $p \in {{\operatorname}{\mathcal{Z}}(\Gamma(G))}$, then $p\leq t$ by Lemma \[angraph\], and so $r>\frac{n}{2}>k\geq t\geq p$. We claim that $r\notin\pi({{\operatorname}{C}_{G}(P)})$, for $P\in{{\operatorname}{Syl}_{p}\left(G\right)}$. Let suppose first that $p\neq 2$. Assume that there exists an element $x\in G$ of order $r$ such that $P\leq {{\operatorname}{C}_{G}(x)}$. Since ${{\operatorname}{C}_{G}(x)}$ is isomorphic to a subgroup of $C_r\times \Sigma_{n-r}$ and $p\neq r$, then ${\ensuremath{\left| P \right|}}$ divides ${\ensuremath{\left| \Sigma_{n-r} \right|}}$, and so ${\ensuremath{\left| \Sigma_n:\Sigma_{n-r} \right|}}= n(n-1)\cdots (n-r+1)$ should be a $p'$-number. But this is a contradiction, since $p<r$. If $p=k=2$, then, by Lemma \[angraph\], it should be $G=\Sigma_n$, so the above reasonings work as well. Finally, if $p=2$ and $k\geq 3$, then $r\geq 5$ and we can argue as above to get a contradiction since ${\ensuremath{\left| \Sigma_n:\Sigma_{n-r} \right|}}= n(n-1)\cdots (n-r+1)$ is divisible by $4$. If $n=6$, by Lemma \[a6\], the only case to be considered is $G={\operatorname}{Aut}(N)$, and since ${{\operatorname}{\mathcal{Z}}(\Gamma(G))}=\{2\}$ it should be $p=2$. But a Sylow $2$-subgroup of $G$ is self-centralising, so we get a contradiction. \[notsp\] $N$ is not an sporadic group. Assume that $N$ is an sporadic group. Since $p \in {{\operatorname}{\mathcal{Z}}(\Gamma(G))}$, we may assume, by Lemma \[sporgraph\], that either $N= J_2$ or $N=McL$, $G={{\operatorname}{\textup{Aut}}({N})}$, and $p=2$. Now Lemma \[cent\] implies that $2$ is adjacent in $N$ to any prime $r\neq 2$, but this is a contradiction since $N$ has a self-centralising Sylow $s$-subgroup (take $s=7$ for $N=J_2$ and $s=11$ for $N=McL$; see [@Atl]). $N$ is not a simple group of Lie type. If $N$ is a simple group of Lie type of characteristic $t$, first notice that the prime $p$ such that $\pi(G)=\pi(C_G(P))$ should be different from $t$, because it is well known that a Sylow $t$-subgroup is self-centralising in $G$. Moreover, since ${\ensuremath{\left| N \right|}}{\ensuremath{\left| A\cap B \right|}}={\ensuremath{\left| \frac{G}{N} \right|}}{\ensuremath{\left| N\cap A \right|}}{\ensuremath{\left| N\cap B \right|}}$ and $|N|_t > |{{\operatorname}{\textup{Out}}({N})}|_t$, we get that $t \in \pi((N \cap A) \cup (N \cap B)) \subseteq \pi(C_{N}(P))$. This means that $t$ should be adjacent to $p$ in $\Gamma(N)$. Now, we derive from Lemmas \[class\] and \[excep\] that, apart from some exceptional cases that we consider below, either there exist a Sylow $s$-subgroup of $N$ of order $s \not \in \pi({{\operatorname}{\textup{Out}}({N})})$ which is self-centralising in $N$, or there exist two primes $r, s \in \pi (N)\setminus \pi(G/N)$, and two maximal tori $T_1$ and $T_2$ of $N$ such that $r \in \pi(T_1)$, $s\in \pi(T_2)$, and $(|T_1|, |T_2|)=1$. But from Lemma \[cent\], $p$ is a prime which is adjacent both to $r$ and $s$ in $\Gamma(N)$, and therefore $p \in \pi(T_1) \cap \pi(T_2)$, which gives a contradiction. For $N=L_2(q)$, $q=t^e$, the fact that $C_N(x)$ is a $t$-group for any $t$-element $x \in N$, implies that $t$ is not adjacent in $\Gamma(N)$ to any other prime in $\pi(N)$, a contradiction. If $N=L_3(q)$ or $N=U_3(q)$, $q=t^e$, the assertion in Lemma \[class\] on the corresponding maximal torus $T$ in each case guarantees that $p \in \pi(T)$ is a primitive prime divisor of $q^3-1$ (respectively $q^6-1$) and $p$ is not adjacent to the prime $t$ in $\Gamma(N)$. In fact, $p$ is not adjacent in $\Gamma(N)$ to any prime $s \not \in \pi(T)$, which gives a contradiction. The Main Theorem is proved.\ **Acknowledgements.** Research supported by Proyecto PGC2018-096872-B-I00 from the Ministerio de Ciencia, Innovación y Universidades, Spain, and FEDER. The second author is also supported by Project VIP-008 of Yaroslavl P. Demidov State University and the third author by Proyecto PROMETEO/2017/057 from the Generalitat Valenciana, Spain. [9]{} B. Amberg, S. Franciosi and F. de Giovanni, *Products of groups*, Oxford University Press Inc., New York, 1992. A. Ballester-Bolinches, J. Cossey and Y. Li, Mutually permutable products and conjugacy classes, [*Monatsh. Math.*]{} **170** (2013), 305–310. A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, *Products of finite groups*, vol. 53 of de Gruyter Expositions in Mathematics, Berlin, 2010. A. R. Camina, R. D. Camina. The influence of conjugacy class sizes on the structure of finite groups: a survey, [*Asian-Eur. J. Math.*]{} [**4**]{} (2011), 559–588. R. W. Carter, Centralizers of semisimple elements in the finite classical groups, *Proc. London Math. Soc.* [**42**]{} (1981), 1–41. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, [*Atlas of finite groups*]{}, Clarendon Press (1985). J. Cossey and Y. Li, On the structure of a mutually permutable product of finite groups. [*Acta Math. Hungar.*]{} [**154**]{} (2018), 525–529. K. Doerk and T. Hawkes, *Finite Soluble Groups*, vol. 4 of de Gruyter Expositions in Mathematics, Berlin, 1992. S. Dolfi, E. Pacifici, L. Sanus and P. Spiga, On the orders of zeros of irreducible characters, [*J. Algebra*]{} **321** (2009), 345–352. B. Fein, W. M. Kantor and M. Schacher, Relative Brauer groups II, [*J. Reine Angew. Math.*]{} **328** (1981), 39–57. M.J. Felipe, A. Martínez-Pastor and V.M. Ortiz-Sotomayor, Square-free class sizes in products of groups, [*J. Algebra*]{} [**491**]{} (2017), 190–206. M. J. Felipe, A. Martínez-Pastor and V. M. Ortiz-Sotomayor, Prime power indices in factorised groups, [*Mediterr. J. Math.*]{} **14** (2017), article: 225. M. J. Felipe, A. Martínez-Pastor and V. M. Ortiz-Sotomayor, Zeros of irreducible characters in factorised groups, [*Ann. Mat. Pura Appl.*]{} **198** (2019), 129–142. M.J. Felipe, A. Martínez-Pastor and V.M. Ortiz-Sotomayor, Products of groups and class sizes of $\pi$-elements, [*Mediterr. J. Math.*]{} **17** (2020), article: 15. H. He and W. Shi, A Note on the Adjacency Criterion for the Prime Graph and the Characterization of $C_p (3)$, [*Algebra Colloq.*]{} [**19**]{}(3) (2012), 553–562. L. S. Kazarin, A. Martínez-Pastor and M. D. Pérez-Ramos, A reduction theorem for a conjecture on products of two $\pi$-decomposable groups, [*J. Algebra*]{} [**379**]{} (2013), 301–313. A. S. Kondrat’iev, On prime graph components of finite simple groups. [*Math. Sb.*]{} [**67**]{}(1) (1990), 235–247. Translation from *Mat. Sb*. **180**(6) (1989), 787–797. M. Liebeck, C. E. Praeger and J. Saxl, The maximal factorizations of the finite simple groups and their automorphism groups, [*Mem. Amer. Math. Soc.*]{} [**86**]{}, no. 432, (Amer. Math. Soc., Providence, RI, 1990). X. Liu, Y. Wang and H. Wei, Notes on the length of conjugacy classes of finite groups, [*J. Pure Appl. Algebra*]{} [**196**]{} (2005), 111–117. M. S. Lucido, Prime graph components of finite almost simple groups. [*Rend. Semin. Mat. Univ. Padova*]{}, [**102**]{} (1999), 1–22. Addendum in: [*Rend. Semin. Mat. Univ. Padova*]{} [**107**]{} (2002), 1–2 A. V. Vasiliev, E. P. Vdovin, An adjacency criterion for the prime graph of a finite simple group, *Algebra and Logic* [**44**]{}(6) (2005), 381–406. A. V. Vasiliev, E. P. Vdovin, Cocliques of maximal size in the prime graph of a finite simple group, *Algebra and Logic* [**50**]{}(4) (2011), 291–322. J. S. Williams, Prime graph components of finite groups. [*J. Algebra*]{} [**69**]{} (1981), 487–513. A.V. Zavarnitsin, Recognition of the simple groups $L_3 (q)$ by element orders, [*J. Group Theory*]{} **7** (2004), 81-97. X.H. Zhao, X.Y. Guo, J.Y. Shi, On the conjugacy class sizes of prime power order $\pi$-elements, [*South. Asian Bull. Math.*]{} [**35**]{} (2011), 735–740. K. Zsigmondy, Zur Theorie der Potenzreste, [*Monatsh. Math. Phys.*]{} [**3**]{} (1892), 265-284. [^1]: Instituto Universitario de Matemática Pura y Aplicada (IUMPA), Universitat Politècnica de València, Camino de Vera, s/n, 46022, Valencia, Spain. [^2]: Department of Mathematics, Yaroslavl P. Demidov State University, Sovetskaya Str 14, 150014 Yaroslavl, Russia. mfelipe@mat.upv.es, Kazarin@uniyar.ac.ru, anamarti@mat.upv.es, vicorso@doctor.upv.es

--- abstract: 'We compute the algebra of left and right currents for a principal chiral model with arbitrary Wess-Zumino term on supergroups with zero Killing form. We define primary fields for the current algebra that match the affine primaries at the Wess-Zumino-Witten points. The Maurer-Cartan equation together with current conservation tightly constrain the current-current and current-primary operator product expansions. The Hilbert space of the theory is generated by acting with the currents on primary fields. We compute the conformal dimensions of a subset of these states in the large radius limit. The current algebra is shown to be consistent with the quantum integrability of these models to several orders in perturbation theory.' author: - 'Raphael Benichou$^{a}$ and Jan Troost$^{b}$' --- $^{a}$ Theoretische Natuurkunde, Vrije Universiteit Brussel,\ Pleinlaan 2, B-1050 Brussels, Belgium\ $^{b}$Laboratoire de Physique Théorique\ Unité Mixte du CRNS et de l’École Normale Supérieure\ associée à l’Université Pierre et Marie Curie 6\ UMR 8549 [^1]\ École Normale Supérieure\ $24$ Rue Lhomond Paris $75005$, France Introduction ============ Principal chiral models with Wess-Zumino term on supergroups (and their cosets) arise in many contexts including string theory on Anti-de Sitter backgrounds with Ramond-Ramond fluxes, the integer quantum hall effect, quenched disorder systems, polymers, as well as other domains in physics. When the supergroup has zero Killing form, the model is perturbatively conformal [@Berkovits:1999im; @Bershadsky:1999hk; @Babichenko:2006uc]. Thus, these models provide us with a two-parameter family of two-dimensional conformal field theories with supergroup symmetry. They exhibit a current algebra which is conformal and non-chiral [@Ashok:2009xx]. Since these models fall into a class which exhibits integrability at least classically and most likely quantum mechanically, these two-dimensional conformal field theories may allow for an exact determination of their spectrum. Steps towards solving these models were made using various techniques. For particular supergroups the Wess-Zumino-Witten points are well-understood [@Rozansky:1992rx][@Schomerus:2005bf][@Gotz:2006qp]. The bulk spectrum was computed at some special points of the moduli space in [@Read:2001pz]. The spectrum for states living on particular boundaries can be obtained at any point of the moduli space [@Quella:2007sg; @Mitev:2008yt; @Candu:2009ep]. Methods to compute a subset of correlation functions were recently proposed in [@Candu:2010yg]. Despite these successes, the determination of the full bulk spectrum of the conformal field theories on supergroups remains an open problem. A strong motivation for determining the spectrum of these models, and their cosets, is the prospect of solving string theory in $AdS$ space-times in conformal gauge, which via holography [@Maldacena:1997re] may lead to a neater formulation of the solution of gauge theories at large $N$ [@Gromov:2009tv]. Our attitude in attacking this problem is to first attempt to solve for the spectrum in conformal gauge on supergroups (relevant to $AdS_3$ string theory for instance), and then for the spectrum on supercosets (relevant to $AdS_5$ string theory for example). In this paper, we take a further step in our understanding of the symmetry, the integrability and the Hilbert space and spectrum of these models. In section \[worldsheetcurrentalgebra\], we review the conformal current algebra [@Ashok:2009xx] obeyed by the conserved current associated to the left action of the supergroup on itself. We will determine further terms at order zero in the current algebra. In section \[left-right\] we compute the interplay between the left and the right conformal current algebra, as well as with the adjoint primary operator. In section \[primaries\] we define the primary fields for the current algebra. These fields correspond to the affine primaries at the Wess-Zumino-Witten points. We show that current primaries are also Virasoro primaries and compute their conformal dimension at large radius. In section \[bootstrap\] we explain how to compute the current-current and current-primary OPEs order by order in perturbation theory by demanding consistency with current conservation and the Maurer-Cartan equation. In section \[confdimcomp\] we compute conformal dimensions of operators that are composites of a current and a primary to first order in semi-classical perturbation theory. We argue that the Hilbert space is generated by composites of currents and primary fields and show how to compute the conformal dimension of such operators in semi-classical perturbation theory. In section \[integrability\] we comment on the classical and quantum integrability of the model, and its consistency with the conformal current algebra. We conclude in section \[conclusions\]. We have gathered many technical details in the appendices. In appendix \[compositeOPEs\] we give a prescription to compute OPEs involving composite operators. In appendix \[XXOPEs\] we compute the behavior at large radius for the coefficients appearing in the current-current and current-primary OPEs. In appendix \[consistentPertOPEs\] we prove the consistency of the perturbative algorithm used to compute the current-current and current-primary OPEs. Appendix \[AppCurrents\] contains further consistency checks of the current algebra as well as details of the computation of the current algebra. In appendix \[AppPrimaries\] we detail calculations involving the primary fields. In appendix \[commutators\] we translate the current-current OPEs into (anti-)commutation relations for the modes of the currents when the theory is defined on a cylinder. Finally, classical integrability of the model is proven in appendix \[classint\] The conformal current algebra {#worldsheetcurrentalgebra} ============================= Setting {#setting .unnumbered} ------- We study a non-linear sigma-model on a supergroup $G$ with zero Killing form, including a kinetic term and a Wess-Zumino term with arbitrary coefficient. The model is conformal and has a global symmetry group corresponding to the left and right action of the group on itself. In this section we review and complement the analysis of the algebra of current components associated to the left group action [@Ashok:2009xx]. The action of the non-linear sigma-model on the supergroup is: $$\begin{aligned} \label{ourmodel} S &= S_{kin} + S_{WZ}\cr S_{kin} &= \frac{1}{ 16 \pi f^2}\int d^2 z Tr'[- \partial^\mu g^{-1} \partial_\mu g] \cr S_{WZ} &= - \frac{ik}{24 \pi} \int_B d^3 y \epsilon^{\alpha \beta \gamma} Tr' (g^{-1} \partial_\alpha g g^{-1} \partial_\beta g g^{-1} \partial_\gamma g )\end{aligned}$$ where $g$ takes values in the supergroup $G$ and $Tr'$ indicates the non-degenerate bi-invariant metric. We will use the normalization and results of [@Ashok:2009xx]. The Wess-Zumino-Witten points are given by the equation $1/f^2 = |k|$. Note that the action is invariant under group inversion $g \leftrightarrow g^{-1}$ and simultaneous orientation reversal $z \leftrightarrow \bar{z}$. The conformal current algebra {#the-conformal-current-algebra .unnumbered} ----------------------------- [From]{} the action we can calculate the classical currents associated to the invariance of the theory under left multiplication of the field $g$ by a group element in $G_L$ and right multiplication by a group element in $G_R$. The classical $G_L$ currents are given by $$\begin{aligned} \label{normeqn} j_{L,z} &= c_+ \partial g g^{-1}\cr j_{L,\bar{z}} &= c_- \bar{\partial} g g^{-1} \,,\end{aligned}$$ where the constant $c_+$ and $c_-$ are given in terms of the couplings by: \[c+-\] c\_ = - . Similarly, we also have the left-invariant currents that generate right multiplication: $$\begin{aligned} j_{R,z} &= -c_- g^{-1} \p g \cr j_{R,\bar z} &= -c_+ g^{-1} \bar \p g\, .\end{aligned}$$ The operator product expansions (OPEs) satisfied by the left currents have been derived in [@Ashok:2009xx]. They read: $$\begin{aligned} \label{euclidOPEs} j_{L,z}^a (z) &j_{L,z}^b (0) \sim \ \kappa^{ab} \frac{c_1}{z^2} + {f^{ab}}_c \left[ \frac{c_2}{z} j_{L,z}^c(0)+ (c_2-g) \frac{\bar{z}}{z^2} j_{L,\bar{z}}^c(0) \right] \cr & + {f^{ab}}_c \left[-\frac{g}{4}\frac{\bar z}{z}(\partial_z j_{\bar z}^c(0)-\partial_{\bar z}j_z^c(0)) + \frac{c_2}{2} \partial_z j_{L,z}^c(0)+ \frac{c_2-g}{2} \frac{\bar{z}^2}{z^2} \partial_{\bar z}j_{L,\bar{z}}^c(0) \right] \cr & + :j_z^a j_z^b:(0) + {{A}^{ab}}_{cd} \frac{1}{2} \frac{\bar z^2}{z^2}:j_{\bar z}^{ c} j_{\bar z}^{d }:(0) + {{B}^{ab}}_{cd} \frac{\bar z}{z} :j_z^{ c} j_{\bar z}^{d }:(0) - {{C}^{ab}}_{cd} \log |z|^2 :j^{ c}_z j_z^{e } :(0) \cr & + ... \cr % j_{L,\bar{z}}^a (z)& j_{L,\bar{z}}^b (0) \sim \ \kappa^{ab} c_3 \frac{1}{\bar{z}^2} + {f^{ab}}_c \left[ \frac{c_4}{\bar{z}} j_{L,\bar{z}}^c(0) + \frac{(c_4-g)z}{\bar{z}^2} j_{L,z}^c(0)\right] \cr & + {f^{ab}}_c \left[\frac{g}{4}\frac{z}{\bar z}(\partial_z j_{\bar z}^c(0)-\partial_{\bar z}j_z^c(0)) + \frac{c_4}{2} \partial_{\bar z} j_{L,\bar z}^c(0)+ \frac{c_4-g}{2} \frac{z^2}{\bar z^2} \partial_{z}j_{L,z}^c(0) \right] \cr & + :j_{\bar z}^a j_{\bar z}^b:(0) - {{A}^{ab}}_{cd} \log |z|^2 :j_{\bar z}^{ c} j_{\bar z}^{d }:(0) + {{B}^{ab}}_{cd}\frac{ z}{\bar z} :j_z^{ c} j_{\bar z}^{d }:(0) + {{C}^{ab}}_{cd} \frac{1}{2} \frac{z^2}{\bar z^2} :j^{ c}_z j_z^{d }:(0) \cr & + ... \cr % j_{L,z}^a (z) &j_{L,\bar{z}}^b(0) \sim \ \tilde{c}\kappa^{ab} 2\pi \delta^{(2)}(z-w) + {f^{ab}}_c \left[ \frac{(c_4-g)}{\bar{z}} j_{L,z}^c(0) + \frac{(c_2-g) }{z} j_{L,\bar{z}}^c(0) \right] \cr & + {f^{ab}}_c \left[ -\frac{g}{4} \log |z|^2 (\partial_z j_{\bar z}^c(0)-\partial_{\bar z}j_z^c(0)) + \frac{(c_4-g)z}{\bar{z}}\partial_z j_{L,z}^c(0) \right] \cr & + :j_z^a j_{\bar z}^b:(0) + {{A}^{ab}}_{cd} \frac{\bar z}{z}:j_{\bar z}^{ c} j_{\bar z}^{d }:(0) - {{B}^{ab}}_{cd} \log |z|^2 :j_z^{ c} j_{\bar z}^{d }:(0) + {{C}^{ab}}_{cd} \frac{z}{\bar z} :j^{ c}_z j_z^{d }:(0) \cr & + ... \cr\end{aligned}$$ Compared to [@Ashok:2009xx], we have added a few terms at order zero in the distance between the insertion points of the two current components[^2]. The ellipses refer to subleading terms in the expansion in the distance between the two insertion points (which includes logarithms). The right current components $j_{R,z}$ and $j_{R,\zbar}$ satisfy similar operator product expansions amongst themselves, with the holomorphic coordinates replaced by anti-holomorphic ones. This can be proven by using the $\mathbb{Z}_2$ symmetry that we noted before. Associativity of the current algebra is discussed in appendix \[associativity\]. For the supergroup non-linear sigma-model in equation , the coefficients of the second and first order poles in the conformal current algebra, expressed purely in terms of $c_{\pm}$, are given by [@Ashok:2009xx] $$\begin{aligned} \label{candg} c_1 &= -\frac{c_+^2}{c_++c_-} \qquad\qquad\qquad c_3 = -\frac{c_-^2}{c_++c_-} \cr c_2 &= i \frac{c_+(c_++2c_-)}{(c_++c_-)^2} \qquad\qquad c_4 = i \frac{c_-(2c_++c_-)}{(c_++c_-)^2} \cr g &= i \frac{2c_+c_-}{(c_++c_-)^2} \qquad\qquad\qquad \tilde{c} = \frac{ c_+ c_-}{c_++c_-} \,,\end{aligned}$$ where the coefficients $c_{\pm}$ are the factors defined in equation as the normalization of the currents. The coefficients $c_i$ are exact. The current algebra defined by equation is compatible with both current conservation and the Maurer-Cartan equation : \[CC\] |j\_[L,z]{}\^a + j\_[L,|z]{}\^a = 0 \[MC\] c\_- |j\_[L,z]{}\^a - c\_+ j\_[L,|z]{}\^a - i [f\^a]{}\_[bc]{} :j\_[L, z]{}\^c j\_[L,|z]{}\^b : = 0. Indeed the OPE of a current with the left-hand side of the current conservation equation (respectively the Maurer-Cartan equation ) gives zero up to contact terms (respectively exactly zero). Moreover, demanding compatibility of the current algebra with both equations and is a way to determine all the other subleading terms in the current algebra, order by order in semi-classical perturbation theory, namely for small $f^2$ (at fixed $kf^2$). This is explained in section \[bootstrap\]. As we will see, the assumption of the validity of current conservation and especially the Maurer-Cartan equation in the quantum theory, determines a tightly constrained and interesting algebraic structure associated to supergroups with vanishing Killing form. This hypothesis is tightly linked to the quantum integrability of the model, as we discuss in section \[integrability\]. We can use this perturbative technique to compute the coefficients of the current bilinears that appear in equation , up to order $f^2$. This computation is detailed in appendix \[jMCOPE\] and it leads to the results: \_[cd]{} &=& ([f\^b]{}\_[cg]{} [f\^[ag]{}]{}\_d (-1)\^[cd]{} + [f\^b]{}\_[dg]{} [f\^[ag]{}]{}\_c) +(f\^4)\_[cd]{} &=& ([f\^b]{}\_[cg]{} [f\^[ag]{}]{}\_d (-1)\^[cd]{} + [f\^b]{}\_[dg]{} [f\^[ag]{}]{}\_c) + (f\^4)\_[cd]{} &=& ([f\^b]{}\_[cg]{} [f\^[ag]{}]{}\_d (-1)\^[cd]{} + [f\^b]{}\_[dg]{} [f\^[ag]{}]{}\_c)+(f\^4). \[ABC\] The fact that the same tensors appear in the three different current-current OPEs is a consequence of current conservation. The four-tensors $A,B,C$ are (graded) symmetric in their two upper indices. This follows from the interchangeability of the current components on the left hand side of the first two OPEs in . Equation shows that these four-tensors are also (graded) symmetric in their two lower indices. Thus they are linear maps from graded symmetric tensors onto graded symmetric tensors. They partially code higher order corrections to equation (see appendix \[jMCOPE\]). In appendix \[jMCOPE\], we have included a careful discussion of minus signs arising due to the graded statistics of the supergroup. For the remainder of the paper however, we will not be careful about minus signs arising due to the grading of operators. Since we use only universal group and (super) Lie algebra properties in our calculations, all signs can be consistently restored. The Virasoro algebra from the current algebra {#the-virasoro-algebra-from-the-current-algebra .unnumbered} --------------------------------------------- In [@Ashok:2009xx] it was shown that the left and right Virasoro algebra emerge from the current algebra via the Sugawara construction. For instance the holomorphic stress-tensor : T(z) = \_[ba]{} :j\^a\_[L,z]{} j\^b\_[L,z]{}: satisfies the following OPEs : \[Tj\] T(z) j\^a\_[L,z]{}(w) = + + (z-w)\^0 \[Tjbar\] T(z) j\^a\_[L,|z]{}(w) = + (z-w)\^0 \[TT\] T(z) T(w) = + + + (z-w)\^0. In appendix \[TjandTT\] we give more details of the proof of equation . In particular it is shown that the terms of order zero in equation (as well as any of the other subleading terms) do not modify this OPE. We also checked through explicit computation that the invariant contractions of the structure constants and the metric with the four-tensors appearing in the energy-momentum tensor/current OPE give zero for the algebra $psl(2|2)$. The left-right current algebra {#left-right} ============================== In this section, we compute the operator product expansions of currents associated to the left and the right action of the group upon itself. The primary adjoint operator ---------------------------- The right current components can be rewritten in terms of the adjoint group action on the left currents: \[jLAdj=jR0\] j\_[R,z]{} = -c\_- g\^[-1]{} g = - Ad\_[g\^[-1]{}]{} (j\_[L,z]{}) \[jLAdjB=jRB0\] j\_[R,|z]{} = -c\_+ g\^[-1]{} |g = - Ad\_g (j\_[L,|[z]{}]{}) . In the quantum theory the adjoint group action is generated by an operator that we call the primary adjoint operator : \[PrimAdj\] \^[a |a]{}= x Str (g\^[-1]{} t\^a g t\^[|a]{}).Here $x$ is some normalization factor. This operator transforms in the adjoint representation with respect to both the left and the right algebras. In the following unbarred (respectively barred) indices refer to the left (respectively right) adjoint representation. We recall that this field is also useful in writing down the Lagrangian of the model, and that its anomalous dimension is proportional to the beta-function of the model (which is zero in the case under study) [@Knizhnik:1984nr]. Special properties of the primary adjoint operator in non-linear sigma models on supergroup with vanishing Killing form were also discussed in [@Quella:2007sg]. We can rewrite equations and as: \[jLAdj=jR\] j\_[R,z]{}\^[|b]{} = - \_[ba]{}:j\^a\_[L,z]{} \^[b |b]{}: \[jLAdjB=jRB\] j\_[R,|z]{}\^[|b]{} = - \_[ba]{}:j\^a\_[L,|z]{} \^[b |b]{}:. Using the $\mathbb{Z}_2$ symmetry of the theory we have also: \[jRAdj=jL\] j\^[b]{}\_[L,z]{} = - \_[|b |a]{}:j\^[|a]{}\_[R,z]{} \^[b |b]{}:(z) \[jRAdjB=jLB\] j\^[b]{}\_[L,|z]{} = - \_[|b |a]{}:j\^[|a]{}\_[R,|z]{} \^[b |b]{}:(z) These relations fix a normalization for the operator $\mathcal{A}^{a \bar a}$. They are compatible if the following relations hold : \[AdjAdj=Ibar\] \_[ba]{} \^[a |a]{} \^[b |b]{} = \^[|a |b]{}I \[AdjAdj=I\] \_[|b |a]{} \^[a |a]{} \^[b |b]{} = \^[a b]{}I where $I$ is the identity at least as acting upon the current algebra. One can argue more generically that these bilinears are proportional to the unit operator by using the definition of the primary adjoint in terms of the supertrace, and using completeness of the Lie algebra generators. Remember also that the left and right conformal dimensions of the adjoint operator $\mathcal{A}^{a \bar a}$ vanish since they are proportional to the dual Coxeter number of the Lie superalgebra. The action of the zero modes of the currents generates the group transformations. Since the structure constants are the generators of the Lie superalgebra in the adjoint representation, the OPE between a current and the primary adjoint operator reads : $$\begin{aligned} \label{jAdj} & j^a_{L,z}(z) \mathcal{A}^{b \bar b}(w) = \frac{c_+}{c_++c_-} \frac{i{f^{ab}}_c \mathcal{A}^{c \bar b}}{z-w} + ... \cr % & j^a_{L,\bar z}(z) \mathcal{A}^{b \bar b}(w) = \frac{c_-}{c_++c_-} \frac{i{f^{ab}}_c \mathcal{A}^{c \bar b}}{\bar z- \bar w} +... \cr % & j^{\bar a}_{R,z}(z) \mathcal{A}^{b \bar b}(w) = \frac{c_-}{c_++c_-} \frac{i{f^{\bar a \bar b}}_{\bar c} \mathcal{A}^{b \bar c}}{z-w} +... \cr % & j^{\bar a}_{R,\bar z}(z) \mathcal{A}^{b \bar b}(w) = \frac{c_+}{c_++c_-} \frac{i{f^{\bar a \bar b}}_{\bar c} \mathcal{A}^{b \bar c}}{\bar z- \bar w} +... \end{aligned}$$ In section \[primaries\] the concept of primary field will be defined precisely. The coefficients appearing in the previous OPE will be explained, and we will compute the first subleading terms (see equation ). Moreover, we propose that the following equations hold in the model under consideration: \[dAdj\] \^[a |a]{} = -:j\^c\_[L,z]{} \^[b |a]{}: = -:j\^[|c]{}\_[R,z]{} \^[a |b]{}: \[dbarAdj\] |\^[a |a]{} = -:j\^c\_[L,|z]{} \^[b |a]{}:= -:j\^[|c]{}\_[R,|z]{} \^[a |b]{}:. One argument for the previous equations is the following. We start with the definition of the adjoint operator in terms of the group element , and compute its derivative: \^[a |a]{} &=& x STr(g\^[-1]{} t\^a g t\^[|a]{}) &=& x STr ( - g\^[-1]{} g g\^[-1]{} t\^a g t\^[|a]{} + g\^[-1]{} t\^a g g\^[-1]{} g t\^[|a]{} ) &=& x STr ( g\^[-1]{}\[t\^a,t\^d\] g t\^[|a]{} ) &=& - i[f\^a]{}\_[bc]{} \^[b |a]{}. We have left out the normal ordering symbols from the above classical calculation. The properties used in the calculation are that the supertrace is graded cyclic and the fact that the equation $g g^{-1}=1$ and its derivative hold true. We assume that the quantum theory is consistent with these two rules. In section \[primaries\] we will give a generic proof of equations and , valid up to a certain order in a semi-classical expansion (see equation ). Notice that the relations and imply that $\partial(\kappa_{ab} \mathcal{A}^{a \bar a} \mathcal{A}^{b \bar b})=0 =\bar \partial(\kappa_{ab} \mathcal{A}^{a \bar a} \mathcal{A}^{b \bar b})$ (and identical equations with the barred indices contracted), and thus are compatible with the equations relating the adjoint primary to the identity and . The left current - right current OPEs ------------------------------------- We have collected the tools to calculate the left/right current operator product expansions. Thanks to equations and we only need the left current self OPEs as well as the OPE between the left current and the adjoint primary operator . As an example, we will explicitly compute the OPE $j^a_{L,z}(z) j^{\bar a}_{R,z}(w)$ at the order of the poles. We use the prescription of appendix \[compositeOPEs\]: $$\begin{aligned} \label{jLjR1stStep} j^a_{L,z}&(z) j^{\bar a}_{R,z}(w) = -j^a_{L,z}(z) \frac{c_-}{c_+} \kappa_{cb}:j^b_{L,z} \mathcal{A}^{c \bar a}:(w) \cr % & = -\frac{c_-}{c_+} \kappa_{cb} \lim_{:x \to w:} \left[ j^a_{L,z}(z) j^b_{L,z}(x) \mathcal{A}^{c \bar a}(w) \right] \cr % & = -\frac{c_-}{c_+} \kappa_{cb} \lim_{:x \to w:} \left[ \left( \frac{c_1 \kappa^{ab}}{(z-x)^2} + \frac{c_2 {f^{ab}}_d j^d_{L,z}(x)}{z-x} + \frac{(c_2-g) {f^{ab}}_d j^d_{L,\bar z}(x)(\bar z - \bar x)}{(z-x)^2} + ... \right) \mathcal{A}^{c \bar a}(w) \right. \cr & \qquad + \left. j^b_{L,z}(x) \left( \frac{c_+}{c_++c_-} \frac{i{f^{ac}}_d \mathcal{A}^{d \bar a}(w)}{z-w} + ... \right) \right] \cr % & = -\frac{c_-}{c_+} \left[ \frac{c_1 \mathcal{A}^{a \bar a}(w)}{(z-w)^2} + \left(-c_2 + \frac{i c_+}{c_++c_-} \right) \frac{{f^a}_{bc}:j^b_{L,z}\mathcal{A}^{c \bar a}:(w)}{z-w} \right. \cr & \qquad \left. - (c_2-g) \frac{{f^a}_{bc}:j^b_{L,\bar z}\mathcal{A}^{c \bar a}:(w)(\bar z - \bar w)}{(z-w)^2} + ... \right]\end{aligned}$$ In principle the second- and first-order poles that we obtain in the last line may receive corrections from the lower-order terms that we neglected in the penultimate line. We will now argue that it is not the case. Let us consider the first term in the last line (the second-order pole). This term may receive corrections of the form ${T^a}_b \mathcal{A}^{b \bar a}$, where the tensor ${T^a}_b$ contains at least one structure constant. Such a tensor vanishes by using properties of the Lie super algebras under consideration [@Bershadsky:1999hk]. Let us now consider the second term (the holomorphic simple pole). It could receive corrections of the form ${T^a}_{bc}:j^b_{L,z}\mathcal{A}^{c \bar a}$, where ${T^a}_{bc}$ contains at least two structure constants. Again, according to [@Bershadsky:1999hk], this tensor vanishes because traceless four-tensors invariantly contracted with structure constants over two indices vanish. The third term receives no higher order corrections for the same reason. Thus the terms written in the last line of are not corrected. Using equations and we finally obtain: \[jLjR1\] j\^a\_[L,z]{}(z) j\^[|a]{}\_[R,z]{}(w) = ( + + ) + ... where the ellipses refer to terms of order zero or more in the distance between the two operators. Similarly we can compute: $$\begin{aligned} \label{jLjR2} j^a_{L,\bar z}(z) j^{\bar a}_{R,\bar z}(w) &= \frac{c_+ c_-}{c_++c_-}\left( \frac{\mathcal{A}^{a \bar a}(w)}{(\bar z-\bar w)^2} + \frac{c_+}{c_++c_-} \frac{\partial \mathcal{A}^{a \bar a}(w)(z-w)}{(\bar z-\bar w)^2} + \frac{c_+}{c_++c_-} \frac{\bar \partial \mathcal{A}^{a \bar a}(w)}{\bar z-\bar w} \right)+ ... \cr % j^a_{L,z}(z) j^{\bar a}_{R,\bar z}(w) &= -\frac{c^2_+}{c_++c_-} \left(\mathcal{A}^{a \bar a}(w) 2\pi \delta^{(2)}(z-w) - \frac{c_-}{c_++c_-} \frac{\partial \mathcal{A}^{a \bar a}(w)}{\bar z - \bar w} + \frac{c_-}{c_++c_-} \frac{\bar \partial \mathcal{A}^{a \bar a}(w)}{z - w}\right) + ...\cr % j^a_{L,\bar z}(z) j^{\bar a}_{R, z}(w) &= -\frac{c^2_-}{c_++c_-} \left(\mathcal{A}^{a \bar a}(w) 2\pi \delta^{(2)}(z-w) + \frac{c_+}{c_++c_-} \frac{\partial \mathcal{A}^{a \bar a}(w)}{\bar z - \bar w} - \frac{c_+}{c_++c_-} \frac{\bar \partial \mathcal{A}^{a \bar a}(w)}{z - w}\right)+ ...\end{aligned}$$ The first two OPEs can be written in the alternative form: j\^a\_[L,z]{}(z) j\^[|a]{}\_[R,z]{}(w) = + ... j\^a\_[L,|z]{}(z) j\^[|a]{}\_[R,|z]{}(w) = + ...It is straightforward to show that the OPEs are compatible with current conservation and the Maurer-Cartan equation. These OPEs are also compatible with the fact that the stress-tensor can be written either in terms of the left-current or in terms of the right currents. As an example of these consistency checks, it is shown in appendix \[TRJL\] that when we express the energy-momentum tensor in terms of right currents, it satisfies the expected OPE with the left current: T(z) j\^a\_[L,z]{}(w) = \_[|c |b]{}:j\^[|b]{}\_[R,z]{}j\^[|c]{}\_[R,z]{}:(z) j\^a\_[L,z]{}(w) = + + ((z-w)\^0) When the theory is defined on a cylinder we can Fourier expand the currents along the angular coordinate, at a given time. It was shown in [@Ashok:2009xx] that the modes of the time components of the left (or the right) currents generate an affine Lie algebra at level $k$. The full commutator algebra computed in appendix \[commutators\] shows that these two affine Lie algebras commute. Summary {#summary .unnumbered} ------- In this section we have determined the pole order parts of the left and right current operator product expansions. The algebra closes on the current components and the adjoint field. Under the assumptions on the quantum theory stated above, the coefficients of the algebra are exact[^3]. We now move from the determination of the left-right symmetry algebra of the model to the study of the vertex operators. The primaries {#primaries} ============= In this section we define the concept of current algebra primaries. These fields can be understood as the elementary vertex operators of the conformal field theory. We compute the operator product expansion between a primary field and a current perturbatively, and deduce the OPE between a primary field and the stress-tensor. In particular we derive the OPEs used in [@Ashok:2009jw]. Left current algebra primaries {#left-current-algebra-primaries .unnumbered} ------------------------------ Given a representation $\mathcal{R}$ of the group $G_L$ we define a left primary field $\phi_\mathcal{R}$ with respect to the left current algebra as a field satisfying the operator product expansions: $$\begin{aligned} \label{defPrimaries} j_{L,z}^a(z,\bar z) \phi_\mathcal{R}(w,\bar w) &= - \frac{c_+}{c_+ + c_-} t^a \frac{\phi_\mathcal{R}(w,\bar w)}{z-w} + \text{order zero} \cr j_{L,\bar z}^a(z,\bar z) \phi_\mathcal{R}(w,\bar w) &=- \frac{c_-}{c_+ + c_-} t^a \frac{\phi_\mathcal{R}(w,\bar w)}{\bar z-\bar w} + \text{order zero} \end{aligned}$$ where the matrices $t^a$ are the generators of the Lie super-algebra taken in the representation $\mathcal{R}$ associated to the primary field $\phi_\mathcal{R}$. If one assumes the above form for the operator product expansions, then the coefficients of the poles are fixed by the Ward identity for the symmetry $G_L$ and the demand that the contact term vanishes in the operator product expansion between the field $\phi$ and the Maurer-Cartan operator . The Ward identity implies compatibility of the OPEs with current conservation . An example of a left current primary is the adjoint primary we discussed in the previous section. In appendix \[WZWaffine\] it is shown that a current primary field at a given point of the moduli space remains a current primary field after deformation of the kinetic term in the action. Thus one can consistently think of the current algebra primaries as the group element $g$ taken in the representation $\mathcal{R}$. It also implies that at the WZW points the current primaries are the affine primary fields. As argued in section \[bootstrap\], we can compute the less singular terms in the current-primary OPE order by order in $f^2$, by using the current conservation and the Maurer-Cartan equation. Performing the calculation of higher order terms to order $f^2$, we find the OPE: $$\begin{aligned} \label{jPhiO1} j_{L,z}^a(z,\bar z) \phi(w,\bar w) = & \ - \frac{c_+}{c_+ + c_-} t^a \frac{\phi(w,\bar w)}{z-w} + :j^a_{L,z} \phi:(w,\bar w) \cr & + {A^a}_c \log |z-w|^2 :j^c_{L,z}\phi:(w,\bar w) + {B^a}_c\frac{\bar z - \bar w}{z-w}:j^c_{L,\bar z}\phi:(w,\bar w) +... \cr % j_{L,\bar z}^a(z,\bar z) \phi(w,\bar w) = & \ - \frac{c_-}{c_+ + c_-} t^a \frac{\phi(w,\bar w)}{\bar z-\bar w} + :j^a_{L,\bar z} \phi:(w,\bar w) \cr & - {A^a}_c \frac{z-w}{\bar z - \bar w} :j^c_{L,z}\phi: - {B^a}_c \log |z-w|^2 :j^c_{L,\bar z}\phi:(w,\bar w) +... \end{aligned}$$ where we dropped the subscript $\mathcal{R}$. The coefficients read: \[ABorder1\] [A\^a]{}\_c = i [f\^a]{}\_[cb]{} t\^b + (f\^4) ; \_c = i [f\^a]{}\_[cb]{} t\^b + (f\^4). The details of the calculation are given in appendix \[AppjPhi\]. Note that the coefficients of the simple poles are unmodified. Current primaries are Virasoro primaries {#current-primaries-are-virasoro-primaries .unnumbered} ---------------------------------------- We will now show that a primary field with respect to the left-current algebra is also a primary field with respect to the Virasoro algebra. The holomorphic worldsheet stress tensor is: T(z) = \_[ba]{}:j\^a\_[L,z]{} j\^b\_[L,z]{}:(z). Let us consider the OPE between a left-primary field $\phi$ and the holomorphic stress-tensor: $$\begin{aligned} \phi(z) 2 c_1 T(w) &= \lim_{:x \to w:}\phi(z) j^a_{L,z}(x) j^b_{L,z}(w) \kappa_{ba}.\end{aligned}$$ [From]{} the structure of the OPE , and from the fact that all operators appearing in this OPE are assumed to be composites of currents and of the operator $\phi$, it follows that the most singular term that may appear in this OPE is a double pole, multiplying the operator $\phi$. As a consequence all the positive modes $L_{n>0}$ of the holomorphic stress-tensor annihilate the operator $\phi$. Thus this operator is a Virasoro primary. Furthermore, with the knowledge of the current-primary OPE up to order $f^2$, we can evaluate the stress-tensor/primary OPE up to the same order. Details about this computation are given in appendix \[AppTphi\]. We obtain : $$\begin{aligned} \label{Tphi} T(w) \phi(z) &= \frac{f^2}{2} \frac{t^a t^b \kappa_{ba} \phi(z)}{(z-w)^2} +\frac{1}{c_+} \frac{\kappa_{ba}t^a :j^b_{L,z}\phi:(z)}{w-z}+ \mathcal{O}(z-w)^{0}+ \mathcal{O}(f^4).\end{aligned}$$ The same computation can be performed with the anti-holomorphic stress-tensor. We obtain: $$\begin{aligned} \label{Tbarphi} \bar T(\bar w) \phi(z) &= \frac{f^2}{2} \frac{t^a t^b \kappa_{ba} \phi(z)}{(\bar z-\bar w)^2} +\frac{1}{c_-} \frac{\kappa_{ba}t^a :j^b_{L,\bar z}\phi:(z)}{\bar w-\bar z}+ \mathcal{O}(\bar z-\bar w)^{0}+ \mathcal{O}(f^4).\end{aligned}$$ On general grounds the OPE between the stress-tensor and the primary field $\phi$ reads: T(w) (z) = + + ((z-w)\^[0]{}), where $\Delta_{\phi}$ is the left conformal dimension of the operator $\phi$. Thus we deduce the conformal dimensions of the primary field $\phi$: \_ = |\_ = t\^a t\^b \_[ba]{}+ (f\^4). \[conformaldimension\] The semi-classical result for the conformal dimension of a current primary is as expected. It is equal to the quadratic Casimir of the representation in which the field transforms, times the inverse radius of the group manifold squared. For generic current primaries, there could be corrections of order $f^4$ to this formula. These corrections were conjectured to be absent in [@Bershadsky:1999hk]. This was proven to be the case to all orders in perturbation theory if the superdimension of the representation of the primary is non-zero (i.e. for short multiplets). For example for the short, discrete representation crucial to the calculation in [@Ashok:2009jw], there are no corrections. Notice that the stress-energy tensor can also be written in terms of the right currents. Equation implies that a primary field transforms under the left- and right-action of the group in representations that have the same eigenvalue of the quadratic Casimir operator. The simple poles in and also give the relations: \[dPhi=JPhi\] (z) = \_[ba]{}t\^a :j\^b\_[L,z]{}:(z)+ (f\^4) \[dbarPhi=JbarPhi\] | (z) = \_[ba]{}t\^a :j\^b\_[L,|z]{}:(z)+ (f\^4). ### Remark about the atypical sector {#remark-about-the-atypical-sector .unnumbered} Some of the primary fields are associated to atypical Kac modules, that are reducible but indecomposable [@Gotz:2006qp]. In that case the matrices $t^a$ that appear in equation are not invertible. Moreover the quadratic operator $\kappa_{ba}t^a t^b$ can then be written in an upper-triangular form, with zeros on the diagonal (which is the generalized eigenvalue of the quadratic casimir for atypical representations of e.g. the $psl(n|n)$ superalgebra). Equation tells us that the operator $L_0$ is proportional to this quadratic operator $\kappa_{ba}t^a t^b$ when acting on a primary field. This implies that $L_0$ is non-diagonalizable, which betrays the logarithmic nature of the theory (see [@Gotz:2006qp] for a similar argument in the case of $psl(2|2)$, and [@Gaberdiel:2001tr],[@Flohr:2001zs] for an introduction to logarithmic CFTs). Let us remark here that the fact that the current component $j_{L,z}$ has dimensions $(1,0)$, but is not holomorphic also codes the logarithmic nature of the conformal field theory [@Read:2001pz]. A recursive bootstrap for the elementary operator algebra {#bootstrap} ========================================================== In this section we will explain how to compute the current-current and current-primary OPEs order by order in a semi-classical expansion. We will show that the knowledge of the poles in these OPEs is enough to fix all the subleading terms. The idea driving the bootstrap is to ask for the compatibility of the elementary OPEs with both current conservation and the Maurer-Cartan equation. Current-current OPEs {#current-current-opes .unnumbered} -------------------- First let us consider the current-current OPEs. Current conservation gives the first constraints: j\^a\_[L,z]{}(z) = 0 j\^a\_[L,|z]{}(z) = 0. The first line implies a one-to-one correspondence between the terms in the $j^a_{L,z} j^b_{L,z}$ and $j^a_{L,z} j^b_{L,\bar z}$ OPEs. The second line then links the $j^a_{L,\bar z} j^b_{L,\bar z}$ and the $j^a_{L,z} j^b_{L,\bar z}$ OPEs. These OPEs are expected to vanish up to contact terms. Indeed the same OPEs code the Ward identity for the global symmetry $G_L$. It follows that the contact terms in these OPEs are given by the transformation properties of the left current under the left action of the group on itself [^4]. The second constraint comes from the Maurer-Cartan equation : j\^a\_[L,z]{}(z) = 0. Contact terms in this OPE should vanish. Using current conservation and the fact that $c_++c_-=-f^{-2}$ we rewrite this constraint as : \[jmodMC\] j\^a\_[L,z]{}(z) |j\^b\_[L,z]{}(w) = f\^2 j\^a\_[L,z]{}(z) i [f\^b]{}\_[cd]{}:j\^d\_[L,z]{} j\^c\_[L,|z]{}:(w). Thanks to the factor of $f^2$ on the right-hand side of the previous equation, it becomes manifest that the knowledge of the current algebra at a given order in $f^2$ will also determine the current algebra at the next order. The discussion of appendix \[XXOPEs\] shows that the terms appearing in the current-current OPEs at order $f^{2n}$ are composites of at most $n+1$ currents. This allows us to make an ansatz for the current-current OPE at higher-order. Then equation fixes the coefficients in this ansatz. This method is illustrated in appendix \[jMCOPE\] where we compute the current algebra up to order $f^2$. Current-primary OPEs {#current-primary-opes .unnumbered} -------------------- The same logic applies to the computation of the current-primary OPEs. Current conservation links the $j^a_{L,z} \phi$ and $j^a_{L,\bar z} \phi$ OPEs : \[phiCC\] (z) = 0. When the above equation is valid, the Maurer-Cartan constraint can be rewritten as: \[phimodMC\] (z) |j\^a\_[L,z]{}(w) = f\^2 (z) i [f\^a]{}\_[cd]{}:j\^d\_[L,z]{} j\^c\_[L,|z]{}:(w). Again the discussion of appendix \[XXOPEs\] gives an ansatz for the current-primary OPE at a given order in $f^2$: the terms appearing in the current-primary OPE at order $f^{2n}$ are composites of at most $n$ currents with the primary field $\phi$. When we plug this ansatz in equation we obtain the value of the coefficients. This method is illustrated in appendix \[AppjPhi\] where we compute the current-primary OPE up to order $f^2$. Further remarks {#further-remarks .unnumbered} --------------- This perturbative approach squares well with the observation that the most singular terms in the current-current and current-primary OPEs come with the lower power of $f^2$. This is explained in appendix \[XXOPEs\]. Thus performing a computation up to a certain order in $f^2$ allows to truncate the current-current and current-primary OPEs at a certain order in the distance between the insertion points of the operators. The consistency of this perturbative approach demands that the addition of higher-order terms to the elementary OPEs does not spoil their compatibility both with current conservation and with the Maurer-Cartan equation at lower order in $f^2$. That this is the case is proven in appendix \[consistentPertOPEs\]. One may hope to obtain a closed formula for the full current-current and current-primary OPEs thanks to this algebraic bootstrap. Composite operators and their conformal dimension {#confdimcomp} ================================================= In this section we consider operators that are composites of one or more currents with a primary operator. We are mostly interested in the computation of the conformal dimension of such operators as a function of the two parameters $(k,f)$ of the supergroup sigma-model. At the WZW point these operators are descendants in the highest-weight representations of the left affine Lie algebra. Operators of the form $:j_L \phi :$ {#operators-of-the-form-j_l-phi .unnumbered} ----------------------------------- Let us consider the operator $:j_{L,z}^a \phi:$ defined as the regular term in the OPE between the operators $j_{L,z}^a$ and $\phi$. To compute the holomorphic dimension of this operator we compute its OPE with the stress-tensor, and look at the second order pole. The computation is done following the method described in appendix \[compositeOPEs\]. The fact that the stress-tensor is holomorphic simplifies the calculation. We find: $$\begin{aligned} T(z) :j^a_{L,z}\phi:(w) & = \lim_{:x \to w:} T(z) j^a_{L,z}(x) \phi(w) \cr % & = \lim_{:x \to w:} \left \{ \left( \frac{j^a_{L,z}(x)}{(z-x)^2} + \frac{\p j^a_{L,z}(x)}{z-x} \right) \phi(w) %\right. \cr %& \qquad \left. + j^a_{L,z}(x) \left( \frac{\Delta_{\phi} \phi(w)}{(z-w)^2} + \frac{\p \phi(w)}{z-w} \right) \right\} \cr % &= \lim_{:x \to w:} \left \{ \frac{1}{(z-x)^2}\left( -\frac{c_+}{c_++c_-} \frac{t^a \phi(w)}{x-w} + :j^a_{L,z} \phi:(w) \right.\right. \cr &\qquad \qquad \left. \left. + {A^a}_c \log|x-w|^2 :j^c_{L,z} \phi:(w) + {B^a}_c \frac{\bar x - \bar w}{x-w} :j^c_{L,\bar z} \phi:(w) + ... \right) \right. \cr & \quad + \frac{1}{z-x}\left( \frac{c_+}{c_++c_-} \frac{t^a \phi(w)}{(x-w)^2} + :\p j^a_{L,z} \phi:(w) \right. \cr &\qquad \qquad \left. \left. + {A^a}_c \frac{1}{x-w} :j^c_{L,z} \phi:(w) - {B^a}_c \frac{\bar x - \bar w}{(x-w)^2} :j^c_{L,\bar z} \phi:(w) + ... \right) \right. \cr & \quad \left. + \frac{\Delta_{\phi}:j^a_{L,z} \phi:(w)}{(z-w)^2} + \frac{:j^a_{L,z}\p \phi:(w)}{z-w} \right\} \cr % &= -\frac{2}{(z-w)^3} \frac{c_+}{c_++c_-} t^a \phi(w) + \frac{:j^a_{L,z} \phi:(w)}{(z-w)^2} \cr &\qquad + \frac{1}{(z-w)^3} \frac{c_+}{c_++c_-} t^a \phi(w) + \frac{:\p j^a_{L,z} \phi:(w)}{z-w} + {A^a}_c \frac{1}{(z-w)^2} :j^c_{L,z} \phi:(w) \cr & \quad + \frac{\Delta_{\phi}:j^a_{L,z} \phi:(w)}{(z-w)^2} + \frac{:j^a_{L,z}\p \phi:(w)}{z-w} + \mathcal{O}(z-w)^0\end{aligned}$$ Using equation we obtain : $$\begin{aligned} \label{TjphiStep1} T(z) :j^a_{L,z}\phi:(w) &= -\frac{c_+}{c_++c_-}\frac{t^a \phi(w)}{(z-w)^3} +\frac{(\Delta_\phi +1):j^a_{L,z}\phi:(w) + \frac{c_-}{(c_++c_-)^2}i{f^a}_{cb}t^b:j^c_{L,z}\phi:(w)}{(z-w)^2} \cr &\quad +\frac{\partial :j^a_{L,z}\phi:(w)}{z-w} + \mathcal{O}(f^4)+ \mathcal{O}(z-w)^0.\end{aligned}$$ The matrices $t^a$ are the generators of the Lie algebra in the representation in which the operator $\phi$ transforms. Since one has a non-vanishing third-order pole, not all of the operators $:j^a_{L,z}\phi:$ are Virasoro primary. Indeed we know from equation that the operator $L_{-1}\phi = \partial \phi$, which is a Virasoro descendant, is a linear combination of these operators. However the remaining ones are all Virasoro primaries. In the case where the quadratic Casimir of the representation $\mathcal{R}$ associated to the representation of the operator $\phi$ is non-zero, it is straightforward to check that in the OPE between the stress-tensor and the operator $ c^{(2)}_\mathcal{R} :j^a_{L,z}\phi: - t^a t_b :j^b_{L,z}\phi: $, the third-order pole vanishes. We adopt the notation $c^{(2)}_{\mathcal{R}}$ both for the (generalized) quadratic Casimir operator and for its eigenvalues in the irreducible representation or reducible indecomposable structure $\mathcal{R}$. [From]{} the double pole in equation we can read off the action of the scaling operator $L_0$ on the operator $:j^a_{L,z}\phi:$: L\_0 :j\^a\_[L,z]{}: = (\_+1):j\^a\_[L,z]{}: + i[f\^a]{}\_[cb]{}t\^b:j\^c\_[L,z]{}:.The operators $:j^a_{L,z}\phi:$ do not diagonalize the scaling operator $L_0$. In order to extract the conformal dimensions of these operators we have to compute the eigenvalues of the following operator : \[offDiagL0\] [f\^a]{}\_[cb]{} [[(t\^b)]{}\_]{}\^[ ]{} where we wrote explicitly the indices $\alpha$, $\beta$ associated to the representation $\mathcal{R}$ in which the primary field $\phi$ transforms. This operator is an endomorphism acting on the vector space associated to the tensor product of the adjoint and the representation $\mathcal{R}$, namely $Adj \otimes \mathcal{R}$. Since the structure constants are the generators of the Lie super-algebra in the adjoint representation, the operator can be rewritten as: \_[bd]{} t\^d\_[Adj]{} t\^b\_where this time we kept the external indices implicit. The generators of the Lie super-algebra in the (reducible) representation $Adj \otimes \mathcal{R}$ read : t\^a\_[Adj]{} Id + Id t\^a\_. Hence the quadratic Casimir operator in the tensor product of representations is : c\^[(2)]{}\_[Adj ]{} &=& \_[ba]{} (t\^a\_[Adj]{} Id + Id t\^a\_) (t\^b\_[Adj]{} Id + Id t\^b\_) &=& c\^[(2)]{}\_ + c\^[(2)]{}\_[Adj]{} + 2 \_[ba]{} t\^a\_[Adj]{} t\^b\_.We deduce that the operator that we want to diagonalize reads: \_[ba]{} t\^a\_[Adj]{} t\^b\_ = ( c\^[(2)]{}\_[Adj ]{} - c\^[(2)]{}\_ - c\^[(2)]{}\_[Adj]{} ). Recall that the quadratic Casimir vanishes in the adjoint representation: $c^{(2)}_{Adj}=0$. In the tensor product $Adj \otimes \mathcal{R}_\phi$, reducible indecomposable structures may appear. The Casimir operator is not diagonalizable on these structures, but we can still define its generalized eigenvalues. Finally we obtain the conformal dimension of the operators $:j^a_{L,z} \phi:$. Let us denote by $\tilde{\mathcal{R}}$ a representation that appears in the tensor product $Adj \otimes \mathcal{R}$, and by $[ :j^a_{L,z} \phi: ]_{\tilde{\mathcal{R}}}$ a linear combination of the operators $:j^a_{L,z} \phi:$ that transforms in the representation $\tilde{\mathcal{R}}$. We have shown: $$\begin{aligned} h \left(\left[:j^a_{L,z} \phi:\right]_{\tilde{\mathcal{R}}} \right) &=& \frac{f^2}{2} c^{(2)}_\mathcal{R} + 1 + \frac{f^2}{2} (1-k f^2) (c^{(2)}_{\tilde{\mathcal{R}}}-c^{(2)}_\mathcal{R})+ \mathcal{O}(f^4). %\cr\end{aligned}$$ The interpretation of this semi-classical result is as follows. At zero Wess-Zumino coupling $k=0$, we find that the conformal dimension at leading order is $f^2 c^{(2)}_{\tilde{\mathcal{R}}}/2+1$, namely the quadratic Casimir of the representation in which the total wave-function $j \phi$ transforms times the inverse radius squared, plus one for the fact that we are looking at a descendant state. That is as for a naive evaluation of the conformal dimension of the derivative of an ordinary point-particle wave function in representation $\tilde{\mathcal{R}}$. Note that at the WZW point $k f^2 = 1$, we also recuperate the usual behavior, which is that only the representation of the primary state $\phi$ counts for the basic conformal dimension, while currents add precisely one to the conformal dimension, independent of the representation in which the descendant state transforms. Thus the formula interpolates between these two intuitive behaviors, linearly in $kf^2$. Notice that the corrections to the dimension at the WZW point come from the logarithmic term in the current-primary OPE . This result illustrates the fact that the bulk partition function will split into a sum over (mini-superspace) representations of the supergroup with conformal dimensions depending on the representation in question. That demonstrates that this behavior, observed in boundary partition functions [@Quella:2007sg], extends to bulk partition functions. This structure carries over to both left and right conformal dimensions simultaneously. Indeed, let us turn to the calculation of the anti-holomorphic conformal dimension of the operator $:j^a_{L,z} \phi:$. As previously we compute the OPE between the anti-holomorphic stress-tensor and the operator: $$\begin{aligned} \bar T(\bar z) &:j^a_{L,z}\phi:(w) = \lim_{:x \to w:} \bar T(\bar z) j^a_{L,z}(x) \phi(w) \cr % & = \lim_{:x \to w:} \left \{ \left( \frac{\bar \p j^a_{L,z}(x)}{\bar z-\bar x} \right) \phi(w) + j^a_{L,z}(x) \left( \frac{\bar \Delta_{\phi} \phi(w)}{(\bar z-\bar w)^2} + \frac{\bar \p \phi(w)}{\bar z-\bar w} \right) \right\} \cr % %&= \lim_{:x \to w:} \left \{ % \frac{1}{\bar z-\bar x}\left( %-\frac{c_+}{c_++c_-} t^a \phi(w) 2\phi \delta^{(2)}(x-w) + :\bar \p j^a_{L,z} \phi:(w) \right. \right. \cr %&\qquad \qquad \qquad \left. %+ {A^a}_c \frac{1}{\bar x-\bar w} :j^c_{L,z} \phi:(w) + {B^a}_c \frac{1}{x-w} :j^c_{L,\bar z} \phi:(w) + \mathcal{O}(f^4) %\right) \cr %& \quad \left. %+ \frac{\bar \Delta_{\phi}:j^a_{L,z} \phi:(w)}{(\bar z-\bar w)^2} + \frac{:j^a_{L,z}\bar \p \phi:(w)}{\bar z-\bar w} %\right\} \cr % &= \frac{:\bar \p j^a_{L,z} \phi:(w)}{\bar z-\bar w} + {A^a}_c \frac{1}{(\bar z-\bar w)^2} :j^c_{L,z} \phi:(w) \cr & \quad + \frac{\bar \Delta_{\phi}:j^a_{L,z} \phi:(w)}{(\bar z-\bar w)^2} + \frac{:j^a_{L,z}\bar \p \phi:(w)}{\bar z-\bar w} \end{aligned}$$ Hence we have: $$\begin{aligned} \bar T(\bar z) :j^a_{L,z}\phi:(w) &= \frac{\bar \Delta_{\phi}:j^a_{L,z}\phi:(w) - \frac{c_-}{(c_++c_-)^2}i{f^a}_{bc}t^b:j^c_{L,z}\phi:(w)}{(\bar z-\bar w)^2} + \frac{\bar \partial :j^a_{L,z}\phi:(w)}{\bar z-\bar w} + \mathcal{O}(f^4).\end{aligned}$$ That leads to the conformal dimension: $$\begin{aligned} \bar h \left(\left[:j^a_{L,z} \phi:\right]_{\tilde{\mathcal{R}}} \right) &=& \frac{f^2}{2} c^{(2)}_\mathcal{R} + \frac{f^2}{2} (1-k f^2) (c^{(2)}_{\tilde{\mathcal{R}}}-c^{(2)}_{\mathcal{R}})+ \mathcal{O}(f^4). \nonumber % \cr %\end{aligned}$$ This is identical to the previous result, except for the lack of shift by one (since we are acting with the holomorpic component of the left current). Finally one can perform the same computation for the operators $:j^a_{L,\bar z} \phi:$. One finds : $$\begin{aligned} h \left(\left[:j^a_{L,\bar z} \phi:\right]_{\tilde{\mathcal{R}}} \right) &=& \frac{f^2}{2} c^{(2)}_\mathcal{R} + \frac{f^2}{2} (1+k f^2)( c^{(2)}_{\tilde{\mathcal{R}}}-c^{(2)}_{\mathcal{R}})+ \mathcal{O}(f^4) \cr \bar h \left(\left[:j^a_{L,\bar z} \phi:\right]_{\tilde{\mathcal{R}}} \right) &=& \frac{f^2}{2} c^{(2)}_\mathcal{R} + 1 + \frac{f^2}{2} (1+k f^2) (c^{(2)}_{\tilde{\mathcal{R}}}-c^{(2)}_{\mathcal{R}})+ \mathcal{O}(f^4) \end{aligned}$$ One can perform similar computations for operators that are composites of a right-current and a primary operator. The conformal dimensions of these operators can also be deduced from the symmetry of the model under the simultaneous interchanges $g \leftrightarrow g^{-1}$ and $z \leftrightarrow \bar z$. A basis of operators {#spectrum .unnumbered} -------------------- At the Wess-Zumino-Witten points $k f^2 = 1$ the $\bar z$-component (respectively $z$-component) of the left current (respectively right current) vanishes, and the left current (respectively right current) is holomorphic (respectively anti-holomorphic). Thus we can expand the currents in a Laurent expansion. The spectrum is generated by acting on the affine primary fields with the modes of the current. It is spanned by the operators : { j\^[a\_1]{}\_[-n\_1]{}j\^[a\_2]{}\_[-n\_2]{}...j\^[a\_p]{}\_[-n\_p]{} } where $\phi$ is an affine primary operator and the currents $j^{a_i}$ can be either the left-current $J^{a_i}$ or the right-current $\bar J^{a_i}$. In fact, all the negative modes of the currents $J^a_{-n}$ (respectively $\bar J^a_{-n}$) can be generated by successive commutations of the first negative mode $J^a_{-1}$ (respectively $\bar J^a_{-1}$). This is most easily seen by working in the Chevalley basis for the generators of the bosonic subalgebra. As a consequence the spectrum is also spanned by the operators : { j\^[a\_1]{}\_[-1]{}j\^[a\_2]{}\_[-1]{}...j\^[a\_p]{}\_[-1]{} }. Finally, we notice that for any operator $\chi$, the operator $J^a_{-1} \chi$ is the regular term in the OPE between the current $J^a$ and the operator $\chi$. Thus we can rewrite $J^a_{-1} \chi = :J^a \chi:$. So the previous set of operators spanning the spectrum can be rewritten as: { :j\^[a\_1]{}:j\^[a\_2]{}...:j\^[a\_p]{} :...:: }. We wrote the spectrum in this unusual form since it has the advantage that these operators are also defined away from the WZW point. At a generic point of the moduli space both the left- and the right-currents have two non-vanishing components. Since both left and right invariant one-forms generate a basis for the cotangent bundle in spacetime, the sets of operators generated by acting with left-currents or with right-currents on primary fields are isomorphic. This indicates that we have two (overcomplete) bases of operators : \[proposalSpectrum\] { :j\^[a\_1]{}\_L:j\^[a\_2]{}\_L...:j\^[a\_p]{}\_L :...:: } = { :j\^[a\_1]{}\_R:j\^[a\_2]{}\_R...:j\^[a\_p]{}\_R :...:: } where $\phi$ is a primary field as defined in section \[primaries\], and $j^{a_i}_L$ (respectively $j^{a_i}_R$) can be either the $z$- or $\bar z$-component of the left current (respectively right current). Of course, a mixture of left and right current components is also an allowed choice. We can compute the conformal dimensions of the operators of the sets by following the computation given at the beginning of this section. The knowledge of the current-current OPEs and of the current-primary OPEs up to terms of order $f^4$ allows the computation of the conformal dimensions up to terms of order $f^4$. Following the logic of section \[bootstrap\] it is then possible to compute order by order in $f^2$ the current-current OPEs, the current-primary OPEs and finally the conformal dimensions of the operators . The recursive calculation may allow for a closed solution. Let us stress that the spectrum can be generated by acting with the currents on a rather small set of primary operators. The current primaries at any point of the moduli space are in one-to-one correspondence with the affine primaries at the WZW points. In particular the set of current primaries is much smaller than the set of Virasoro primaries. Using the current algebra allows to take advantage of the extension of the symmetry algebra at particular points of the moduli space, namely the WZW points. In other words, in the scheme proposed here, we attempt to maximally exploit the presence of WZW lines in the two-dimensional moduli space of $G_L \times G_R$ invariant supergroup sigma-models. The classical and quantum integrability {#integrability} ======================================= The two-dimensional field theory under consideration is classically integrable in the sense that one can code the equations of motion in the demand that a connection depending on a spectral parameter is flat, thus leading to an infinite set of non-local conserved charges. We give the proof of this fact for a generic principal chiral model with Wess-Zumino term in appendix \[classint\]. For the model to be quantum integrable, there needs to be an infinite set of conserved charges in the quantum theory. There are circumstances in which anomalies prevent the lifting of the charges from the classical to the quantum theory. It is important to argue that this is not the case for the supergroup sigma-models under consideration here. Beyond the usual conserved charges $Q^a_{(0)}$ associated to the group action(s) on itself, a first set of non-local conserved charges can be defined as [@Luscher:1977uq]: $$\begin{aligned} \label{Q1} Q^a_{(1)} &=& N \int d \sigma j_\sigma^a + \int d \sigma_1 d \sigma_2 \epsilon(\sigma_1-\sigma_2) {f^a}_{bc} j^c_\tau(\tau,\sigma_1) j^b_\tau(\tau,\sigma_2),\end{aligned}$$ where $\tau,\sigma$ are time- and space-coordinates, the factor $N$ is an appropriate normalization constant, and the function $\epsilon$ takes the values $\pm 1$ depending on the sign of its argument. The non-local charges exists for both left and right currents. The proof of conservation of the non-local charge runs through the fact that the current $j$ is conserved, and the validity of the Maurer-Cartan equation. When both equations are preserved in the quantum theory, the (normal ordered) non-local charges survive (since from the first non-local charges, all others can be generated through commutation with the charges associated to the global symmetries). It should be clear now that we can view the fixing of higher order terms in the current-current operator products by demanding the vanishing of Maurer-Cartan operator as demanding OPEs compatible with the quantum integrability of the model. Conversely, the fact that one can find such OPEs in this model (using the special algebraic properties of the supergroup) lend credence to this hypothesis. It would be useful to make the link between the existence of the Yangian and the form of the current-current operator product expansions even more manifest. The main threat to the existence of the non-local charge comes from the UV-divergence in its definition. In the quadratic term, the current components are both integrated, and the integration involves a region in which the currents come very close to one another, thus necessitating a UV regulator that could potentially render the non-local charges anomalous. We will now show that in the models at hand, these potential UV divergences are absent, at least to the first few orders in perturbation theory, and presumably to all orders. [From]{} the current algebra , we see that : \_[cb]{} j\^b\_[L]{}(z) j\^c\_[L]{}(w) &=& [f\^a]{}\_[cb]{} :j\^b\_[L]{}(z) j\^c\_[L]{}(w): + (f\^4) which is true for the $z$ and $\bar z$ components of the currents. This follows from the fact that the tensors $\kappa^{bc}$, ${A^{bc}}_{de}$, ${B^{bc}}_{de}$ and ${C^{bc}}_{de}$ appearing in are graded-symmetric in the indices $b,c$. Moreover the double contraction of structure constants (the Killing form) also vanishes. This is a proof of the consistency of the current algebra with quantum integrability to second order. It is a strong suggestion of quantum integrability to all orders, a property which is closely tied to quantum conformal invariance. Conclusions =========== In this paper we continued the investigation of the conformal current algebra in non-linear sigma models on supergroups. The left and right current algebra closes on itself and a primary adjoint operator. The current algebra as well as the current-primary OPEs are tightly constrained by the Maurer-Cartan equation and current conservation, and can be computed order by order in a semi-classical expansion. We argued that one can view the Hilbert space of the theory as generated by currents acting on primaries, since WZW lines exist in the moduli space of theory. We initiated the (perturbative) computation of the spectrum, and argued for the possibility of a recursive bootstrap. We discussed the quantum integrability of the model, and tied it to properties of the current algebra. We hope our analysis contributes to the determination of an explicit solution to the full bulk spectrum of two-dimensional conformal field theories on supergroups and their cosets. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Sujay Ashok, Costas Bachas, Denis Bernard, Vladimir Fateev, Frank Ferrari, Matthias Gaberdiel, Bernard Julia, Anatoly Konechny, Thomas Quella, Sylvain Ribault and Walter Troost for useful questions and helpful discussions. J.T. would like to acknowledge support by ANR grant ANR-09-BLAN-0157-02. The work of R.B. is supported in part by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole IAP VI/11 and by FWO-Vlaanderen through project G011410N. Operator products involving composite operators {#compositeOPEs} =============================================== In this appendix we discuss the computation of OPEs involving composite operators. We consider the following OPE: \[A:BC:\] \_[zw]{} A(z) :BC:(w). The composite operator $:BC:(w)$ is defined as the term multiplied by $(x-w)^0 (\bar x - \bar w)^0$ in the OPE between the operators $B(x)$ and $C(w)$. To compute the OPE we use a point splitting procedure. We denote the extraction of the normal ordered term by the limit $:BC:(w) = \lim_{:x \to w:}B(x) C(w) $. This symbolizes that at the end of the calculation we take the limit $x \to w$, and discard all terms that are singular in $x-w$ in this limit. To compute the operator product of the operator $A$ with the composite operator $:BC:$ we proceed as follows. On the one hand we perform the OPE of the operators $A$ and $B$, and then we perform the OPE of the result with $C$. On the other hand we perform the OPE of the operators $A$ and $C$, and then we perform the OPE of the result with $B$. Eventually take the regular limit $:x \to w:$ and add up the two terms. Additional details about these operations follow. - First let us consider the OPE between $A(z)$ and $B(x)$. We evaluate the result at the point $x$ – otherwise taking the regular limit $:x \to w:$ would become cumbersome. Let us consider one term in the OPE between $A(z)$ and $B(x)$: \[A:BC:step1\] A(z) B(x) = ... + (z-x)\^[\_D - \_A - \_B]{} (|z-|x)\^[|\_D - |\_A - |\_B]{} D(x) + ... where $\Delta_O$ (respectively $\bar \Delta_O$) stands for the holomorphic (respectively anti-holomorphic) conformal dimension of an operator $O$. For simplicity we consider a term in which no logarithm appears, but the generalization is straightforward. We have to perform the OPE of the right-hand side with the operator $C(w)$. Let us consider one term in the result: $$\begin{aligned} &(z-x)^{\Delta_D - \Delta_A - \Delta_B}(\bar z-\bar x)^{\bar\Delta_D - \bar\Delta_A - \bar \Delta_B} D(x) C(w) =\cr &...+ (x-w)^{\Delta_E - \Delta_D - \Delta_C}(\bar x-\bar w)^{\bar\Delta_E - \bar\Delta_D - \bar \Delta_C}(z-x)^{\Delta_D - \Delta_A - \Delta_B}(\bar z-\bar x)^{\bar\Delta_D - \bar\Delta_A - \bar \Delta_B} E(w) +... \nonumber \end{aligned}$$ Now to take the normal ordered limit $:x \to w:$, we expand the functions depending on $x$ in the neighborhood of $w$, namely, we write: (z-x)\^= (z-w)\^-(x-w) (z-w)\^[-1]{} + ... and we keep only the terms that end up with no factor of $(x-w)$. The same manipulations have to be done for the anti-holomorphic factors. If both $\Delta_E - \Delta_D - \Delta_C$ and $\bar\Delta_E - \bar\Delta_D - \bar \Delta_C$ are non-positive integers, then the term we isolated in the previous steps contributes to the OPE as: $$\begin{aligned} \label{A:BC:step4}\lim_{z\to w}& A(z) :BC:(w) = ... + \# (z-w)^{\Delta_E - \Delta_A - \Delta_B-\Delta_C} (\bar z - \bar w)^{\bar\Delta_E - \bar\Delta_A - \bar\Delta_B-\bar\Delta_C} E(w)\end{aligned}$$ with numerical coefficient: $$\begin{aligned} \label{A:BC:step4coef}\# &= (-1)^{-\Delta_E + \Delta_D + \Delta_C}(-1)^{-\bar \Delta_E + \bar \Delta_D + \bar \Delta_C} \cr & \times \frac{(\Delta_D - \Delta_A - \Delta_B)(\Delta_D - \Delta_A - \Delta_B-1)...(\Delta_E - \Delta_A - \Delta_B-\Delta_C+1)}{(-\Delta_E + \Delta_D + \Delta_C)!} \cr & \times \frac{(\bar\Delta_D - \bar\Delta_A - \bar\Delta_B)(\bar\Delta_D - \bar\Delta_A - \bar\Delta_B-1)...(\bar\Delta_E - \bar\Delta_A - \bar\Delta_B-\bar\Delta_C+1)}{(-\bar\Delta_E + \bar\Delta_D + \bar\Delta_C)!} .\end{aligned}$$ Let us stress that a given term in the result of the OPE may receive contributions from an $infinite$ number of terms in the OPE between $A$ and $B$. This makes the computation of OPEs involving composite operators rather involved. One may need to resort to perturbation theory in a small parameter to render the calculation manageable. The perturbation theory that we use in the bulk of the paper is explained in section \[bootstrap\] and in the appendices \[XXOPEs\] and \[consistentPertOPEs\]. - Let us turn to the OPE between $A(z)$ and $C(w)$, which we evaluate at the point $w$. This second step is simpler than the first. Again, we concentrate on one term in this OPE: A(z) C(w) = ... + (z-w)\^[\_F - \_A - \_B]{}(|z-|w)\^[|\_F - |\_A - |\_B]{} F(w) + ...We then have to perform the OPE between the right-hand side and the operator $B(x)$. We evaluate the result at the point $w$. Let us write down one term in the result: $$\begin{aligned} &(z-w)^{\Delta_F - \Delta_A - \Delta_B}(\bar z-\bar w)^{\bar\Delta_F - \bar\Delta_A - \bar \Delta_B} B(x) F(w) =\cr &...+ (z-w)^{\Delta_F - \Delta_A - \Delta_B}(\bar z-\bar w)^{\bar\Delta_F - \bar\Delta_A - \bar \Delta_B}(x-w)^{\Delta_G - \Delta_B - \Delta_F}(\bar x-\bar w)^{\bar\Delta_G - \bar\Delta_B - \bar \Delta_F} G(w) +... \nonumber\end{aligned}$$ Finally we take the straightforward normal ordered limit $:x\to w:$, that discards all the terms except for the one with $\Delta_G - \Delta_B - \Delta_F=\bar\Delta_G - \bar\Delta_B - \bar \Delta_F=0$. Thus only the regular term $:BF:(w)$ in the OPE between $B(x)$ and $F(w)$ survives. We obtain the following contribution to the OPE : $$\begin{aligned} \lim_{z\to w}& A(z) :BC:(w) = ... + (z-w)^{\Delta_F + \Delta_B- \Delta_A -\Delta_C} (\bar z - \bar w)^{\bar\Delta_F+\bar \Delta_B - \bar\Delta_A -\bar\Delta_C} :BF:(w). \nonumber\end{aligned}$$ Simplification in the case of a holomorphic operator {#simplification-in-the-case-of-a-holomorphic-operator .unnumbered} ---------------------------------------------------- The computation of the singular terms in the OPE simplifies if the operator $A(z)$ is holomorphic. Let us consider a term of the form . Since the operator $A$ is holomorphic there is no dependence on $\bar z$, so $\bar\Delta_D - \bar\Delta_A - \bar \Delta_B = 0$. Let us also assume that $\Delta_D - \Delta_A - \Delta_B$ is an integer. The question is whether such a term may contribute to a pole in the OPE , i.e. a term of the form with $\Delta_E - \Delta_A - \Delta_B-\Delta_C$ a negative integer (and $\bar\Delta_E - \bar\Delta_A - \bar\Delta_B-\bar\Delta_C=0$). But this is only possible if $\Delta_D - \Delta_A - \Delta_B$ is already a negative integer, since otherwise the coefficient vanishes. It follows from the previous discussion that under the assumption that only integer powers of $(z-x)$ appear in the OPE between the operators $A(z)$ and $B(x)$, then in the computation of singular terms in the OPE one can truncate the OPE between $A(z)$ and $B(x)$ to the singular terms only (i.e. keep only the poles in $(z-x)$). That specific feature of this special case is put to good use in some standard calculations in two-dimensional conformal field theory [@yellowbook]. The semi-classical behavior of the OPE coefficients {#XXOPEs} =================================================== At large radius, namely in the limit $f^2 \to 0$ (either at fixed level $k$ or at fixed $kf^2$), the target space flattens and the worldsheet theory becomes free. More precisely we obtain a theory of $d$ free bosons, where $d$ is the dimension of the adjoint representation of the super Lie algebra. Among these bosons, some are commuting and some are anti-commuting, depending on whether they can be associated to bosonic or fermionic coordinates of target space. At fixed $kf^2$ the $f^2 \to 0$ limit is the semi-classical limit of the model. Our goal in this appendix is to evaluate the behavior at large radius (small $f^2$) of the terms appearing in the current-current and current-primary OPEs. Let us start with the action of the model: $$\begin{aligned} S &= S_{kin} + S_{WZ}\cr S_{kin} &= \frac{1}{ 16 \pi f^2}\int d^2 x Tr'[- \partial^\mu g^{-1} \partial_\mu g] \cr S_{WZ} &= - \frac{ik}{24 \pi} \int_B d^3 y \epsilon^{\alpha \beta \gamma} Tr' (g^{-1} \partial_\alpha g g^{-1} \partial_\beta g g^{-1} \partial_\gamma g ).\end{aligned}$$ We write the group element as: g=e\^[f X]{}=e\^[i f X\_a t\^a]{} where the $X_a$ are coordinates on the supergroup and the matrices $t^a$ are the generators of the Lie superalgebra. The kinetic term and the Wess-Zumino term become: $$\begin{aligned} \label{action(X)} S_{kin} &=& \frac{1}{4 \pi} \int d^2 z \left( \partial X_a \bar{\partial} X^a - \frac{f^2}{12} {f^a}_{fe} {f}_{acb} X^b \partial X^c X^e \bar{\partial} X^f + ...\right) \nonumber \\ S_{WZ} &=& -\frac{kf^2 }{12 \pi} \int d^2 z \left( f f_{abc} X^c \partial X^b \bar{\partial} X^a + ... \right).\end{aligned}$$ Written in this way the theory describes a set of interacting bosons (some of which are anti-commuting). The quadratic terms in the action give rise to the free propagator: $$\begin{aligned} \label{freeProp} X^a(z,\bar{z}) X^b(w,\bar{w}) &=& - \kappa^{ab} \log \mu^2 |z-w|^2,\end{aligned}$$ where $\mu$ is an infrared regulator. The propagator behaves like $\mathcal{O}(f^0)$, whereas a vertex with $p+2$ legs (i.e. Lie algebra indices) behaves as $\mathcal{O}(f^p)$. It follows that the theory reduces to a theory of free bosons in the semi-classical limit, as anticipated. At fixed $kf^2$ and for each interaction vertex, the power of the coupling constant $f$ is equal to the number of structure constants that appear. Since we are interested in computing OPEs involving the currents and the primary fields, let us write these fields in terms of the bosons $X^a$: $$\begin{aligned} \label{current(X)} \frac{j^a_{L,z}}{c_+} &=& (\partial g g^{-1})^a = i (f \partial X^a + f^2 \frac{{f^{a}}_{bc}}{2} X^c \partial X^b + \frac{f^3}{6} {f^a}_{bc} {f^c}_{de} \partial X^e X^d X^b +...) \nonumber \\ \frac{j^a_{L,\bar z}}{c_-} &=& (\bar{\partial} g g^{-1})^a = i (f \bar{\partial} X^a + f^2 \frac{{f^{a}}_{cb}}{2} X^b \bar{\partial} X^c + \frac{f^3}{6} {f^a}_{bc} {f^c}_{de} \bar \partial X^e X^d X^b +...),\end{aligned}$$ \[phi(X)\] = e\^[i f X\_a t\^a]{} = i f X\_a t\^a - f\^2 X\_a t\^a X\_b t\^b +... where in the last line the generators $t^a$ are taken in the representation associated to the primary field $\phi$. The semi-classical behavior of the current-current OPE {#the-semi-classical-behavior-of-the-current-current-ope .unnumbered} ------------------------------------------------------ We study the semi-classical behavior of the OPE between two $z$-components of the left-current. The discussion generalizes straightforwardly to other current-current OPEs. We assume that the only operators that appear in the result of this OPE are composites of (derivatives of) left currents. This is true at the WZW point, and can presumably be proven at any point using conformal perturbation theory. Let us isolate one term in this OPE : \[jjOneTerm\] j\^a\_[L,z]{}(z) j\^b\_[L,z]{}(w) = ... + [A\^[ab]{}]{}\_[a\_p a\_[p-1]{}...a\_[2]{} a\_1]{}(z-w, |z - |w) :j\^[a\_1]{}\_[L,z]{}:j\^[a\_2]{}\_[L,z]{}...:j\^[a\_[p-1]{}]{}\_[L,z]{}j\^[a\_p]{}\_[L,z]{}:...::(w) +...Our goal is to evaluate the behavior of the tensor ${A^{ab}}_{a_p...a_1}(z-w, \bar z - \bar w)$ when the parameter $f$ is small. The reasoning will not depend on the particular current component, nor on the presence of further derivative operators. To proceed we use the expression of the currents in terms of the bosonic fields $X^a$. First let us focus on the leading term in the expansion . We consider the OPE: \[dXdXOneTerm\] X\^a(z) X\^b(w) = ... + [\^[ab]{}]{} \_[a\_p a\_[p-1]{}...a\_[2]{} a\_1]{}(z-w, |z - |w) :X\^[a\_1]{}:X\^[a\_2]{}...:X\^[a\_[p-1]{}]{}X\^[a\_p]{}:...::(w) +...The behavior of the tensor ${\tilde{A}^{ab}} {}_{a_p a_{p-1}...a_{2} a_1}$ as a function of the parameter $f$ will give the behavior of the tensor ${A^{ab}}_{a_p...a_1}(z-w, \bar z - \bar w)$ defined in equation . As a first step let us consider the following three-point function: \[FeynmanDiag\] X\^a(z) X\^b(w) :X\^[a\_1]{}:X\^[a\_2]{}...:X\^[a\_[p-1]{}]{}X\^[a\_p]{}:...::(x) \_[connected]{} We consider only the contribution of connected Feynman diagrams to this correlation function. Indeed, if the external operators $\p X^a(z)$ and $\p X^b(w)$ are contracted on different pieces of a disconnected Feynman diagram, then the result contributes to the regular term $:\p X^a(x) \p X^b(w):$ on the right-hand side of the OPE . Thus to compute the non-trivial terms in this OPE one needs to consider only the Feynman diagrams for which the external operators $\p X^a(z)$ and $\p X^b(w)$ are connected. But this in turn implies that the Feynman diagram is fully connected. Indeed, if this were not the case then one connected piece of the Feynman diagram has for external lines operators coming from the composite operator $ :\p X^{a_1}:\p X^{a_2}...:\p X^{a_{p-1}}\p X^{a_p}:...::(x)$ only. Such a piece would depend on the coordinate $x$ only, and would necessarily be zero by translation invariance. This shows that we need to consider only fully connected Feynman diagrams. Now let us evaluate the $f$-dependence of a connected Feynman diagram contributing to . We will show by induction the following statement: a connected Feynman diagram in the theory with $p+2$ external legs behaves like $\mathcal{O}(f^p)$. This is the case for $p=0$ since the propagator is of order $f^0$. Now let us assume that the statement has been proven for $p <n+2$, and consider a Feynman diagram with $n+2$ external lines. We isolate $m$ of these external legs that are contracted on the same vertex with $m+1$ legs. This piece is of order $f^{m-1}$. The other piece of the Feynman diagram has $n+2-m+1$ external lines, and by induction is of order $f^{n-m+1}$. Thus the result is of order $f^{n}$, and the proof is completed. We deduce that: X\^a(z) X\^b(w) :X\^[a\_1]{}:X\^[a\_2]{}...:X\^[a\_[p-1]{}]{}X\^[a\_p]{}:...::(x) \_[connected]{} = (f\^p).Since two-point functions of (composites of) the fields $X^a$ behave at least as $\mathcal{O}(f^0)$, we can now combine the previous result with equation to evaluate the order of the term in the current OPE under consideration[^5]: = ... + (f\^p) ::...::...::(w) +...Given that $f c_+ = \mathcal{O}(f^{-1})$, we obtain: j\^a\_[L,z]{}(z) j\^b\_[L,z]{}(w) = ... + (f\^[2p-2]{}) :j\^[a\_1]{}\_[L,z]{}:j\^[a\_2]{}\_[L,z]{}...:j\^[a\_[p-1]{}]{}\_[L,z]{}j\^[a\_p]{}\_[L,z]{}:...::(w) +...This is a property we repeatedly confirm as well as use in the bulk of the paper. The semi-classical behavior of the current-primary OPE {#the-semi-classical-behavior-of-the-current-primary-ope .unnumbered} ------------------------------------------------------ We can perform a similar analysis to determine the behavior of the terms in the current-primary OPE at large radius. Let us consider a primary field $\phi$. We assume that all the terms that appear in the OPE between a left current and this primary field are composite operators including an arbitrary number of left currents and one field $\phi$ only. This is the case at the WZW point. Then by continuously deforming the OPEs away from the WZW point, this is the case over the whole moduli space of the theory. Let us isolate one term in the OPE between the left current $j^a_{L,z}$ and the primary field $\phi$: j\^a\_[L,z]{}(z) (w) = ... + [B\^[a]{}]{}\_[a\_p a\_[p-1]{}... a\_1]{}(z-w, |z - |w) :j\^[a\_1]{}\_[L,z]{}:j\^[a\_2]{}\_[L,z]{}...:j\^[a\_p]{}\_[L,z]{} :...::(w) +...Our goal is to evaluate the behavior of the tensor ${B^{a}}_{a_p...a_1}(z-w, \bar z - \bar w)$ when the parameter $f^2$ is small. The composite operator we wrote down does not have any derivative and contains only $z$-components of the left current, but the result would be the same for a more general operator. Only the number $p$ of currents will be relevant. Following the previous reasoning one can show that : X\^a(z) X\^b(w) :X\^[a\_1]{}:X\^[a\_2]{}...:X\^[a\_[p]{}]{} X\^[a\_[p+1]{}]{}:...::(x) \_[connected]{} = (f\^[p+1]{}).Combining this result together with equations and and the fact that two-points functions are of order $\mathcal{O}(f^0)$ we get: = ... + (f\^[p+1]{}) ::...: :...::(w) +...which we rewrite as: j\^a\_[L,z]{}(z) (w) = ... + (f\^[2p]{}) :j\^[a\_1]{}\_[L,z]{}:j\^[a\_2]{}\_[L,z]{}...:j\^[a\_p]{}\_[L,z]{} :...::(w) +...This result on the order of magnitude of the operator product is confirmed and used in the bulk of the paper. Consistency of perturbation theory {#consistentPertOPEs} ================================== Current-current OPE {#current-current-ope .unnumbered} ------------------- In section \[bootstrap\] we explained how to compute the current-current OPEs order by order in a semi-classical expansion. The idea is to ask for the vanishing of the OPE between a current and both current conservation and the Maurer-Cartan equation, order by order in $f^2$. These two constraints can be combined as : \[AppjMC\] j\^a\_[L]{}(z) (|j\^b\_[L,z]{}(w)-i f\^2 [f\^b]{}\_[cd]{}:j\^d\_[L,z]{} j\^c\_[L,|z]{}:(w)) = 0. For this perturbative method to be consistent a term of order $f^{2n}$ in the current-current OPEs should not spoil the vanishing of the previous OPE up to order $f^{2n-2}$. The subtlety lies in the computation of the OPE involving the composite operator in equation . Indeed the fact that the leading singularity in the current-current OPE has a coefficient of order $f^{-2}$ threatens to generates terms of low order in $f^2$ in this computation. In this appendix we will show that a term of order $f^{2n}$ in the current-current OPE does produce terms of order $f^{2n}$ in the OPE between a current and the composite operator appearing in equation , namely $f^2 {f^b}_{cd}:j^d_{L,z} j^c_{L,\bar z}:$. As a preliminary step let us prove the following useful lemma. We consider a composite of $p$ currents $:j:j:j...j:...::$ that we write symbolically $:j^p:$. Then the OPE of this operator with one current $j$ is at most of order $f^{-2}$: \[lemmaf-2\] j(z) :j\^p:(w) = (f\^[-2]{}). To prove this property we rewrite the current in terms of the bosons $X^a$ using equation . Schematically we have: j = f\^[-2]{} \_[n=0]{}\^ \# f \^[n+1]{} :X\^[n+1]{}: where we kept the numerical factors, possible derivatives acting on the fields $X$, and the index structures implicit to simplify the formula. Similarly the composite operator $:j^p:$ is written as: :j\^p: = f\^[-2p]{} \_[m=0]{}\^ \# f\^[m+p]{} :X\^[m+p]{}: To evaluate the OPE between the current $j$ and the composite operator $:j^p:$ we need to evaluate the OPE between operators of the form $:X^{q}:$. Remember that the propagator for the field $X$ is of order $f^0$, and that the $n$-point vertex is of order $f^{n-2}$. We deduce: :X\^[q\_1]{}:(z) :X\^[q\_2]{}:(w) = \_[q=0]{}\^ (f\^[|q\_1-q\_2|-q]{}) :X\^[q]{}:(z) In the previous equation the estimation of the order of the terms is rough (especially for large $q$) but it will be sufficient for our purposes. The proof is similar to the argument given below (except that in the present case disconnected Feynman diagrams contribute). We deduce an estimation for the order of the terms in the OPE \[j.j\^p\] j(z) :j\^p:(w) = f\^[-2p-2]{} \_[n,m=0]{}\^ f\^[n+m+p+1]{} \_[q=0]{}\^ (f\^[|n+1-m-p|-q]{}) :X\^[q]{}: The operators that appear in the OPE are themselves (composites of) currents. Let us evaluate the coefficient of a composite operator of the form $:j^r:$. According to equation the leading-order term in this composite operator written in terms of $X$’s is : :j\^r: = f\^[-r]{} :X\^[r]{}: + (:X\^[r+1]{}:). So to get the order of the coefficient that multiplies and operator $:j^r:$, it is enough to look for the coefficient of the terms multiplying $f^{-r}:X^r:$ in the OPE . These terms have a coefficient of order: f\^[-2p-2+n+m+p+1+|n+1-m-p|]{}={ [lll]{} f\^[2(n+1-p)-2]{} & if & n+1 m+p\ f\^[2m-2]{} & if & n+1 m+p. . Thus this coefficient is of order $\mathcal{O}(f^{-2})$. This completes the proof of . Now let us come back to the evaluation of the OPE between a current and the composite operator in equation : \[modjMC3\] j\^a\_[L,z]{}(z) i f\^2 [f\^b]{}\_[cd]{}:j\^d\_[L,z]{} j\^c\_[L,|z]{}:(w) Let us consider one term of order $f^{2n}$ in the OPE between the operators $j^a_{L,z}$ and $j^d_{L,z}$, that we write schematically $f^{2n}:j^p:$. To complete the computation we have to perform the OPE of this operator with the remaining current $j^c_{L,\bar z}$. According to the previous lemma, this OPE produces terms with coefficients of order $f^{-2}$. So we have proven that terms of order $f^{2n}$ in the current-current OPE produce in the OPE terms of order $f^{2 + 2n - 2} = f^{2n}$. This proves the consistency of the algorithm to compute the current-current OPE order by order in $f^2$. Current-primary OPE {#current-primary-ope .unnumbered} ------------------- As explained in section \[bootstrap\] the same logic allows us to perturbatively compute the operator product expansion between a current and a primary operator. The Maurer-Cartan equation can be combined with current conservation to give the constraint : \[phiModMC\] (z) (|j\^b\_[L,z]{}(w)+i f\^2 [f\^b]{}\_[cd]{}:j\^d\_[L,z]{} j\^c\_[L,|z]{}:(w)) = 0 This allows the computation of the $j^a_{L,z}.\phi$ OPE order by order in $f^2$. The consistency of this algorithm is ensured by a slight generalization of lemma , namely: \[lemmaf-2bis\] j(z) :j\^p :(w) = (f\^[-2]{}). The proof is similar to the proof of formula . Conformal current algebra: precisions {#AppCurrents} ===================================== In this appendix we gather various technical results related to the current algebra . The current algebra at order $f^2$ {#jMCOPE} ---------------------------------- In [@Ashok:2009xx] the current algebra was computed at the order of the poles. The discussion of section \[bootstrap\] shows that we can compute the less-singular terms by demanding consistency with current conservation and the Maurer-Cartan equation. In this appendix we will give details of this computation, and derive in particular the value of the new coefficients in the current algebra . In this particular calculation, we show how to restore various signs that are associated to the fact that we deal with a super Lie algebra. Since we use the special algebraic structure of supergroups with zero Killing form, these signs are crucial. To set up the problem, we establish conventions for the metric inverse and the contraction of indices: $$\begin{aligned} \kappa_{ab} \kappa^{cb} &=& {\delta_a}^c \nonumber \\ j_a &=& \kappa_{ab} j^b \nonumber \\ {[} t_a, t_b {]} &=& i t_c {f^c}_{ab}.\end{aligned}$$ We contract indices south-west north-east[^6]. As explained in section \[bootstrap\] current conservation implies that the tensors $A,B,C$ that appear in each one of the three OPEs are equal. To compute them we ask for the vanishing of the OPE between a current and the Maurer-Cartan operator : $$\begin{aligned} c_- \partial_{\bar{z}} j_{L,z}^c - c_+ \partial_z j^c_{L,\bar z} - i {f^c}_{de} :j_{L,z}^e j_{L,\bar z}^d:.\end{aligned}$$ Below we compute the OPE between the (left) current $j_{\bar z}^a$ and the Maurer-Cartan operator. For ease of writing, we will separate various terms in the calculation. We first calculate the operator product of the current with the first term: $$\begin{aligned} \mbox{Term 1} &=& j_{\bar z}^a (z) \cdot c_- \partial_{\bar{w}} j_z^c (w) \nonumber \\ & \sim & c_- \partial_{\bar w} ( \tilde{c} \kappa^{ac} 2 \pi \delta(z-w) \nonumber \\ & & + {f^{ac}}_g (\frac{c_4-g}{\bar{z}-\bar{w}} j^g_z (z) + \frac{(c_2-g)}{z-w} j_{\bar{z}}^g (z) \nonumber \\ & & + \frac{g}{4} \log |z-w|^2 (\partial_z j_{\bar{z}}^g(z) - \partial_{\bar z} j^g_z(z))) \nonumber \\ & & + (-1)^{ac} : j_z^c j_{\bar z}^a :(z) \nonumber \\ & & + ( {(A)^{ac}}_{gh} \frac{\bar z - \bar w}{z-w} :j^{g}_{\bar z} j^{h}_{\bar z}: (z) - ( {(B)^{ac}}_{gh} \log |z-w|^2 :j^{g}_{z} j^{h}_{\bar z}: (z) \nonumber \\ & & + ( {(C)^{ac}}_{gh} \frac{z-w}{\bar{z}-\bar{w}} : j_z^{g} j_z^{h}:(z)))+... %\nonumber\end{aligned}$$ The second term we take into account comes from contracting the current with the second term in the Maurer-Cartan operator: $$\begin{aligned} \mbox{Term 2} &=& j_{\bar z}^a (z) \cdot (-)c_+ \partial_{w} j_{\bar z}^c (w) \nonumber \\ & \sim &- c_+ \partial_{w} (c_3 \kappa^{ac} \frac{1}{(\bar z - \bar w)^2} \nonumber \\ & & + {f^{ac}}_g (\frac{c_4}{\bar{z}-\bar{w}} j^g_{\bar z} (w) + \frac{(c_4-g)(z-w)}{(\bar z- \bar w)^2} j^g_{z} (w) \nonumber \\ & & + \frac{g}{4} \frac{z-w}{\bar z - \bar w} (\partial_z j_{\bar{z}}^g(w) - \partial_{\bar z} j^g_z(w)) +\frac{c_4}{2} \partial_{\bar z} j^g_{\bar z}(w) + \frac{c_4-g}{2} \frac{(z-w)^2}{(\bar z - \bar w)^2} \partial_z j^g_z (w)) \nonumber \\ & & + : j_{\bar z}^a j_{\bar z}^c :(w) \nonumber \\ & & + (- {(A)^{ac}}_{gh} \log |z-w|^2 :j^g_{\bar z} j^h_{\bar z}:+ {(B)^{ac}}_{gh} \frac{z-w}{\bar{z}-\bar{w}} :j^{g}_{z} j^{h}_{\bar z}: \nonumber \\ & & + {(C)^{ac}}_{gh} \frac{(z-w)^2}{(\bar{z}-\bar{w})^2} : j_z^g j_z^h:(w)))+... %\nonumber \end{aligned}$$ Furthermore we have the contractions with the composite piece of the Maurer-Cartan operator. Following appendix \[compositeOPEs\] we use a point-splitting procedure and write ${f^c}_{de} :j_z^e j_{\bar z}^d:(w) = \lim_{:x \to w:} {f^c}_{de} j_z^e(x) j_{\bar z}^d(w)$. Then we distinguish two terms. The simplest is the term where we contract the current component $j_{\bar z}^a$ with the part at $w$ of the split operator. We then still need to contract further while eliminating singularities as $x$ goes to $w$, but this is easily done: only regular terms survive. We obtain: $$\begin{aligned} \mbox{Term 3} &=& (-i) (-1)^{ea} {f^c}_{de} ( (c_3 \kappa^{ad} \frac{1}{(\bar z - \bar w})^2 j^e_z (w) \nonumber \\ & & + {f^{ad}}_g (\frac{c_4}{\bar{z}-\bar{w}} :j^e_z j^g_{\bar z}: (w) + \frac{(c_4-g)(z-w)}{(\bar z- \bar w)^2} :j^e_z j^g_{z}: (w) \nonumber \\ & & + \mbox{order zero in the separation.}\end{aligned}$$ There is also the more involved term where we contract first with $j^e_z(x)$, and then further with $j^d_{\bar z}(w)$: $$\begin{aligned} \mbox{Term 4} &= & \lim_{:x \to w:} X^{ae}(z,x) (-i) {f^c}_{de} j^d_{\bar z}(w)\end{aligned}$$ where $$\begin{aligned} X^{ae}(z,x) & \sim & \tilde{c} \kappa^{ae} 2 \pi \delta(z-x) \nonumber \\ & & + {f^{ae}}_g (\frac{c_4-g}{\bar{z}-\bar{x}} j^g_z (z) + \frac{(c_2-g)}{z-x} j_{\bar{z}}^g (z) \nonumber \\ & & + \frac{g}{4} \log |z-x|^2 (\partial_z j_{\bar{z}}^g(z) - \partial_{\bar z} j^g_z(z))) \nonumber \\ & & + (-1)^{ae} : j_z^e j_{\bar z}^a :(z) \nonumber \\ & & + {A_{}^{ac}}_{gh} \frac{\bar z - \bar x}{z-x} :j^{g}_{\bar z} j^{h}_{\bar z}: (z) - {B_{}^{ac}}_{gh} \log |z-x|^2 :j^{g}_{z} j^{h}_{\bar z}: (z)\nonumber \\ & & + {C_{}^{ac}}_{gh} \frac{z-x}{\bar{z}-\bar{x}} : j_z^{g} j_z^{h}:(z) \nonumber \\ & & + \mbox{order 1 in the separation and higher order in the parameter $f^2$.} \nonumber\end{aligned}$$ Let’s sum these four terms and discuss the vanishing of the total operator product order by order. The contact terms and double pole terms were already treated in [@Ashok:2009xx]. We cancel them as follows: 1\. There are terms proportional to $\partial_{\bar w} 2 \pi \delta(z-w)$. These have coefficients: $$\begin{aligned} c_- \tilde{c} \kappa^{ac} +c_+ c_3 \kappa^{ac} \end{aligned}$$ which vanishes since the coefficients of the current algebra satisfy : $$\begin{aligned} c_- \tilde{c} &=&- c_+ c_3\end{aligned}$$ 2\. There are terms proportional to $2 \pi \delta(z-w)$ with coefficient: $$\begin{aligned} - c_- & {f^{ac}}_g (c_2-g) j_{\bar z}^g (w)+ c_+ {f^{ac}}_g c_4 j_{\bar z}^g(w) -i {f^c}_{de} \tilde{c} \kappa^{ae} j_{\bar z}^d (w) \nonumber \\ & = - c_- {f^{ac}}_g (c_2-g) j_{\bar z}^g (w)+ c_+ {f^{ac}}_g c_4 j_{\bar z}^g(w) -i (-1)^a (-1)^a {f^{ac}}_{g} \tilde{c} j_{\bar z}^g (w)\end{aligned}$$ which also vanishes thanks to the relation : $$\begin{aligned} -c_- (c_2-g) + c_+ c_4 - i \tilde{c} &=& 0.\end{aligned}$$ 3\. There are terms proportional to $1/(\bar z - \bar w)^2$ with coefficients: $$\begin{aligned} c_- (c_4-g) {f^{ac}}_g j^g_z + c_+ (c_4-g) {f^{ac}}_g j_z^g -i (-1)^{ea} {f^c}_{de} c_3 \kappa^{ad} j_z^e -i {f^c}_{de} {f^{ae}}_g (c_4-g)^2 {f^{gd}}_h j^h_z \nonumber\end{aligned}$$ where the last term arises from expanding $1/(z-x)$ and taking into account the further contraction in Term 4. This last term vanishes thanks to the super-Jacobi identity combined with the vanishing of the Killing form. Note that this implies that the second line in Term 4 does not contribute when the contraction between $j^g_z$ and $j^d_{\bar z}$ gives rise to either a metric or structure constant. Thus, it can potentially contribute starting at order zero in the separation only. The coefficient of the terms under consideration then vanishes since the coefficient satisfies the relation : $$\begin{aligned} (c_-+c_+)(c_4-g) +i c_3 &=& 0 .\end{aligned}$$ 4\. We now turn to the calculation which is new compared to [@Ashok:2009xx]. In the operator product expansion the simple pole in $1/(\bar z - \bar w)$ comes with the coefficient : $$\begin{aligned} c_-& {f^{ac}}_g (c_4-g) \partial_{\bar z} j_z^g(w) -c_+ {f^{ac}}_g c_4 \partial_{z} j_{\bar z}^g (w) \nonumber \\ & -c_- \frac{g}{4} {f^{ac}}_g (\partial_z j_{\bar z}^g- \partial_{\bar z} j_{ z}^g) +c_- {B^{ac}}_{gh} : j_z^{g} j^{h}_{\bar z}:(w) \nonumber \\ & +c_+ \frac{g}{4} {f^{ac}}_g (\partial_z j_{\bar z}^g- \partial_{\bar z} j_{ z}^g) +c_+ {B^{ac}}_{gh} : j_z^{g} j^{h}_{\bar z}:(w) \nonumber \\ & -i (-1)^{ea} {f^{c}}_{de} {f^{ad}}_g c_4 : j_z^e j_{\bar z}^g:(w) \nonumber \\ & -i {f^{c}}_{de} {f^{ae}}_g (c_4-g) :j_z^g j^d_{\bar z} \nonumber \\ & - i {f^{c}}_{de} {f^{ae}}_g (c_2-g) {{B}^{gd}}_{xy} :j^{x}_{z} j^{y}_{\bar z}: \nonumber \\ & + \mathcal{O}(f^2) %\mbox{higher order in $f^2$} \end{aligned}$$ We use current conservation and the Maurer-Cartan equation to write: $$\begin{aligned} +i &(c_4-\frac{g}{2}) {f^{ac}}_g {f^g}_{de} :j^e_z j^d_{\bar z}: +c_- {B^{ac}}_{ed} : j_z^{e} j^{d}_{\bar z}:(w) \nonumber \\ & +c_+ {B^{ac}}_{ed} : j_z^{e} j^{d}_{\bar z}:(w) \nonumber \\ & +(c_4-g/2) ( i {f^c}_{eg} {f^{ag}}_d (-1)^{ed} - i {f^c}_{dg} {f^{ag}}_e ) : j_z^e j_{\bar z}^d:(w) \nonumber \\ & + g/2 (i {f^c}_{eg} {f^{ag}}_d (-1)^{ed} + i {f^c}_{dg} {f^{ag}}_e ): j_z^e j_{\bar z}^d:(w) \nonumber \\ & - i {f^{c}}_{hx} {f^{ax}}_g (c_2-g) ){{B}^{gh}}_{ed} :j^{e}_{z} j^{d}_{\bar z}: \nonumber \\ & + \mathcal{O}(f^2)\end{aligned}$$ where we have separated out (graded) symmetric and anti-symmetric terms. We now apply the super Jacobi identity to the first term in the third line and note that: $$\begin{aligned} {f^{ce}}_{g} {f^{ag}}_d &=& {f^{ce}}_g {f^{ga}}_d (-1)^{1+a + ad} \nonumber \\ &=& {f^{ec}}_g {f^{ga}}_d (-1)^{a + ad+ec} \nonumber \\ &=& - (-1)^{a+ad+ec + cd} ( (-1)^{ac} { f^{ea}}_g {{f^g}_d}^c + (-1)^{ad} {f^e}_{dg} f^{gca}),\end{aligned}$$ which leads to: $$\begin{aligned} {f^c}_{eg} {f^{ag}}_d (-1)^{ed} - {f^c}_{dg} {f^{ag}}_e &=&(-1)^{1+a+ad+ec+cd+ed+ac+e+g+g+1+cd} {f^c}_{dg} {f^{ag}_e} \nonumber \\ & & + (-1)^{1+a+ad+ec+cd+ad+g+ed+g+g+ca+ed} {f^{ac}}_g {f^g}_{de} \nonumber \\ & & - {f^c}_{dg} {f^{ag}}_e \nonumber \\ &=& - {f^{ac}}_g {f^g}_{de}.\end{aligned}$$ Therefore, the third line cancels the first term in the first line and we are left with: $$\begin{aligned} (c_- & {(B)^{ac}}_{ed}+ c_+ {(B)^{ac}}_{ed}) : j_z^{e} j^{d}_{\bar z}:(w) \nonumber \\ & + g/2 (i {f^c}_{eg} {f^{ag}}_d (-1)^{ed} + i {f^c}_{dg} {f^{ag}}_e ) : j_z^e j_{\bar z}^d:(w) \nonumber \\ & - i {f^{c}}_{hx} {f^{ax}}_g (c_2-g){{B}^{gh}}_{ed} :j^{e}_{z} j^{d}_{\bar z}: + \mathcal{O}(f^2). \label{final}\end{aligned}$$ As expected the demand of the vanishing of this term gives the value of the tensor $B$ at the first non-trivial order in $f^2$ : $$\begin{aligned} {B^{ac}}_{ed} &=& -i \frac{g}{2(c_++c_-)} ({f^c}_{eg} {f^{ag}}_d (-1)^{ed} + {f^c}_{dg} {f^{ag}}_e) + O(f^4). \end{aligned}$$ 5\. A similar analysis for the other two first-order poles proportional respectively to $1/( z - w)$ and $( z - w)/(\bar z - \bar w)^2$ gives respectively the tensors $A$ and $C$ in equations (\[euclidOPEs\], \[ABC\]). The details of the calculation are very similar to the calculation we just discussed. Remarks on higher order terms in $f^2$ {#remarks-on-higher-order-terms-in-f2 .unnumbered} -------------------------------------- To discuss a few aspects of the higher order terms that we encountered, it is useful to define the following tensor: $$\begin{aligned} {S^{ac}}_{ed} &=& {f^c}_{eg} {f^{ag}}_d (-1)^{ed} + {f^c}_{dg} {f^{ag}}_e.\end{aligned}$$ It is manifestly graded symmetric in the lower indices. Let’s also check that it is graded symmetric in the upper indices: $$\begin{aligned} {S^{ca}}_{ed} &=& {f^a}_{eg} {f^{cg}}_d (-1)^{ed} + {f^a}_{dg} {f^{cg}}_e \nonumber \\ &=& {f^{ag}}_e (-1)^{g+eg+1+ed} {f^c}_{dg} (-1)^{gd+1} + (-1)^{g+1+gd+1+eg} {f^c}_{eg} {f^{ag}}_d \nonumber \\ &=& {f^{ag}}_e (-1)^{ac} {f^c}_{dg} + (-1)^{ac} {f^c}_{eg} {f^{ag}}_d \nonumber \\ &=& (-1)^{ac} {S^{ac}}_{ed}.\end{aligned}$$ Therefore, $S$ is a linear operator that acts on the space of (graded) symmetric two-tensors. The higher order term in the last line in the above explicit calculation gives rise to the square of the linear operator $S$. We computed it for $psl(2|2)$ for which it simplifies to $$\begin{aligned} {S^{ac}}_{gh} {S^{gh}}_{ed} &=& 8 ( \kappa^{ac} \kappa_{de} + ( \delta^a_e \delta^c_d + (-1)^{ed} \delta^a_d \delta^c_e)).\end{aligned}$$ We also have the equality $S^3=16S$. When we take a supertrace of $S^2$, it can be shown to be zero because the superdimension of $psl(2|2)$ is $-2$. Using some of these properties, it is clear that at higher order the structure of a pole in the $j^a \cdot MC^c$ OPE will look like: $$\begin{aligned} \dots \kappa^{ac} :j_{e\bar z} j_{z}^e : + \dots (:j^a_{\bar z} j^c_{ z} : + (-1)^{ac} :j^c_{\bar z} j^a_{ z} :).\end{aligned}$$ The first term is proportional to a component of the energy-momentum tensor (and to the kinetic term in the Lagrangian). The other term indicates that at higher order, we need a new four-tensor index structure in the current-current operator product expansion. At the same time, the special properties of the linear operator given above show that only few four-tensors will appear. It is certainly feasible to push the above calculation, and therefore the other calculations in the bulk of the paper to higher order. The Virasoro algebra from the current algebra {#TjandTT} --------------------------------------------- In [@Ashok:2009xx] it was shown that the Virasoro algebra emerges from the current algebra via the Sugawara construction. More precisely it was argued that the normal ordered classical expression for the stress tensor : T = :j\_[L,zb]{} j\^[b]{}\_[L,z]{}: satisfies the OPEs : \[TjApp\] T(z) j\^a\_[L,z]{}(w) = + + (z-w)\^0 \[TTApp\] T(z) T(w) = + + + (z-w)\^0. In this section, we fill a gap in the demonstration of equation . We reconsider the OPE between a current and the bilinear operator $:j_{L,zb} j^{b}_{L,z}:$. To perform this computation in [@Ashok:2009xx] we truncated the current algebra at the order of the poles. We obtained : \[j:jj:\] j\_[L,z]{}\^a(z) :j\_[L,zb]{} j\^b\_[L,z]{}:(w) &=& 2c\_1 + c\_2 ( (-1)\^[bc]{} :j\^b\_[L,z]{} j\^c\_[L,z]{}:+ :j\^c\_[L,z]{} j\^b\_[L,z]{}:(w)) &&+ (c\_2-g) [f\^a]{}\_[bc]{} ( (-1)\^[bc]{}:j\^b\_[L,z]{} j\^c\_[L,|z]{}:+: j\^c\_[L,|z]{} j\^b\_[L,z]{}:(w) )\ & & +... where the ellipses contain terms of order zero in the distance between the insertion points $z$ and $w$. We will now show that the subleading terms in the current algebra do not modify this result. Let us divide these terms into two sets. First we have the regular terms and the terms that multiply an $n^{th}$-derivative of a single current. These terms were already considered in [@Ashok:2009xx] and it is straightforward to show that they do not modify . The second set contains the terms that multiply composites of (derivatives of) several currents (not including the regular terms). This includes for instance the current bilinears in equation . The crucial point is that all these terms come with a coefficient that contains at least two structure constants. This is a consequence of the discussion in appendix \[XXOPEs\]. In full generality, a term in this second set may lead to the following type of contribution to : $$\begin{aligned} \label{j:jj:Mod} %j_{L,z}^a(z) :j_{b,L,z} j^b_{L,z}:(w) &=& & \frac{{T^a}_b j_{L,z}^b(w)}{(z-w)^2} +\frac{{U^a}_b \p j_{L,z}^b(w)}{z-w} + \frac{{V^a}_b \bar \p j_{L,z}^b(w)(\bar z - \bar w)}{(z-w)^2}+\frac{{\bar T^a}_b j_{L,\bar z}^b(w)(\bar z - \bar w)}{(z-w)^3} \cr & +\frac{{\bar U^a}_b \p j_{L,\bar z}^b(w)(\bar z - \bar w)}{(z-w)^2} + \frac{{\bar V^a}_b \bar \p j_{L,\bar z}^b(w)(\bar z - \bar w)^2}{(z-w)^3}+\frac{ {W^a}_{bc}:j^c_{L,z} j^b_{L,z}:(w) }{z-w}\cr & +\frac{{X^a}_{bc}:j^c_{L,\bar z} j^b_{L,z}:(w)(\bar z - \bar w)}{(z-w)^2} +\frac{{Y^a}_{bc}:j^c_{L,\bar z} j^b_{L,\bar z}:(w)(\bar z - \bar w)^2}{(z-w)^3} \end{aligned}$$ where the tensors ${T^a}_b$, etc. are invariant two- and three-tensors made of contractions of structure constants. According to the argument of [@Bershadsky:1999hk], any invariant two-tensor obtained by contracting at least one structure constant vanishes. Moreover any invariant three-tensor obtained by contracting at least two structure constant also vanishes. Since all tensors appearing in contain at least two structure constants that come from the current-current OPE, all these terms vanish. This completes the proof of equation . Currents as a primary fields of dimension one revisited {#TRJL} ------------------------------------------------------- The stress-energy tensor can be written either in terms of the left or of the right currents. As a consistency check on our formalism, we will compute in this appendix the OPE between the stress-energy tensor and the current components $j_{L,z}$ using the expression of the energy-momentum tensor $T$ in terms of the right currents: T(z) = :j\^[|b]{}\_[R,z]{}j\^[|c]{}\_[R,z]{}:(w)\_[|c |b]{} To proceed we use the OPEs between left and right currents , , as well as the OPEs between a current and the primary adjoint operator . Notice that the latter OPE may receive higher-order corrections in $f^2$. In the following we keep track only of the leading-order terms in $f^2$. The computation goes as follows: $$\begin{aligned} j^a_{L,z}(z)& :j^{\bar b}_{R,z}j^{\bar c}_{R,z}:(w)\kappa_{\bar c \bar b} = \frac{c_+ c_-}{c_++c_-} \left( \frac{[:\phi^{a \bar b}j^{\bar c}_{R,z}:(w) + :j^{\bar b}_{R,z} \phi^{a \bar c}:(w)]\kappa_{\bar c \bar b}}{(z-w)^2} \right. \cr &\qquad + \frac{c_-}{c_++c_-}\frac{[:\partial \phi^{a \bar b}j^{\bar c}_{R,z}:(w) + :j^{\bar b}_{R,z} \partial \phi^{a \bar c}:(w)]\kappa_{\bar c \bar b}}{z-w} \cr & \left.\qquad + \frac{c_-}{c_++c_-}\frac{[:\bar \partial \phi^{a \bar b}j^{\bar c}_{R,z}:(w) + :j^{\bar b}_{R,z} \bar \partial \phi^{a \bar c}:(w)]\kappa_{\bar c \bar b}(\bar z-\bar w)}{(z-w)^2} \right)\end{aligned}$$ where a triple pole vanishes since it is proportional to the contraction of a structure constant with the metric. We have to treat carefully the normal-ordered operators appearing in the previous expression. The central point is the property: \[jphi-phij\] :j\^[|a]{}\_R \^[b |b]{}: - :\^[b |b]{} j\^[|a]{}\_R: \_[|c]{}and similarly for the left currents. This property follows from the OPE between the current and the adjoint primary . Thus we can deal with the first line easily, and using equation we obtain : = - . Now let us consider the second line. We use equation : \^[a |a]{} = - j\^[|c]{}\_[R,z]{} \^[a |b]{} .Notice that we do not need the normal ordering symbol on the right-hand side since (at leading order in $f^2$) there is no singular term to discard in the OPEs . Thus we rewrite the second line as: $$\begin{aligned} \frac{c_-}{c_++c_-} & \frac{[ :\partial \phi^{a \bar b} j^{\bar c}_{R,z}: (w) + :j^{\bar b}_{R,z} \partial \phi^{a \bar c}:(w)]\kappa_{\bar b \bar c}}{z-w} \cr &= \frac{-i}{c_++c_-} \frac{[:{f^{\bar b}}_{\bar d \bar e} j^{\bar e}_{R,z} \phi^{a \bar d} j^{\bar c}_{R,z}:(w) + :j^{\bar b}_{R,z} {f^{\bar c}}_{\bar d \bar e} j^{\bar e}_{R,z} \phi^{a \bar d}:(w)]\kappa_{\bar b \bar c}}{z-w} \cr & = \frac{-i f_{\bar c \bar d \bar e}}{c_++c_-} \frac{:j^{\bar e}_{R,z} j^{\bar c}_{R,z} \phi^{a \bar d}:(w) + :j^{\bar c}_{R,z} j^{\bar e}_{R,z} \phi^{a \bar d}:(w)}{z-w}\end{aligned}$$ where we used the property again in the last step to commute the adjoint operator and the current in the normal ordered triple operator. Now thanks to the anti-symmetry of the structure constants this term vanishes. We can perform similar manipulations on the third line: $$\begin{aligned} \frac{c_-}{c_++c_-} & \frac{[:\bar \partial \phi^{a \bar b}j^{\bar c}_{R,z}:(w) + :j^{\bar b}_{R,z} \bar \partial \phi^{a \bar c}:(w)]\kappa_{\bar b \bar c}(\bar z-\bar w)}{(z-w)^2}.\end{aligned}$$ The first operator can be rewritten as: :|\^[a |b]{}j\^[|c]{}\_[R,z]{}:(w)\_[|b |c]{} = - : j\^[|e]{}\_[R,|z]{} j\^[|c]{}\_[R,z]{} \^[a |d]{}:(w) and the second one as: :j\^[|b]{}\_[R,z]{} |\^[a |c]{}:(w)\_[|b |c]{} = - : j\^[|c]{}\_[R,z]{} j\^[|e]{}\_[R,|z]{} \^[a |d]{}:(w) = :|\^[a |b]{}j\^[|c]{}\_[R,z]{}:(w)\_[|b |c]{} where in the last step we used that $f_{\bar c \bar d \bar e}: j^{\bar c}_{R,z} j^{\bar e}_{R,\bar z}:= f_{\bar c \bar d \bar e}:j^{\bar e}_{R,\bar z}j^{\bar c}_{R,z} :$. Therefore the two operators present on the third line are the same. Now we can use the Maurer-Cartan equation and current conservation for the right currents to rewrite them as: - : j\^[|e]{}\_[R,|z]{} j\^[|c]{}\_[R,z]{} \^[a |d]{}:(w) = - :|j\^[|b]{}\_[R,z]{} \^[a |d]{}:(w)\_[|b |d]{}. Therefore, we can rewrite the third line as: = -2 where we used equation once more. Gathering all terms, we obtain: j\^a\_[L,z]{}(z) :j\^[|b]{}\_[R,z]{}j\^[|c]{}\_[R,z]{}:(w)\_[|b |c]{} = -( + )which we can finally rewrite in the expected form: T(w) j\^a\_[L,z]{}(z) = :j\^[|b]{}\_[R,z]{}j\^[|c]{}\_[R,z]{}:(w)\_[|b |c]{} j\^a\_[L,z]{}(z) = + , thus completing our consistency check. The associativity of the current algebra {#associativity} ---------------------------------------- In this appendix we address the issue of the associativity of the current algebra . We will prove the associativity of this current algebra at the first non-trivial order in $f^2$. The OPE $j^a_{L,z}(z) j^b_{L,z}(w) j^c_{L,z}(x)$ {#the-ope-ja_lzz-jb_lzw-jc_lzx .unnumbered} ------------------------------------------------ First we consider the OPE between three $z$-components of the left-current: j\^a\_[L,z]{}(z) j\^b\_[L,z]{}(w) j\^c\_[L,z]{}(x). We will compute this OPE using the current algebra at the order of the poles. Moreover we will only compute the lowest-order terms in the $f^2$ expansion. In this case these are terms of order $f^{-2}$. To prove associativity we will first compute the OPE between the first two currents, then compute the OPE of the result with the third current, and show that the result is invariant under permutation of the currents. We start out with: $$\begin{aligned} j^a_{L,z}(z) & j^b_{L,z}(w) j^c_{L,z}(x) = \left( \frac{c_1 \kappa^{ab}}{(z-w)^2} + \frac{c_2 {f^{ab}}_d j^d_{L,z}(w)}{z-w}+ \frac{(c_2-g) {f^{ab}}_d j^d_{L,\bar z}(w)(\bar z - \bar w)}{(z-w)^2} \right. \cr & \left. \qquad + :j^a_{L,z}(z) j^b_{L,z}(w): + ... \right) j^c_{L,z}(x).\end{aligned}$$ The ellipses stand for lower-order terms in the OPEs, that we do not keep track of. We obtain : $$\begin{aligned} j^a_{L,z}(z) & j^b_{L,z}(w) j^c_{L,z}(x) = \frac{c_1 \kappa^{ab}j^c_{L,z}(x)}{(z-w)^2} + \frac{c_1 c_2 {f^{abc}}}{(z-w)(w-x)^2} + :j^a_{L,z}(z) j^b_{L,z}(w):j^c_{L,z}(x) + ...\end{aligned}$$ up to a contact terms. We now have to compute the OPE involving the regular operator $:j^a_{L,z}(z) j^b_{L,z}(w):$. In order to use the techniques presented in appendix \[compositeOPEs\] we rewrite both currents as being evaluated at the point $w$: :j\^a\_[L,z]{}(z) j\^b\_[L,z]{}(w): = \_[n,|n=0]{}\^ :(\^n |\^[|n]{} j\^a\_[L,z]{}) j\^b\_[L,z]{}:(w). Let us now consider the OPE of one of these composite operators with the current $j^c_{L,z}(x)$: $$\begin{aligned} j^c_{L,z}(x) & :(\p^n \bar \p ^{\bar n} j^a_{L,z}) j^b_{L,z}:(w) = j^c_{L,z}(x) \lim_{:y \to w:} \p_y^n \bar \p_y ^{\bar n} j^a_{L,z}(y) j^b_{L,z}(w) \cr % & = \lim_{:y \to w:} \p_y^n \bar \p_y ^{\bar n} \left[ \left ( \frac{c_1 \kappa^{ca}}{(x-y)^2} + \frac{c_2 {f^{ca}}_d j^d_{L,z}(y)}{x-y}+ \frac{(c_2-g) {f^{ca}}_d j^d_{L,\bar z}(y)(\bar x - \bar y)}{(x-y)^2} \right.\right. \cr & \left. \qquad \qquad + \sum_{m,\bar m=0}^{\infty} \frac{(x-y)^m }{m ! }\frac{ (\bar x - \bar y)^{\bar m}}{ \bar m !} :(\p^m \bar \p ^{\bar m} j^c_{L,z}) j^a_{L,z}:(y) + \mathcal{O}(f^2) \right) j^b_{L,z}(w) \cr & \left. \qquad + j^a_{L,z}(y) \left ( \frac{c_1 \kappa^{cb}}{(x-w)^2} + ... \right ) \right] \cr % & = \lim_{:y \to w:} \p_y^n \bar \p_y ^{\bar n} \left[ \frac{c_1 \kappa^{ca}j^b_{L,z}(w)}{(x-y)^2} + \frac{c_1 c_2 {f^{cab}} }{(x-y)(y-w)^2} + \frac{c_1 \kappa^{cb}j^a_{L,z}(y)}{(x-w)^2} +.. \right] \end{aligned}$$ where the ellipses in the last line contains singular terms that comes from the OPE between the regular operators and the current in the third line of the previous computation. These terms in this OPE will be removed by the regular limit $:y \to w:$. In order to compute the action of the derivatives more conveniently, we rewrite the second term in the last line as: = c\_1 c\_2 [f\^[cab]{}]{} \_[p=0]{}\^ Thus we obtain: $$\begin{aligned} j^c_{L,z}&(x) :(\p^n \bar \p ^{\bar n} j^a_{L,z}) j^b_{L,z}:(w) = \lim_{:y \to w:} \left[ \delta_{\bar n,0}\frac{(n+1)!}{(x-y)^{n+2}} c_1 \kappa^{ca}j^b_{L,z}(w) \right. \cr & \left. \qquad + \delta_{\bar n,0}\sum_{p=0}^{\infty} \frac{(p-2)...(p-2-n+1)(y-w)^{p-2-n}}{(x-w)^{p+1}} c_1 c_2 {f^{cab}} + \frac{c_1 \kappa^{cb} \p^n \bar \p ^{\bar n} j^a_{L,z}(y)}{(x-w)^2}+... \right] \cr % & = \delta_{\bar n,0}\frac{(n+1)!}{(x-w)^{n+2}} c_1 \kappa^{ca}j^b_{L,z}(w) + \delta_{\bar n,0} \frac{n!}{(x-w)^{n+3}} c_1 c_2 {f^{cab}} + \frac{c_1 \kappa^{cb} \p^n \bar \p ^{\bar n} j^a_{L,z}(w)}{(x-w)^2}+ ... \end{aligned}$$ Resumming the series, we get: $$\begin{aligned} :j^a_{L,z}(z) & j^b_{L,z}(w): j^c_{L,z}(x) = \sum_{n,\bar n=0}^{\infty} \frac{(z-w)^n }{n ! }\frac{ (\bar z - \bar w)^{\bar n}}{ \bar n !} \left[ \delta_{\bar n,0}\frac{(n+1)!}{(x-w)^{n+2}} c_1 \kappa^{ca}j^b_{L,z}(w) \right. \cr & \qquad \left. + \delta_{\bar n,0} \frac{n!}{(x-w)^{n+3}} c_1 c_2 {f^{cab}} + \frac{c_1 \kappa^{cb} \p^n \bar \p ^{\bar n} j^a_{L,z}(w)}{(x-w)^2}+ ... \right] \cr & = \frac{c_1 \kappa^{ca}j^b_{L,z}(w)}{(x-z)^2} + \frac{c_1 c_2 {f^{cab}}}{(x-z)(x-w)^2} + \frac{c_1 \kappa^{cb} j^a_{L,z}(z)}{(x-w)^2} +...\end{aligned}$$ After gathering all terms, we obtain: $$\begin{aligned} j^a_{L,z}(z) & j^b_{L,z}(w) j^c_{L,z}(x) = \frac{c_1 \kappa^{ab}j^c_{L,z}(x)}{(z-w)^2} + \frac{c_1 c_2 {f^{abc}}}{(z-w)(w-x)^2} + \frac{c_1 \kappa^{ca}j^b_{L,z}(w)}{(x-z)^2} \cr & \qquad + \frac{c_1 c_2 {f^{cab}}}{(x-z)(x-w)^2} + \frac{c_1 \kappa^{cb} j^a_{L,z}(z)}{(x-w)^2} +... \cr & = \frac{c_1 c_2 {f^{abc}}}{(z-x)(x-w)(w-z)} + \frac{c_1 \kappa^{ab}j^c_{L,z}(x)}{(z-w)^2}+ \frac{c_1 \kappa^{ca}j^b_{L,z}(w)}{(x-z)^2} +\frac{c_1 \kappa^{cb} j^a_{L,z}(z)}{(x-w)^2}+ \mathcal{O}(f^0) +... \nonumber\end{aligned}$$ which is manifestly invariant under permutation of the currents. The OPE $j^a_{L,z}(z) j^b_{L,z}(w) j^c_{L,\bar z}(x)$ {#the-ope-ja_lzz-jb_lzw-jc_lbar-zx .unnumbered} ----------------------------------------------------- We now consider the OPE involving two $z$-components and one $\bar z$-component of the left current: j\^a\_[L,z]{}(z) j\^b\_[L,z]{}(w) j\^c\_[L,|z]{}(x). First we will take first the OPE between the two $z$-components of the current: $$\begin{aligned} [ j^a_{L,z}(z) & j^b_{L,z}(w) ] j^c_{L,\bar z}(x) = \left( \frac{c_1 \kappa^{ab}}{(z-w)^2} + \frac{c_2 {f^{ab}}_d j^d_{L,z}(w)}{z-w}+ \frac{(c_2-g) {f^{ab}}_d j^d_{L,\bar z}(w)(\bar z - \bar w)}{(z-w)^2} \right. \cr & \left. \qquad + :j^a_{L,z}(z) j^b_{L,z}(w): + ... \right) j^c_{L,\bar z}(x) \cr % & = \frac{c_1 \kappa^{ab}j^c_{L,\bar z}(x)}{(z-w)^2} + \frac{c_3(c_2-g) {f^{abc}} (\bar z - \bar w)}{(z-w)^2(\bar w - \bar x)^2} + :j^a_{L,z}(z) j^b_{L,z}(w):j^c_{L,\bar z}(x) + ...\end{aligned}$$ The OPE involving the composite operator does not produces any term of order $f^{-2}$, thus we obtain : j\^c\_[L,|z]{}(x) = + + (f\^0) +... Now let us perform the same computation taking first the OPE between one $z$-component and one $\bar z$-component of the current: $$\begin{aligned} j^a_{L,z}(z) & [j^b_{L,z}(w) j^c_{L,\bar z}(x)] \cr & = j^a_{L,z}(z) \left( \frac{(c_4-g) {f^{bc}}_d j^d_{L,z}(x)}{\bar w-\bar x}+ \frac{(c_2-g) {f^{bc}}_d j^d_{L,\bar z}(x)}{(w-x)}+ :j^b_{L,z}(w) j^c_{L,\bar z}(x): +... \right) \cr % & = \frac{c_1(c_4-g) {f^{abc}}}{(z-w)^2(\bar w - \bar x)} + j^a_{L,z}(z):j^b_{L,z}(w) j^c_{L,\bar z}(x): +... \cr % & = \frac{c_1(c_4-g) {f^{abc}}}{(z-w)^2(\bar w - \bar x)} + \frac{c_1 \kappa^{ab}j^c_{L,\bar z}(x)}{(z-w)^2} +...\end{aligned}$$ Thanks to the relations between the coefficients of the current algebra : c\_1(c\_4-g) = c\_3(c\_2-g) we find that the current algebra is indeed associative at the order at which we performed the computation. The coordinate dependence does not match exactly since we did not take into account the terms containing derivatives of the currents that appear in the current algebra as subleading terms. It is interesting to pursue the full proof of associativity. The holomorphy of the stress-tensor ----------------------------------- In this appendix we address the issue of the holomorphy of the stress-tensor[^7]: T(z) = \_[ab]{} :j\_[L,z]{}\^b j\_[L,z]{}\^a:(z). Since the $z$-component of the left-current is not holomorphic away from the WZW point, it is not obvious that the stress-tensor will be holomorphic in the quantum theory. The anti-holomorphic derivative of the stress-tensor reads: |T(z) = \_[ab]{} ( :|j\_[L,z]{}\^b j\_[L,z]{}\^a:(z) + :j\_[L,z]{}\^b |j\_[L,z]{}\^a:(z) ). To continue the computation we combine current conservation with the Maurer-Cartan equation to write the anti-holomorphic derivative of the $z$-component of the current in terms of a bilinear : |j\_[L,z]{}\^a = -i f\^2 [f\^a]{}\_[bc]{} :j\^c\_[L,z]{} j\^b\_[L,|z]{}:. Since all the poles in the OPE between $j^c_{L,z}$ and $ j^b_{L,\bar z}$ vanish when contracted with the structure constant ${f^a}_{bc}$, we can also write : |j\_[L,z]{}\^a = -i f\^2 [f\^a]{}\_[bc]{} :j\^b\_[L,|z]{} j\^c\_[L,z]{}:. Thus using successively the last two equations we obtain: |T(z) = f\_[abc]{} ( : :j\^c\_[L,z]{} j\^b\_[L,|z]{}: j\_[L,z]{}\^a:(z) + :j\_[L,z]{}\^a :j\^b\_[L,|z]{} j\^c\_[L,z]{} ::(z) ). Now let us consider the composite operator $::j^c_{L,z} j^b_{L,\bar z}: j_{L,z}^a:(z)$. It is defined as the regular term in the OPE between $ :j^c_{L,z} j^b_{L,\bar z}:$ and $j_{L,z}^a$. We will show that we have : \[f::::=f::\] f\_[abc]{}::j\^c\_[L,z]{} j\^b\_[L,|z]{}: j\_[L,z]{}\^a:(z) = f\_[abc]{}:j\^c\_[L,z]{} j\^b\_[L,|z]{} j\_[L,z]{}\^a:(z) where the operator $:j^c_{L,z} j^b_{L,\bar z} j_{L,z}^a:$ is defined as the regular term in the OPE of the three currents $j^c_{L,z}$, $j^b_{L,\bar z}$ and $j_{L,z}^a$. The difference between the operators on the left-hand side and the right-hand side of equation comes from the non-regular terms in the OPE between $j^c_{L,z}$ and $j^b_{L,\bar z}$. The crucial point is that all these terms vanish when contracted with the structure constant $f_{abc}$: f\_[abc]{} \[j\^c\_[L,z]{}(z) j\^b\_[L,|z]{}(w) - :j\^c\_[L,z]{}(z) j\^b\_[L,|z]{}(w):\] = 0. This can be checked via the current algebra OPEs order by order in $f^2$. In equation the current algebra is given up to terms of order $f^4$, and thus one can prove the previous statement up to terms of order $f^4$. Indeed, all tensors that appear in the current algebra vanish upon double contraction with a structure constant: f\_[abc]{} = 0 The non-degenerate metric $\kappa^{cb}$ and the tensors ${A^{cb}}_{de},\ {B^{cb}}_{de},\ {C^{cb}}_{de}$ are graded-symmetric in the indices $c,b$. Moreover the double contraction of the structure constant vanishes since the dual Coxeter number of the Lie super algebra vanishes. This concludes the proof of equation up to terms of order $f^4$. Let us mention that the same equation also guaranties the quantum integrability of the model up to this order, as discussed in section \[integrability\]. The same argument leads to the equality: f\_[abc]{} :j\_[L,z]{}\^a :j\^b\_[L,|z]{} j\^c\_[L,z]{} ::(z) = f\_[abc]{} :j\_[L,z]{}\^a j\^b\_[L,|z]{} j\^c\_[L,z]{} :(z). Thus we have: |T(z) = f\_[abc]{} ( : j\^c\_[L,z]{} j\^b\_[L,|z]{} j\_[L,z]{}\^a:(z) + :j\_[L,z]{}\^a j\^b\_[L,|z]{} j\^c\_[L,z]{} :(z) ) =0 which vanishes thanks to the (graded) anti-symmetry of the structure constants. It would be interesting to have a non-perturbative understanding of the consistency of the normal-ordering and the holomorphy of the energy-momentum tensor. Details on primary operators {#AppPrimaries} ============================ Behavior of current primaries under perturbation of the kinetic term {#WZWaffine} -------------------------------------------------------------------- In this section we will show that current primaries at a given point of moduli space remain current primaries after perturbation of the kinetic term. More precisely we will show that if an operator $\phi$ satisfies the OPEs at a given point of moduli space, then it also satisfies the same OPEs after exactly marginal deformation of the theory. This implies that it is consistent to think of a current primary as being the group element taken in a given representation, at any point of the moduli space. It also proves the claim in section \[primaries\] that the affine primary fields at the WZW points become current primaries after deformation of the theory. For convenience let us recall the OPEs that define a primary operator $\phi$ : $$\begin{aligned} \label{defPrimariesBis} j^a_{L,z}(z) \phi(w) &= - \frac{c_+}{c_++c_-} t^a \frac{\phi(w)}{z-w} + \text{less singular} \cr j^a_{L,\bar z}(z) \phi(w) &= - \frac{c_-}{c_++c_-} t^a \frac{\phi(w)}{\bar z-\bar w} + \text{less singular.} \end{aligned}$$ We assume that these OPEs hold at a given point of moduli space $(f^2,k)$. Then we perturb the kinetic term : $f^2 \to f^2 + \epsilon$ and we compute the way the OPEs are modified. A procedure to compute OPEs in conformal perturbation theory was given in [@Ashok:2009xx]. Here we will only compute the deformation of the OPEs up to first order in $\epsilon$. The prescription is to compute first the OPE between the current and the perturbation of the action, and then to compute the OPE of the result with the field $\phi$. We begin with the first step of this procedure, for the first OPE in . The OPE between the current and the marginal operator can be computed thanks to the current algebra : $$\begin{aligned} \label{jMarg} j^a_{L,z}(z)& \frac{\epsilon}{4\pi f^4} \int d^2 x \kappa_{cb}:\frac{j^b_{L,z}}{c_+} \frac{j^c_{L, \bar z}}{c_-}:(x) \cr & = \frac{\epsilon}{4\pi f^4 c_+ c_-} \int d^2 x \left( c_1 \frac{j^a_{L,\bar z}(x)}{(z-x)^2} + \tilde c j^a_{L,z}(x) 2\pi \delta^{(2)}(z-x) +... \right).\end{aligned}$$ The ellipses contains higher-order terms both in $f^2$ and in the distance between $z$ and $x$. We will not keep track of these terms for the time being, and we will comment on their relevance at the end of the computation. We now have to take the OPE of the previous result with the primary field $\phi$. We obtain : $$\begin{aligned} \label{jMargPhi} \frac{\epsilon}{4\pi f^4 c_+ c_-}& \int d^2 x \left( - \frac{c_1 c_-}{c_++c_-} t^a \frac{\phi(w)}{(z-x)^2(\bar x - \bar w)} - \frac{\tilde{c} c_+}{c_++c_-} t^a \frac{\phi(w)}{x-w} \delta^{(2)}(z-x) + ... \right) \cr & = \frac{\epsilon}{2 f^4 c_+ c_-}\left(\frac{c_1 c_-}{c_++c_-}-\frac{\tilde{c} c_+}{c_++c_-}\right) t^a \frac{\phi(w)}{z-w}+ ... \cr & = -\epsilon c_+ t^a \frac{\phi(w)}{z-w}+ \text{less singular,}\end{aligned}$$ where we used the explicit value of the coefficients . As claimed, the structure of the OPEs is unaltered after perturbation of the kinetic term. It is also straightforward to check that (taking into account the renormalization of the currents) the perturbation $f^2 \to f^2 + \epsilon$ induces a deformation of the coefficients in that matches the result obtained at first order in $\epsilon$. Now let us come back to the terms we discarded in equation . They contain the contribution to this computation from the poles and less singular terms in the current algebra . All these terms are (composites of) currents. It follows from and from dimensional analysis that in the OPE between any one of these terms and the primary field $\phi$, the most singular term that may arise multiplies the operator $\phi$. Here we assume that all terms appearing in the OPE can be written as composites of currents with the field $\phi$. Thus if any of these terms has any effect on the previous computation, it may at worse modify the coefficient obtained in . On the other hand, as was mentioned in section \[primaries\], the coefficients in are fixed by demanding compatibility with current conservation and the Maurer-Cartan equation. Since these coefficients were already recovered in it follows that the term we discarded in equation indeed has no effect on the result of the computation. This can also be checked by hand for the terms that are explicitly given in equation . Current-primary OPE at order $f^2$ {#AppjPhi} ---------------------------------- Equation gives the OPE between a current and a primary field at leading order. According to the discussion of section \[bootstrap\] it is possible to compute the higher-order terms thanks to current conservation and the Maurer-Cartan equation. In this appendix we perform the computation of the first correction to the OPE , which leads to the OPE in the bulk of the paper. The terms on the right-hand side of the OPE are of order $f^0$. We will now compute the current-primary OPE at order $f^2$. Following the discussion of appendix \[XXOPEs\] we make the following educated ansatz for the OPEs between the left-currents and a primary field $\phi$: $$\begin{aligned} j^a_{L,z}(z) \phi(w) = & -\frac{c_+}{c_++c_-} \frac{t^a \phi(w)}{z-w} + :j^a_{L,z} \phi:(w) \cr & + {A^a}_c \log|z-w|^2 :j^c_{L,z} \phi:(w) + {B^a}_c \frac{\bar z - \bar w}{z-w} :j^c_{L,\bar z} \phi:(w) + \mathcal{O}(f^4) \cr j^a_{L,\bar z}(z) \phi(w) = & -\frac{c_-}{c_++c_-} \frac{t^a \phi(w)}{\bar z-\bar w} + :j^a_{L,\bar z} \phi:(w) \cr & + {D^a}_c \log|z-w|^2 :j^c_{L,\bar z} \phi:(w) + {C^a}_c \frac{z-w}{\bar z - \bar w} :j^c_{L, z} \phi:(w) + \mathcal{O}(f^4).\end{aligned}$$ We expect the coefficients ${A^a}_c$, ${C^a}_c$, ${B^a}_c$, ${D^a}_c$ to be of order $f^2$. We will check that the coefficient of the first-order poles are not modified. As explained in section \[bootstrap\] the demand of consistency with current conservation imposes that the terms in the $j^a_{L,\bar z}(z) \phi(w)$ OPE can be deduced from the terms in the $j^a_{L,z}(z) \phi(w)$: \_c + [C\^a]{}\_c = 0 = [B\^a]{}\_c + [D\^a]{}\_c. To get further constraints on the tensors ${A^a}_c$ and ${B^a}_c$ we ask for the vanishing of the first-order poles in the OPE between the operator $\phi$ and the Maurer-Cartan operator, that we write as in : \[MC.phi=0\] \[ |j\^a\_[L,z]{}(z) + i f\^2 [f\^a]{}\_[bc]{} :j\^c\_[L,z]{} j\^b\_[L,|z]{}:(z)\](w) = 0. The first part of this OPE is: $$\begin{aligned} \bar \p j^a_{L,z}(z) \phi(w) = & {A^a}_c \frac{:j^c_{L,z}\phi:(w)}{\bar z - \bar w} + {B^a}_c \frac{:j^c_{L,\bar z}\phi:(w)}{z -w} + \mathcal{O}(f^4).\end{aligned}$$ The simple poles in the previous expression should be canceled by the simple poles in the OPE between the composite operator $ i f^2 {f^a}_{bc} :j^c_{L,z} j^b_{L,\bar z}:$ and the operator $\phi$. Notice that because of the factors $f^{2}$ multiplying the composite operator, we only need to compute the OPE at order $f^0$. We calculate : $$\begin{aligned} \phi(w)&[ i f^2 {f^a}_{bc} :j^c_{L,z} j^b_{L,\bar z}:(z)] = i f^2 {f^a}_{bc} \lim_{:x \to z:} \phi(w)j^c_{L,z}(x) j^b_{L,\bar z}(z) \cr = & i f^2 {f^a}_{bc} \lim_{:x \to z:} \left \{ \left[ - \frac{c_+}{c_++c_-} \frac{t^c \phi(w)}{x-w} + :j^c_{L,z} \phi:(w) %\right. \right. \cr %& \quad \quad \left. + {A^c}_d \log|x-w|^2 :j^d_{L,z}\phi:(w) + {B^c}_d \frac{\bar x - \bar w}{x-w} :j^d_{L,z}\phi:(w) +%AA \mathcal{O}(f^2) ... \right] j^b_{L,\bar z}(z) \right. \cr & \quad \left. + j^c_{L,z}(x) \left[ -\frac{c_-}{c_++c_-}\frac{t^b \phi(w)}{\bar z - \bar w} + :j^b_{L,\bar z} \phi:(w) + %AA\mathcal{O}(f^2) ...\right] \right\} \end{aligned}$$ To proceed according to the prescription of appendix \[compositeOPEs\] we have to expand the fields in the first line (respectively the second line) in the neighborhood of the point $x$ (respectively $z$). Then we have to perform the remaining OPEs between the currents and the (derivatives of) the primary field $\phi$. Notice however that all the terms proportional to ${f^a}_{cb} t^b t^c = \frac{i}{2} {f^a}_{cb} {f^{bc}}_d t^d$ do vanish. Only the regular term in the current-primary OPE will contribute to the result at order $f^0$. Moreover it is straightforward to check that the terms proportional to ${A^a}_c$ and ${B^a}_c$ in the previous OPE do not contribute at order $f^0$. We obtain: (w)\[ i f\^2 [f\^a]{}\_[bc]{} :j\^c\_[L,z]{} j\^b\_[L,|z]{}:(z)\] = -i f\^2 [f\^a]{}\_[bc]{} ( + + ... ). where the ellipses contains terms of order $f^4$ as well as terms of order zero in the distance between $z$ and $w$. Gathering terms, we conclude that we have the equalities: $$\begin{aligned} {A^a}_c &=& \frac{c_-}{(c_++c_-)^2} i {f^a}_{cb} t^b + \mathcal{O}(f^4) \nonumber \\ {B^a}_c &=& \frac{c_+}{(c_++c_-)^2} i {f^a}_{cb} t^b + \mathcal{O}(f^4).\end{aligned}$$ We note that one can reach the same conclusion by computing the OPE between a current and both sides of the equation , i.e. by demanding compatibility with the proportionality relation between the operators $\p \phi$ and $ t_a :j^a_{L,z} \phi:$. Stress-tensor-primary OPE at order $f^2$ {#AppTphi} ---------------------------------------- Here we present the computation of the OPE between the stress-energy tensor and a primary field $\phi$. This computation relies on the prescription of appendix \[compositeOPEs\], and on the current-current and current-primary OPEs and .Since we computed these OPEs up to order $f^2$, we will also obtain the stress-tensor OPE up to order $f^2$. $$\begin{aligned} \label{phiT} \phi(z) 2 c_1 T(w) &= \lim_{:x \to w:}\phi(z) j^a_{L,z}(x) j^b_{L,z}(w) \kappa_{ba} \cr &= \kappa_{ab} \lim_{:x \to w:} \left[ \left( -\frac{c_+}{c_++c_-} \frac{t^a \phi(z)}{x-z} + :j^a_{L,z} \phi:(x) \right. \right. \cr & \left. + {A^a}_c \log |z-x|^2 :j^c_{L,z}\phi:(x) + {B^a}_c\frac{\bar z - \bar x}{z-x}:j^c_{L,\bar z}\phi:(x) + ... %\mathcal{O}(f^4) \right) j^b_{L,z}(w) \cr & + \left. j^a_{L,z}(x) \left( - \frac{c_+}{c_++c_-} \frac{t^b \phi(w)}{w-z} + \mathcal{O}\left( (z-w)^0 \right) \right) \right]\end{aligned}$$ Let us first consider the first term in the previous expression. According to the prescription given in appendix \[compositeOPEs\], we have to evaluate the operator $\phi(z)$ at the point $x$ before we take the OPE with the remaining current $j^b_{L,z}(w)$. So we rewrite this term as: $$\begin{aligned} \kappa_{ab} & \lim_{:x \to w:} \left( -\frac{c_+}{c_++c_-} \frac{t^a}{x-z} \sum_{n,\bar n=0}^{\infty} \frac{(z-x)^n}{n!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \p^n \bar \p^{\bar n} \phi(x) \right) j^b_{L,z}(w) \cr % & = \kappa_{ab} t^a \frac{c_+}{c_++c_-} \lim_{:x \to w:}\sum_{n,\bar n=0}^{\infty}\frac{(z-x)^{n-1}}{n!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \p_x^n \bar \p_x^{\bar n} \left( -\frac{c_+}{c_++c_-} \frac{t^b \phi(x)}{w-x} - :j^b_{L,z} \phi:(w) + ... \right) \cr % & = -\kappa_{ab} t^a t^b \left(\frac{c_+}{c_++c_-} \right)^2 \lim_{:x \to w:}\sum_{n,\bar n=0}^{\infty}\frac{(z-x)^{n-1}}{n!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \p_x^n \bar \p_x^{\bar n} \cr & \qquad \left(\frac{1}{w-x} \sum_{m,\bar m=0}^{\infty}\frac{(x-w)^{m}}{m!}\frac{(\bar x-\bar w)^{\bar m}}{\bar m!} \p^m \bar \p^{\bar m}\phi(w) \right) - \kappa_{ab} t^a \frac{c_+}{c_++c_-} \frac{:j^b_{L,z} \phi:(w)}{w-z} + ... \nonumber\end{aligned}$$ In the previous lines we only kept track of the operators that will lead to poles in the final result. We evaluated the operator $\phi$ at the point $w$ so that the action of the derivatives is easier to take care of: $$\begin{aligned} = -\kappa_{ab}& t^a t^b \left(\frac{c_+}{c_++c_-} \right)^2 \lim_{:x \to w:}\sum_{n,\bar n=0}^{\infty}\frac{(z-x)^{n-1}}{n!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \cr & \left( \sum_{m,\bar m=0}^{\infty} (-1) \frac{(x-w)^{m-n-1}}{m\ (m-n-1)!} \frac{(\bar x - \bar w)^{\bar m - \bar n}}{(\bar m - \bar n)!}\p^m \bar \p^{\bar m}\phi(w) \right)- \kappa_{ab} t^a \frac{c_+}{c_++c_-} \frac{:j^b_{L,z} \phi:(w)}{w-z} + ... \nonumber\end{aligned}$$ The regular limit gives a non-zero result for the anti-holomorphic factor only if $\bar m - \bar n = 0$. For the holomorphic factor, one needs $m-n-1=0$. Notice that the terms with $n=m=0$ also contributes with a non-vanishing term. Eventually we obtain: $$\begin{aligned} \kappa_{ab}& t^a t^b \left(\frac{c_+}{c_++c_-} \right)^2 \left( \sum_{n,\bar n=0}^{\infty}\frac{(z-w)^{n-1}}{(n+1)!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \p^{n+1} \bar \p^{\bar n}\phi(w) \right. \cr & \qquad \left. + \sum_{\bar n=0}^{\infty}\frac{1}{(z-w)^2} \frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \bar \p^{\bar n}\phi(w) \right)- \kappa_{ab} t^a \frac{c_+}{c_++c_-} \frac{:j^b_{L,z} \phi:(w)}{w-z} + ...\cr % & = \kappa_{ab} t^a t^b \left(\frac{c_+}{c_++c_-} \right)^2 \frac{\phi(z)}{(z-w)^2} - \kappa_{ab} t^a \frac{c_+}{c_++c_-} \frac{:j^b_{L,z} \phi:(w)}{w-z} + ... \end{aligned}$$ This completes the evaluation of the first term in . The other terms are much easier to deal with. The only non-trivial part is the computation the OPE between a current $j^b_{L,z}(w)$ and the composite operators $:j^c_{L,z}\phi:(z)$ and $:j^c_{L,\bar z}\phi:(z)$. Since the coefficients ${A^a}_c$ and ${B^a}_c$ already are of order $f^2$, we only need to know these OPEs at order $f^0$. We find: $$\begin{aligned} j^b_{L,z}(w):j^c_{L,z}\phi:(z) = & \left(c_1 \kappa^{bc} + \frac{c_-(c_2-g)-c_+c_2}{c_++c_-}{f^{bc}}_d t^d \right)\frac{\phi(z)}{(w-z)^2} \cr & \quad - \frac{c_+}{c_++c_-} \frac{t^b :j^c_{L,z}\phi:(z)}{w-z} + \mathcal{O}(f^2) \\ % j^b_{L,z}(w):j^c_{L,\bar z}\phi:(z) = & \left(\tilde{c} \kappa^{bc} + \frac{c_-(c_2-g)+c_+(c_4-g)}{c_++c_-}{f^{bc}}_d t^d \right)\phi(z)2\pi \delta^{(2)}(w-z) \cr & \quad - \frac{c_+}{c_++c_-} \frac{t^b :j^c_{L,\bar z}\phi:(z)}{w-z} + \mathcal{O}(f^2). \end{aligned}$$ All the terms that appear in these OPEs give zero once contracted either with ${A^a}_c \kappa_{ab}$ or with ${B^a}_c \kappa_{ab}$. In particular factors of the form ${f^a}_{bc}t^c t^b$ vanish since the dual Coxeter number is zero. Gathering everything we obtain: $$\begin{aligned} \phi(z) 2 c_1 T(w) &= \frac{c_+^2}{(c_++c_-)^2}t^a t^b \kappa_{ab} \frac{\phi(z)}{(z-w)^2} -\frac{2 c_+}{c_++c_-} \frac{\kappa_{ab}t^a :j^b_{L,z}\phi:(z)}{w-z}+ \mathcal{O}(z-w)^{0}+ \mathcal{O}(f^2).\end{aligned}$$ The previous result is true only up to terms of order $f^2$, since a term of order $f^4$ in the current-primary OPE may give a term of order $f^2$ once contracted with an additional current (see lemma ). We rewrite the result as: $$\begin{aligned} T(w) \phi(z) &= \frac{f^2}{2} \frac{t^a t^b \kappa_{ab} \phi(z)}{(z-w)^2} +\frac{1}{c_+} \frac{\kappa_{ab}t^a :j^b_{L,z}\phi:(z)}{w-z}+ \mathcal{O}(z-w)^{0}+ \mathcal{O}(f^4) \end{aligned}$$ This concludes the proof of equation in section \[primaries\]. The mode expansion on the cylinder {#commutators} ================================== When the theory is defined on a cylinder we can expand the operators in modes by means of a Fourier transform along the compact coordinate. Then we can convert the current-current OPEs into graded commutation relations for the modes of the currents. This was done for the current algebra in [@Ashok:2009xx]. In this appendix we give the translation of the left current - right current OPEs (\[jLjR1\], \[jLjR2\]) in terms of commutation relations. We use the same techniques as in section 5 of [@Ashok:2009xx]. To simplify the notation we do not write explicitly the subscript $L$ or $R$ on the currents since it is redundant with the different notation for the left and right adjoint representations. We expand the currents and the adjoint operator in modes: j\^a\_[z]{}(,) &=& +i \_[n Z]{} e\^[-in]{}j\^a\_[z,n]{}() j\^a\_[| z]{}(,) &=& -i \_[n Z]{} e\^[-in]{}j\^a\_[| z,n]{}() j\^[|a]{}\_[z]{}(,) &=& +i \_[n Z]{} e\^[-in]{}j\^[|a]{}\_[z,n]{}() j\^[|a]{}\_[| z]{}(,) &=& -i \_[n Z]{} e\^[-in]{}j\^[|a]{}\_[| z,n]{}() \^[a |a]{}(,) &=& \_[n Z]{} e\^[-in]{} \^[a |a]{}(). We obtain the commutation relations: &=& + \^[a | a]{}\_[n+m]{} &=& - \^[a |a]{}\_[n+m]{} &=& - (m+n) \^[a |a]{}\_[n+m]{} &=& + (m+n) \^[a |a]{}\_[n+m]{}, as well as the standard commutation relations between the modes of the currents and the left-right adjoint primary (as determined by their OPE), and the left-left commutation relations calculated in [@Ashok:2009xx]. In [@Ashok:2009xx] it was shown that the combination of left current components $j^a_{z,n}-j^a_{\bar z,n}$ generate a Kac-Moody algebra at integer level $k$. This is also the case for the right combination $j^{\bar a}_{z,m}-j^{\bar a}_{\bar z,m}$. As a consequence of the above commutation relations, we find moreover that the left and right Kac-Moody subalgebras commute: = 0. Only the zero modes of these affine currents commute with the worldsheet Hamiltonian. Classical integrability {#classint} ======================= In this appendix, we will show that principal chiral models with or without Wess-Zumino term are classically integrable. We generalize here the standard calculation to the case with non-zero Wess-Zumino term. The equations of motion $d \ast j = 0$ for the model written in terms of the left current components read: $$\begin{aligned} \bar{\partial} j^a_z + \partial j^a_{\bar{z}} &=& 0,\end{aligned}$$ where we have that: $$\begin{aligned} j_z &=& - \frac{1}{2} ( \frac{1}{f^2} + k) \partial g g^{-1} \nonumber \\ j_{\bar{z}} &=& - \frac{1}{2} ( \frac{1}{f^2} - k) \bar{\partial} g g^{-1}.\end{aligned}$$ As before, the coefficient of the principal chiral model term is $1/f^2$ and the Wess-Zumino term has coefficient $k$. The Maurer-Cartan equation $d (dg g^{-1} ) = dg g^{-1} \wedge dg g^{-1}$ is: $$\begin{aligned} - \frac{1}{2} ( \frac{1}{f^2} - k) \bar{\partial} j^a_z + \frac{1}{2} ( \frac{1}{f^2} + k) \partial j^a_{\bar{z}} - i {f^{a}}_{bc} j_z^c j_{\bar{z}}^b &=& 0. \end{aligned}$$ In this context it is easier to work with the canonical right invariant one-form: $$\begin{aligned} \omega &=& dg g^{-1}\end{aligned}$$ and rewrite the equations of motion in terms of $\omega$ and the coefficients $c_\pm$ defined as in the bulk of the paper: $$\begin{aligned} \bar{\partial} \omega_z &=& -\frac{ c_-}{c_+ + c_-} [\omega_z , \omega_{\bar{z}}] \nonumber \\ \partial \omega_{\bar{z}} &=& + \frac{c_+}{c_++c_-} [\omega_z , \omega_{\bar{z}}]. \label{EOMs}\end{aligned}$$ Now consider a connection which is a function of a spectral parameter $\lambda$: $$\begin{aligned} A(\lambda) &=& -\frac{2}{1+\lambda} \frac{c_+}{c_++c_-} \omega_z dz - \frac{2}{1-\lambda} \frac{ c_-}{c_++c_-} \omega_{\bar{z}} d \bar{z}\end{aligned}$$ and compute the curvature of the connection: $$\begin{aligned} %F &=& d A + A \wedge A %\nonumber \\ %F_{ \bar{z} z} &=& \frac{2}{1+\lambda} \frac{c_+}{c_++c_-} \bar{\partial} \omega_z - % \frac{2}{1-\lambda} \frac{ c_-}{c_++c_-} \partial \omega_{\bar{z}} - % \frac{c_+}{c_++c_-} \frac{ c_-}{c_++c_-} \frac{2}{1+\lambda} \frac{2}{1-\lambda} %[\omega_z , \omega_{\bar{z}}] %\nonumber \\ F_{\bar{z} z} &=& -\frac{2}{1+\lambda} \frac{c_+}{c_++c_-} \bar{\partial} \omega_z + \frac{2}{1-\lambda} \frac{ c_-}{c_++c_-} \partial \omega_{\bar{z}} - \frac{c_+}{c_++c_-} \frac{ c_-}{c_++c_-} 2 (\frac{1}{1+\lambda}+ \frac{1}{1-\lambda}) [\omega_z ,\omega_{\bar{z}}]. \nonumber\end{aligned}$$ Flatness of the connection for all values of the spectral parameter $\lambda$ is equivalent to the validity of the equations of motion (\[EOMs\]). Using the on-shell flat connection, we can define an infinite set of conserved charges, for instance by calculating the traced holonomy for the model on a circle times time, and expanding in the spectral parameter. The infinite set of conserved charges renders the theory classically integrable. The theory can then be studied using the powerful tools of integrability. [99]{} N. Berkovits, C. Vafa and E. Witten, “Conformal field theory of AdS background with Ramond-Ramond flux,” JHEP [**9903**]{} (1999) 018 \[arXiv:hep-th/9902098\]. M. Bershadsky, S. Zhukov and A. Vaintrob, “PSL(n|n) sigma model as a conformal field theory,” Nucl. Phys.  B [**559**]{} (1999) 205 \[arXiv:hep-th/9902180\]. A. Babichenko, “Conformal invariance and quantum integrability of sigma models on symmetric superspaces,” Phys. Lett.  B [**648**]{}, 254 (2007) \[arXiv:hep-th/0611214\]. S. K. Ashok, R. Benichou and J. Troost, “Conformal Current Algebra in Two Dimensions,” arXiv:0903.4277 \[hep-th\]. L. Rozansky and H. Saleur, “Quantum field theory for the multivariable Alexander-Conway polynomial,” Nucl. Phys.  B [**376**]{} (1992) 461. V. Schomerus and H. Saleur, “The GL(1|1) WZW model: [From]{} supergeometry to logarithmic CFT,” Nucl. Phys.  B [**734**]{} (2006) 221 \[arXiv:hep-th/0510032\]. G. Gotz, T. Quella and V. Schomerus, “The WZNW model on PSU(1,1 2),” JHEP [**0703**]{}, 003 (2007) \[arXiv:hep-th/0610070\]. N. Read and H. Saleur, “Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions,” Nucl. Phys.  B [**613**]{}, 409 (2001) \[arXiv:hep-th/0106124\]. T. Quella, V. Schomerus and T. Creutzig, “Boundary Spectra in Superspace Sigma-Models,” JHEP [**0810**]{} (2008) 024 \[arXiv:0712.3549 \[hep-th\]\]. V. Mitev, T. Quella and V. Schomerus, “Principal Chiral Model on Superspheres,” JHEP [**0811**]{} (2008) 086 \[arXiv:0809.1046 \[hep-th\]\]. C. Candu, V. Mitev, T. Quella, H. Saleur and V. Schomerus, “The Sigma Model on Complex Projective Superspaces,” arXiv:0908.0878 \[hep-th\]. C. Candu, T. Creutzig, V. Mitev and V. Schomerus, “Cohomological Reduction of Sigma Models,” arXiv:1001.1344 \[hep-th\]. J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys.  [**2**]{} (1998) 231 \[Int. J. Theor. Phys.  [**38**]{} (1999) 1113\] \[arXiv:hep-th/9711200\]. N. Gromov, V. Kazakov and P. Vieira, “Exact Spectrum of Anomalous Dimensions of Planar N=4 Supersymmetric Yang-Mills Theory,” Phys. Rev. Lett.  [**103**]{}, 131601 (2009) \[arXiv:0901.3753 \[hep-th\]\]. S. K. Ashok, R. Benichou and J. Troost, “Asymptotic Symmetries of String Theory on AdS3 X S3 with Ramond-Ramond JHEP [**0910**]{} (2009) 051 \[arXiv:0907.1242 \[hep-th\]\]. V. G. Knizhnik and A. B. Zamolodchikov, “Current algebra and Wess-Zumino model in two dimensions,” Nucl. Phys.  B [**247**]{} (1984) 83. P. Di Francesco, P. Mathieu, D. Senechal, “Conformal Field Theory," Springer, 1997. M. R. Gaberdiel, “An algebraic approach to logarithmic conformal field theory,” Int. J. Mod. Phys.  A [**18**]{}, 4593 (2003) \[arXiv:hep-th/0111260\]. M. Flohr, “Bits and pieces in logarithmic conformal field theory,” Int. J. Mod. Phys.  A [**18**]{} (2003) 4497 \[arXiv:hep-th/0111228\]. M. Luscher, “Quantum Nonlocal Charges And Absence Of Particle Production In The Two-Dimensional Nonlinear Sigma Model,” Nucl. Phys.  B [**135**]{} (1978) 1. [^1]: Preprint LPTENS-10/11. [^2]: We would like to thank Anatoly Konechny for stressing the importance of these terms, and for sharing his insights in these terms in perturbation theory near Wess-Zumino-Witten points. [^3]: If the assumptions are not valid, the coefficients will receive higher order corrections in $f^2$. The results in the rest of the paper are independent of these possible corrections. [^4]: These contact terms allow for the computation of the holomorphic (respectively anti-holomorphic) poles in the $j^a_{L,z} j^b_{L,z}$ (respectively $j^a_{L,\bar z} j^b_{L,\bar z}$) OPE. These poles were already computed to all orders in $f^2$ in [@Ashok:2009xx] using different methods. [^5]: Using similar methods it can be shown that the subleading terms in equation do not modify this conclusion [^6]: These conventions differ only slightly from those in [@Ashok:2009xx]. [^7]: We would like to thank Matthias Gaberdiel for raising the issue.

--- abstract: 'It is shown that the Hall, Hu and Marron \[Hall, P., Hu, T., and Marron J.S. (1995), Improved Variable Window Kernel Estimates of Probability Densities, [*Annals of Statistics*]{}, 23, 1–-10\] modification of Abramson’s \[Abramson, I. (1982), On Bandwidth Variation in Kernel Estimates – A Square-root Law, [*Annals of Statistics*]{}, 10, 1217–-1223\] variable bandwidth kernel density estimator satisfies the optimal asymptotic properties for estimating densities with four uniformly continuous derivatives, uniformly on bounded sets where the preliminary estimator of the density is bounded away from zero.' author: - | [Evarist Giné$^{a}$[^1]  and Hailin Sang$^b$]{}\ *$^a$University of Connecticut* and *$^b$University of Cincinnati* date: January 2009 title: Uniform asymptotics for kernel density estimators with variable bandwidths --- *MSC 2000 subject classification*: Primary: 62G07. *Key words and phrases: kernel density estimator, variable bandwidth, spatial adaptation, square root law, sup-norm loss, law of the logarithm, rates of convergence.* =1truein Introduction and statement of the main result ============================================= Let $f$ be a density on the real line and let $X_i$, $i\in\mathbb N$, be independent, identically distributed random variables with distribution of density $f$. Abramson (1982) discovered that if in the usual kernel density estimator one allows the bandwidth $h_n$ to vary with the data according to the ‘square root law’, that is, if one takes $$\label{abr1} f_n(t)=\frac{1}{n}\sum_{i=1}^n\frac{ f^{1/2}_t(X_i)}{h_n}K\left(\frac{t-X_i}{h_n} f^{1/2}_t(X_i)\right)$$ instead of the classical estimators with the same sequence $h_n\to 0$, where $ f_t(x)=f(x)\vee (f(t)/10)$, then a [*bias reduction*]{} phenomenon occurs. This has been used by Hall, Hu and Marron (1995) (following Hall and Marron (1988), corr. (1992)), McKay (1993) and Novak (1999), to propose density estimators which are [*non-negative*]{} at all points and which estimate $f(t)$ at any given $t$ at the $L_2$-norm loss minimax rate of $n^{-4/9}$ if the density $f$ is four times differentiable with continuous and bounded derivatives. Of course, the expression (\[abr1\]) is not an estimator of $f$ as it depends on the unknown $f$ through $f_t$, but it becomes one if $f$ is replaced by a preliminary estimator (based on the same data, or on an independent set of data). As in the mentioned papers, expressions such as (\[abr1\]) will be referred to here as ‘ideal’ estimators. It was once believed that $f_t$ in (\[abr1\]) could be replaced by $f$, but Terrell and Scott (1992) showed that in this case the bias reduction at a single $t$ depends heavily on the tail of $f$ and becomes negligible in the normal case (see also Hall, Hu and Marron (1995) and McKay (1993)). Taking $f_t$ instead of $f$ as Abramson did constitutes a way to deal with the tail effects on the localities $t$. Hall, Hu and Marron (1995), McKay (1993) and Novak (1999) also devised other ways of dealing with the problem. In particular, Hall, Hu and Marron proposed the ideal estimator $$\label{ideal0} \bar f_n(t)=\frac{1}{n h_n}\sum_{i=1}^{n}K\left(\frac{t-X_i}{h_n} f^{1/2}(X_i)\right) f^{1/2}(X_i)I(|t-X_{i}|<h_nB),$$ for some $B>0$. Novak replaces $h_n$ in the indicator by $h_n/f^{1/2}(t)$ and considers powers other than 1/2 as well, and McKay replaces $ f^{1/2}_t(x)$ in (\[abr1\]) by a smooth function $\alpha (x) =cv^{1/2}(f(x)/c^2)$ with $v(t)=t$ for all $t\ge t_0\ge 1$ with the first four derivatives of $v$ vanishing at zero. We will focus our attention only on the simplest of these ideal estimators, which is (\[ideal0\]), although our results should hold for the other versions as well. The ideal estimator will only be a means to study the ‘true’ estimator, obtained from the ideal by replacement of $f$ by a preliminary estimator. Specifically, in this article we study the uniform approximation of a density $f$ by estimators of the form $$\label{realest0} \hat f(t;h_{1,n}, h_{2,n})=\frac{1}{n h_{2,n}}\sum_{i=1}^{n}K\left(\frac{t-X_i}{h_{2,n}}\hat f^{1/2}(X_i;h_{1,n})\right)\hat f^{1/2}(X_i;h_{1,n})I(|t-X_{i}|<h_{2,n}B),$$ where $\hat f(x;h_{1,n})$ is the classical kernel density estimator $$\hat f(x;h_{1,n})=\frac{1}{n h_{1,n}}\sum_{i=1}^{n}K\left(\frac{x-X_i}{h_{1,n}}\right)$$ and $h_{i,n}$ are two sequences of bandwidths that tend to zero as $n\to\infty$. Ideally, we would like to prove results for $\|\hat f(t;h_{1,n}, h_{2,n})-f(t)\|_\infty$, however controlling the bias part of this error, $|E\hat f(t;h_{1,n}, h_{2,n})-f(t)|$, seems to require that $f(t)$ be bounded away from zero, so, we will consider instead the supremum of the estimation error on the ‘ideal’ regions $$\label{region0} D_r=D_r(f):=\{t: f(t)>r, |t|<1/r\},\ \ r>0,$$ and will eventually replace $D_r$ by a region that depends on the data only and that can be made arbitrarily close to the positivity set of $f$. (We will not display the argument $f$ in $D_r(f)$ unless confusion is possible.) It is known (Hall, Hu and Marron (1995), Novak (1999)) that the bias reduction does hold for $f$ and $K$ four times differentiable and that then one has a bias of the order of $h_{2,n}^4$. This leads almost immediately in the case of the ideal estimator, and with some relatively hard work in the case of the real estimator, to an a.s. rate of convergence of $\hat f(t;h_{1,n}, h_{2,n})-f(t)$ (fixed $t$) of the order of $n^{-4/9}$ if we take $h_{2,n}\simeq n^{-1/9}$ and $h_{1,n}\simeq n^{-2/9}$ or of a smaller order, and this is best possible ($n^{-4/9}$ is the minimax rate in the $L_2(P)$ norm for estimating $f(t)$ four times differentiable with continuity). The minimax rate for the sup norm in this case is $(n/\log n)^{-4/9}$ and we show in this article that this rate is achieved by the estimator (\[realest0\]) uniformly in $D_r$ and in a similar data-dependent region. (See e.g. Efromovich (1999) for minimax rates.) Concretely, we prove the following theorem, in fact, as explained below, a [*uniform* ]{}version of it. \[main0\] Assume the density $f$ and its first four derivatives are uniformly continuous and bounded, that the same is true for the kernel $K$, which, moreover is non-negative, has support contained in $[-T,T]$, $T<\infty$, integrates to 1 and is symmetric about zero. Set $h_{2,n}= ((\log n)/n)^{1/9}$ and $h_{1,n}=n^{-2/9}$ (or $h_{1,n}=n^{-(2+\eta)/9}$ for some $0\le\eta<1$), $n\in\mathbb N$. Then, for all $r>0$ and constant $B\ge T/r^{1/2}$ in the definition of $\hat f(t;h_{1,n}, h_{2,n})$ in (\[realest0\]), we have $$\label{main1} \sup_{t\in D_r}\left|\hat f(t;h_{1,n}, h_{2,n})-f(t)\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right).$$ If $\hat D_r^n$ is defined as $$\label{region1} \hat D_r^n=\left\{t: \hat f(t;h_{1,n})>2r, |t|<1/r\right\},$$ then we also have $$\label{main2} \sup_{t\in \hat D_r^n}\left|\hat f(t;h_{1,n}, h_{2,n})-f(t)\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right).$$ (Actually, $h_{2,n}$ needs only be asymptotically of the order $((\log n)/n)^{1/9}$ in the sense that $0\le \liminf_n \frac{h_{2,n}}{((\log n)/n)^{1/9}}\le \limsup_n\frac{h_{2,n}}{((\log n)/n)^{1/9}}<\infty$, and the same comment applies to $h_{1,n}$, but for simplicity we will work with exact values.) We note that, by the zero-one law, statement (\[main1\]) is equivalent to the existence of a finite constant $C$ such that $$\label{main1a} \limsup_n\left(\frac{n}{\log n}\right)^{4/9}\sup_{t\in D_r}\left|\hat f(t;h_{1,n}, h_{2,n})-f(t)\right|=C\ \ {\rm a.s.}$$ and likewise for (\[main2\]). And (\[main1a\]) holds for some $C<\infty$ if and only if there is $C'<\infty$ such that $$\lim_{k\to\infty}\Pr\left\{\sup_{n\ge k}\left(\frac{n}{\log n}\right)^{4/9}\sup_{t\in D_r}\left|\hat f(t;h_{1,n}, h_{2,n})-f(t)\right|>C'\right\}=0.$$ So, the following definition is justified (it is similar to the definition of uniform Glivenko-Cantelli classes of functions in Dudley, Giné and Zinn (1991)): For each $n\in\mathbb N$, let $Z_n(x_1,\dots,x_n;f)$ be functions of $n$ real variables $x_1,\dots, x_n$ and of the density $f$, $f\in\cal D$, where $\cal D$ is a collection of densities. We say that the collection of random variables $Z_n(X_1,\dots, X_n,f)$, $f\in{\cal D}$, $n\in\mathbb N$, is a.s. asymptotically of the order of $a_n$ uniformly in $f\in {\cal D}$, $$Z_n(X_1,\dots,X_n,f)=O_{\rm a.s.}(a_n)\ \ {\rm uniformly\ in}\ f\in{\cal D},$$ if there exists $C<\infty$ such that $$\label{defunif} \lim_{k\to\infty}\sup_{f\in {\cal D}}{\Pr}_f\left\{\sup_{n\ge k}\frac{1}{a_n}|Z_n(X_1,\dots,X_n,f)|>C\right\}=0,$$ and $o_{\rm a.s.}(a_n)$ uniformly in $f\in\cal D$ if the limit (\[defunif\]) holds for every $C>0$. For $0<C<\infty$ and non-negative function $z$ such that $z(\delta)\searrow 0$ as $\delta\searrow 0$, define the class of densities $${\cal D}_{C,z}:=\Bigg\{f: f\ {\rm is\ a\ density,}\ \|f^{(k)}\|_\infty\le C, \ 0\le k\le 4,~~~~~~~~~~~~~~~~~~~~~~$$ $$\label{De} ~~~~~~~~~~~~~~~~~~~~~~~{\rm and}\ \sup_{{t\in\mathbb R}\atop {|u|\le \delta}}\left|f^{(4)}(t+u)-f^{(4)}(t)\right|\le z(\delta), \ 0<\delta\le 1\Bigg\}$$ Here is the stronger version of Theorem \[main0\] that we prove in this article. \[mainu\] Under the hypotheses of Theorem \[main0\] we have $$\label{main1'} \sup_{t\in D_r(f)}\left|\hat f(t;h_{1,n}, h_{2,n})-f(t)\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ f\in{\cal D}_{C,z}$$ and $$\label{main2'} \sup_{t\in \hat D_r^n}\left|\hat f(t;h_{1,n}, h_{2,n})-f(t)\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ f\in{\cal D}_{C,z}$$ for all $0<C<\infty$ and function $z\ge 0$ such that $z(h)\searrow 0$ as $h\searrow 0$. It is natural that, as shown by Hall, Hu and Marron (1995), the estimator (\[realest0\]) be locally (that is, at each point $t$) asymptotically better than the classical kernel estimator that it modifies because, after all, it is obtained from the classical one by [*local or spatial adaptation*]{} of the bandwidth. This theorem shows that, up to a logarithmic factor, the improvement is not only local but holds uniformly over all $t$ for which $f(t)$ is slightly above zero, and uniformly as well over large classes of densities with four continuous derivatives. This may seem surprising and is certainly desirable. See the comments by Donoho, Johnstone, Kerkyacharian and Picard (1995) about the scarcity of theoretical results on ‘spatially adaptive’ estimators. We do not know of any other non-negative estimators of a density that achieve such good rates in sup-norm loss (although Abramson’s or Novak’s may). Thresholding wavelet density estimators (Donoho, Johnstone, Kerkyacharian and Picard (1996)) constitutes also a kind of adaptation to the local behavior of $f$ since wavelets pick up local behavior; these estimators may not be non-negative on the whole domain, but are rate adaptive to the smoothness of $f$ in sup-norm loss, in particular satisfying Theorem \[mainu\] -but also attaining the rate $((\log n)/n)^{t/(2t+1)}$ uniformly on densities in the unit ball of $C^t({\mathbb R})$ (Giné and Nickl (2008)). See also Giné and Nickl (2009) for estimators with this property based on convolution kernels of higher order and Lepski’s method. We first prove Theorem \[mainu\] for the ideal estimator and then show that the supremum over $D_r$ of the difference between the true and the ideal estimators is of the order of $\left((\log n)/n\right)^{4/9}$. For this we use empirical process and U-process techniques: basically, the classes of functions involved in the supremum in (\[main1\]) and in other suprema appearing in the proofs are of Vapnik-Červonenkis type (see e.g. de la Peña and Giné (1999)) and therefore we can use the appropriate version of Talagrand’s exponential inequality for empirical processes (as in Einmahl and Mason (2000) and Giné and Guillou (2002)), and an inequality due to Major (2006) for $U$-processes. We relegate to an appendix proving that the relevant classes of functions are of VC type, so that we get this technicality out of the way in the main proofs. Since we use empirical processes, in order to avoid measurability problems and without loss of generality, we assume throughout that the variables $X_i$ are the coordinate functions on $\Omega={\mathbb R}^{\mathbb N}$, equipped with the product $\sigma$-algebra and the probability measure $\Pr=P^{\mathbb N}$, $dP(x)=f(x)dx$, that we will denote as ${\Pr}_f$ if (and only if) we need to distinguish among several densities. The ideal estimator =================== In this section we obtain the asymptotic size of the uniform deviation of the ideal estimator (\[ideal0\]) from the density $f$, that is, we will consider the a.s. asymptotic size of $$\sup_{t\in D_r} |\bar f(t;h_n)-f(t)|:=\|\bar f(t;h_n)-f(t)\|_{D_r}$$ As usual this quantity is divided into the bias part, $\|E\bar f(t;h_n)-f(t)\|_{D_r}$, and the stochastic part or variance part $\|\bar f(t;h_n)-E\bar f(t;h_n)\|_{D_r}$. Each is studied in a different subsection. There is no problem with extending the supremum for the variance part over the whole of $\mathbb R$; the problem is, as mentioned above, with the bias. We will use the shorthand notations $$\bar f_n(t;h)=\bar f_n(t)=\bar f(t;h_n)$$ so that we display only either $h_n$ or $n$ but not both; the first expression is used in this section and the second in the next. Stochastic part of the ‘ideal’ estimator ---------------------------------------- In this subsection we assume: \[ass1\] The sequence $h_n$ will satisfy the following classical conditions: $$\label{band} h_{n}\searrow0, \;\; \frac{nh_{n}}{|\log h_{n}|}\rightarrow\infty, \;\; \frac{|\log h_{n}|}{\log\log{n}} \rightarrow\infty,\;\; and\:\: nh_n\nearrow,$$ as $n\to\infty$. The kernel $K$ will be a non-negative left or right continuous function, bounded, with support contained in $[-T,T]$ for some $T<\infty$, and of bounded variation. $f$ is a bounded density. The proof of the following proposition is patterned after the proof of a similar theorem in Giné and Guillou (2002), and it consists of blocking and application of Talagrand’s inequality (\[tal\]). It extends to the variable bandwidth estimator a well known uniform rate for the usual kernel estimator (Silverman (1978), formula (9)). \[varid\] Under the hypotheses in Assumptions \[ass1\], $$||\bar{f}_n-E\bar{f}_n||_{\infty}=O_{\rm a.s.}\left(\sqrt{\frac{\log h_{n}^{-1}}{n h_{n}}}\right)$$ uniformly over all densities $f$ such that $\|f\|_\infty\le C$, for any $0<C<\infty$. We block the terms between dyadic integers as follows, where, for ease of notation we set $\textbf{1}_{i,n}(t):=I(|t-X_i|<h_nB)$ and $\textbf{1}_{ih}(t)=I(|t-X_{i}|<hB)$: $$\begin{aligned} \label{eq1} &&\Pr\left\{\max_{2^{k-1}<n\le 2^{k}}\sqrt{\frac{n h_{n}}{\log h_{n}^{-1}}} ||\bar{f}_n-E\bar{f}_n||_{\infty}>\lambda\right\} \notag\\ &&~~~\le\Pr\Biggr\{\max_{2^{k-1}<n\le 2^{k}}\sqrt{\frac{1}{2^{k-1} h_{2^k}\log h_{2^k}^{-1}}}\sup_{t\in \mathbb{R}}\left|\sum_{i=1}^{n}\left[ K\left(\frac{t-X_{i}}{h_{n}}f^{1/2}(X_{i})\right)f^{1/2}(X_{i})\textbf{1}_{i,n}(t) \right.\right.{}\nonumber\\ &&~~~~~~~~~~~~~~~~~~~~~~~\left.\left.{} -EK \left(\frac{t-X_{i}}{h_{n}}f^{1/2}(X_{i})\right)f^{1/2}(X_{i})\textbf{1}_{i,n}(t)\right]\right| >\lambda\Biggr\} \nonumber \\ &&~~~\le\Pr\Biggr\{\max_{2^{k-1}<n\le 2^{k}} \sup_{ {t\in \mathbb{R}}\atop {h_{2^{k}}\le h<h_{2^{k-1}}} }\left|\sum_{i=1}^{n}\left[ K\left(\frac{t-X_{i}}{h}f^{1/2}(X_{i})\right)f^{1/2}(X_{i})\textbf{1}_{ih}(t)- \right.\right.\nonumber\\ &&~~~~~~~~~~~~~~~~~~~~~~~\left.\left.-EK\left(\frac{t-X_{i}}{h}f^{1/2}(X_{i})\right)f^{1/2}(X_{i})\textbf{1}_{ih}(t)\right]\right| >\lambda\sqrt{2^{k-1} h_{2^k}\log h_{2^k}^{-1}}\Biggr\}\end{aligned}$$ for any $\lambda>0$, where we used that $h_n$ decreases and that the function $x\log x^{-1}$ is decreasing for $x\le 1/e$. As we see in the Appendix the class of functions $$\mathcal{F}=\left\{K\left(\frac{t- \cdot}{h}f^{1/2}(\cdot)\right)f^{1/2}(\cdot)I( |t-\cdot|<hB):t\in\mathbb{R}, h>0 \right\} \label{entr0}$$ is a bounded VC class of measurable functions with respect to the constant envelope $W:=\|K\|_V||f||_{\infty}^{1/2}$, where $||K||_V$ is the total variation norm of $K$. Hence, the subclasses $$\mathcal{F}_{k}=\left\{K\left(\frac{t-\cdot}{h}f^{1/2}(\cdot)\right)f^{1/2}(\cdot)I( |t-\cdot|<hB):t\in\mathbb{R}, h_{2^{k}}\le h<h_{2^{k-1}} \right\} \label{entr3}$$ are VC classes of functions with respect to $U_{k}=W$ also and with the same characteristics $A(v)$ and $v$ as $\cal F$. Next, in order to apply Talagrand’s inequality (\[tal\]), we obtain a sensible bound $\sigma^2_k$ for the maximum variance of the functions in ${\cal F}_k$: $$\begin{aligned} &&\frac{1}{h}\int_{\mathbb{R}}K^{2}\left(\frac{t-x}{h}f^{1/2}(x)\right) I( |t-x|<hB)f^2(x)dx \le\frac{1}{h}\int_{\mathbb{R}}K^{2}\left(\frac{t-x}{h}f^{1/2}(x)\right) f^{2}(x)dx\nonumber\\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\int_{\mathbb{R}}K^{2}\left(uf^{1/2}(t-hu)\right)f^{2}(t-hu)du \nonumber\\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\le\int_{\mathbb{R}}(||K||_{\infty}^{2}||f||_{\infty}^{2})\wedge (||K||_{\infty}^{2} (T/|u|)^4)du \notag\\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=2||K||_{\infty}^{2}\left[||f||_{\infty}^{2}\int_{0}^{T/||f||_{\infty}^{1/2}}du +\int_{T/||f||_{\infty}^{1/2}}^{\infty} \left(\frac{T}{u}\right)^4du\right]\nonumber\\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{8}{3}T||K||_{\infty}^{2}||f||_{\infty}^{3/2}.\end{aligned}$$ So, we can take $\sigma_k^2:=\frac{8}{3}T||K||_{\infty}^{2}||f||_{\infty}^{3/2}2h_{2^{k}}$ (using the fourth condition in (\[band\])). $U_k=W$ is eventually much larger than $\sigma_k$ and $$\sqrt{2^k}\sigma_k\sqrt{\log \frac{U_k}{\sigma_k}}<<2^k\sigma_k^2$$ by the second condition in (\[band\]) (here and elsewhere, the sign $<<$ should be read as ‘of smaller order than’ when the indexing variable, in this case $k$, tends to infinity). If $\lambda$ in (\[eq1\]) is taken to be large enough so that $$\label{tcond} C_1\sqrt{2^k}\sigma_k\sqrt{\log \frac{RU_k}{\sigma_k}}<\lambda\sqrt{2^{k-1} h_{2^k}\log h_{2^k}^{-1}}<<2^k\sigma_k^2,$$ where $C_1$ is one of the constants in Talagrand’s inequality (\[tal\]), then, this inequality applied to the inequalities (\[eq1\]), gives $$\label{tineq} \Pr\left\{\max_{2^{k-1}<n\le 2^{k}}\sqrt{\frac{n h_{n}}{\log h_{n}^{-1}}} ||\bar{f}_n-E\bar{f}_n||_{\infty}>\lambda\right\}\le C_2\exp\left(-\frac{C_3\lambda^2 2^{k-1} h_{2^k}\log h_{2^k}^{-1}}{2^k\frac{16}{3}T||K||_{\infty}^{2}||f||_{\infty}^{3/2}h_{2^k}}\right).$$ Set $\lambda=L\sqrt{T}\|K\|_\infty C^{3/4}$. Then we can choose $L$ large enough such that inequality (\[tcond\]) is satisfied for all $k>k_0$, $k_0$ depending on $K$ only, and for this $\lambda$ inequality (\[tineq\]) becomes $$\sup_{f:\|f\|_\infty\le C} {\Pr}\left\{\max_{2^{k-1}<n\le 2^{k}}\sqrt{\frac{n h_{n}}{\log h_{n}^{-1}}} ||\bar{f}_n-E\bar{f}_n||_{\infty}>\lambda\right\}\le C_2\exp\left(-\frac{3C_3L^2 h_{2^k}\log h_{2^k}^{-1}}{2^5}\right),$$ where the term at the right hand side is the general term of a convergent series because $(\log h_{2^k}^{-1})/\log k\to\infty$ by the third inequality in (\[band\]). This proves the proposition. This result, which is good enough for our purposes, can possibly be made more precise for each particular density $f$: for instance, Sang (2008) proves $$\lim_{n\rightarrow \infty}\sqrt{\frac{n h_{n}}{\log h_{n}^{-1}}} ||\bar{f}_n-E\bar{f}_n||_{\infty}=\|K\|_2\|f\|_\infty^{3/4}\;\;a.s.$$ if the ideal Hall, Hu, Marron estimator is replaced by the ideal Novak estimator with $\alpha=1/2$, and under some additional, natural assumptions. This suggests that the rate in Proposition 1 is optimal. Also, Theorem \[varid\] admits more general and stronger versions: see Mason and Swanepoel (2008) for a recent result along the lines of the previous theorem, with uniformity in bandwidth added, and for a general class of estimators that includes ours. Bias of the ‘ideal’ estimator ----------------------------- The assumptions on $f$, $K$ and $h_n$ in this section are as follows: \[ass2\] We assume that the densities $f$ and the kernel $K$ as well as their first four derivatives are bounded and uniformly continuous, and moreover that $K$ has support contained in $[-T,T]$, $T<\infty$, it integrates to 1 and is symmetric about zero. We also assume $h_n\to0$ as $n\to\infty$ (and $h_n>0$). We set $$\begin{aligned} \label{ftilde} \tilde f(t;h)&:=&E\bar f_n(t;h)=\frac{1}{ h}\int f^{3/2}(x)K\left({x-t\over h}f^{1/2}(x)\right)I(|x-t|< Bh)dx\notag\\ &=&\int_{-B}^{B} f^{3/2}(t+hw)K(wf^{1/2}(t+hw))dw=\int_{-B}^Bg_{t,w}(hw)dw,\end{aligned}$$ where, for $t$ and $w$ fixed, $$\label{g} g_{t,w}(u)=f^{3/2}(t+u)K(wf^{1/2}(t+u)).$$ If no confusion may arise, we drop the subindices $t,w$ from $g$. To estimate the bias of the ideal estimator, $\tilde f(t;h)-f(t)$, one develops $g(hw)$ about zero and integrates. For further reference, we record the first four derivatives of $g(u)$: by direct computation or e.g. from Novak (1999), we have, with $r(u)=f^{3/2}(t+u)$ and $s(u)=wf^{1/2}(t+u)$, $$g(u)=r(u)K(s(u)),\ \ g'(u)=r'(u)K(s(u))+r(u)s'(u)K'(s(u)),$$ and, dropping the arguments for simplicity, $$\begin{aligned} \label{deriv} g''&=&r''K +(2r's'+rs'')K'+r(s')^2K'',\notag\\ g'''&=&r'''K+(3r''s'+3r's''+rs''')K'+3(r'(s')^2+rs's'')K''+r(s')^3K'''\notag\\ g^{(4)}&=&r^{(4)}K+(4r'''s'+6r''s''+4r's'''+rs^{(4)})K'+(6r''(s')^2+12r's's''+4rs's''' \notag\\ &&~~~~~~~~~~~~~~+3r(s'')^2)K''+ (4r'(s')^3+6r(s')^2s'')K'''+r(s')^4K^{(4)}.\end{aligned}$$ \[biasid\] Under the hypotheses in Assumptions \[ass2\], if the constant $B$ in the definition of $\tilde f_n(t;h_n)$ satisfies $B\ge T/r^{1/2}$, then, for all $0<C<\infty$ and functions $z\ge0$ with $z(h)\searrow 0$ as $h\searrow 0$, we have $$\label{unibias} \lim_{n\to\infty}\sup_{f\in{\cal D}_{C,z}}\sup_{t\in D_r}\left|\frac{\tilde f(t;h_n)-f(t)}{h_n^4}-H(t,f,K)\right|=0$$ and $$\label{H} \sup_{f\in{\cal D}_{C,z}}\sup_{t\in D_r}|H(t,f,K)|<\infty,$$ where $$H(t,f,K)=\left[\frac{(f')^4(t)}{ f^5(t)}-\frac{3(f')^2(t)f''(t)}{ 2f^4(t)}+\frac{4f'(t)f'''(t)+3(f'')^2(t)}{ 12f^3(t)}-\frac{f^{(4)}(t)}{24 f^2(t)}\right]\int v^4K(v)dv.$$ Since $f$ and $K$ and their first four derivatives are continuous and $f>r/2$ on a neighborhood of $D_r$, it follows that, if $g_{tw}$ is as defined in (\[g\]), there exists $n_0<\infty$ such that, for all $t\in D_r$ and for all $w\in{\mathbb R}$, $g_{t,w}^{(4)}$ is continuous on $[-Bh_n,Bh_n]$ for all $n\ge n_0$: note that $g^{(4)}(u)$ is a linear combination of $K$ and its first four derivatives at $wf^{1/2}(t+u)$ whose coefficients are fractions that have products of powers of $w$ and powers of $f(t+u)$ and its derivatives in the numerator, and powers of $f(t+u)$ in the denominator (see(\[deriv\])). Therefore, Taylor expansion gives $$\label{taylor} g(u)=\sum_{k=0}^3g^{(i)}(0)\frac{u^i}{ i! }+\frac{u^4}{ 4!}E_\tau g^{(4)}(\tau u)$$ where $\tau$ is a random variable with density $\lambda(x)=4(1-x)^3$, $0\le x\le 1,$ that does not depend on $t$, $w$ or $u$, and $E_\tau$ denotes expectation with respect to this variable. Equation (\[taylor\]) can be easily verified by integration by parts in $\int_0^14(1-t)^3g^{(4)}(tu)dt$. Next note that $$\label{firstt} \int_{-B}^Bg_{t,w}(0)dw=\int_{-B}^Bf^{3/2}(t)K(wf^{1/2}(t))dw=\int_{-Bf^{1/2}(t)}^{Bf^{1/2}(t)}f(t)K(v)dv=f(t)$$ since the support of $K$ is contained in $[-Bf^{1/2}(t),Bf^{1/2}(t)]$ by the hypothesis on $B$ and since $K$ integrates to 1. Further, since $s'$ contains a $w$ factor, there are functions $c_i(f,t)$, $i=1,2$, such that $$\int_{-B}^Bwg_{t,w}'(0)dw=\int_{-Bf^{1/2}(t)}^{Bf^{1/2}(t)}(c_1(f,t)vK(v)+c_2(f,t)v^2K'(v))dv=0$$ because $K$ is even and $K'$ is odd. Similarly (that is, using only the symmetry properties of $K$ and its derivatives), we also get $\int_{-B}^B w^3g_{t,w}'''(0)dw=0.$ That these two integrals vanish is obvious and not surprising; what is remarkable is that also $\int_{-B}^Bw^2g_{t,w}''(0)dw=0$, and this fact is the main reason for the bias reduction achieved by Abramson’s (1982) ‘inverse square root rule’. We sketch an argument for completeness. Note first that, from the expression for $g''$ in (\[deriv\]), integrating by parts, $$\begin{aligned} \int_{-B}^Bw^2[r(s')^2K''(s)](0)dw&=&{1\over 4}f^{1/2}(t)(f')^2(t)\int_{-B}^Bw^4K''(wf^{1/2}(t))dw\\ &=&-(f')^2(t)\int_{-B}^Bw^3K'(wf^{1/2}(t))dw.\end{aligned}$$ Collecting terms in $K'$, this gives $$\begin{aligned} \int_{-B}^Bw^2g_{t,w}''(0)dw&=&\left[{3\over 4}f^{-1/2}(t)(f')^2(t)+{3\over 2}f^{1/2}(t)f''(t)\right]\int_{-B}^Bw^2K(wf^{1/2}(t))dw\\ &&~~~~~~~~+\left[{1\over 4}(f')^2(t)+{1\over 2}f(t)f''(t)\right]\int_{-B}^Bw^3K'(wf^{1/2}(t))dw,\end{aligned}$$ and, integrating by parts the second integral, we get zero. \[See Novak (1999) for a proof that, if one replaces $f^{1/2}$ by $f^\alpha$ (and $f^{3/2}$ by $f^{\alpha +1}$) in the definition of $\hat f_n(t;h)$ the only $\alpha$ for which $\int_{-B}^Bw^2g''(0)dw=0$ for all $f$ twice differentiable with $f(t)\ne0$ is $\alpha=1/2$.\]. Thus, we have $$\int_{-B}^Bw^2g^{(i)}_{t,w}(0)dw=0\ \ {\rm for}\ \ i=1,2,3,$$ and we conclude, from this, (\[ftilde\]), (\[taylor\]) and (\[firstt\]), that $$\label{ftilde2} \tilde f(t;h)=\int_{-B}^Bg_{t,w}(hw)dw=f(t)+\frac{h^4}{4!}\int_{-B}^Bw^4E_\tau g^{(4)}(\tau h w)dw.$$ Using the formula for $g^{(4)}$ in (\[deriv\]), integrating by parts and collecting terms, it is tedious but straightforward to check that $$\begin{aligned} \label{rem} &&\int_{-B}^Bw^4g_{t,w}^{(4)}(0)dw\\ &&~~~~~~~~=\left[\frac{24(f')^4(t)}{f^5(t)}-\frac{36(f')^2(t)f''(t)}{ f^4(t)}+\frac{8f'(t)f'''(t)+6(f'')^2(t)}{f^3(t)}-\frac{f^{(4)}(t)}{ f^2(t)}\right]\int v^4K(v)dv,\notag\end{aligned}$$ and to note that $$\label{rem2} \sup_{f\in {\cal D}_{C,z}}\sup_{t\in D_r}\left|\int_{-B}^Bw^4g_{t,w}^{(4)}(0)dw\right|<\infty.$$ Now, the boundedness and uniform continuity of $K$ and its four derivatives and the facts that, for $f\in{\cal D}_{C,z}$, $f$ and its first three derivatives are Lipschitz with common constant $C$ and the fourth derivatives $f^{(4)}$ have all the same modulus of continuity $z$ at all $t$, and that $f$ is bounded away from zero in a neighborhood of $D_r$, imply that $$\label{comp} \lim_{n\to\infty}\sup_{f\in {\cal D}_{C,z}}\sup_{0\le \tau\le 1}\sup_{w\in[-B,B]}\sup_{t\in D_r}|g_{t,w}^{(4)}(\tau h_nw)-g_{t,w}^{(4)}(0)|=0.$$ Therefore, $$\lim_{n\to\infty}\sup_{f\in {\cal D}_{C,z}}\sup_{t\in D_r}\left|\int_{-B}^Bw^4E_\tau (g_{t,w}^{(4)}(\tau h_nw)-g_{t,w}^{(4)}(0))dw\right|=0$$ and we have from this and (\[ftilde2\]) that $$\begin{aligned} &&\sup_{f\in {\cal D}_{C,z}}\sup_{t\in D_r} \left|h_n^{-4}(\tilde f(t;h_n)-f(t)) -{1\over 4!}\int_{-B}^Bw^4g_{t,w}^{(4)}(0)dw\right|\\ &&~~~~~~~~~~~=\sup_{f\in {\cal D}_{C,z}}\sup_{t\in D_r}\left| {1\over 4!}E_\tau\int_{-B}^Bw^4 (g_{t,w}^{(4)}(\tau h_nw)-g_{t,w}^{(4)}(0))dw\right|\to 0\end{aligned}$$ as $n\to\infty$. This, together with (\[rem\]) and (\[rem2\]) prove the proposition. .1truein This proposition is similar to Theorem 3.1 of Hall, Hu and Marron (1995) and to Theorem 1 of Novak (1999), who do not consider uniformity in $t$ or $f$, and our proof is somewhat adapted from the latter reference (which deals with a slightly different estimator). See also Hall (1990), Terrell and Scott (1992) and McKay (1993). Combining Propositions \[varid\] and \[biasid\] we obtain the following result for the ‘ideal’ estimator. \[unifidealthm\] Under the Assumptions \[ass2\] and with $h_n= ((\log n)/n)^{1/9}$, we have, for every $0<C<\infty$ and function $Z$ such that $z(h)\searrow 0$ as $h\searrow 0$, for all $r>0$ and constant $B\ge T/r^{1/2}$ in the definition of $\bar f_n(t;h)$ in (\[ideal0\]), $$\label{main4} \sup_{t\in D_r}\left|\bar f(t;h_n)-f(t)\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly \ in}\ f\in{\cal D}_{C,z}.$$ [The limit (\[comp\]) is straightforward, but lengthy to compute. By way of illustration we indicate how to prove a ‘small piece’ of it. Let us consider, for example, the term in $f^{(4)}$ from the first summand $r^{(4)}K$ in the expression for $g^{(4)}$ in (\[deriv\]). It is $(3/2)f^{1/2}(t+u)f^{(4)}(t+u)K(wf^{1/2}(t+u))$. Then, $$|f^{1/2}(t+u)f^{(4)}(t+u)K(s(t+u))-f^{3/2}(t)f^{(4)}(t)K(s(t))| \le \|f\|_\infty^{1/2}\|K\|_\infty|f^{(4)}(t+u)-f^{(4)}(t)|$$ $$+\|f^{(4)}\|_\infty\|K\|_\infty|f^{1/2}(t+u)-f^{1/2}(t)|+\|f^{(4)}\|_\infty \|f\|_\infty^{1/2}|K(s(t+u))-K(s(t))|.$$ And we have, for the first summand, $$|f^{(4)}(t+\tau h_nw)-f^{(4)}(t)|\le z(Bh_n)\to 0$$ uniformly in $t$ and $f$ (recall $|\tau|\le 1$, $|w|\le B$). For the second summand, for $n$ large enough, $$|f^{1/2}(t+u\tau h_nw)-f^{1/2}(t)|\le\frac{|f(t+u\tau h_nw)-f(t)|}{r^{1/2}}\le\frac{CBh_n}{r^{1/2}}\to0,$$ and the limit zero for the third follows directly by uniform continuity of $K$ and the common Lipschitz constant $C$ for all $f\in{\cal D}_{C,z}$.]{} Comparison between the ideal and the true estimators ==================================================== In this section we make the following assumptions on the kernel $K$, the densities $f$ and the band sequences: \[ass3\] We assume that $K$ is supported by $[-T,T]$ for some $T<\infty$ and that it has a uniformly bounded second derivative. We also assume that the densities $f$ are bounded and have at least two bounded derivatives, $${pc} f\in{\cal P}_C:=\{f\ {\rm is\ a\ density}:\|f^{(k)}\|_\infty\le C, 0\le k\le 2\}$$ for some $C<\infty$. We set $h_{1,n}=n^{-2/9}$ and $h_{2,n}=((\log n)/n)^{1/9}$, $n\in\mathbb N$. Let $$\label{realest} \hat f(t;h_{1,n}, h_{2,n})=\frac{1}{n h_{2,n}}\sum_{i=1}^{n}K\left(\frac{t-X_i}{h_{2,n}}\hat f^{1/2}(X_i;h_{1,n})\right)\hat f^{1/2}(X_i;h_{1,n})I(|t-X_{i}|<h_{2,n}B),$$ where $\hat f(x;h_{1,n})$ is the classical kernel density estimator $$\label{real} \hat f(x;h_{1,n})=\frac{1}{n h_{1,n}}\sum_{i=1}^{n}K\left(\frac{x-X_i}{h_{1,n}}\right).$$ The object of this section consists in proving that $$\label{diff} \hat f(t;h_{1,n}, h_{2,n})-\bar f(t;h_{2,n})$$ is asymptotically almost surely of the order of $\sqrt{(\log h_{2,n}^{-1})/(nh_{2,n})}$ uniformly in $t$ on the region $D_r$ defined in (\[region0\]), for any $r>0$, if we take $h_{2,n}=\left((\log n)/n\right)^{1/9}$ and $h_{1,n}=n^{-2/9}$. Note that $h_{2,n}$ is the optimal rate ‘up to a log’ given the order of the bias, whereas the preliminary estimator has a bandwidth sensibly smaller than the optimal $n^{-1/5}$ (it is less smooth than the optimal, ‘undersmoothed’) and therefore its bias will be negligible with respect to its variance term. The main result of this paper will follow from this analysis and the result from the ‘ideal’ estimator. We follow the pattern in Hall and Marron (1988) and Hall, Hu and Marron (1995) for the linearization of (\[diff\]), with significant differences in order to account for the uniformity in $t$. For instance, they do not necessarily undersmooth the preliminary estimator (whereas we believe one should) and, moreover, we are required to use empirical and U-process theory. We adhere to their notation as much as possible. The first step is to notice that, if we define $\delta_n(t)$ by the equation $$\label{delta} \delta_n(t)=\frac{\hat f^{1/2}(t;h_{1,n})-f^{1/2}(t)}{f^{1/2}(t)}=\frac{\hat f(t;h_{1,n})-f(t)}{(\hat f^{1/2}(t;h_{1,n})+f^{1/2}(t))f^{1/2}(t)},$$ so that $\hat f^{1/2}(t;h_{1,n})=f^{1/2}(t)(1+\delta_n(t))$, then we have $$\label{zero} \sup_{t\in D_r^\varepsilon}\delta_n(t)=o_{\rm a.s.}(1)\ \ {\rm uniformly\ in}\ f\ {\rm such\ that}\ \|f\|_\infty\le C,$$ where $D_r^\varepsilon$ denotes the $\varepsilon$-neighborhood of $D_r$ for $\varepsilon$ such that $f(t)>r/2$ in $D_r^\varepsilon$ ($f$ is uniformly continuous). We drop the subindex $n$ from $\delta$ from now on. Set $$D(t;h_{1,n})=\hat f(t;h_{1,n})-E\hat f(t;h_{1,n})\ \ {\rm and}\ \ b(t;h_{1,n})=E\hat f(t;h_{1,n})-f(t)$$ and note that $$\label{classic1} \|D(\cdot;h_{1,n})\|_\infty=O_{a.s.}\left(\sqrt{\frac{\log h_{1,n}^{-1}}{nh_{1,n}}}\right)\ \ {\rm uniformly\ in}\ f\ {\rm such\ that}\ \|f\|_\infty\le C$$ for all $0<C<\infty$ by a result in Deheuvels (2000) and in Giné and Guillou (2002), and that $$\label{classic2} \|b(\cdot;h_{1,n})\|_\infty\le \left(\int K(u)u^2du\right)\|f''\|_\infty h_{1,n}^2$$ by the classical bias computation for symmetric kernels. Since the numerator in the expression at the right hand side (\[delta\]) is just $D(t)+b(t)$ and the denominator is not smaller than $f(t)$ which is in turn larger than $r/2$, (\[zero\]) follows from (\[classic1\]) and (\[classic2\]). Define $$L_1(z)=zK'(z)\ \ {\rm and}\ \ L(z)=K(z)+zK'(z),\ \ z\in\mathbb R.$$ We then have $$\begin{aligned} K\left(\frac{t-X_i}{h_{2,n}}\hat f^{1/2}(X_i;h_{1,n})\right) &=&K\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)+ \frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\delta (X_i)\right)\\ &=&K\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)\\ &&~~~+K^\prime\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) \frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\delta (X_i)+\delta_2(t,X_i)\\ &=&K\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)+L_1\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) \delta (X_i)+\delta_2(t,X_i),\end{aligned}$$ where $$\label{delta2} \delta_2(t,X_i)=\frac{K^{\prime\prime}(\xi)}{2} \frac{(t-X_i)^2}{h_{2,n}^2}f(X_i)\delta^2(X_i),$$ $\xi$ being a (random) number between $\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)$ and $\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)+\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\delta (X_i).$ Then, plugging this development and that of $\hat f^{1/2}$ in the definition (\[realest\]) of $\hat f$, we obtain $$\begin{aligned} \hat f(t;h_{1,n}, h_{2,n})\!\!\!&=&\!\!\!\bar f(t;h_{2,n})\nonumber\\ &&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L_1\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) f^{1/2}(X_i)\delta (X_i)I(|t-X_{i}|<h_{2,n}B)\nonumber\\ &&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}K\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) f^{1/2}(X_i)\delta (X_i)I(|t-X_{i}|<h_{2,n}B)\nonumber\\ &&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}f^{1/2}(X_i)\delta_2 (t,X_i))I(|t-X_{i}|<h_{2,n}B)\label{e1}\\ &&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L_1\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) f^{1/2}(X_i)\delta^2 (X_i)I(|t-X_{i}|<h_{2,n}B)\label{e2}\\ &&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n} f^{1/2}(X_i)\delta (X_i)\delta_2 (t,X_i))I(|t-X_{i}|<h_{2,n}B)\label{e3}\\ \!\!\!&=&\!\!\!\bar{f}(t;h_{2,n})+\delta_3(t)\nonumber\\ &&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) f^{1/2}(X_i)\delta (X_i)I(|t-X_{i}|<h_{2,n}B),\label{e4}\end{aligned}$$ where $\delta_3(t)$ is the sum of the terms (\[e1\]), (\[e2\]) and (\[e3\]), which are of a smaller order than the term (\[e4\]) by (\[zero\]) and (\[delta2\]) for $t\in D_r$ (as we will readily check). Since by (\[classic1\]) and (\[classic2\]), $D(y;h_{1,n})$ dominates $b(y;h_{1,n})$ uniformly in $\mathbb R$, we should further decompose (\[e4\]) to display its $D$-part and its $b$-part. By the definitions of $\delta$, $D$ and $b$, we have $$\begin{aligned} \delta(t)&=&\frac{D(t;h_{1,n})}{2f(t)}+\frac{b(t;h_{1,n})}{2f(t)}+\frac{D(t;h_{1,n})+b(t;h_{1,n})}{2f(t)} \frac{f^{1/2}(t)-\hat f^{1/2}(t;h_{1,n})}{\hat f^{1/2}(t;h_{1,n})+f^{1/2}(t)}\\ &:=&\frac{D(t;h_{1,n})}{2f(t)}+\frac{b(t;h_{1,n})}{2f(t)}+\delta_4(t),\end{aligned}$$ (where $\delta_4$ depends on $n$, but we do not display this dependence) and note that (again using $(a^{1/2}-b^{1/2}=(a-b)/(a^{1/2}+b^{1/2})$), $$\label{delta4} \sup_{t\in D_r^\varepsilon}|\delta_4(t)|\le \frac{1}{3r^{3/2}}\sup_{t\in D_r^\varepsilon}\left[D(t;h_{1,n})+b(t;h_{1,n})\right]^2$$ which is small by (\[zero\]) (note that $\delta_n$ is $D+b$ divided by a quantity which is bounded away from zero on $D_r$) . Setting $$\label{eps1} \varepsilon_1(t,h_{1,n},h_{2,n}):=\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) f^{-1/2}(X_i)D(X_i;h_{1,n})I(|t-X_{i}|<h_{2,n}B),$$ $$\label{eps2} \varepsilon_2(t,h_{1,n},h_{2,n}):=\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) f^{-1/2}(X_i)b(X_i;h_{1,n})I(|t-X_{i}|<h_{2,n}B),$$ and $$\label{eps3} \varepsilon_3(t,h_{1,n},h_{2,n}):=\delta_3(t)+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) f^{1/2}(X_i)\delta_4(X_i;h_{1,n})\textbf{1}\{|t-X_{i}|<h_{2,n}B\},$$ we obtain (from (\[e1\])-(\[e4\])), $$\label{expans} \hat f(t;h_{1,n}, h_{2,n})=\bar f(t;h_{2,n})+\frac{1}{2}\varepsilon_1(t)+\frac{1}{2}\varepsilon_2(t)+\varepsilon_3(t).$$ By the comments above, the $\varepsilon_2$ and $\varepsilon_3$ terms will be of smaller order than $\varepsilon_1$. $\varepsilon_1$ itself has a $U$-process structure, and the linear term in its Hoeffding decomposition will be the dominant term. This is the content of the lemmas that follow. \[lemma1\] For $i=2,3$, $$\sup_{t\in D_r^\varepsilon}|\varepsilon_i(t,h_{1,n},h_{2,n})|=O_{\rm a.s.}(n^{-4/9}) \ \ uniformly\ in\ \ f\in{\cal P}_C$$ for all $C<\infty$. We begin with $i=2$. Because the function $L$ is of bounded variation and $b(t;h_{1,n})$ satisfies inequality (\[classic2\]), it follows (see the Appendix) that the classes of functions $$\label{qu} {\cal Q}_n:=\left\{Q(x)=L\left(\frac{t-x}{h_{2,n}}f^{1/2}(x)\right) f^{-1/2}(x)b(x;h_{1,n})I(|t-x|<h_{2,n}B):t\in D_r\right\}$$ are of VC type with the same characteristics $A$ and $v$, for envelopes of the order of $M(K,r)\|f''\|_\infty h_{1,n}^2$, where $M$ depends on $r$ and $K$ only (in particular, through $L$). If we set $$Q_i(t)=L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right) f^{-1/2}(X_i)b(X_i;h_{1,n})I(|t-X_i|<h_{2,n}B)$$ it then follows (by the bound (\[classic2\]) on $b$, boundedness of $L$ and boundedness away from zero of $f$ on $D_r$), that $$\sup_{t\in D_r}E|Q_i(t)|\lessim \|f''\|_\infty h_{1,n}^2h_{2,n}=\|f''\|_\infty n^{-5/9}(\log n)^{1/9},$$ $$\sup_{t\in D_r}EQ_i^2(t)\lessim \|f''\|^2_\infty h_{1,n}^4h_{2,n}\le \|f''\|^2_\infty n^{-1}(\log n)^{1/9},\ \ \sup_{t\in D_r}|Q_i(t)|\lessim \|f''\|_\infty h_{1,n}^2=\|f''\|_\infty n^{-4/9},$$ where in these bounds we ignore multiplicative constants that do not depend on $f$. We have $$\begin{aligned} \sup_{t\in D_r}\left|\epsilon_2(t;h_{1,n}, h_{2,n})\right| &\le &\sup_{t\in D_r}\left|\frac{1}{nh_{2,n}}\sum_{i=1}^{n} [Q_i(t)-EQ_i(t)]\right|+ \sup_{t\in D_r}\frac{1}{h_{2,n}}|EQ_1(t)|\\ &\lessim & \sup_{t\in D_r}\left|\frac{1}{nh_{2,n}}\sum_{i=1}^{n} [Q_i(t)-EQ_i(t)] \right|+\|f''\|_\infty n^{-4/9},\end{aligned}$$ and Talagrand’s inequality (\[tal\]) gives that for $0<\delta\le 4/9$, $$\sum_n\sup_{f\in{\cal P}_C}{\Pr}_f\left\{\sup_{t\in D_r}\left|\sum_{i=1}^{n} [Q_i(t)-EQ_i(t)] \right|\ge n^{\delta}\right\}\le C_2\sum_n\exp\left(-\frac{C_3n^{2\delta}}{C^2(\log n)^{1/9}}\right)<\infty.$$ Since $n^{\delta}<nh_{2,n}n^{-4/9}$, we conclude $$\sup_{t\in D_r}\left|\epsilon_2(t;h_{1,n}, h_{2,n})\right|=O_{\rm a.s.}(n^{-4/9})\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C$$ proving the lemma for $\varepsilon_2$. Note that $h_{1,n}^2\simeq n^{-4/9}$ plays a critical role in this estimation. Next, from (\[eps3\]) we see that $\varepsilon_3$ consists of four sums, the three that define $\delta_3$ and one involving $\delta_4$ (multiplied by bounded terms and by the indicator of $|X_i-t|\le h_{2,n}B$). The three terms from $\delta_3$ involve, instead of $\delta_4$, respectively $\delta_2$, $\delta^2$ and $\delta_2\delta$ (see (\[e1\])-(\[e3\])). We have from (\[delta\]), (\[classic1\]) and (\[classic2\]) that $$\sup_{t\in D_r^\varepsilon}\delta^2_n=O_{\rm a.s.}\left(n^{-7/9}\log n\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C,$$ that the same is true for $\delta_4$ by (\[delta4\]), and, moreover, by (\[delta2\]), $$|\delta_2(t,X_i)|I(|t-X_i|\le h_{2,n}B)\le \frac{\|K''\|_\infty}{2}B^2\|f\|_\infty\delta^2(X_i)=O_{\rm a.s.}\left(n^{-7/9}\log n\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ then, if we define $\tilde Q_i(t)$ by $$\varepsilon_3(t,h_{1,n},h_{2,n})=\frac{1}{nh_{2,n}}\sum_{i=1}^n\tilde Q_i(t),$$ we have $$\sup_{t\in D_r}|\tilde Q_i(t)|=O_{\rm a.s.}\left(n^{-7/9}\log n\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C,$$ and therefore, $$\sup_{t\in D_r}\left|\epsilon_3(t;h_{1,n}, h_{2,n})\right|=O_{\rm a.s.}( h_{2,n}^{-1}n^{-7/9}\log n)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C,$$ proving the lemma for $\varepsilon_3$ as $h_{2,n}^{-1}n^{-7/9}\log n<< n^{-4/9}$. \[eps1-t\] Let $$\begin{aligned} \label{tien} &&T(t; h_{1,n}, h_{2,n})=\\ &&\frac{1}{nh_{1,n}h_{2,n}}\sum_{i=1}^{n} E_{X}\Big[f^{-1/2}(X)\Big\{K\Big(\frac{X-X_i}{h_{1,n}}\Big)-E_YK\Big(\frac{X-Y}{h_{1,n}}\Big)\Big\} L\Big(\frac{t-X}{h_{2,n}}f^{1/2}(X)\Big)I(|t-X|\le h_{2,n}B)\Big],\notag\end{aligned}$$ where $L(z)=K(z)+zK^\prime(z)$. Then, $$\sup_{t\in D_r}\left|\varepsilon_1(t,h_{1,n},h_{2,n})-T(t;h_{1,n},h_{2,n})\right|=o_{\rm a.s.}(n^{-4/9})\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ Given a function $H$ of two variables, and two i.i.d. random variables $X$ and $Y$ such that $H(X,Y)$ is integrable, we recall that the second order Hoeffding projection of $H(X,Y)$ is $$\pi_2(H)(X,Y)=H(X,Y)-E_XH(X,Y)-E_YH(X,Y)+EH.$$ We also recall the $U$-statistic notation $$U_n(H)=\frac{1}{n(n-1)}\sum_{1\le i\ne j\le n}H(X_i,X_j),$$ where the variables $X_i$ are i.i.d. Set $$H_t(X,Y):=L\left(\frac{t-X}{h_{2,n}}f^{1/2}(X)\right)f^{-1/2}(X)K\left(\frac{X-Y}{h_{1,n}}\right)I(|t-X|\le h_{2,n}B).$$ Then, $$\label{hoef1} \frac{n^2h_{1,n}h_{2,n}}{n(n-1)}\varepsilon_1(t,h_{1,n},h_{2,n})=\frac{1}{n(n-1)}\sum_{i=1}^n (H_t(X_i,X_i)-E_YH_t(X_i,Y))+U_n(H_t-E_YH_t(\cdot,Y))$$ (decomposition of a $V$-statistic into the diagonal term and a $U$-statistic). Now, notice that $$\begin{aligned} \label{hoef2} U_n(H_t-E_YH_t(X_i,Y))&=&U_n\left(\pi_2(H_t(\cdot,\cdot))+(E_XH_t(X,\cdot)-EH)\right)\notag\\ &=&U_n(\pi_2(H_t(\cdot,\cdot))+h_{1,n}h_{2,n}T(t;h_{1,n},h_{2,n})\end{aligned}$$ So, we now must handle the diagonal term, a completely centered or canonical $U$-process and (in the next lemma) the empirical process $T$. [*Diagonal term.*]{} Note that if we define $\bar Q_i$ such that $$\frac{1}{n^2h_{1,n}h_{2,n}}\sum_{i=1}^n (H_t(X_i,X_i)-E_YH_t(X_i,Y)):=\frac{1}{n^2h_{1,n}h_{2,n}}\sum_{i=1}^n\bar Q_i(t),$$ then we have $$\sup_{t\in D_r}|E\bar Q_1(t)|\lessim h_{2,n},\ \ \sup_{t\in D_r}E\bar Q_1^2(t)\lessim h_{2,n}, \ \ \sup_{t\in D_r}|\bar Q_1(t)|\lessim 1,$$ where as usual we overlook multiplicative constants that do not depend on $f$, and the last bound does not depend on $n$. So, $$\sup_{t\in D_r}\frac{1}{n^2h_{1,n}h_{2,n}}\left|\sum_{i=1}^n\bar Q_i(t)\right|\lessim\frac{1}{n^2h_{1,n}h_{2,n}}\sup_{t\in D_r}\left|\sum_{i=1}^n(\bar Q_i(t)-E\bar Q_1(t))\right|+\frac{1}{nh_{1,n}}.$$ The supremum part correspond to the empirical process over the class of functions of $x$ $$\label{qubar} \bar{\cal Q}_n=\left\{L\left(\frac{t-x}{h_{2,n}}f^{1/2}(x)\right)f^{-1/2}(x)\left(K(0)-EK\left(\frac{x-X}{h_{1,n}}\right)\right) I(|t-x|\le h_{2,n}B): t\in D_r\right\}.$$ These classes are VC type with the same characteristics $A$ and $v$ that do not depend on $f$, and with the same envelope, that depends only on $K$ and $r$ (see the Appendix). Then, Talagrand’s inequality gives, as in previous instances, that, for some $\delta>0$, $$\sum_n\sup_f{\Pr}_f\left\{\sup_{t\in D_r}\left|\sum_{i=1}^n(\bar Q_i(t)-E\bar Q_1(t))\right|>n^{(4+\delta)/9}\right\}\le C_2\sum_n\exp\left(-C_3\frac{n^{(8+2\delta)/9}}{nh_{2,n}}\right)<\infty,$$ which, since $n^2h_{1,n}h_{2,n}n^{-4/9}>n^{(4+\delta)/9}$ and since $nh_{1,n}>> n^{4/9}$, yields $$\label{diag} \sup_{t\in D_r}\frac{1}{n^2h_{1,n}h_{2,n}}\left|\sum_{i=1}^n (H_t(X_i,X_i)-E_YH_t(X_i,Y))\right|=o_{a.s.}(n^{-4/9})\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ [*The canonical $U$-statistic term.*]{} We will use Major’s exponential bound (\[major\]) for canonical $U$-processes over VC type classes of functions. In our case, since the class of functions $\{H_t:t\in D_r\}$ is uniformly bounded and of VC type (see the Appendix) we can apply Major’s exponential bound to $\sup_{t\in D_r}|U_n(\pi_2(H_t))|$. Since, as is easy to check, $$EH_t^2(X,Y)\le 2B\|L\|_\infty^2\|f\|_\infty\|K\|_2^2 h_{1,n}h_{2,n},$$ we can take, for $C$ such that $\|f\|_\infty\le C$, $\sigma^2\simeq C h_{1,n}h_{2,n}$ and $t=C^{1/2}n^{1+\delta}\sqrt{h_{1,n}h_{2,n}}$ for a small $\delta>0$, to have, from (\[major\]), $$\sum_n\sup_{f:\|f\|_\infty\le C}{\Pr}_f\left\{\sup_{t\in D_r}|U_n(\pi_2(H_t))|>Cn^{\delta-1}\sqrt{h_{1,n}h_{2,n}}\right\}\le C_2\sum_n\exp\left(-C_3n^\delta\right)<\infty.$$ Since $\frac{n^{\delta -1}}{\sqrt{h_{1,n}h_{2,n}}}<<n^{-4/9}$ (we can take $\delta$ so that this is true), we obtain $$\label{ust} \sup_{t\in D_r}\frac{1}{h_{1,n}h_{2,n}}|U_n(\pi_2(H_t))|=o_{\rm a.s.}(n^{-4/9})\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ The following lemma will conclude the analysis of (\[diff\]). \[lemma3\] With $T$ as defined in Lemma \[eps1-t\], we have $$\sup_{t\in D_r}|T(t;h_{1,n},h_{2,n})|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ Note that $$T(t;h_{1,n},h_{2,n})=\frac{1}{nh_{1,n}h_{2,n}}\sum_{i=1}^{n}(g(t,X_i)-Eg(t,X))$$ where $$\label{ge} g(t,x)=E_{X}\Big[f^{-1/2}(X)K\Big(\frac{X-x}{h_{1,n}}\Big) L\Big(\frac{t-X}{h_{2,n}}f^{1/2}(X)\Big)I(|t-X|\le h_{2,n}B)\Big].$$ By (\[entg\]) in the Appendix, the class of functions $\{g(t,\cdot):t\in D_r\}$ is of VC type for the envelope $\|L\|_V\|K\|_2h_{1,n}^{1/2}$ and the characteristics $A=R$ and $v=22$, and the lemma will follow by application of Talagrand’s inequality. We just need to estimate $Eg^2(t,X)$. We have, making several natural changes of variables, $$\begin{aligned} Eg^2(t,X_1)&=& E\Big\{\int f^{1/2}(x)K\Big(\frac{x-X_1}{h_{1,n}}\Big)L\Big(\frac{t-x}{h_{2,n}}f^{1/2}(x)\Big) I(|t-x|<h_{2,n}B)dx\nonumber\\ &&~~~~~~~~~~\times \int f^{1/2}(y)K\Big(\frac{y-X_1}{h_{1,n}}\Big)L\Big(\frac{t-y}{h_{2,n}}f^{1/2}(y) \Big)I(|t-y|<h_{2,n}B) dy\Big\}\nonumber\\ &=&h_{1,n}^2\int\int\int f^{1/2}(t-h_{1,n}v_1)L\Big(\frac{h_{1,n}}{h_{2,n}}v_1f^{1/2}(t-h_{1,n}v_1)\Big) f^{1/2}(t-h_{1,n}v_2)\nonumber\\ &&~~~~~~~~~~\times L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big) K\Big(\frac{t-u}{h_{1,n}}-v_1\Big)K \Big(\frac{t-u}{h_{1,n}}-v_2\Big)\nonumber\\ &&~~~~~~~~~~~~~~~\times I\Big(\frac{h_{1,n}}{h_{2,n}}|v_1|<B\Big) I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)f(u)dudv_1dv_2\nonumber\\ &=&h_{1,n}^3\int\int\int f^{1/2}(t-h_{1,n}v_1)L\Big(\frac{h_{1,n}}{h_{2,n}}v_1f^{1/2}(t-h_{1,n}v_1)\Big) f^{1/2}(t-h_{1,n}v_2)\nonumber\\ &&~~~~~~~~~~\times L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)K(v)K(v+v_1-v_2)\nonumber\\ &&~~~~~~~~~~~~~~~\times I\Big(\frac{h_{1,n}}{h_{2,n}}|v_1|<B\Big) I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)f(t-h_{1,n}v_1-h_{1,n}v)dvdv_1dv_2\nonumber\\ &\le& h_{1,n}^3||f||_\infty^2\int\int\int \left|L\Big(\frac{h_{1,n}}{h_{2,n}}v_1f^{1/2}(t-h_{1,n}v_1)\Big) L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)\right|\notag\\ &&~~~~~~~~~~\times K(v)K(v+v_1-v_2) I\Big(\frac{h_{1,n}}{h_{2,n}}|v_1|<B\Big) I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)dvdv_1dv_2\nonumber\\ &= &h_{1,n}^3||f||_\infty^2\int\int\int \left|L\Big(\frac{h_{1,n}}{h_{2,n}}(w+v_2)f^{1/2}(t-h_{1,n}w-h_{1,n}v_2)\Big) L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)\right|\nonumber\\ &&~~~~~~~~~~\times K(v)K(v+w)I\Big(\frac{h_{1,n}}{h_{2,n}}|w+v_2|<B\Big) I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)dvdwdv_2\nonumber\\ &=&h_{1,n}^2h_{2,n}||f||_\infty^2\int\int\int\left|L\Big((\frac{h_{1,n}}{h_{2,n}}w+z) f^{1/2}(t-h_{1,n}w-h_{2,n}z)\Big)\right|\nonumber\\ &&~~~~~~~~~~\times \left|L\Big(z f^{1/2}(t-h_{2,n}z)\Big)\right|K(v)K(v+w)I\Big(|\frac{h_{1,n}}{h_{2,n}}w+z|<B\Big) I\big(|z|<B\big)dvdwdz\nonumber\\ &\le& 2h_{1,n}^2h_{2,n}||f||_\infty^2 B(||K||_\infty+B\|f\|_\infty^{1/2}||K^\prime||_\infty)^2.\end{aligned}$$ So we can take $\sigma^2=c_2^2(1\vee\|f\|_\infty^3)h_{1,n}^2h_{2,n}$, where $c_2$ depends only on $K$. Since, as indicated above, the collection of functions $g(t,\cdot)$, $t\in D_r$, is VC for an envelope of the order $h_{1,n}$, Talagrand’s inequality (\[tal\]) implies that there exist finite positive constants $c_0,c_1$ such that, with $C_i$ as in (\[tal\]), if $$C_1(1\vee\|f\|_\infty^{3/2})\sqrt{n}h_{1,n}h_{2,n}^{1/2}\sqrt{\log \frac{c_0h_{1,n}^{1/2}}{c_2h_{1,n}h_{2,n}^{1/2}}}<u<C_2\frac{n(1\vee\|f\|_\infty^3)c_2^2h_{1,n}^2h_{2,n}}{c_1h_{1,n}^{1/2}}$$ then $${\Pr}_f\left\{\left\|\sum_{i=1}^n(g(t,X_i)-Eg(t,X))\right\|_{D_r}\ge u\right\}\le C_2\exp\left(-\frac{C_3u^2}{c_2^2(1\vee\|f\|_\infty^3)nh_{1,n}^2h_{2,n}}\right).$$ The condition on $u$ can be written as $$C'_1(1\vee\|f\|_\infty^{3/2}) n^{2/9}(\log n)^{5/9}<u<C_2'(1\vee\|f\|_\infty^3)n^{5/9}(\log n)^{1/9},$$ and if we take $u=M(1\vee\|f\|_\infty^{3/2})n^{2/9}(\log n)^{5/9}$ for some large enough $M$, then $$\sum \exp\left(-\frac{C_3u^2}{c_2^2(1\vee\|f\|_\infty^3)nh_{1,n}^2h_{2,n}}\right)= \sum e^{-M^2C_3(\log n)/c_2^2}<\infty$$ uniformly in $f$. Hence, $$\sum\sup_{f\in{\cal P}_C}{\Pr}_f\left\{\left\|\sum_{i=1}^n(g(t,X_i)-Eg(t,X))\right\|_{D_r}\ge M(1\vee C^{3/2})n^{2/9}(\log n)^{5/9}\right\}$$ $$\le\sum\sum e^{-M^2C_3(\log n)/c_2^2}<\infty$$ This shows that $T(t;h_{1,n}h_{2,n})$ is asymptotically a.s. of the order of $n^{2/9}(\log n)^{5/9}/(nh_{1,n}h_{2,n})=[(\log n)/n]^{4/9}$ uniformly in $f\in{\cal P}_C$. From (\[expans\]) and Lemmas \[lemma1\], \[eps1-t\] and \[lemma3\], we obtain: \[real-ideal\] Under the Assumptions \[ass3\], for any $C<\infty$ we have: $$\sup_{t\in D_r}\left|\hat f(t;h_{1,n},h_{2,n})-\bar f(t;h_{2,n})-T(t;h_{1;n},h_{2,n})\right|=o_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ and in particular, $$\sup_{t\in D_r}\left|\hat f(t;h_{1,n},h_{2,n})-\bar f(t;h_{2,n})\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ [ a) We should remark that if we undersmooth the preliminary estimator a little more, by taking $h_{1,n}=n^{-(2+\eta)/9}$ with $0<\eta< 2$, then the three lemmas above are true and moreover we have $\sup_{t\in D_r^\varepsilon}|\varepsilon_i(t,h_{1,n},h_{2,n})|=o_{\rm a.s.}(n^{-4/9})$ in Lemma \[lemma1\]. So, for such $h_{1,n}$ the order of the first term in Proposition \[real-ideal\] is actually $o_{\rm a.s.}\left(n^{-4/9}\right)$. This is at odds with condition (9) in Hall, Hu and Marron (1995), as their condition does not necessarily imply undersmoothing of the preliminary estimator. b) It is worth mentioning that Proposition \[real-ideal\] does require that the indicators $I(|t-X_i|\le h_{2,n}B)$ be part of the definition of (\[ideal0\]) and (\[realest0\]): in fact none of the three lemmas in its proof seem to go through without it. This condition is required as well for the bias of the ideal estimator, but it is not necessary for its variance part. ]{} Now we can complete the proof of the main theorems \[main0\] and \[mainu\]. Only the stronger Theorem \[mainu\] requires proof: [**Proof of Theorem \[mainu\].**]{} Proposition \[real-ideal\] and Theorem \[unifidealthm\] together give (\[main1’\]). The limit (\[main2’\]) can be easily derived from (\[main1’\]), as follows. By (\[classic1\]) and (\[classic2\]), the preliminary estimator satisfies $$\label{prelimunif} \sup_{t\in D_r}|\hat f(t;h_{1,n})-f(t)|=O_{\rm a.s. }\left(\frac{\sqrt{\log n}}{n^{7/18}} \right)\ \ {\rm uniformity\ in}\ {\cal D}_{C,z}$$ for all $C<\infty$, $z$ and $r$. Now, for all $n$ large enough, on the event $$\left\{\sup_{n\ge k}\frac{n^{7/18}}{\sqrt{\log n}}||\hat f(t;h_{1,n})-f(t)||_\infty\le \lambda_1\right\}$$ we have $\hat D_r^n\subset D_r$ for all $n\ge k$, and therefore, $$\begin{aligned} &&{\Pr}_f\left\{\sup_{n\ge k}\big(\frac{ n}{\log n}\big)^{4/9}\|\hat f(t;h_{1,n}, h_{2,n},\omega)-f(t)\|_{\hat D_r^n}>\lambda_2\right\}\\ &&\le{\Pr}_f\left\{\sup_{n\ge k}\big(\frac{ n}{\log n}\big)^{4/9}\|\hat f(t;h_{1,n}, h_{2,n},\omega)-f(t)\|_{ D_r}>\lambda_2\right\}\\ &&~~~~~~~~~~+{\Pr}_f\left\{\sup_{n\ge k}\frac{n^{7/18}}{\sqrt{\log n}}||\hat f(t;h_{1,n})-f(t)||_\infty> \lambda_1\right\}.\end{aligned}$$ Now, there exist $\lambda_1$ and $\lambda_2$ such that the limit of the sup over ${\cal D}_{C,z}$ of the first probabilities is zero by (\[main1’\]), and the limit of the sup of the second ones over the same set is also zero by (\[prelimunif\]), proving (\[main2’\]). $\blacksquare$ Appendix: Some Vapnik-Červonenkis classes of functions and their exponential bounds =================================================================================== Let $\cal F$ be a collection of uniformly bounded measurable functions on $(S,{\cal S})$. We say that $\cal F$ is of VC type with respect to an envelope $F$ if there exist constants $A$, $v$ positive such that for all probability measures $Q$ on $\cal S$, $$N({\cal F}, L_2(Q),\varepsilon)\le\left(\frac{A\|F\|_{L_2(Q)}}{\varepsilon}\right)^v,\ \ 0<\varepsilon<1,$$ where $F\ge |f|$ for all $f\in \cal F$ and $N({\cal F},L_2(Q),\varepsilon)$ denotes the smallest number of $L_2(Q)$-balls of radius at most $\varepsilon$ required to cover $\cal F$. (See e.g., de la Pena and Giné (1999).) It turns out that empirical processes or $U$-processes indexed by these classes of functions are very well behaved, particularly if $F$ is uniformly bounded and if the class $\cal F$ is countable. For instance, we have the following version of an exponential inequality of Talagrand (1996) from Einmahl and Mason (2000) and Giné and Guillou (2001, 2002). Let $P$ be a probability measure on $S$ and let $X_i:S^{\mathbb N}\mapsto S$ be the coordinate functions of $S^{\mathbb N}$, which are i.i.d. (P), and set $\Pr=P^{\mathbb N}$. If the class $\cal F$ is VC type, bounded and countable, then there exist $0<C_i<\infty$, $1\le i\le 3$, depending on $v$ and $A$ such that, for all $t$ satisfying $$C_1\sqrt{n}\sigma\sqrt{\log\frac{2\|F\|_\infty}{\sigma}}\le t\le \frac{n\sigma^2}{\|F\|_\infty},$$ we have $$\label{tal} \Pr\left\{\max_{1\le k\le n}\left\|\sum_{i=1}^k(f(X_i)-Pf)\right\|_{\cal F}>t\right\}\le C_2\exp\left(-C_3\frac{t^2}{n\sigma^2}\right),$$ where $$\|F\|_\infty\ge \sigma^2\ge\|{\rm Var}_P(f)\|_{\cal F}.$$ (Talagrand (1996) states his inequality only for the sum over $n$, but the same works for the maximum of the partial sums up to $n$ by a (sub)martingale argument that can be carried out because these inequalities are obtained by integrating bounds on the moment generating function -see e.g., Ledoux (2001).) Major (2006) also has a similar inequality for classes of functions of several variables. We will state his inequality for bounded VC type classes of functions of two variables only. Let $\cal F$ be such a class of functions and let $\|F\|_\infty^2\ge \sigma^2\ge \|{\rm Var}(f(X_1,X_2))\|_{\cal F}$. Let $\pi_2^P(f)(x,y)=f(x,y)-Ef(X,y)-Ef(x,X)+Ef(X,Y)$. Then, if $\cal F$ is a uniformly bounded, countable class of VC type, there exist $0<C_i<\infty$, $1\le i\le 3$, depending on $v$ and $A$ such that, for all $t$ satisfying $$C_1n\sigma\log\frac{2\|F\|_\infty}{\sigma}\le t\le \frac{n^2\sigma^3}{\|F\|_\infty^2}$$ we have $$\label{major} \Pr\left\{\left\|\sum\sum_{1\le i\ne j\le n}\pi_2^Pf(X_i,X_j)\right\|_{\cal F}>t\right\}\le C_2\exp\left(-C_3\frac{t}{n\sigma}\right).$$ Major states the theorem for $\{\pi_2^Pf\}$ of VC type, but it is easy to see that if $\cal F$ is VC type for $F$ then $\{\pi_2^Pf:f\in {\cal F}\}$ is VC type for the envelope $4F$. It is also worth mentioning that (much easier to prove) moment bounds for the above quantities are also available (e.g. in Giné and Mason (2007) and references therein) and that they can be used instead of Talagrand and Major’s inequalities if one is only interested in the ‘in probability’ version of Theorems \[main0\] and \[mainu\]. We now show that the classes of functions appearing in the previous sections are of VC type, and the suprema countable. We will do this in all detail for the class $\cal F$ in (\[entr0\]), and will give indications for the rest of the classes of functions used. First we observe that the sup inside the probability bound in (\[eq1\]) is actually a supremum over the set $\{t\in\mathbb Q, h\in \mathbb Q\cap[h_{2^k},h_{2^{k-1}})\}$ by the continuity properties of $K$ and the indicator of $|t-X_i|<h B$. This observation applies to all the other classes of functions in the previous two sections. \[vc1\] Let $\cal F$ be as in (\[entr0\]) with $K$ and $f$ satisfying Assumptions \[ass1\]. Then, there exists a universal constant $R$ such that for every Borel probability measure $Q$ on $\mathbb R$, $$\label{ivc1} N({\cal F},L_2(Q),\varepsilon)\le\left(\frac{R\|K\|_V\|f\|_\infty^{1/2}}{\varepsilon}\right)^{22}$$ where $\|K\|_V$ is the total variation norm of $K$, that is, $\cal F$ is of VC type with envelope $\|K\|_V\|f\|_\infty^{1/2}$ with $A=R$ independent of $K$ and $f$ and $v=22$. By adding an arbitrarily small strictly increasing function to the positive and negative variation functions of $K$, we have $K=K_1-K_2$ with $K_i$ strictly increasing, positive and bounded, with $\|K_1\|_\infty$ ($\|K_2\|_\infty$) arbitrarily close to the positive (negative) variation of $K$. Let ${\cal K}_1$ be the class of functions obtained from $\cal F$ by replacing $K$ by $K_1$ and deleting the indicator in each of the functions in the class. Then, if we assume $f(x)>0$ for all $x$, the subgraphs of the functions in the class ${\cal K}_1$ have the form $$\left\{(x,u):K_{1}\left(\frac{t-x}{h}f^{1/2}(x)\right)f^{1/2}(x) \ge u \right\} =\left\{(x,u):\frac{t-x}{h}f^{1/2}(x)\ge K_{1}^{-1}(u/f^{1/2}(x))\right\}$$ $$=\left\{(x,u):\frac{tf^{1/2}(x)}{h}-\frac{xf^{1/2}(x)}{h}-K_{1}^{-1}(u/f^{1/2}(x)) \ge0\right\},$$ and so they are the positivity sets of functions from the linear space of functions of the two variables $u$ and $x$ spanned by $f^{1/2}(x)$, $xf^{1/2}(x)$ and $K_1^{-1}(u/f^{1/2}(x))$. Hence, by a result of Dudley (e.g. Proposition 5.1.12 in de la Peña and Giné (1999)) the subgraphs of ${\cal K}_1$ are VC of index 4. If the set $\{x:f(x)= 0\}$ is not empty, the same argument above shows that the class of subsets of $S=\{x:f(x)>0\}\times\mathbb R$, $\left\{(x,u)\in S:K_{1}\left(\frac{t-x}{h}f^{1/2}(x)\right)f^{1/2}(x) \ge u \right\}$ is VC of index 4, and therefore so is the class of subgraphs of ${\cal K}|_1$, which is obtained from this one by taking the union of each of these sets with the set $\{x:f(x)=0\}\times\{u\le 0\}$ (which is disjoint with all of them). Therefore, in either case, by the Dudley-Pollard entropy theorem for VC-subgraph classes (e.g., loc. cit. Theorem 5.1.5), we have $$N({\cal K}_1, L_2(P), \varepsilon)\le \left(\frac{A\|K_1\|_\infty\|f\|_\infty^{1/2}}{\varepsilon}\right)^8,\ \ 0<\varepsilon\le \|K_1\|_\infty\|f\|_\infty^{1/2}$$ where $A$ is a universal constant, hence, $$\label{vc2} N({\cal K}_1, L_2(P), \varepsilon)\le \left(\frac{A\|K\|_+\|f\|_\infty^{1/2}}{\varepsilon}\right)^8,\ \ 0<\varepsilon\le \|K\|_+\|f\|_\infty^{1/2}$$ where $\|K\|_+$ is the positive variation seminorm of $K$. The analogous bound holds for ${\cal K}_2$, defined with $K_2$ replacing $K$ in $\cal F$. Since, as is well known, the set ${\cal J}$ of all indicator functions of intervals in $\mathbb R$ is $VC$ of order 3, we also have $$\label{vc3} N({\cal J}, L_2(P),\varepsilon)\le\left(\frac{\bar A }{\varepsilon}\right)^6, \ \ 0<\varepsilon\le 1.$$ for another universal constant $\bar A$. Now, any $H\in\cal F$ can be written as $H=k_1g-k_2g$ for $k_i\in{\cal K}_i$ and $g\in{\cal J}$, so that, for any probability measure $Q$ we have $$\begin{aligned} Q(H-\bar H)^2&=&Q((k_1-k_2)g-(\bar k_1-\bar k_2)\bar g)^2\\ &\le&4Q(k_1-\bar k_1)^2+4Q(k_2-\bar k_2)^2+2\|K\|_V^2\|f\|_\infty Q(g-\bar g)^2.\end{aligned}$$ Given $\varepsilon>0$ let $\delta_1=\varepsilon/4$ and $\delta_2=\varepsilon/(2\|K\|_V\|f\|^{1/2})$. Then, if the collections of functions $k_1^{(1)},\dots, k_{N_1}^{(1)}$ and $k_1^{(2)},\dots, k_{N_2}^{(2)}$ are $L_2(Q)$ $\delta_1$-dense respectively in the classes ${\cal K}_1$, ${\cal K}_2$, and $g_1,\dots,g_{N_3}$ are $L_1(Q)$ $\delta_2$-dense in the class $\cal J$, with optimal cardinalities $N_i=N({\cal K}_i, L_2(Q),\delta_1)$, $i=1,2$, and $N_3=N({\cal J},L_2(Q),\delta_2)$, then, by the previous inequality, the functions $(k_i^{(1)}-k_j^{(2)})g_l$ are $L_2(Q)$ $\varepsilon$-dense in $\cal F$ . Since there are at most $N_1N_2N_3$ such functions (this estimate may not be optimal), the inequality (\[ivc1\]) follows. A similar result holds for the classes ${\cal Q}_n$ defined by (\[qu\]) in the proof of Lemma \[lemma1\], the classes of functions $\bar{\cal Q}_n$ defined by (\[qubar\]) and the classes $\{H_t(x,y):t\in D_r\}$ in the proof of Lemma \[eps1-t\], as all these classes have the same structure as $\cal F$ in Lemma \[vc1\]. The class of functions ${\cal G}:=\{g(t,\cdot):t\in D_r\}$ where $g$ is defined in (\[ge\]) in the proof of Lemma \[lemma3\], requires some extra considerations. Let $Q$ be any probability measure on the line and let $s,t\in D_r$. Then, using Hölder, we have $$E_Q(g(t,x)-g(s,x))^2\le \int E_X\left(f(X)^{-1}K^2\left(\frac{X-x}{h_{1,n}}\right)\right)\times$$ $$\times E_X\left(L\Big(\frac{t-X}{h_{2,n}}f^{1/2}(X)\Big) I(|t-X|<h_{2,n}B)-L\Big(\frac{s-X}{h_{2,n}}f^{1/2}(X)\Big) I(|s-X|<h_{2,n}B)\right)^2dQ(x)$$ $$= h_{1,n}\|K\|_2^2\int \left(L\Big(\frac{t-y}{h_{2,n}}f^{1/2}(y)\Big) I(|t-y|<h_{2,n}B)-L\Big(\frac{s-y}{h_{2,n}}f^{1/2}(y)\Big) I(|s-y|<h_{2,n}B)\right)^2f(y)dy$$ $$=h_{1,n}\|K\|_2^2E_f(\ell_t-\ell_s)^2$$ where $\ell_{s}$ and $\ell_{t}$ are functions from the class $${\cal L}:=\left\{L\left(\frac{t- \cdot}{h}f^{1/2}(\cdot)\right)I( |t-\cdot|<hB):t\in\mathbb{R}, h>0 \right\}$$ which is VC with a constant envelope by Lemma \[vc1\]. This lemma then proves that for all $Q$, $$\label{entg} N({\cal G}, L_2(Q),\varepsilon)\le\left(\frac{R\|L\|_V\|K\|_2h_{1,n}^{1/2}}{\varepsilon}\right)^{22},\ \ 0<\varepsilon<\|L\|_V\|K\|_2h_{1,n}^{1/2},$$ in particular, $\cal G$ is VC for the constant envelope $\|L\|_V\|K\|_2h_{1,n}^{1/2}$, with characteristics $A=R$ and $v=22$. [**Acknowledgement.**]{} We thank Richard Nickl for several useful conversations on the subject of this article. [99]{} I. Abramson, [*On bandwidth variation in kernel estimates - a square-root law*]{}, Ann. Statist. 10 (1982), pp. 1217-1223. V. de la Peña and E. Giné, [*Decoupling, from Dependence to Independence*]{}, Springer-Verlag, New York, (1999). P. Deheuvels, [/it Uniform limit laws for kernel density estimators on possible unbounded intervals]{}, In: N. Limnios, M. Nikulin (Eds.), Recent Advances in Reliability Theory: Methodology, Practice and Inference, Birkhauser, Boston, (2000), pp. 477-492. D.L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, [*Wavelet Shrinkage: Asymptopia?*]{}, J. R. Statist. Soc. Ser. B (1995) 57, pp. 301-369. D.L. Donoho, I.M. Johnstone, G. Kerkyacharian, and D. Picard, [*Density estimation by wavelet thresholding*]{}, Ann. Statist. 24 (1996), pp. 508-539. R.M. Dudley, E. Giné, and J. Zinn, [*Uniform and universal Glivenko-Cantelli classes*]{}, J. Theoret. Probab. 4 (1991), pp. 485-510. S. Efromovich, [*Nonparametric curve estimation: Methods, theory, and applications*]{}, Springer-Verlag, New York, (1999). U. Einmahl and D. M. Mason, [*An empirical process approach to the uniform consistency of kernel-type function estimators*]{}, J. Theoret. Probab. 13 (2000), pp. 1-37. E. Giné and A. Guillou, [*On consistency of kernel density estimators for randomly censored data: rates holding uniformly over adaptive intervals*]{}, Ann. I. H. Poincaré-Pr 37 (2001), pp.503-522. E. Giné and A. Guillou, [*Rates of strong uniform consistency for multivariate kernel density estimators*]{}, Ann. I. H. Poincaré-Pr 38 (2002), pp. 907-921. E. Giné and D.M. Mason, [*Laws of the iterated logarithm for the local U-statistic process*]{}, J. Theoret. Probab. 20 (2007), pp. 457–485. P. Hall, [*On the bias of variable bandwidth kernel estimators*]{}, Biometrika 77 (1990), pp. 529-535. E. Giné and R. Nickl, [*An exponential inequality for the distribution function of the kernel density estimator with applications to adaptive estimation*]{}, Probab. Theory Rel. Fields 143 (2009), pp. 569-596. E. Giné and R. Nickl, [*Uniform limit theorems for wavelet density estimators*]{}, Ann. Probab., to appear, (2008). P. Hall and J. S. Marron, [*Variable Window Width Kernel Estimates of Probability Densities*]{}, Probab. Th. Rel. Fields 80 (1988), pp. 37-49. Erratum: Probab. Th. Rel. Fields 91, pp. 133. P. Hall, T. Hu, and J. S. Marron, [*Improved Variable Window Kernel Estimates of Probability Densities*]{}, Ann. Statist. 23 (1995), pp.1-10. M. Ledoux, [*The concentration of measure phenomenon*]{}, Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI. P. Major, [*An estimate on the supremum of a nice class of stochastic integrals and $U$-statistics*]{}, Probab. Theory Related Fields 134 (2006), pp. 489–537. D.M. Mason and J. Swanepoel, [*A general result on the uniform in bandwidth consistency of kernel-type function estimators*]{}, Preprint (2008). I. J. McKay, [*A note on bias reduction in variable kernel density estimates*]{}, Canad. J. Statist. 21 (1993), pp. 367-375. S. Novak, [*Generalized kernel density estimator*]{}, Theory Probab. Appl. 44 (1999), pp. 570-583. H. Sang, [*Asymptotic properties of generalized kernel density estimators*]{}, Ph.D. Dissertation, University of Connecticut. B. Silverman, [*Weak and strong uniform consistency of the kernel estimate of a density and its derivatives*]{}, Ann. Statist. 6 (1978), pp. 177-184. M. Talagrand, [*New concentration inequalities in product spaces*]{},Invent. Math. 126 (1996), pp. 505-563. G. R. Terrell and D. Scott, [*Variable kernel density estimation*]{}, Ann. Statist. 20 (1992), pp. 1236-1265. E. GinéDepartment of Mathematics, U-3009University of ConnecticutStorrs, CT 06269gine@math.uconn.edu H. SangDepartment of MathematicsUniversity of CincinnatiCincinnati, OH 45221 sanghn@ucmail.uc.edu [^1]: $^a$Corresponding author. Email: gine@math.uconn.edu

--- abstract: 'We show two-sided bounds between the traditional quantum Rényi divergences and the new notion of Rényi divergences introduced recently in Müller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. **54**, 122203, (2013), and Wilde, Winter, Yang, arXiv:1306.1586. The bounds imply that the two versions can be used interchangeably near $\alpha=1$, and hence one can benefit from the best properties of both when proving coding theorems in the case of asymptotically vanishing error. We illustrate this by giving short and simple proofs of the quantum Stein’s lemma with composite null-hypothesis, universal source compression, and the achievability part of the classical capacity of compound quantum channels. Apart from the above interchangeability, we benefit from a weak quasi-concavity property of the new Rényi divergences that we also establish here.' --- Milán Mosonyi Física Teòrica: Informació i Fenomens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain. Mathematical Institute, Budapest University of Technology and Economics\ Egry József u 1., Budapest, 1111 Hungary Introduction ============ Rényi introduced a generalization of the Kullback-Leibler divergence (relative entropy) in [@Renyi]. According to his definition, the $\alpha$-divergence of two probability distributions $p$ and $q$ on a finite set $\X$ for a parameter $\alpha\in[0,+\infty)\setminus\{1\}$ is given by $$\begin{aligned} \label{Renyi def} {D_{\alpha}\bz p\|q\jz}:= \frac{1}{\alpha-1}\log\sum_{x\in\X}p(x)^{\alpha}q(x)^{1-\alpha}.\end{aligned}$$ The limit $\alpha\to 1$ yields the standard relative entropy. These quantities turned out to play a central role in information theory and statistics; indeed, the Rényi divergences quantify the trade-off between the exponents of the relevant quantities in many information-theoretic tasks, including hypothesis testing, source coding and noisy channel coding; see, e.g. [@Csiszar] for an overview of these results. It was also shown in [@Csiszar] that the Rényi relative entropies, and other related quantities, like the Rényi entropies and the Rényi capacities, have direct operational interpretations as so-called generalized cutoff rates in the corresponding information-theoretic tasks. In quantum theory, the state of a system is described by a density operator instead of a probability distribution, and the definition can be extended for pairs of density operators in various inequivalent ways, due to the non-commutativity of operators. The traditional way to define the Rényi divergence of two density operators is $$\label{old Renyi} {D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}:= \frac{1}{\alpha-1}\log\operatorname{Tr}\rho^{\alpha}\sigma^{1-\alpha}.$$ It has been shown in [@MH] that, similarly to the classical case, the Rényi $\alpha$-divergences $D_{\alpha}^{\mathrm{(old)}}$ with $\alpha\in(0,1)$ have a direct operational interpretation as generalized cutoff rates in the so-called direct domain of binary state discrimination. This is a consequence of another, indirect, operational interpretation in the setting of the quantum Hoeffding bound [@ANSzV; @Hayashi; @HMO2; @Nagaoka]. Recently, a new quantum extension of the Rényi $\alpha$-divergences has been proposed in [@Renyi_new; @WWY], defined as $$\label{new Renyi def} {D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz}:= \frac{1}{\alpha-1}\log\operatorname{Tr}\bz\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\jz^{\alpha}.$$ This definition was introduced in [@Renyi_new] as a parametric family that connects the min- and max-relative entropies [@Datta; @RennerPhD] and Umegaki’s relative entropy [@Umegaki]. In [@WWY], the corresponding generalized Holevo capacities were used to establish the strong converse property for the classical capacities of entanglement-breaking and Hadamard channels. It was shown in [@MO] that these new Rényi divergences play the same role in the (strong) converse problem of binary state discrimination as the traditional Rényi divergences in the direct problem. In particular, the strong converse exponent was expressed as a function of the new Rényi divergences, and from that a direct operational interpretation was derived for them as generalized cutoff rates in the sense of [@Csiszar]. The above results suggest that, somewhat surprisingly, one should use two different quantum extensions of the classical Rényi divergences: for the direct part, corresponding to $\alpha\in(0,1)$, the “right” definition is the one given in , while for the converse part, corresponding to $\alpha>1$, the “right” definition is the one in . Although coding theorems supporting this separation have only been shown for binary state discrimination so far, it seems reasonable to expect the same separation in the case of other information-theoretic tasks. We remark that, in line with this expectation, lower bounds on the classical capacity of quantum channels can be obtained in terms of the traditional Rényi divergences [@MD], while upper bounds were found in terms of the new Rényi divergences in [@WWY]. On the other hand, the above two quantum Rényi divergences have different mathematical properties, which might make them better or worse suited for certain mathematical manipulations, and therefore it might be beneficial to use the new Rényi divergences in the direct part of coding problems, and the traditional ones in converse parts, despite the “real” quantities being the opposite. The problem that one faces then is how to arrive back to the natural quantity of the given problem. As it turns out, this is possible, at least if one’s aim is to study the case of asymptotically vanishing error, corresponding to $\alpha\to 1$; this is thanks to the well-known Araki-Lieb-Thirring inequality, and its complement due to Audenaert [@Aud-ALT]. We explain this in detail in Section \[sec:ALT\]. Convexity properties of these divergences are of particular importance for applications. As it was shown in [@FL; @WWY], both versions of the Rényi divergences are jointly quasi-convex around $\alpha=1$. In Section \[sec:convexity\] we show a certain converse to this quasi-convexity in the form of a weak partial quasi-concavity (Corollary \[cor:new renyi superadd\] and Proposition \[prop:old Renyi quasiconcavity\]), which is still strong enough to be useful for applications, as we illustrate on various examples in Section \[sec:applications\]. Coding theorems for the problems considered in Section \[sec:applications\] have been established in [@BDKSSSz; @Notzel] for Stein’s lemma with composite null-hypothesis, in [@JHHH] for universal source compression, and in [@BB; @DD] for the classical capacity of compound and averaged channels. Here we provide alternative proofs for these coding theorems, using the following general approach: 1. We take a single-shot coding theorem that bounds the relevant error probability in terms of a Rényi divergence. In the case of Stein’s lemma and source compression, this is Audenaert’s inequality [@Aud], while in the case of channel coding, we use the random coding theorem due to Hayashi and Nagaoka [@HN]. The bounds are given in terms of $D_{\alpha}\old$. 2. We use lemma \[lemma:old-new bounds\] to switch from the old to the new Rényi divergences in the upper bound to the error probability, and then we use the weak partial quasi-concavity properties of the Rényi divergences, given in Corollary \[cor:new renyi superadd\] and Proposition \[prop:old Renyi quasiconcavity\], to decouple the upper bound into a sum of individual Rényi divergences. 3. If necessary, we use again lemma \[lemma:old-new bounds\] to return to $D_{\alpha}\old$ in the upper bound. 4. We use the additivity of the relevant Rényi quantities (divergences, entropies, generalized Holevo quantities) to obtain the asymptotics. The advantage the above approach is that it only uses very general arguments that are largely independent of the concrete model in consideration. Once the single-shot coding theorems are available, the coding theorems for the composite cases follow essentially by the same amount of effort as for the simple cases (simple null-hypothesis, single source, single channel), using only very general properties of the Rényi divergences. This makes the proofs considerably shorter and simpler than e.g., in [@BDKSSSz; @BB; @DD]. Moreover, this approach is very easy to generalize to non-i.i.d. compound problems, unlike the methods of [@JHHH; @Notzel], which are based on the method of types. Notations {#sec:notations} ========= For a finite-dimensional Hilbert space $\hil$, let $\B(\hil)_+$ denote the set of all non-zero positive semidefinite operators on $\hil$, and let $\S(\hil):=\{\rho\in\B(\hil)_+\,;\,\operatorname{Tr}\rho=1\}$ be the set of all density operators (states) on $\hil$. We define the powers of a positive semidefinite operator $A$ only on its support; that is, if $\lambda_1,\ldots,\lambda_r$ are the strictly positive eigenvalues of $A$, with corresponding spectral projections $P_1,\ldots,P_r$, then we define $A^{\alpha}:=\sum_{i=1}^r \lambda_i^{\alpha}P_i$ for all $\alpha\in\bR$. In particular, $A^0=\sum_{i=1}^rP_i$ is the projection onto the support of $A$. We will use the convention $\log 0:=-\infty$ and $\log +\infty:=+\infty$. Rényi divergences {#sec:Renyi} ================= Two definitions {#sec:ALT} --------------- For non-zero positive semidefinite operators $\rho,\sigma$, the [@Renyi] of $\rho$ w.r.t. $\sigma$ with parameter $\alpha\in(0,+\infty)\setminus\{1\}$ is traditionally defined as $$\begin{aligned} {D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}&:= \begin{cases} \frac{1}{\alpha-1}\log\operatorname{Tr}\rho^{\alpha}\sigma^{1-\alpha}-\frac{1}{\alpha-1}\log\operatorname{Tr}\rho, & \alpha\in(0,1){\mbox{ }\mbox{ }}\text{or}{\mbox{ }\mbox{ }}\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma,\\ +\infty,&\text{otherwise}. \end{cases}\end{aligned}$$ For the mathematical properties of $D_{\alpha}\old$, see, e.g. [@Lieb; @MH; @Petz]. Recently, a new notion of Rényi divergence has been introduced in [@Renyi_new; @WWY], defined as $$\begin{aligned} {D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz}&:= \begin{cases} \frac{1}{\alpha-1}\log\operatorname{Tr}\bz\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\jz^{\alpha} -\frac{1}{\alpha-1}\log\operatorname{Tr}\rho, & \alpha\in(0,1){\mbox{ }\mbox{ }}\text{or}{\mbox{ }\mbox{ }}\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma,\\ +\infty,&\text{otherwise}. \end{cases}\end{aligned}$$ For the mathematical properties of $D_{\alpha}\nw$, see, e.g. [@Beigi; @FL; @MO; @Renyi_new; @WWY]. It is easy to see that for non-zero $\rho$, we have $\lim_{\sigma\to 0}{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}=\lim_{\sigma\to 0}{D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz}=+\infty$, and hence we define ${D^{\mathrm{(old)}}_{\alpha}\bz \rho\|0\jz}:={D^{\mathrm{(new)}}_{\alpha}\bz \rho\|0\jz}:=+\infty$ when $\rho\ne 0$. On the other hand, for non-zero $\sigma$, the limits $\lim_{\rho\to 0}{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}$ and $\lim_{\rho\to 0}{D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz}$ don’t exist, and hence we don’t define the values of ${D^{\mathrm{(old)}}_{\alpha}\bz 0\|\sigma\jz}$ and ${D^{\mathrm{(new)}}_{\alpha}\bz 0\|\sigma\jz}$. To see the latter, one can consider $\rho_n:=\frac{1}{n}{{|0\rangle\langle 0|}}+\frac{1}{n^{\beta}}{{|1\rangle\langle 1|}}$, and $\sigma:={{|1\rangle\langle 1|}}$, where ${{|0\rangle\langle 0|}}$ and ${{|1\rangle\langle 1|}}$ are orthogonal rank $1$ projections. It is easy to see that for $\alpha<1$, $\lim_{n\to +\infty}{D^{\mathrm{(old)}}_{\alpha}\bz \rho_n\|\sigma\jz}= \lim_{n\to +\infty}{D^{\mathrm{(new)}}_{\alpha}\bz \rho_n\|\sigma\jz}= \lim_{n\to +\infty}\frac{1}{\alpha-1}\log\frac{n^{1-\beta\alpha}}{1+n^{1-\beta}}$ depends on the value of $\beta$. A similar example can be used for $\alpha>1$. \[rem:limits\] Note that the definition of $D_{\alpha}\old$ makes sense also for $\alpha=0$, and we get ${D_{0}\bz \rho\|\sigma\jz}=-\log\operatorname{Tr}\rho^0\sigma$. It is easy to see that if $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$ then $${D^{\mathrm{(old)}}_{\infty}\bz \rho\|\sigma\jz}:=\lim_{\alpha\to+\infty}{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}= \max\{r/s:\,\operatorname{Tr}P_{\rho}(\{r\})P_{\sigma}(\{s\})>0\},$$ where $P_{\rho}(\{r\})$ and $P_{\sigma}(\{s\})$ are the spectral projections of $\rho$ and $\sigma$ corresponding to $r$ and $s$, respectively. If $\operatorname{supp}\rho\nsubseteq\operatorname{supp}\sigma$ then obviously ${D^{\mathrm{(old)}}_{\infty}\bz \rho\|\sigma\jz}=+\infty$. In the case of $D_{\alpha}\nw$, it was shown in [@Renyi_new] that $${D^{\mathrm{(new)}}_{\infty}\bz \rho\|\sigma\jz}:=\lim_{\alpha\to+\infty}{D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz}= {D_{\mathrm{max}}\bz \rho\|\sigma\jz}:=\log\inf\{\gamma:\,\rho\le\gamma\sigma\},$$ where $D_{\max}$ is the max-relative entropy [@Datta; @RennerPhD]. The limit ${D^{\mathrm{(new)}}_{0}\bz \rho\|\sigma\jz}:=\lim_{\alpha\to 0}D_{\alpha}\nw(\rho\|\sigma)$ is in general different from $D_0\old(\rho\|\sigma)$; see, e.g., [@AD; @DL]. According to the Araki-Lieb-Thirring inequality [@Araki; @LT], for any positive semidefinite operators $A,B$, $$\label{ALT} \operatorname{Tr}A^{\alpha}B^{\alpha}A^{\alpha}\le\operatorname{Tr}(ABA)^{\alpha}$$ for $\alpha\in(0,1)$, and the inequality holds in the converse direction for $\alpha>1$. A converse to the Araki-Lieb-Thirring inequality was given in [@Aud-ALT], where it was shown that $$\label{converse ALT1} \operatorname{Tr}(ABA)^{\alpha}\le \bz{\left\| B\right\|}^{\alpha}\operatorname{Tr}A^{2\alpha}\jz^{1-\alpha} \bz\operatorname{Tr}A^{\alpha}B^{\alpha}A^{\alpha}\jz^{\alpha}$$ for $\alpha\in(0,1)$, and the inequality holds in the converse direction for $\alpha>1$. Applying and to $A:=\rho^{\half}$ and $B:=\sigma^{\frac{1-\alpha}{\alpha}}$, we get $$\begin{aligned} \label{old-new bounds0} \operatorname{Tr}\rho^{\alpha}\sigma^{1-\alpha} \le \operatorname{Tr}\bz\rho^{\half}\sigma^{\frac{1-\alpha}{\alpha}}\rho^{\half}\jz^{\alpha} \le {\left\| \sigma\right\|}^{(1-\alpha)^2}\bz\operatorname{Tr}\rho^{\alpha}\jz^{1-\alpha}\bz\operatorname{Tr}\rho^{\alpha}\sigma^{1-\alpha}\jz^{\alpha}\end{aligned}$$ for $\alpha\in(0,1)$, and the inequalities hold in the converse direction for $\alpha>1$. This immediately yields the following: \[lemma:old-new bounds\] For any $\rho,\sigma\in\B(\hil)_+$ and $\alpha\in[0,+\infty)\setminus\{1\}$, $$\begin{aligned} \label{old-new bounds} {D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz} \ge {D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz} \ge \alpha{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}+\log\operatorname{Tr}\rho -\log\operatorname{Tr}\rho^{\alpha}+(\alpha-1)\log{\left\| \sigma\right\|}.\end{aligned}$$ The first inequality in has already been noted in [@WWY] for $\alpha>1$. It is straightforward to verify that $D_{\alpha}\old$ yields Umegaki’s relative entropy in the limit $\alpha\to 1$; i.e., for any $\rho,\sigma\in\B(\hil)_+$, $$\begin{aligned} \label{1 limit} {D_{1}\bz \rho\|\sigma\jz}&:=\lim_{\alpha\to 1}{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz} = \begin{cases} \frac{1}{\operatorname{Tr}\rho}\operatorname{Tr}\rho(\log\rho-\log\sigma),&\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma,\\ +\infty,&\text{otherwise}. \end{cases}\end{aligned}$$ This, together with lemma \[lemma:old-new bounds\], yields immediately the following: \[cor:1 limit\] For any two non-zero positive semidefinite operator $\rho,\sigma$, $$\begin{aligned} \label{1 limit new} \lim_{\alpha\to 1}{D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz}= {D_{1}\bz \rho\|\sigma\jz}.\end{aligned}$$ Taking into account – and Remark \[rem:limits\], we finally have the definitions of $D_{\alpha}\old$ and $D_{\alpha}\nw$ for every parameter value $\alpha\in[0,+\infty]$. The limit relation has been shown in [@Renyi_new], and in [@WWY] for $\alpha\searrow 1$, by explicitly computing the derivative of $\alpha\mapsto\log\operatorname{Tr}\bz\rho^{\half}\sigma^{\frac{1-\alpha}{\alpha}}\rho^{\half}\jz^{\alpha}$ at $\alpha=1$. It is easy to see (by computing its second derivative) that $\psi\old(\alpha):=\log\operatorname{Tr}\rho^{\alpha}\sigma^{1-\alpha}$ is a convex function of $\alpha$, which yields immediately that ${D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}$ is a monotonic increasing function of $\alpha$ for any fixed $\rho$ and $\sigma$. The following Proposition, due to [@TCR] and [@Tomamichel], complements this monotonicity property around $\alpha=1$, and in the same time gives a quantitative version of : \[prop:TCR\] Let $\rho,\sigma\in\B(\hil)_+$ be such that $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$, let $\eta:=1+\operatorname{Tr}\rho^{3/2}\sigma^{-1/2}+\operatorname{Tr}\rho^{1/2}\sigma^{1/2}$, let $c>0$, and $\delta:=\min\left\{\half, \frac{c}{2\log\eta}\right\}$. Then $$\begin{aligned} {D_{1}\bz \rho\|\sigma\jz}&\ge{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz} \ge {D_{1}\bz \rho\|\sigma\jz}-4(1-\alpha)(\log\eta)^2\cosh c,{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}1-\delta<\alpha<1,\\ {D_{1}\bz \rho\|\sigma\jz}&\le{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz} \le {D_{1}\bz \rho\|\sigma\jz}-4(1-\alpha)(\log\eta)^2\cosh c,{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}1<\alpha<1+\delta.\end{aligned}$$ The new Rényi divergences $D_{\alpha}\nw(\rho\|\sigma)$ are also monotonic increasing in $\alpha$, as was shown in in Theorem 6 of [@Renyi_new] (see also [@MO] for a different proof for the case $\alpha>1$). Combining Proposition \[prop:TCR\] with lemma \[lemma:old-new bounds\], we obtain the following: \[cor:TCR\] In the setting of Proposition \[prop:TCR\], we have $$\begin{aligned} {D_{1}\bz \rho\|\sigma\jz}\ge{D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz} \ge& \alpha{D_{1}\bz \rho\|\sigma\jz}-4\alpha(1-\alpha)(\log\eta)^2\cosh c\\ &+\log\operatorname{Tr}\rho -\log\operatorname{Tr}\rho^{\alpha}+(1-\alpha)\log{\left\| \sigma\right\|}\inv ,& 1-\delta<\alpha<1,\\ {D_{1}\bz \rho\|\sigma\jz}\le{D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz} \le& {D_{1}\bz \rho\|\sigma\jz}-4(1-\alpha)(\log\eta)^2\cosh c,& 1<\alpha<1+\delta.\end{aligned}$$ The inequalities in the second line above have already appeared in [@WWY]. Finally, we consider Lemma \[lemma:old-new bounds\] in some special cases. Note that the monotonicity of the Rényi divergences in $\alpha$ yields that the Rényi entropies $$S_{\alpha}(\rho):=-{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|I\jz}=-{D^{\mathrm{(new)}}_{\alpha}\bz \rho\|I\jz}=\frac{1}{1-\alpha}\log \operatorname{Tr}\rho^{\alpha}-\frac{1}{1-\alpha}\log\operatorname{Tr}\rho$$ are monotonic decreasing in $\alpha$ for any fixed $\rho$, and hence, $$\begin{aligned} \label{power bound} \operatorname{Tr}\rho^{\alpha}\le(\operatorname{Tr}\rho^0)^{(1-\alpha)}(\operatorname{Tr}\rho)^{\alpha}\end{aligned}$$ for every $\alpha\in(0,1)$, and the inequality holds in the converse direction for $\alpha>1$. Assume that $\alpha\in(0,1)$. Plugging into , we get that for any $\rho,\sigma\in\B(\hil)_+$, $$\begin{aligned} \label{old-new bounds3} \operatorname{Tr}\rho^{\alpha}\sigma^{1-\alpha} \le \operatorname{Tr}\bz\rho^{\half}\sigma^{\frac{1-\alpha}{\alpha}}\rho^{\half}\jz^{\alpha} \le {\left\| \sigma\right\|}^{(1-\alpha)^2}\bz\operatorname{Tr}\rho^{0}\jz^{(1-\alpha)^2}\bz\operatorname{Tr}\rho\jz^{\alpha(1-\alpha)}\bz\operatorname{Tr}\rho^{\alpha}\sigma^{1-\alpha}\jz^{\alpha} $$ for every $\alpha\in(0,1)$. This in turn yields that for every $\alpha\in(0,1)$, $$\begin{aligned} {D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz} \ge \alpha{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}+ (1-\alpha)\bz\log\operatorname{Tr}\rho-\log\operatorname{Tr}\rho^{0}-\log{\left\| \sigma\right\|}\jz.$$ In particular, if ${\left\| \sigma\right\|}\le 1$ then $$\begin{aligned} \label{old-new bounds5} {D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz} \ge \alpha{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}+ (1-\alpha)\bz\log\operatorname{Tr}\rho-\log\operatorname{Tr}\rho^{0}\jz.\end{aligned}$$ Assume now that $\alpha>1$. Then $\operatorname{Tr}\bz\rho/{\left\| \rho\right\|}\jz^{\alpha}\le\operatorname{Tr}\bz\rho/{\left\| \rho\right\|}\jz$, and plugging it into yields $$\begin{aligned} {D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz} \ge \alpha{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}+(\alpha-1)\bz\log{\left\| \sigma\right\|}-\log{\left\| \rho\right\|}\jz.\end{aligned}$$ In particular, if ${\left\| \rho\right\|}\le 1$ then $\operatorname{Tr}\sigma\le{\left\| \sigma\right\|}\operatorname{Tr}\sigma^0$ yields $$\begin{aligned} \label{old-new bounds6} {D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz} \ge \alpha{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}+(\alpha-1)\bz\log\operatorname{Tr}\sigma-\log\operatorname{Tr}\sigma^0\jz.\end{aligned}$$ \[cor:old-new bounds for states\] Let $\rho,\sigma\in\S(\hil)$ be density operators. For every $\alpha\in[0,+\infty)$, $$\begin{aligned} {D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz} \ge {D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz} \ge \alpha{D^{\mathrm{(old)}}_{\alpha}\bz \rho\|\sigma\jz}-|\alpha-1|\log(\dim\hil).\end{aligned}$$ Immediate from Lemma \[lemma:old-new bounds\], and . Corollary \[cor:old-new bounds for states\] together with Proposition \[prop:TCR\] yield the following version of Corollary \[cor:TCR\] when $\rho$ and $\sigma$ are states: \[cor:lower bound for states\] Let $\rho,\sigma\in\S(\hil)$ be density operators. With the notations of Proposition \[prop:TCR\], we have $$\begin{aligned} {D_{1}\bz \rho\|\sigma\jz}& \ge {D^{\mathrm{(new)}}_{\alpha}\bz \rho\|\sigma\jz} \ge \alpha{D_{1}\bz \rho\|\sigma\jz}-(1-\alpha)\left[ 4\alpha(\log\eta)^2\cosh c+\log(\dim\hil)\right].\end{aligned}$$ for every $1-\delta<\alpha<1$. Convexity properties {#sec:convexity} -------------------- The general concavity result in [@Hiai Theorem 2.1] implies as a special case that the quantity $$\label{new Q} Q\nw_{\alpha}(\rho\|\sigma):= \operatorname{Tr}\bz\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\jz^{\alpha} = \operatorname{Tr}\bz\rho^{\half}\sigma^{\frac{1-\alpha}{\alpha}}\rho^{\half}\jz^{\alpha}$$ is jointly concave for $\alpha\in[1/2,1)$. (See also [@FL] for a different proof of this). In [@Renyi_new; @WWY], joint convexity of $Q\nw_{\alpha}$ was shown for $\alpha\in[1,2]$, which was later extended in [@FL], using a different proof method, to all $\alpha>1$. That is, if $\rho_i,\sigma_i\in\B(\hil)_+,\,i=1,\ldots,r$, and $\gamma_1,\ldots,\gamma_r$ is a probability distribution on $[r]:=\{1,\ldots,r\}$, then $$\begin{aligned} Q\nw_{\alpha}\bz\sum_i\gamma_i\rho_i\Big\|\sum_i\gamma_i\sigma_i\jz&\ge \sum_i\gamma_iQ\nw_{\alpha}(\rho_i\|\sigma_i),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\half\le\alpha<1,\label{joint concavity}\\ Q\nw_{\alpha}\bz\sum_i\gamma_i\rho_i\Big\|\sum_i\gamma_i\sigma_i\jz&\le \sum_i\gamma_iQ\nw_{\alpha}(\rho_i\|\sigma_i),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}1<\alpha. \label{joint convexity}\end{aligned}$$ (For the second inequality one also has to assume that $\operatorname{supp}\rho_i\subseteq\operatorname{supp}\sigma_i$ for all $i$.) This yields immediately that the Rényi divergences $D_{\alpha}\nw$ are jointly quasi-convex for $\alpha>1$ (see [@WWY] for $\alpha\in(1,2]$), and jointly convex for $\alpha\in[1/2,1)$ when restricted to $\{\rho\in\B(\hil)_+\,:\,\operatorname{Tr}\rho=t\}\times\B(\hil)_+$ for any fixed $t>0$ [@FL]. Our goal here is to complement these inequalities to some extent. The following lemma is a special case of the famous Rotfel’d inequality (see, e.g., Section 4.5 in [@Hiaibook]). Below we provide an elementary proof for $\alpha\in[0,2]$. \[lemma:subadditivity1\] The function $A\mapsto \operatorname{Tr}A^{\alpha}$ is subadditive on positive semidefinite operators for every $\alpha\in[0,1]$, and superadditive for $\alpha\ge 1$. That is, if $A,B\in\B(\hil)_+$ then $$\begin{aligned} \operatorname{Tr}(A+B)^{\alpha}&\le\operatorname{Tr}A^{\alpha}+\operatorname{Tr}B^{\alpha},{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\alpha\in[0,1],\label{subadditivity inequality1}\\ \operatorname{Tr}(A+B)^{\alpha}&\ge\operatorname{Tr}A^{\alpha}+\operatorname{Tr}B^{\alpha},{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}1\le \alpha.\label{subadditivity inequality2}\end{aligned}$$ We only prove the case $\alpha\in[0,2]$. Assume first that $A$ and $B$ are invertible and let $\alpha\in(0,1)$. Then $$\begin{aligned} \operatorname{Tr}(A+B)^{\alpha}-\operatorname{Tr}A^{\alpha}&= \int_{0}^1\frac{d}{dt}\operatorname{Tr}(A+tB)^{\alpha}\,dt = \int_{0}^1\alpha\operatorname{Tr}B(A+tB)^{\alpha-1}\,dt\\ &\le \int_{0}^1\alpha\operatorname{Tr}B(tB)^{\alpha-1}\,dt =\operatorname{Tr}B^{\alpha}\int_{0}^1\alpha t^{\alpha-1}\,dt =\operatorname{Tr}B^{\alpha},\end{aligned}$$ where in the first line we used the identity $(d/dt)\operatorname{Tr}f(A+tB)=\operatorname{Tr}Bf'(A+tB)$, and the inequality follows from the fact that $x\mapsto x^{\alpha-1}$ is operator monotone decreasing on $(0,+\infty)$ for $\alpha\in(0,1)$. This proves for invertible $A$ and $B$, and the general case follows by continuity. The proof for the case $\alpha\in(1,2]$ goes the same way, using the fact that $x\mapsto x^{\alpha-1}$ is operator monotone increasing on $(0,+\infty)$ for $\alpha\in(1,2]$. The case $\alpha=1$ is trivial, and the case $\alpha=0$ follows by taking the limit $\alpha\to 0$ in . \[prop:complements\] Let $\sigma,\rho_1,\ldots,\rho_r\in\B(\hil)_+$, and $\gamma_1,\ldots,\gamma_r$ be a probability distribution on $[r]$. We have $$\begin{aligned} \sum_i\gamma_iQ\nw_{\alpha}(\rho_i\|\sigma)&\le Q\nw_{\alpha}\bz\sum_i\gamma_i\rho_i\Big\|\sigma\jz\le \sum_i\gamma_i^{\alpha}Q\nw_{\alpha}(\rho_i\|\sigma),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}0<\alpha<1,\label{concavity complement}\\ \sum_i\gamma_iQ\nw_{\alpha}(\rho_i\|\sigma)&\ge Q\nw_{\alpha}\bz\sum_i\gamma_i\rho_i\Big\|\sigma\jz\ge \sum_i\gamma_i^{\alpha}Q\nw_{\alpha}(\rho_i\|\sigma),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}1<\alpha.\label{convexity complement}\end{aligned}$$ Moreover, the second inequalities in and are valid for arbitrary non-negative $\gamma_1,\ldots,\gamma_r$ with $\gamma_1+\ldots+\gamma_r>0$. By lemma \[lemma:subadditivity1\], we have $$\begin{aligned} \operatorname{Tr}\bz\sigma^{\frac{1-\alpha}{2\alpha}}\bz\sum_{i=1}^r\gamma_i\rho_i\jz\sigma^{\frac{1-\alpha}{2\alpha}}\jz^{\alpha} &\le \sum_{i=1}^r\operatorname{Tr}\bz\sigma^{\frac{1-\alpha}{2\alpha}}\gamma_i\rho_i\sigma^{\frac{1-\alpha}{2\alpha}}\jz^{\alpha} = \sum_{i=1}^r\gamma_i^{\alpha}\operatorname{Tr}\bz\sigma^{\frac{1-\alpha}{2\alpha}}\rho_i\sigma^{\frac{1-\alpha}{2\alpha}}\jz^{\alpha} $$ for $\alpha\in(0,1)$, and the inequality is reversed for $\alpha>1$, which proves the second inequalities in and . The first inequalities follow the same way, by noting that $A\mapsto\operatorname{Tr}A^{\alpha}$ is concave for $\alpha\in(0,1)$ and convex for $\alpha>1$. Note that the first inequality in follows from the joint cconvexity of $Q_{\alpha}\nw$, and the first inequality in can be obtained from the joint concavity of $Q_{\alpha}\nw$ for $1/2\le\alpha<1$; however, not for the range $0<\alpha<1/2$, where joint concavity fails [@Renyi_new]. \[cor:new renyi superadd\] Let $\sigma,\rho_1,\ldots,\rho_r\in\B(\hil)_+$, and $\gamma_1,\ldots,\gamma_r$ be a probability distribution on $[r]$. For every $\alpha\in[0,+\infty]$, $$\begin{aligned} \min_i{D^{\mathrm{(new)}}_{\alpha}\bz \rho_i\|\sigma\jz}+\log \min_i\gamma_i \le {D^{\mathrm{(new)}}_{\alpha}\bz \sum_{i=1}^r\gamma_i\rho_i\Big\|\sigma\jz} \le \max_i{D^{\mathrm{(new)}}_{\alpha}\bz \rho_i\|\sigma\jz}.\end{aligned}$$ We prove the inequalities for $\alpha\in(1,+\infty)$; the proof for $\alpha\in(0,1)$ goes exactly the same way, and the cases $\alpha=0,1,+\infty$ follow by taking the corresponding limit in $\alpha$. By the first inequality in , we have $$\begin{aligned} {D^{\mathrm{(new)}}_{\alpha}\bz \sum_{i=1}^r\gamma_i\rho_i\Big\|\sigma\jz}&= \frac{1}{\alpha-1}\log\frac{Q\nw_{\alpha}\bz\sum_i\gamma_i\rho_i\Big\|\sigma\jz}{\sum_i\gamma_i\operatorname{Tr}\rho_i} \le \frac{1}{\alpha-1}\log\frac{\sum_i\gamma_iQ\nw_{\alpha}\bz\rho_i\|\sigma\jz}{\sum_i\gamma_i\operatorname{Tr}\rho_i}\\ &\le \frac{1}{\alpha-1}\log\min_i\frac{Q\nw_{\alpha}\bz\rho_i\|\sigma\jz}{\operatorname{Tr}\rho_i},\end{aligned}$$ proving the second inequality of the assertion. The second inequality in yields $$\begin{aligned} {D^{\mathrm{(new)}}_{\alpha}\bz \sum_{i=1}^r\gamma_i\rho_i\Big\|\sigma\jz}&= \frac{1}{\alpha-1}\log \frac{Q\nw_{\alpha}\bz\sum_i\gamma_i\rho_i\Big\|\sigma\jz}{\operatorname{Tr}\sum_i\gamma_i\rho_i} \ge \frac{1}{\alpha-1}\log \frac{\sum_i\gamma_i^{\alpha} Q\nw_{\alpha}\bz\rho_i\|\sigma\jz}{\sum_i\gamma_i\operatorname{Tr}\rho_i}. $$ We have $$\begin{aligned} \gamma_i^{\alpha} Q\nw_{\alpha}\bz\rho_i\|\sigma\jz &\ge (\gamma_i\operatorname{Tr}\rho_i)\gamma_i^{\alpha-1} \min_j\frac{\gamma_j^{\alpha} Q\nw_{\alpha}\bz\rho_j\|\sigma\jz}{\gamma_j^{\alpha}\operatorname{Tr}\rho_j} \ge \gamma_i\operatorname{Tr}\rho_i\bz\min_j\gamma_j^{\alpha-1}\jz \min_j\frac{Q\nw_{\alpha}\bz\rho_j\|\sigma\jz}{\operatorname{Tr}\rho_j},\end{aligned}$$ and summing over $i$ yields that $$\begin{aligned} \frac{1}{\alpha-1}\log \frac{\sum_i\gamma_i^{\alpha} Q\nw_{\alpha}\bz\rho_i\|\sigma\jz}{\operatorname{Tr}\sum_i\gamma_i\rho_i} \ge \frac{1}{\alpha-1}\log \min_j\frac{Q\nw_{\alpha}\bz\rho_j\|\sigma\jz}{\operatorname{Tr}\rho_j}+\log\min_j\gamma_j,\end{aligned}$$ as required. Note that the inequalities in and express joint concavity/convexity, whereas in the complements given in Proposition \[prop:complements\] and Corollary \[cor:new renyi superadd\] we only took a convex combination in the first variable and not in the second. It is easy to see that this restriction is in fact necessary. Indeed, let $\rho_1:=\sigma_2:={{|x\rangle\langle x|}}$ and $\rho_2:=\sigma_1:={{|y\rangle\langle y|}}$, where $x$ and $y$ are orthogonal unit vectors in some Hilbert space. If we choose $\gamma_1=\gamma_2=1/2$ then $\sum_{i}\gamma_i\rho_i=\sum_i\gamma_i\sigma_i$, and hence $${D^{\mathrm{(new)}}_{\alpha}\bz \sum_{i=1}^r\gamma_i\rho_i\Big\|\sum_{i=1}^r\gamma_i\sigma_i\jz}=0,{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\text{while}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{D^{\mathrm{(new)}}_{\alpha}\bz \rho_1\|\sigma_1\jz}={D^{\mathrm{(new)}}_{\alpha}\bz \rho_2\|\sigma_2\jz}=+\infty,$$ and hence no inequality of the form ${D^{\mathrm{(new)}}_{\alpha}\bz \sum_{i=1}^r\gamma_i\rho_i\Big\|\sum_{i=1}^r\gamma_i\sigma_i\jz}\ge c_1\min_i{D^{\mathrm{(new)}}_{\alpha}\bz \rho_i\|\sigma_i\jz}-c_2$ can hold for any positive constants $c_1$ and $c_2$. The quantity $$Q_{\alpha}\old(\rho\|\sigma):=\operatorname{Tr}\rho^{\alpha}\sigma^{1-\alpha}$$ is jointly concave for $\alpha\in(0,1)$ according to Lieb’s concavity theorem [@Lieb], and jointly convex for $\alpha\in(1,2]$ according to Ando’s convexity theorem [@Ando]; see also [@Petz] for a different proof of both. That is, if $\rho_i,\sigma_i\in\B(\hil)_+,\,i=1,\ldots,r$, and $\gamma_1,\ldots,\gamma_r$ is a probability distribution on $[r]:=\{1,\ldots,r\}$, then $$\begin{aligned} Q\old_{\alpha}\bz\sum_i\gamma_i\rho_i\Big\|\sum_i\gamma_i\sigma_i\jz&\ge \sum_i\gamma_iQ\old_{\alpha}(\rho_i\|\sigma_i),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}0\le\alpha<1,\\ Q\old_{\alpha}\bz\sum_i\gamma_i\rho_i\Big\|\sum_i\gamma_i\sigma_i\jz&\le \sum_i\gamma_iQ\old_{\alpha}(\rho_i\|\sigma_i),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}1<\alpha\le 2.\end{aligned}$$ (For the second inequality, one has to assume that $\operatorname{supp}\rho_i\subseteq\operatorname{supp}\sigma$ for all $i$.) Note the difference in the ranges of joint convexity/concavity as compared to and . This yields immediately that $D_{\alpha}\old$ is jointly convex for $\alpha\in(0,1)$ when restricted to $\{\rho\in\B(\hil)_+\,:\,\operatorname{Tr}\rho=t\}\times\B(\hil)_+$ for any fixed $t>0$, and it is jointly quasi-convex for $\alpha\in(1,2]$. Moreover, it is convex in its second argument for $\alpha\in(1,2]$, according to Theorem II.1 in [@MH]; see also Proposition 1.1 in [@AH]. It is not clear whether a subadditivity argument can be used to complement the above concavity/convexity properties. However, one can use the bounds for $Q_{\alpha}\nw$ and $D_{\alpha}\nw$ together with lemma \[lemma:old-new bounds\] to obtain the following: \[prop:old Renyi quasiconcavity\] Let $\sigma,\rho_1,\ldots,\rho_r\in\B(\hil)_+$, and $\gamma_1,\ldots,\gamma_r$ be a probability distribution on $[r]$. We have $$\begin{aligned} \label{complement old1} Q\old_{\alpha}\bz\sum_i\gamma_i\rho_i\|\sigma\jz \le \sum_i\gamma_i^{\alpha}Q\old_{\alpha}(\rho_i\|\sigma)^{\alpha} {\left\| \sigma\right\|}^{(1-\alpha)^2}(\operatorname{Tr}\rho_i^{\alpha})^{1-\alpha}\end{aligned}$$ for $\alpha\in(0,1)$, and the inequality holds in the converse direction for $\alpha>1$. As a conseqence, $$\begin{aligned} \label{complement old2} D\old_{\alpha}\bz\sum_i\gamma_i\rho_i\|\sigma\jz \ge \alpha\min_i D\old_{\alpha}(\rho_i\|\sigma) +(\alpha-1)\log{\left\| \sigma\right\|} +\log\min_i\left\{\gamma_i\frac{\operatorname{Tr}\rho_i}{\operatorname{Tr}\rho_i^{\alpha}}\right\}\end{aligned}$$ for all $\alpha\in(0,+\infty)\setminus\{1\}$. The inequality in is immediate from and Proposition \[prop:complements\]. The same argument as in the proof of Corollary \[cor:new renyi superadd\] yields . For $\alpha\in(0,1)$, we can use to further bound the RHS of from below and get $$\begin{aligned} &D\old_{\alpha}\bz\sum_i\gamma_i\rho_i\|\sigma\jz\nonumber\\ &{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\ge \alpha\min_i D\old_{\alpha}(\rho_i\|\sigma) +(\alpha-1)\log{\left\| \sigma\right\|} +\log\min_i\left\{\gamma_i(\operatorname{Tr}\rho_i)^{1-\alpha}(\operatorname{Tr}\rho_i^0)^{\alpha-1}\right\}.\end{aligned}$$ Rényi capacities ---------------- By a channel $W$ we mean a map $W:\,\X\to\S(\hil)$, where $\X$ is some input alphabet (which can be an arbitrary non-empty set) and $\hil$ is a finite-dimensional Hilbert space. We recover the usual notion of a quantum channel when $\X=\S(\kil)$ for some Hilbert space $\kil$, and $W$ is a completely positive trace-preserving linear map. For an input alphabet $\X$, let $\{\delta_x\}_{x\in\X}$ be a set of rank-$1$ orthogonal projections in some Hilbert space $\hil_{\X}$, and for every channel $W:\,\X\to\S(\hil)$ define $$\what W:\,x\mapsto \delta_x\otimes W_x,$$ which is a channel from $\X$ to $\S(\hil_{\X}\otimes\hil)$. Let $\P_f(\X)$ denote the set of finitely supported probability measures on $\X$. The channels $W$ and $\what W$ can naturally be extended to convex maps $W:\,\P_f(\X)\to\S(\hil)$ and $\what W:\,\P_f(\X)\to\S(\hil_{\X}\otimes\hil)$, as $$\begin{aligned} W(p):=\sum_{x\in\X}p(x)W(x),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\what W(p):=\sum_{x\in\X}p(x)\what W(p)=\sum_{x\in\X}p(x)\delta_x\otimes W(x).\end{aligned}$$ Note that $\what W(p)$ is a classical-quantum state, and the marginals of $\what W(p)$ are given by $$\operatorname{Tr}_{\hil}\what W(p)=\hat p:=\sum_x p(x)\delta_x{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\text{and}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\operatorname{Tr}_{\hil_{\X}}\what W(p)=W(p).$$ Let $D$ be a function on pairs of positive semidefinite operators. For a channel $W:\,\X\to\S(\hil)$, we define its corresponding $D$-capacity as $$\begin{aligned} {\hat\chi_{_{D}}}(W):=\sup_{p\in\P_f(\X)}\chi_D(W,p),\end{aligned}$$ where $$\begin{aligned} {\chi_{_{D}}}(W,p):=\inf_{\sigma\in\S(\hil)} D\bz\what W(p)\|\hat p\otimes \sigma\jz,{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}p\in\P_f(\X).\end{aligned}$$ For the cases $D=D\old_{\alpha}$ and $D=D\nw_{\alpha}$, we use the shorthand notations ${\chi_{_{\alpha}}}\old(W,p)$, ${\hat\chi_{_{\alpha}}}\old(W)$ and ${\chi_{_{\alpha}}}\nw(W,p)$, ${\hat\chi_{_{\alpha}}}\nw(W)$, respectively. Note that these quantities generalize the $$\begin{aligned} \label{Holevo} \chi(W,p)&:={\chi_{_{1}}}\old(W,p)={\chi_{_{1}}}\nw(W,p)= \inf_{\sigma\in\S(\hil)}{D_{1}\bz \what W(p)\|\hat p\otimes \sigma\jz}\nonumber\\ &={D_{1}\bz \what W(p)\|\hat p\otimes W(p)\jz}\end{aligned}$$ and the $$\begin{aligned} \label{Holevo cap} \hat\chi(W)&:=\sup_{p\in\P_f(\X)}\chi(W,p),\end{aligned}$$ and hence we refer to them as generalized Holevo quantities for a general $D$, and generalized $\alpha$-Holevo quantities for the $\alpha$-divergences. As it was pointed out in [@KW; @Sibson], $$\begin{aligned} \label{Sibson} {D^{\mathrm{(old)}}_{\alpha}\bz \what W(p)\|\hat p\otimes \sigma\jz}= \frac{\alpha}{\alpha-1}\log\operatorname{Tr}\omega(p)+D_{\alpha}\old\bz \bar\omega(W,p)\|\sigma\jz\end{aligned}$$ for any state $\sigma$, where $$\label{optimal state} \bar\omega(W,p):=\omega(W,p)/\operatorname{Tr}\omega(W,p),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\omega(W,p):=\bz\sum_x p(x)W(x)^{\alpha}\jz^{\frac{1}{\alpha}}.$$ Since $D_{\alpha}\old$ is non-negative on pairs of density operators, we get $$\label{explicit alpha-Holevo} {\chi_{_{\alpha}}}\old(W,p)=\frac{\alpha}{\alpha-1}\log\operatorname{Tr}\omega(p)= \frac{\alpha}{\alpha-1}\log\operatorname{Tr}\bz\sum_x p(x)W(x)^{\alpha}\jz^{\frac{1}{\alpha}}.$$ However, no such explicit formula is known for ${\chi_{_{\alpha}}}\nw(W,p)$. Note that $\max\{\operatorname{Tr}\what W(p)^{0},\operatorname{Tr}(\hat p\otimes \sigma)^0\}\le|\operatorname{supp}p|\dim\hil$, where $|\operatorname{supp}p|$ denotes the cardinality of the support of $p$, and Lemma \[lemma:old-new bounds\] with and yields that $$\label{weak capacity bounds} {\chi_{_{\alpha}}}\old(W,p) \ge {\chi_{_{\alpha}}}\nw(W,p) \ge \alpha{\chi_{_{\alpha}}}\nw(W,p)-|\alpha-1|\log\bz|\operatorname{supp}p|\dim\hil\jz$$ for every $\alpha\in(0,+\infty)$. A more careful application of and yields the following improved bound: \[lemma:chi bounds\] Let $W:\,\X\to\S(\hil)$ be a channel, and $\alpha\in(0,+\infty)$. For any $p\in\P_f(\X)$ and any $\sigma\in\S(\hil)$, we have $$\begin{aligned} {D^{\mathrm{(new)}}_{\alpha}\bz \what W(p)\|\hat p\otimes \sigma\jz} \ge \alpha{D^{\mathrm{(old)}}_{\alpha}\bz \what W(p)\|\hat p\otimes \sigma\jz}-|\alpha-1|\log(\dim\hil),\end{aligned}$$ and hence, $$\begin{aligned} {\chi_{_{\alpha}}}\old(W,p)\ge {\chi_{_{\alpha}}}\nw(W,p)\ge \alpha{\chi_{_{\alpha}}}\old(W,p)-|\alpha-1|\log(\dim\hil).\end{aligned}$$ Assume that $\alpha>1$. By Corollary \[cor:old-new bounds for states\] we have $\operatorname{Tr}\bz W(x)^{\half}\sigma^{\frac{1-\alpha}{\alpha}}W(x)^{\half}\jz^{\alpha}\ge (\dim\hil)^{-(\alpha-1)^2}\bz\operatorname{Tr}W(x)^{\alpha}\sigma^{1-\alpha}\jz^{\alpha}$ for every $x\in\X$, and hence, $$\begin{aligned} {D^{\mathrm{(new)}}_{\alpha}\bz \what W(p)\|\hat p\otimes \sigma\jz}&= \frac{1}{\alpha-1}\log\sum_x p(x)\operatorname{Tr}\bz W(x)^{\half}\sigma^{\frac{1-\alpha}{\alpha}}W(x)^{\half}\jz^{\alpha}\\ &\ge -(\alpha-1)\log(\dim\hil)+\frac{1}{\alpha-1}\log\sum_x p(x)\bz\operatorname{Tr}W(x)^{\alpha}\sigma^{1-\alpha}\jz^{\alpha}\\ &\ge -(\alpha-1)\log(\dim\hil)+\frac{1}{\alpha-1}\log\bz\sum_x p(x)\operatorname{Tr}W(x)^{\alpha}\sigma^{1-\alpha}\jz^{\alpha}\\ &= -(\alpha-1)\log(\dim\hil)+\alpha{D^{\mathrm{(old)}}_{\alpha}\bz \what W(p)\|\hat p\otimes \sigma\jz},\end{aligned}$$ where the second inequality is due to the convexity of $x\mapsto x^{\alpha}$. The proof for $\alpha\in(0,1)$ goes exactly the same way. Monotonicity of the Rényi divergences in $\alpha$ yields that the corresponding quantities ${\chi_{_{\alpha}}}\old(W,p)$ and ${\chi_{_{\alpha}}}\nw(W,p)$ are also monotonic increasing in $\alpha$. A simple minimax argument shows (see, e.g. [@MH Lemma B.3]) that $$\label{chi limit} \lim_{\alpha\to 1}{\chi_{_{\alpha}}}\old(W,p)=\chi(W,p),$$ where $\chi(W,p)$ is the Holevo quantity. This, together with lemma \[lemma:chi bounds\] yields that also $$\lim_{\alpha\to 1}{\chi_{_{\alpha}}}\nw(W,p)=\chi(W,p).$$ Moreover, it was shown in [@MH Proposition B.5] that if $\operatorname{ran}W:=\{W(x)\,:\,x\in\X\}$ is compact then $$\lim_{\alpha\to 1}{\hat\chi_{_{\alpha}}}\old(W)=\hat\chi(W).$$ Applying lemma \[lemma:chi bounds\] to this, we obtain $$\label{new capacity limit} \lim_{\alpha\to 1}{\hat\chi_{_{\alpha}}}\nw(W)=\hat\chi(W).$$ Carathéodory’s theorem and the explicit formula imply that in the definition ${\hat\chi_{_{\alpha}}}\old(W):=\sup_{p\in\P_f(\X)}{\chi_{_{\alpha}}}\old(W,p)$ it is enough to consider probability distributions with $|\operatorname{supp}p|\le(\dim\hil)^2+1$. However, this is not known for ${\hat\chi_{_{\alpha}}}\nw(W)$, and hence is insufficient to derive . Finally, we point out a connection between $\alpha$-capacities and a special case of a famous convexity result by Carlen and Lieb [@CL; @CL2]. For any finite-dimensional Hilbert space $\hil$ and $A_1,\ldots,A_n\in\B(\hil)_+$, define $$\Phi_{\alpha,q}(A_1,\ldots,A_n):=\bz\operatorname{Tr}\left[\bz\sum_{i=1}^n A_i^{\alpha}\jz^{q/\alpha}\right]\jz^{1/q},{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\alpha\ge 0,\,q> 0.$$ Theorem 1.1 in [@CL2] says that for any finite-dimensional Hilbert space $\hil$, $\Phi_{\alpha,q}$ is concave on $\bz\B(\hil)_+\jz^n$ for $0\le \alpha\le q\le 1$, and convex for all $1\le \alpha\le 2$ and $q\ge 1$. Below we give an elementary proof of the following weaker statement: $\Phi_{\alpha,1}^{\alpha}$ is concave for $\alpha\in(0,1)$ and convex for $\alpha\in(1,2]$. For a set $\X$, a finitely supported non-negative function $p:\,\X\to\bR_+$, and a finite-dimensional Hilbert space $\hil$, let $\hat\Phi_{p,\hil,\alpha}:\,\bz\B(\hil)_+\jz^{\X}\to\bR_+$ be defined as $$\hat\Phi_{p,\hil,\alpha}(W):=\bz\operatorname{Tr}\bz\sum_{x\in\X}p(x)W(x)^{\alpha}\jz^{1/\alpha}\jz^\alpha,{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}W\in\bz\B(\hil)_+\jz^{\X}.$$ The following Proposition is equivalent to our assertion: \[prop:CL\] For any $\X,\,p$ and $\hil$, $\hat\Phi_{p,\hil,\alpha}$ is concave on $\bz\B(\hil)_+\jz^{\X}$ for $\alpha\in(0,1)$ and convex for $\alpha\in(1,2]$. Exactly the same way as in –, we can see that $$\begin{aligned} \label{khi representation} \frac{\alpha}{\alpha-1}\log\operatorname{Tr}\bz\sum_x p(x)W(x)^{\alpha}\jz^{\frac{1}{\alpha}}= \min_{\sigma\in\S(\hil)}D_{\alpha}\old\bz\what W(p)\|\hat p\otimes \sigma\jz.\end{aligned}$$ Assume for the rest that $\alpha\in(1,2]$; the proof for the case $\alpha\in(0,1)$ goes exactly the same way. Let $r\in\bN$, $W_1,\ldots,W_r\in(\B(\hil)_+)^{\X}$, and $\gamma_1,\ldots,\gamma_r$ be a probability distribution. Then $$\begin{aligned} \hat\Phi_{p,\hil,\alpha}\bz\sum_i\gamma_i W_i\jz&= \min_{\sigma\in\S(\hil)}Q_{\alpha}\old\bz\sum_i\gamma_i\what W(p)\Big\|\hat p\otimes \sigma\jz\\ &= \min_{\sigma_1,\ldots,\sigma_r\in\S(\hil)}Q_{\alpha}\old\bz\sum_i\gamma_i\what W(p)\Big\|\hat p\otimes \sum_i\gamma_i\sigma_i\jz\\ &\le \min_{\sigma_1,\ldots,\sigma_r\in\S(\hil)}\sum_i\gamma_i Q_{\alpha}\old\bz\what W(p)\|\hat p\otimes\sigma_i\jz\\ &= \sum_i\gamma_i\min_{\sigma_i}Q_{\alpha}\old\bz\what W(p)\|\hat p\otimes\sigma_i\jz\\ &= \sum_i\gamma_i\hat\Phi_{p,\hil,\alpha}\bz W_i\jz,\end{aligned}$$ where the first and the last identities are due to , and the inequality follows from the joint convexity of $Q_{\alpha}\old$ [@Ando; @Petz]. (In the case $\alpha\in(0,1)$, we have to use joint concavity [@Lieb; @Petz].) Applications to coding theorems {#sec:applications} =============================== Preliminaries {#sec:prel} ------------- For a self-adjoint operator $X$, we will use the notation $\{X>0\}$ to denote the spectral projection of $X$ corresponding to the positive half-line $(0,+\infty)$. The spectral projections $\{X\ge 0\},\,\{X<0\}$ and $\{X\le 0\}$ are defined similarly. The positive part $X_+$ and the negative part $X_-$ are defined as $X_+:=X\{X>0\}$ and $X_-:=X\{X<0\}$, respectively, and the absolute value of $X$ is $|X|:=X_++X_-$. The trace-norm of $X$ is ${\left\| X\right\|}_1:=\operatorname{Tr}|X|$. The following lemma is Theorem 1 from [@Aud]; see also Proposition 1.1 in [@JOPS] for a simplifed proof. \[lemma:Aud\] Let $A,B$ be positive semidefinite operators on the same Hilbert space. For any $t\in[0,1]$, $$\operatorname{Tr}A(I-\{A-B>0\})+\operatorname{Tr}B\{A-B>0\}=\half\operatorname{Tr}(A+B)-\half{\left\| A-B\right\|}_1\le\operatorname{Tr}A^tB^{1-t}.$$ The next lemma is a reformulation of Lemma 2.6 in [@MS]. We include the proof for readers’ convenience. \[lemma:state approximation\] Let $(V,{\left\| .\right\|})$ be a finite-dimensional normed vector space, and let $D$ denote its real dimension. Let $\N\subset V$ be a subset. For every $\delta>0$, there exists a finite subset $\N_{\delta}\subset\N$ such that 1\. $\displaystyle{|\N_{\delta}|\le (1+2/\delta)^{D}}$, and 2\. for every $v\in\N$ there exists a $v_{\delta}\in\N_{\delta}$ such that ${\left\| v-v_{\delta}\right\|}<\delta$. For every $\delta>0$, let $\N_{\delta}$ be a maximal set in $\N$ such that ${\left\| v-v'\right\|}\ge \delta$ for every $v,v'\in\N_{\delta}$; then $\N_{\delta}$ clearly satisfies 2. On the other hand, the open ${\left\| {\mbox{ }}\right\|}$-balls with radius $\delta/2$ around the elements of $\N_{\delta}$ are disjoint, and contained in the ${\left\| {\mbox{ }}\right\|}$-ball with radius $1+\delta/2$ and origin $0$. Since the volume of balls scales with their radius on the power $D$, we obtain 1. The of positive semidefinite operators $A$ and $B$ is defined as $F(A,B):=\operatorname{Tr}\bz A^{1/2}BA^{1/2}\jz^{1/2}$. The entanglement fidelity of a state $\rho$ and a completely positive trace-preserving map $\map$ is $F_e(\rho,\map):=F\bz{{|\psi_{\rho}\rangle\langle \psi_{\rho}|}},(\operatorname{id}\otimes \map){{|\psi_{\rho}\rangle\langle \psi_{\rho}|}}\jz$, where $\psi_{\rho}$ is any purification of the state $\rho$; see Chapter 9 in [@NC] for details. Quantum Stein’s lemma with composite null-hypothesis {#sec:Stein} ---------------------------------------------------- Consider the asymptotic hypothesis testing problem with null-hypothesis $H_0:\,\N_n\subset\S(\hil_n)$ and alternative hypothesis $H_1:\,\sigma_n\in\S(\hil_n)$, $n\in\bN$, where $\hil_n$ is some finite-dimensional Hilbert space. Our goal is to decide between these two hypotheses based on the outcome of a binary POVM $(T_n(0),T_n(1))$ on $\hil_n$, where $0$ and $1$ indicate the acceptance of $H_0$ and $H_1$, respectively. Since $T_n(1)=I-T_n(0)$, the POVM is uniquely determined by $T_n=T_n(0)$, and the only constraint on $T_n$ is that $0\le T_n\le I_n$. We will call such operators tests. Given a test $T_n$, the probability of mistaking $H_0$ for $H_1$ (type I error) and the probability of mistaking $H_1$ for $H_0$ (type II error) are given by $$\begin{aligned} \alpha_n(T_n):=\sup_{\rho_n\in\N_n}\operatorname{Tr}\rho_n(I-T_n),{\mbox{ }\mbox{ }}\text{(type I)},{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\text{and}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\beta_n(T_n):=\operatorname{Tr}\sigma_n T_n,{\mbox{ }\mbox{ }}\text{(type II)}.\end{aligned}$$ \[def:direct rate\] We say that a rate $R\ge 0$ is if there exists a sequence of tests $T_n,\,n\in\bN$, with $$\begin{aligned} \lim_{n\to+\infty}\alpha_n(T_n)=0{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\text{and}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\limsup_{n\to+\infty}\frac{1}{n}\log\beta_n(T_n)\le -R.\end{aligned}$$ The largest achievable rate $R(\{\N_n\}_{n\in\bN}\|\{\sigma_n\}_{n\in\bN})$ is the of the hypothesis testing problem. For what follows, we assume that $\hil_n=\hil^{\otimes n},\,n\in\bN$, where $\hil=\hil_1$, and that the alternative hpothesis is i.i.d., i.e., $\sigma_n=\sigma^{\otimes n},\,n\in\bN$, with $\sigma=\sigma_1$. We say that the null-hypothesis is if there exists a set $\N\subset\S(\hil)$ such that for all $n\in\bN$, $\N_n=\N\notimes:=\{\rho^{\otimes n}:\,\rho\in\N\}$. The null-hypothesis is if $\N$ consists of one single element, i.e., $\N=\{\rho\}$ for some $\rho\in\S(\hil)$. According to the quantum Stein’s lemma [@HP; @ON2], the direct rate in the simple i.i.d. case is given by $D_1(\rho\|\sigma)$. The case of the general composite null-hypothesis was treated in [@BDKSSSz] under the name of quantum Sanov theorem. There it was shown that there exists a sequence of tests $\{T_n\}_{n\in\bN}$ such that $\lim_{n\to+\infty}\operatorname{Tr}\rho^{\otimes n}(I-T_n)=0$ for every $\rho\in\N$, and $\limsup_{n\to+\infty}\frac{1}{n}\log\beta_n(T_n)\le -D_1(\N\|\rho)$, where $D_1(\N\|\rho):=\inf_{\rho\in\N}D_1(\rho\|\sigma)$. Note that this is somewhat weaker than $D_1(\N\|\rho)$ being achievable in the sense of Definition \[def:direct rate\]. Achievability in this stronger sense has been shown very recently in [@Notzel], using the representation theory of the symmetric group and the method of types. The proof in both papers followed the approach in [@HP] of reducing the problem to a classical hypothesis testing problem by projecting all states onto the commutative algebra generated by $\{\sigma^{\otimes n}\}_{n\in\bN}$. Below we use a different proof technique to show that $D_1(\N\|\rho)$ is achievable in the sense of Defintion \[def:direct rate\]. Our proof is based solely on Audenaert’s trace inequality [@Aud] and the subadditivity property of $Q_{\alpha}\nw$, given in Proposition \[prop:complements\]. We obtain explicit upper bounds on the error probabilities for any finite $n\in\bN$ for a sequence of Neyman-Pearson types tests. Moreover, if a $\delta$-net can be explicitly constructed for $\N$ for every $\delta>0$ (this is trivially satisfied when $\N$ is finite) then the tests can also be constructed explicitly. In [@BDKSSSz], Stein’s lemma was stated with weak converse, while the results of [@Notzel] imply a strong converse. Here we use Nagaoka’s method to further strengthen the converse part by giving exlicit bounds on the exponential rate with which the worst-case type I success probability goes to zero when the type II error decays with a rate larger than the optimal rate $D_1(\N\|\rho)$. Note that our proof technique doesn’t actually rely on the i.i.d. assumption, as we demonstrate in Theorem \[thm:correlated Stein\], where we give achievability bounds in the general correlated scenario. However, in the most general case we have to restrict to a finite null-hypothesis. We show examples in Remark \[rem:correlated Stein\] where the achievable rate of Theorem \[thm:correlated Stein\] can be expressed as the regularized relative entropy distance of the null-hypothesis and the alternative hypothesis, giving a direct generalization of the i.i.d. case. These results complement those of [@correlated; @Sanov], where it was shown that if $\Theta$ is a set of ergodic states on a spin chain, and $\Phi$ is a state on the spin chain such that for every $\Psi\in\Theta$, Stein’s lemma holds for the simple hypothesis testing problem $H_0:\,\Psi,\,H_1:\,\Phi$, then it also holds for the composite hypothesis testing problem $H_0:\,\Theta,\,H_1:\,\Phi$. This was also extended in [@correlated; @Sanov] to the case where $\Theta$ consists of translation-invariant states, using ergodic decomposition. Now let $\N\subset\S(\hil)$ be a non-empty set of states, and let $\sigma\in\B(\hil)_+$ be a positive semidefinite operator such that $$\label{support condition} \operatorname{supp}\rho\subseteq\operatorname{supp}\sigma,{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\rho\in\N.$$ Note that in hypothesis testing $\sigma$ is usually assumed to be a state on $\hil$; however, the proof for Stein’s lemma works the same way for a general positive semidefinite $\sigma$, and considering this more general case is actually useful e.g., for state compression. Let $$\label{psi} \psi(t):=\sup_{\rho\in\N}\log Q_t\nw(\rho\|\sigma),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}t>0,$$ and for every $a\in\bR$, let $$\label{phi} \vfi(a):=\sup_{0<t\le 1}\{at-\psi(t)\},{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\hat\vfi(a):=\sup_{0<t\le 1}\{a(t-1)-\psi(t)\}=\vfi(a)-a.$$ Note that $\vfi$ is the Legendre-Fenchel transform of $\psi$ on $(0,1]$. \[thm:Stein\] For every $n\in\bN$, let $\N(n)\subset\N$ be a finite subset, and let $\delta(N(n)):=\sup_{\rho\in\N}\inf_{\rho'\in\N(n)}{\left\| \rho-\rho'\right\|}_1$. For every $a\in\bR$, let $S_{n,a}:=\left\{e^{-na}\sum_{\rho\in\N(n)}\rho^{\otimes n}-\sigma^{\otimes n}>0\right\}$ be a Neyman-Pearson test. Then $$\begin{aligned} \sup_{\rho\in\N}\operatorname{Tr}\rho^{\otimes n}(I-S_{n,a})&\le |\N(n)|e^{-n\hat\vfi(a)}+n\delta(N(n)), {\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\text{and}\label{finiten bounds1}\\ \operatorname{Tr}\sigma^{\otimes n}S_{n,a}&\le|\N(n)|e^{-n\vfi(a)}.\label{finiten bounds}\end{aligned}$$ In particular, let $\delta_n:=e^{-n\kappa}$ for some $\kappa>0$, and $\N(n):=\N_{\delta_n}\subset\N$ as in lemma \[lemma:state approximation\]. Then $$\begin{aligned} \limsup_{n\to+\infty}\frac{1}{n}\log\alpha_n(S_{n,a})&\le -\min\{\kappa,\hat\vfi(a)-\kappa D(\hil)\},\label{NP upper1}\\ \limsup_{n\to+\infty}\frac{1}{n}\log\beta_n(S_{n,a})&\le -(\vfi(a)-\kappa D(\hil)).\label{NP upper2}\end{aligned}$$ For every $n\in\bN$, let $\bar\rho_n:=\sum_{\rho\in\N(n)}\rho^{\otimes n}$, $\sigma_n:=\sigma^{\otimes n}$. Applying lemma \[lemma:Aud\] to $A:=e^{-na}\bar\rho_n$ and $B:=\sigma_n$ for some fixed $a\in\bR$, we get $$\begin{aligned} \label{ena upper bound} e_n(a)&:=e^{-na}\operatorname{Tr}\bar\rho_n (I-S_{n,a})+\operatorname{Tr}\sigma_nS_{n,a} \le e^{-nat}\operatorname{Tr}\bar\rho_n^t\sigma_n^{1-t}\end{aligned}$$ for every $t\in[0,1]$. This we can further upper bound as $$\begin{aligned} \operatorname{Tr}\bar\rho_n^t\sigma_n^{1-t}&\le Q_t\nw\bz\bar\rho_n\|\sigma_n\jz \le \sum_{\rho\in\N(n)}Q_t\nw\bz\rho^{\otimes n}\|\sigma^{\otimes n}\jz \le |\N(n)|\sup_{\rho\in\N}Q_t\nw\bz\rho^{\otimes n}\|\sigma^{\otimes n}\jz\nonumber\\ &= |\N(n)|\sup_{\rho\in\N}\bz Q_t\nw\bz\rho\|\sigma\jz\jz^n = |\N(n)|e^{n\psi(t)},\label{Stein proof1}\end{aligned}$$ where the first inequality is due to lemma \[lemma:old-new bounds\], the second inequality is due to , the third inequality is obvious, the succeeding identity follows from the definition , and the last identity is due to the definition of $\psi$. Since holds for every $t\in(0,1]$, together with it yields $e_n(a)\le |\N(n)|e^{-n\vfi(a)}$. Hence we have $\operatorname{Tr}\sigma_nS_{n,a}\le e_n(a)\le |\N(n)|e^{-n\vfi(a)}$, proving . Similarly, $\operatorname{Tr}\bar\rho_n(I-S_{n,a})\le e^{na}e_n(a)$ yields $$\begin{aligned} \label{type I upper bound} \sup_{\rho\in \N(n)}\operatorname{Tr}\rho^{\otimes n}(I-S_{n,a}) \le \operatorname{Tr}\bar\rho_n(I-S_{n,a}) \le e^{na}|\N(n)|e^{-n\vfi(a)} = |\N(n)|e^{-n\hat\vfi(a)}.\end{aligned}$$ The submultiplicativity of the trace-norm on tensor products yields that $\sup_{\rho\in\N}\operatorname{Tr}\rho^{\otimes n}(I-S_{n,a}) \le \sup_{\rho\in\N(n)}\operatorname{Tr}\rho^{\otimes n}(I-S_{n,a})+n\delta(\N(n)))$. Combined with , this yields . The inequalities in – are obvious from the choice of $\delta_n$. \[lemma:positive rate\] We have $\vfi(a)\ge a$, and for every $a<D_1(\N\|\sigma)$, we have $\hat\vfi(a)>0$. Note that for any $t\in(0,1)$, $a(t-1)-\psi(t)=(t-1)[a-\inf_{\rho\in\N}{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}]$. Moreover, by the assumption in , $\rho\mapsto{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}$ is continuous on $\overline{\N}$, and hence, $\inf_{\rho\in\N}{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}=\min_{\rho\in\overline{\N}}{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}$ for every $t\in(0,1)$. Note that $\overline{\N}$ is compact, and for every $\rho\in\overline{\N}$, $t\mapsto {D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}$ is monotone increasing, due to [@Renyi_new Theorem 6]. Applying now the minimax theorem from [@MH Corollary A.2], we get $\sup_{t\in(0,1)}\min_{\rho\in\overline{\N}}{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}= \min_{\rho\in\overline{\N}}\sup_{t\in(0,1)}{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}= \min_{\rho\in\overline{\N}}{D^{\mathrm{(new)}}_{1}\bz \rho\|\sigma\jz}=D_1(\N\|\sigma)$. Thus, for any $a<D_1(\N\|\sigma)$, there exists a $t_a\in(0,1)$ such that $a-\inf_{\rho\in\N}{D^{\mathrm{(new)}}_{t_a}\bz \rho\|\sigma\jz}<0$, and hence $0<(t_a-1)[a-\inf_{\rho\in\N}{D^{\mathrm{(new)}}_{t_a}\bz \rho\|\sigma\jz}]\le \hat\vfi(a)$. Finally, note that assumption yields that $\psi(1)=0$, and hence $\vfi(a)\ge a-\psi(1)=a$. \[thm:Stein2\] The direct rate is lower bounded by $D_1(\N\|\sigma)$, i.e., $$\label{composite rate} R(\{\N\notimes\}_{n\in\bN}\|\{\sigma^{\otimes n}\}_{n\in\bN})\ge D_1(\N\|\sigma).$$ The proposition is trivial when $D_1(\N\|\sigma)=0$, and hence for the rest we assume $D_1(\N\|\sigma)>0$. By lemma \[lemma:positive rate\], for every $0<a<D_1(\N\|\sigma)$ we can find $0<\kappa<\vfi(a)/D(\hil)$, so that – hold. Since we can take $\kappa$ arbitrarily small, and $a$ arbitrarily close to $D_1(\N\|\sigma)$, we see that any rate below $\sup_{0<a<D_1(\N\|\sigma)}\vfi(a)$ is achievable. By lemma \[lemma:positive rate\], $\sup_{0<a<D_1(\N\|\sigma)}\vfi(a)\ge \sup_{0<a<D_1(\N\|\sigma)}a=D_1(\N\|\sigma)$, proving the assertion. The strong converse for the simple i.i.d. case [@ON2] yields immediately the strong converse for the composite i.i.d. case. We include a proof for completeness. \[thm:sc\] If $\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}\sigma^{\otimes n}T_n\le-r$ for some sequence of tests $T_n,\,n\in\bN$, then $$\begin{aligned} \label{sc} \limsup_{n\to+\infty}\frac{1}{n}\log\inf_{\rho\in\N}\operatorname{Tr}\rho^{\otimes n}T_n \le \inf_{t>1}\frac{t-1}{t}\left[-r+\inf_{\rho\in\N}{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}\right].\end{aligned}$$ If $r>D_1(\N\|\sigma)$ then the RHS of is strictly negative, and hence the worst-case success probability $\inf_{\rho\in\N}\operatorname{Tr}\rho^{\otimes n}T_n$ goes to zero exponentially fast. As a consequence, holds as an equality. Following [@Nagaoka2] (see also [@MO]), we can use the monotonicity of the Rényi divergences under measurements for $\alpha>1$ [@FL; @MO; @Renyi_new; @WWY] to obtain that for any sequence of tests $T_n,\,n\in\bN$, any $\rho\in\N$, and any $t>1$, $$\begin{aligned} Q_t\nw(\rho^{\otimes n}\|\sigma^{\otimes n}) &\ge Q_t\nw\bz \left\{\operatorname{Tr}\rho^{\otimes n}T_n,\operatorname{Tr}\rho^{\otimes n}(I_n-T_n)\right\}\| \left\{\operatorname{Tr}\sigma^{\otimes n}T_n,\operatorname{Tr}\sigma^{\otimes n}(I_n-T_n)\right\}\jz\\ &\ge \bz\operatorname{Tr}\rho^{\otimes n}T_n\jz^{t}\bz \operatorname{Tr}\sigma^{\otimes n}T_n\jz^{1-t},\end{aligned}$$ which yields $$\begin{aligned} \frac{1}{n}\log\operatorname{Tr}\rho^{\otimes n}T_n \le \frac{t-1}{t}\left[\frac{1}{n}\log\operatorname{Tr}\sigma^{\otimes n}T_n+{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}\right].\end{aligned}$$ Taking first the infimum in $\rho\in\N$, and then the limsup in $n$, we obtain . Since $\inf_{t>1}\inf_{\rho\in\N}{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}= \inf_{\rho\in\N}\inf_{t>1}{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}=D_1(\N\|\sigma)$, we see that if $r>D_1(\N\|\sigma)$ then there exists a $t>1$ such that $-r+\inf_{t>1}\inf_{\rho\in\N}{D^{\mathrm{(new)}}_{t}\bz \rho\|\sigma\jz}<0$, and hence the RHS of is strictly negative. The rest of the statements follow immediately. Theorem \[thm:Stein2\] shows the existence of a sequence of tests such that the type II error probability decays exponentially fast with rate $D_1(\N\|\sigma)$, while the type I error probability goes to zero. Note that for this statement, it is enough to choose $\delta_n$ polynomially decaying; e.g. $\delta_n:=1/n^2$ does the job, and we get an improved exponent for the type II error, $\limsup_{n\to+\infty}\frac{1}{n}\log\beta_n(S_{n,a})\le -\vfi(a)$. Theorem \[thm:Stein\] yields more detailed information in the sense that it shows that for any rate $r$ below the optimal rate $D_1(\N\|\sigma)$, there exists a sequence of tests along which the type II error decays with the given rate $r$, while the type I error also decays exponentially fast; moreover, – provide a lower bound on the rate of the type I error. Note that if $\N$ is finite then the approximation process can be omitted, and we obtain the bounds $$\limsup_{n\to+\infty}\frac{1}{n}\log\alpha_n(S_{n,a})\le -\hat\vfi(a),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\limsup_{n\to+\infty}\frac{1}{n}\log\beta_n(S_{n,a})\le -\vfi(a).$$ These bounds are not optimal; indeed, in the simple i.i.d. case the quantum Hoeffding bound theorem [@ANSzV; @Hayashi; @HMO2; @Nagaoka2] shows that the above inequalities become equalities with $\vfi$ and $\hat\vfi$ replaced by $\vfi\old(a):=\sup_{0<t\le 1}\{at-\log Q_t\old(\rho\|\sigma\},\,\hat\vfi\old(a):=\vfi\old(a)-a$, and if $\rho$ and $\sigma$ don’t commute then $\vfi\old(a)>\vfi(a)$ and $\hat\vfi\old(a)>\hat\vfi(a)$ for any $0<a<D_1(\rho\|\sigma)$, due to the Araki-Lieb-Thirring inequality [@Araki; @LT]. On the other hand, the RHS of is known to give the exact strong converse rate in the simple i.i.d. case [@MO]. The above arguments can also be used to obtain bounds on the direct rate in the case of states with arbitrary correlations. In this case, however, it may not be possible to find a suitable approximation procedure, and hence we restrict our attention to the case of finite null-hypothesis. Thus, for every $n\in\bN$, our alternative hypothesis $H_1$ is given by some state $\sigma_n\in\S(\hil_n)$, where $\hil_n$ is some finite-dimensional Hilbert space, and the null-hypothesis $H_0$ is given by $\N_n=\{\rho_{1,n},\ldots,\rho_{r,n}\}\subset\S(\hil_n)$, where $r\in\bN$ is some fixed number. We assume that $\operatorname{supp}\rho_{i,n}\subseteq\operatorname{supp}\sigma_n$ for every $i$ and $n$. \[thm:correlated Stein\] In the above setting, we have $$\begin{aligned} \limsup_{n\to+\infty}\frac{1}{n}\log\alpha_n(S_{n,a})&\le -\sup_{0<t<1}\left\{a(t-1)-\max_{1\le i\le r}\limsup_{n\to+\infty}\frac{1}{n}\log Q_t\nw(\rho_{i,n}\|\sigma_n)\right\},\label{correlated Stein1}\\ \limsup_{n\to+\infty}\frac{1}{n}\log\beta_n(S_{n,a})&\le -\sup_{0<t<1}\left\{at-\max_{1\le i\le r}\limsup_{n\to+\infty}\frac{1}{n}\log Q_t\nw(\rho_{i,n}\|\sigma_n)\right\}\le -a,\label{correlated Stein3}\end{aligned}$$ where $S_{n,a}:=\left\{e^{-na}\sum_i\rho_{i,n}-\sigma_n>0\right\}$. As a consequence, the direct rate is lower bounded as $$\begin{aligned} \label{correlated Stein4} R(\{\N_n\}_{n\in\bN}\|\{\sigma_n\}_{n\in\bN})&\ge \sup_{0<t<1}\min_{1\le i\le r}\liminf_{n\to+\infty}\frac{1}{n} D_t\nw(\rho_{i,n}\|\sigma_n).\end{aligned}$$ If $\limsup_{n\to+\infty}\frac{1}{n}\log\dim\hil_n<+\infty$ then we also have $$\begin{aligned} \label{correlated Stein5} R(\{\N_n\}_{n\in\bN}\|\{\sigma_n\}_{n\in\bN})&\ge \min_i\derleft\psi_i\old(1),\end{aligned}$$ where $\derleft$ stands for the left derivative, and $\psi_i\old(t):=\limsup_{n\to+\infty}\frac{1}{n}\log Q_t\old(\rho_{i,n}\|\sigma_n)$. The same argument as in Theorem \[thm:Stein\] yields and , from which follows immediately. Assume now that $L:=\limsup_{n\to+\infty}\frac{1}{n}\log\dim\hil_n<+\infty$. By Corollary \[cor:old-new bounds for states\], we have $$\begin{aligned} \label{correlated Stein2} \limsup_{n\to+\infty}\frac{1}{n}\log Q_t\nw(\rho_{i,n}\|\sigma_n)\le t\psi_i\old(t)+(t-1)^2 L.\end{aligned}$$ Note that $\psi_i\old(t)$ is the pointwise limsup of convex functions, and hence it is convex, too. Moreover, the support condition $\operatorname{supp}\rho_{i,n}\subseteq\operatorname{supp}\sigma_n$ implies $\psi_i\old(1)=0$. Hence, we have $\lim_{t\nearrow 1}\frac{t}{t-1}\psi_i\old(t)=\derleft\psi_i\old(1)$. Combining this with and , we see that $\limsup_{n\to+\infty}\frac{1}{n}\log\alpha_n(S_{n,a})<0$ for all $a<\min_i\derleft\psi_i\old(1)$. Taking into account , we get . \[rem:correlated Stein\] Note that under suitable regularity, we have $\displaystyle{\derleft\psi_i\old(1)=\lim_{n\to+\infty}\frac{1}{n}{D_{1}\bz \rho_{i,n}\|\sigma_n\jz}}$, and hence $$\begin{aligned} \label{correlated Stein7} R(\{\N_n\}_{n\in\bN}\|\{\sigma_n\}_{n\in\bN})&\ge \min_i\lim_{n\to+\infty}\frac{1}{n}{D_{1}\bz \rho_{i,n}\|\sigma_n\jz}.\end{aligned}$$ This is clearly satisfied in the i.i.d. case, and we recover . There are also various important special cases of correlated states where the above holds. This is the case, for instance, if all the $\rho_{i,n}$ and $\sigma_n$ are $n$-site restrictions of gauge-invariant quasi-free states on a fermionic or bosonic chain (the latter type of states are also called Gaussian states). In this case $\lim_{n\to+\infty}\frac{1}{n}{D_{1}\bz \rho_{i,n}\|\sigma_n\jz}$ can be expressed by an explicit formula in terms of the symbols of the states; see [@MHOF; @M] for details. Another class of states where the above holds is when $\rho_{i,n}$ and $\sigma_n$ are group-invariant restrictions of i.i.d. states on a spin chain [@HMH]. In this case one can use the same approximation procedure as in the i.i.d. case, and hence holds for $\N_n:=\{\rho_{i,n}:\,i\in\I\}$, where $\I$ is an arbitrary (not necessearily finite) index set. Finally, we show that the above considerations for the composite null-hypothesis yield the direct rate also for the case. In this setting we have a probability measure $\mu$ on the Borel sets of $\S(\hil)$ such that $\bar\rho_n:=\int_{\S(\hil)}\rho^{\otimes n}\,d\mu$ is well-defined for every $n\in\bN$. The null-hypothesis is given by the sequence $\N_n=\{\bar\rho_n\},\,n\in\bN$. Note that in this case the null-hypotheses is simple, i.e., $\N_n$ consists of one single element, but it is not i.i.d. Let $$D^*:=\sup\left\{\inf_{\rho\in\S(\hil)\setminus H}{D_{1}\bz \rho\|\sigma\jz}:\,H\subset\S(\hil) \text{ Borel set with }\mu(H)=0 \right\},$$ which is essentially the relative entropy distance of $\operatorname{supp}\mu$ from $\sigma$, modulo subsets of zero measure. Assume that $D^*>0$, since otherwise holds trivially. For every $0<a<D^*$, there exists a subset $\N(a)$ such that $a<{D_{1}\bz \N(a)\|\sigma\jz}\le D^*$ and $\mu(\S(\hil)\setminus \N(a))=0$. Applying Theorem \[thm:Stein\] to the composite i.i.d. problem with null-hypothesis $\N(a)$, we get the existence of a sequence of tests $T_n,\,n\in\bN$, such that $$\begin{aligned} &\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}\sigma^{\otimes n}T_n\le -a,\\ &\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}\bar\rho_n(I-T_n) \le \limsup_{n\to+\infty}\frac{1}{n}\log\sup_{\rho\in\N(a)}\operatorname{Tr}\rho^{\otimes n}(I-T_n)<0.\end{aligned}$$ Hence, the direct rate for the averaged i.i.d. problem is lower bounded by $D^*$, i.e., $$\label{averaged rate} R(\{\bar\rho_n\}_{n\in\bN}\|\{\sigma^{\otimes n}\}_{n\in\bN})\ge D^*.$$ Universal state compression --------------------------- Consider a sequence of Hilbert spaces $\hil_n,\,n\in\bN$, and for each $n$, let $\N_n\subset\S(\hil_n)$ be a set of states. An is a sequence $(\C_n,\D_n),\,n\in\bN$, where $\C_n:\,\B(\hil^{\otimes n})\to\B(\kil_n)$ is the compression map, and $\D_n:\,\B(\kil_n)\to\B(\hil^{\otimes n})$ is the decompression. We use two different measures for the fidelity of $(\C_n,\D_n)$, defined as $$F(\C_n,\D_n):=\inf_{\rho_n\in\N_n}F_e(\rho_n,(\D_n\circ\C_n)\rho_n),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\hat F(\C_n,\D_n):=\inf_{\rho_n\in\N_n}F(\rho_n,(\D_n\circ\C_n)\rho_n),$$ where $F$ stands for the fidelity, and $F_e$ for the the entanglement fidelity (see Section \[sec:prel\]). Due to the monotonicity of the fidelity under partial trace, we have $F(\C_n,\D_n)\le\hat F(\C_n,\D_n)$. Let $\left[\C_n(\N_n)\right]$ be the projection onto the subspace generated by the supports of $\C_n(\rho_n),\,\rho_n\in\N_n$. We say that a compression rate $R$ is achievable if there exists an asymptotic compression scheme $(\C_n,\D_n),\,n\in\bN$, such that $$\lim_{n\to+\infty}F(\C_n,\D_n)=1{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\text{and}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}\left[\C_n(\N_n)\right]\le R.$$ The smallest achievable compression rate is the $R(\{\N_n\}_{n\in\bN})$. We say that the compression problem is i.i.d. if $\hil_n=\hil^{\otimes n}$ and $\N_n=\N^{[\otimes n]}:=\{\rho^{\otimes n}:\,\rho\in\N\}$ for every $n\in\bN$, where $\hil=\hil_1$, and $\N\subset\S(\hil)$. It was shown in [@Schumacher] (see also [@JS]) that in the simple i.i.d. case, projecting the state onto its entropy-typical subspace yields the entropy as an achievable coding rate, which is also optimal. In Section 10.3 of [@Hayashibook], Neyman-Pearson type projections were used instead of the typical projections, and exponential bounds were obtained for the error probability for suboptimal coding rates. An extension of the typical projection technique was used in [@JHHH] to obtain universal state compression, i.e., it was shown that for any $s>0$ there exists a coding scheme of rate $s$ that is asymptotically error-free for any state of entropy less than $s$. Theorem \[thm:universal compression\] below shows that the use of Neyman-Pearson projections can also be extended to obtain universal state compression. Since Theorem \[thm:universal compression\] is essentially a special case of Theorems \[thm:Stein\] and \[thm:sc\] with the choice $\sigma:=I$, we omit the proof. The only part that doesn’t follow immediately from Theorems \[thm:Stein\] and \[thm:sc\] is relating the fidelity to the success probability of the generalized state discrimination problem; this, however, is standard and we refer the interested reader to Section 12.2.2 in [@NC]. Let $\psi(t)$, $\vfi(a)$ and $\hat\vfi(a)$ be defined as in –, with $\sigma:=I$. Note that in this case $Q_t\nw(\rho\|\sigma)=Q_t\old(\rho\|\sigma)=\operatorname{Tr}\rho^t$. \[thm:universal compression\] In the i.i.d. case, for every $\kappa>0$, $a\in\bR$, and $n\in\bN$, let $\delta_n:=e^{-n\kappa}$, let $\N_{\delta_n}\subset\N_n$ be a subset as in lemma \[lemma:state approximation\], and let $S_{n,a}:=\left\{e^{-na}\sum_{\rho\in\N_{\delta_n}}\rho^{\otimes n}-I_n>0\right\}$. Define $$\begin{aligned} \C_n(.):=S_{n,a}(.)S_{n,a}+{{|\psi_n\rangle\langle \psi_n|}}\operatorname{Tr}(.)(I-S_{n,a}),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\D_n:=\operatorname{id},\end{aligned}$$ where $\psi_n$ is an arbitrary unit vector in the range of $S_{n,a}$. For every $a\in\bR$ and $\kappa>0$, we have $$\begin{aligned} \limsup_{n\to+\infty}\frac{1}{n}\log(1-F(\C_n,\D_n))&\le -\min\{\kappa,\hat\vfi(a)-\kappa D(\hil)\},\\ \limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}\left[\C_n(\N_n)\right]&\le -\vfi(a)+\kappa D(\hil).\end{aligned}$$ On the other hand, for any coding scheme $(\C_n,\D_n),\,n\in\bN$, we have $$\begin{aligned} \limsup_{n\to+\infty}\frac{1}{n}\log \hat F(\C_n,\D_n)\le \inf_{t>1}\frac{t-1}{t}\left[\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}\left[\C_n(\N_n)\right]-\sup_{\rho\in\N}S_t(\rho)\right].\end{aligned}$$ where $S_t(\rho):=\frac{1}{1-t}\log\operatorname{Tr}\rho^t$ is the Rényi entropy of $\rho$ with parameter $t$. \[cor:universal compression\] The optimal compression rate is equal to the maximum entropy, i.e., $$\begin{aligned} R(\{\N^{[\otimes n]}_{n\in\bN}\})=\sup_{\rho\in\N}S(\rho).\end{aligned}$$ We recover the result of [@JHHH] by choosing $\N:=\{\rho\in\S(\hil):\,S(\rho)\le s\}$. Theorem \[thm:universal compression\] and Corollary \[cor:universal compression\] can be extended to correlated states and averaged states the same way as the analogous results for state discrimination in Section \[sec:Stein\]. Since these extensions are trivial, we omit the details. The simple i.i.d. state compression problem can also be formulated in an ensemble setting, which is in closer resemblance with the usual formulation of classical source coding. In that formulation, a discrete i.i.d. quantum information source is specified by a finite set $\{\rho_x\}_{x\in\X}\subset\S(\hil)$ of states and a probability distribution $p$ on $\X$. Invoking the source $n$ times, we obtain a state $\rho_{{\underline{x}}}:=\rho_{x_1}\otimes\ldots\otimes\rho_{x_n}$ with probability $p_{{\underline{x}}}:=p(x_1)\cdot\ldots\cdot p(x_n)$. The fidelity of a compression-decompression pair $(\C_n,\D_n)$ is then defined as $F(\C_n,\D_n):=\sum_{x\in\X}p(x)F_e\bz\rho_x,\D_n\circ\C_n\jz$. In the classical case the signals $\rho_x$ can be identified with a system of orthogonal rank $1$ projections, $\C_n$ and $\D_n$ are classical stochastic maps, and $F(\C_n,\D_n)$ as defined above gives back the usual expression for the success probability. It follows from standard properties of the fildelity that the optimal compression rate, under the constraint that $F(\C_n,\D_n)$ goes to $1$ asymptotically, only depends on the average state $\rho(p):=\sum_xp(x)\rho_x$, and is equal to $S(\rho(p))$. Theorem \[thm:universal compression\] and Corollary \[cor:universal compression\] thus also provide the optimal compression rate and exponential bounds on the error and success probabilities in the ensemble formulation, for multiple quantum sources. Classical capacity of compound channels {#sec:capacity} --------------------------------------- Recall that by a channel $W$ we mean a map $W:\,\X\to\S(\hil)$, where $\X$ is some input alphabet (which can be an arbitrary non-empty set) and $\hil$ is a finite-dimensional Hilbert space. For a channel $W:\,\X\to\S(\hil)$, we define its $n$-th i.i.d. extension $W^{\otimes n}$ as the channel $W^{\otimes n}:\,\X^n\to\S(\hil^{\otimes n})$, defined as $$W^{\otimes n}(x_1,\ldots,x_n):=W(x_1)\otimes\ldots \otimes W(x_n),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}x_1,\ldots,x_n\in\X.$$ It is obvious from the explicit formula for $\chi_{\alpha}\old$ that $$\label{chi additivity} \chi_{\alpha}\old(W^{\otimes n},p^{\otimes n})=n\chi_{\alpha}\old(W,p),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}n\in\bN,$$ where $p^{\otimes n}\in\P_f(\X^n)$ is the $n$-th i.i.d. extension of $p$, defined as $p^{\otimes n}(x_1,\ldots,x_n):=p(x_1)\cdot\ldots\cdot p(x_n)$, $x_1,\ldots,x_n\in\X$. It is not known whether the same additivity property holds for $\chi_{\alpha}\nw$. Note that in our definition of a channel, we didn’t make any assumption on the cardinality of the input alphabet $\X$, nor did we require any further mathematical properties from $W$, apart from being a function to $\S(\hil)$. The usual notion of a quantum channel is a special case of this definition, where $\X$ is the state space of some Hilbert space and $W$ is a completely positive trace-preserving convex map. In this case, however, our definition of the i.i.d. extensions are more restrictive than the usual definition of the tensor powers of a quantum channel. Indeed, our definition corresponds to the notion of quantum channels with product state encoding. Hence, our definition of the classical capacity below corresponds to the classical capacity of quantum channels with product state encoding. Let $W_i:\,\X\to\S(\hil),\,i\in\I$, be a set of channels with the same input alphabet $\X$ and the same output Hilbert space $\hil$, where $\I$ is any index set. A code $\C=(\C_e,\C_d)$ for $\{W_i\}_{i\in\I}$ consists of an encoding $\C_e:\{1,\ldots,M\}\to\X$ and a decoding $\C_d:\{1,\ldots,M\}\to\B(\hil)_+$, where $\{\C_d(1),\ldots,\C_d(M)\}$ is a POVM on $\hil$, and $M\in\bN$ is the size of the code, which we will denote by $|\C|$. The worst-case average error probability of a code $\C$ is $$\begin{aligned} p_e\bz\{W_i\}_{i\in\I},\C\jz&:= \sup_{i\in\I}\frac{1}{|\C|}\sum_{k=1}^{|\C|}\operatorname{Tr}W_i(\C_e(k))(I-\C_d(k)).\end{aligned}$$ Consider now a sequence $\W:=\{\W_n\}_{n\in\bN}$, where each $\W_n$ is a set of channels with input alphabet $\X^n$ and output space $\hil^{\otimes n}$. The $C(\W)$ of $\W$ is the largest number $R$ such that there exists a sequence of codes $C^{(n)}=\bz C^{(n)}_e,C^{(n)}_d\jz$ with $$\begin{aligned} \lim_{n\to+\infty}p_e(\W_n,\C_n)=0{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\text{and}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\liminf_{n\to+\infty}\frac{1}{n}\log|\C_n|\ge R.\end{aligned}$$ We say that $\W$ is simple i.i.d. if $\W_n$ consists of one single element $W^{\otimes n}$ for every $n\in\bN$ with some fixed channel $W$. In this case we denote the capacity by $C(W)$. The Holevo-Schumacher-Westmoreland theorem [@Holevo; @SW] tells that in this case $$\label{HSW} C(W)\ge\hat\chi(W)=\sup_{p\in\P_f(\X)}\chi(W,p),$$ where $\chi(W,p)$ is the Holevo quantity , and $\hat\chi(W)$ is the Holevo capacity of the channel. It is easy to see that actually holds as an equality, i.e., no sequence of codes with a rate above $\sup_{p\in\P_f(\X)}\chi(W,p)$ can have an asymptotic error equal to zero; this is called the weak converse to the channel coding theorem, while the strong converse theorem [@ON; @Winter] says that such sequences of codes always have an asymptotic error equal to $1$. Here we will consider two generalizations of the simple i.i.d. case: In the case $\W_n=\{W_i^{\otimes n}\}_{i\in\I}$ for some fixed channels $W_i:\,\X\to\S(\hil)$. In the case $\W_n$ consists of the single element $\bar W_n:=\sum_{i\in\I}\gamma_iW_i^{\otimes n}$, where $\I$ is finite, and $\gamma$ is a probability distribution on $\I$. The capacity of finite averaged channels has been shown to be equal to $\sup_{p\in\P_f(\X)}\min_i\chi(W_i,p)$ in [@DD], and the same formula for the capacity of a finite compound channel follows from it in a straightforward way. The protocol used in [@DD] to show the achievability was to use a certain fraction of the communication rounds to guess which channel the parties are actually using, and then code for that channel in the remaining rounds. These results were generalized to arbitray index sets $\I$ in [@BB], using a different approach. The starting point in [@BB] was the following random coding theorem from [@HN] (for the exact form below, see [@MD]). \[thm:error bound\] Let $W:\,\X\to\S(\hil)$ be a channel. For any $M\in\bN$, and any $p\in\P_f(\X)$, there exists a code $\C$ such that $|\C|=M$ and $$p_{e}(W,\C)\le\kappa(c,\alpha) M^{1-\alpha}\operatorname{Tr}\what W(p)^{\alpha}(\hat p\otimes W(p))^{1-\alpha}$$ for every $\alpha\in(0,1)$ and every $c>0$, where $\kappa(c,\alpha):=(1+c)^{\alpha}(2+c+1/c)^{1-\alpha}$. Applying the general properties of the Rényi divergences, established in Section \[sec:Renyi\], together with the single-shot coding theorem of Theorem \[thm:error bound\], we get a very simple proof of the achievability part of the coding theorems in [@DD] and [@BB]. Since our primary interest is the applicability of the inequalities of Section \[sec:Renyi\], we only consider the achievability part and not the converse. The key step of our approach is the following extension of Theorem \[thm:error bound\] to multiple channels. \[thm:capacity\] Let $W_i:\,\X\to\S(\hil),\,i\in \I$, be a set of channels, where $\I$ is a finite index set. For every $R\ge 0$, every $n\in\bN$ and every $p\in\P_f(\X)$, there exists a code $\C_n,\,n\in\bN$, such that for every $\alpha\in(0,1)$, $$\begin{aligned} &|\C_n|\ge \exp(nR),{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\text{and}\nonumber\\ &p_e\bz\{W_i^{\otimes n}\}_{i\in\I},\C_n\jz \le 8|\I|^2\exp\left[ n(\alpha-1)\bz\alpha\min_i\chi_{\alpha}\old(W_i,p)-R-(\alpha-1)\log\dim(\hil)\jz\right]. \label{channel error bound}\end{aligned}$$ Let $M_n:=\lceil \exp(nR)\rceil,\,n\in\bN$ and $\gamma_i:=1/|\I|,\,i\in\I$. Applying Theorem \[thm:error bound\] to $\bar W_n=\sum_{i\in\I}\gamma_iW_i^{\otimes n}$, $M_n$ and $p^{\otimes n}$, we get the existence of a code $\C_n$ with $|\C_n|=M_n$ and $$\begin{aligned} \label{n-shot bound} p_{e}(\bar W_n,\C_n)\le 8M_n^{1-\alpha} Q_{\alpha}\old\bz \sum_{i\in\I}\gamma_i\what W_i^{\otimes n}(p^{\otimes n})\Big\|\hat p^{\otimes n}\otimes \bar W_n(p^{\otimes n})\jz\end{aligned}$$ for every $\alpha\in(0,1)$. Here we chose $c=1$, and used the upper bound $\kappa(1,\alpha)\le 8$. We can further upper bound the RHS above as $$\begin{aligned} &Q_{\alpha}\old\bz \sum_{i\in\I}\gamma_i\what W_i^{\otimes n}(p^{\otimes n})\Big\|\hat p^{\otimes n}\otimes \bar W_n(p^{\otimes n})\jz\nonumber\\ &{\mbox{ }\mbox{ }}\le Q_{\alpha}\nw\bz \sum_{i\in\I}\gamma_i\what W_i^{\otimes n}(p^{\otimes n}) \Big\|\hat p^{\otimes n}\otimes \bar W_n(p^{\otimes n})\jz\label{ineqq1}\\ &{\mbox{ }\mbox{ }}\le \sum_{i\in\I}\gamma_i^{\alpha} Q_{\alpha}\nw\bz \what W_i^{\otimes n}(p^{\otimes n})\big\|\hat p^{\otimes n}\otimes \bar W_n(p^{\otimes n})\jz\label{ineqq2}\\ &{\mbox{ }\mbox{ }}\le \sum_{i\in\I}\gamma_i^{\alpha}\sup_{\sigma\in\S(\hil^{\otimes n})} Q_{\alpha}\nw\bz \what W_i^{\otimes n}(p^{\otimes n})\big\|\hat p^{\otimes n}\otimes \sigma\jz \label{ineqq3}\\ &{\mbox{ }\mbox{ }}\le \sum_{i\in\I}\gamma_i^{\alpha}\sup_{\sigma\in\S(\hil^{\otimes n})} Q_{\alpha}\old\bz \what W_i^{\otimes n}(p^{\otimes n})\big\|\hat p^{\otimes n}\otimes \sigma\jz^{\alpha}(\dim\hil^{\otimes n})^{(\alpha-1)^2}\label{ineqq4}\\ &{\mbox{ }\mbox{ }}= \sum_{i\in\I}\gamma_i^{\alpha}\exp\bz \alpha(\alpha-1)\chi_{\alpha}\old(W_i^{\otimes n},p^{\otimes n}\jz(\dim\hil)^{n(\alpha-1)^2}\label{ineqq5}\\ &{\mbox{ }\mbox{ }}= \sum_{i\in\I}\gamma_i^{\alpha}\exp\bz n\alpha(\alpha-1)\chi_{\alpha}\old(W_i,p)\jz(\dim\hil)^{n(\alpha-1)^2} \label{ineqq6},\\ &{\mbox{ }\mbox{ }}\le |\I|\exp\bz n\alpha(\alpha-1)\min_{i\in\I}\chi_{\alpha}\old(W_i,p)\jz(\dim\hil)^{n(\alpha-1)^2} \label{ineqq7}\end{aligned}$$ where is due to , is due to , is trivial, follows from Corollary \[cor:old-new bounds for states\], and is due to . Note that $$\begin{aligned} p_{e}(\bar W_n,\C_n)&= \frac{1}{|\I|}\sum_{i\in\I}p_e(W_i^{\otimes n},\C_n) \ge \frac{1}{|\I|}\sup_{i\in\I}p_e(W_i^{\otimes n},\C_n).\label{ineqq8}\end{aligned}$$ Combining , , and , we get . \[cor:finite capacity lower bounds\] Let $W_i:\,\X\to\S(\hil),\,i\in \I:=\{1,\ldots,r\}$, be a set of channels, and let $\gamma_1,\ldots,\gamma_r$ be a probability distribution on $\I$ with strictly positive weights. Then $$\begin{aligned} \label{finite capacity} C\bz\left\{\sum\nolimits_i\gamma_i W_i^{\otimes n}\right\}_{n\in\bN}\jz = C\bz\{W_i^{\otimes n}:\,i\in\I\}_{n\in\bN}\jz \ge \sup_{p\in\P_f(\X)}\min_i\chi(W_i,p).\end{aligned}$$ Let $R<\min_i\chi(W_i,p)$, and for every $n\in\bN$, let $\C_n$ be a code as in Theorem \[thm:capacity\]. Then $\liminf_{n\to\infty}\frac{1}{n}\log|\C_n|\ge R$, and $$\begin{aligned} \limsup_{n\to\infty}\frac{1}{n}\log p_e\bz\{W_i^{\otimes n}\}_{i\in\I},\C_n\jz \le (\alpha-1)\bz\alpha\min_i\chi_{\alpha}\old(W_i,p)-R-(\alpha-1)\log\dim(\hil)\jz.\end{aligned}$$ Note that $$\lim_{\alpha\nearrow 1}\bz\alpha\min_i\chi_{\alpha}\old(W_i,p)-R-(\alpha-1)\dim(\hil)\jz =\chi(W,p)-R$$ due to , and hence there exists an $\alpha_0\in(0,1)$ such that the upper bound in goes to zero exponentially fast for every $\alpha\in(\alpha_0,1)$. This proves the inequality in , and the equality of the two capacities is trivial. When the channels are completely positive trace-preserving affine maps on the state space of a Hilbert space, the above results can be extended to the case of infinitely many channels by a simple approximation argument. It is easy to see that the same argument doesn’t work when the channels can be arbitrary maps on an input alphabet. Note that the classical capacity considered in the theorem below is the product-state capacity. Let $\hil\inp$ and $\hil$ be finite-dimensional Hilbert spaces, and $W_i:\,\S(\hil\inp)\to\S(\hil),\,i\in\I$, be completely positive trace-preserving affine maps, where $\I$ is an arbitrary index set. Then $$\begin{aligned} \label{finite capacity} C\bz\{W_i^{\otimes n}:\,i\in\I\}_{n\in\bN}\jz \ge \sup_{p\in\P_f(\X)}\inf_i\chi(W_i,p).\end{aligned}$$ We assume that $\sup_{p\in\P_f(\X)}\inf_i\chi(W_i,p)>0$, since otherwise the assertion is trivial. Let $V$ be the vector space of linear maps from $\B(\hil\inp)$ to $\B(\hil)$, equipped with the norm ${\left\| \map\right\|}:=\sup\{{\left\| \map(X)\right\|}_1:\,{\left\| X\right\|}_1\le 1\}$, and let $D$ denote the real dimension of $V$. Let $\kappa>0$, and for every $n\in\bN$, let $I(n)$ be a finite index set such that $|I(n)|\le (1+2e^{n\kappa})^D$ and $\delta_n:=\sup_{i\in\I}\inf_{j\in\I(n)}{\left\| W_i-W_j\right\|}\le e^{-n\kappa}$. The existence of such index sets is guaranteed by lemma \[lemma:state approximation\]. Let $p\in\P_f(\S(\hil\inp))$ be such that $\inf_i\chi(W_i,p)>0$, and for every $n\in\bN$, let $\C_n$ be a code as in Theorem \[thm:capacity\], with $\I(n)$ in place of $\I$. It is easy to see that $$\begin{aligned} p_e\bz\{W_i^{\otimes n}\}_{i\in\I(n)},\C_n\jz \ge p_e\bz\{W_i^{\otimes n}\}_{i\in\I},\C_n\jz-n\delta_n,\end{aligned}$$ and hence we have $$\begin{aligned} &p_e\bz\{W_i^{\otimes n}\}_{i\in\I},\C_n\jz\\ &{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}{\mbox{ }\mbox{ }}\le 8|\I(n)|^2\exp\left[ n(\alpha-1)\bz\alpha\inf_{i\in\I}\chi_{\alpha}\old(W_i,p)-R-(\alpha-1)\log\dim(\hil)\jz\right] +ne^{-n\kappa}.\end{aligned}$$ Let $0<R<\inf_{i\in\I}\chi_{\alpha}\old(W_i,p)$. By the same argument as in the proof of Corollary \[cor:finite capacity lower bounds\], there exists an $\alpha\in(0,1)$ such that $\vfi:=\alpha\inf_{i\in\I}\chi_{\alpha}\old(W_i,p)-R-(\alpha-1)\log\dim(\hil)>0$. Choosing then $\kappa$ such that $2\kappa D/(1-\alpha)<\vfi$, we see that the error probability goes to zero exponentially fast, while the rate is at least $R$. This shows that $C\bz\{W_i^{\otimes n}:\,i\in\I\}_{n\in\bN}\jz\ge \inf_i\chi(W_i,p)$, and taking the supremum over $p$ yields the assertion. Acknowledgment {#acknowledgment .unnumbered} ============== The author is grateful to Professor Fumio Hiai and Nilanjana Datta for discussions. This research was supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme. The author also acknowledges support by the European Research Council (Advanced Grant “IRQUAT”). Part of this work was done when the author was a Marie Curie research fellow at the School of Mathematics, University of Bristol. [99]{} T. Ando: ; , 203–241 (1979) T. Ando, F. Hiai: H. Araki: ; ; Volume 19, Issue 2, pp. 167–170, (1990) K.M.R. Audenaert, J. Calsamiglia, Ll. Masanes, R. Munoz-Tapia, A. Acin, E. Bagan, F. Verstraete.: ; **98** 160501, (2007) K.M.R. Audenaert, M. Nussbaum, A. Szkoła, F. Verstraete: ; , 251–283 (2008) K.M.R. Audenaert: ; **4**, pp. 78–83, (2008) Koenraad M.R. Audenaert, Nilanjana Datta: , , (2013) Salman Beigi: ; , **54**, 122202 (2013) Igor Bjelakovic, Jean-Dominique Deuschel, Tyll Krüger, Ruedi Seiler, Rainer Siegmund-Schultze, Arleta Szkoła: ; Vol. 260, Issue 3, pp. 659–671, (2005) Igor Bjelakovic, Jean-Dominique Deuschel, Tyll Krüger, Ruedi Seiler, Rainer Siegmund-Schultze, Arleta Szkoła: ; Vol. 279, pp. 559–584, (2008) I. Bjelakovic, H. Boche: ; **55**, 3360–3374, (2009) E.A. Carlen, E.H. Lieb: ; textbf[189]{}, 59–68, (1999) Eric A. Carlen, Elliot H. Lieb: ; **83**, pp. 107–126, (2008) I. Csiszár: ; **41**, 26–34, (1995) N. Datta, T.C. Dorlas: ; , vol. 40, p. 8147, (2007) N. Datta: ; , vol. 55, no. 6, pp. 2816–2826, (2009). Nilanjana Datta and Felix Leditzky: ; **47**, 045304, (2014) Rupert L. Frank and Elliott H. Lieb: ; , (2013). M. Hayashi, H. Nagaoka: ; **49**, (2003) M. Hayashi: [*Quantum Information Theory: An Introduction*]{}; Springer, (2006). M. Hayashi: ; **76**, 062301, (2007) F. Hiai, D. Petz: ; , 99–114 (1991). F. Hiai, M. Mosonyi, T. Ogawa: ; **49**, 032112, (2008) F. Hiai, M. Mosonyi, M. Hayashi, , **50**, 103304 (2009) F. Hiai: (GSIS selected lectures); , 139–248 (2010) F. Hiai: ; **439**, 1568–1589, (2013) A.S. Holevo: ; , vol. 44, no. 1, pp. 269-273, (1998) V. Jaksic, Y. Ogata, C.-A. Pillet, R. Seiringer: ; **24**, no. 6, 1230002, (2012) R. Jozsa, B. Schumacher: ; Volume 41, Issue 12, (1994) R. Jozsa, M. Horodecki, P. Horodecki, R. Horodecki: ; **81**, 1714–1717, (1998) R. König and S. Wehner: ; , vol. 103, no. 7, 070504, (2009) E.H. Lieb: ; , 267–288 (1973) E.H. Lieb, W. Thirring: ; pp. 269–297. Princeton University Press, Princeton, (1976) Vitali D. Milman, Gideon Schechtman: ; , Springer-Verlag Berlin Heidelberg, (1986) M. Mosonyi, F. Hiai, T. Ogawa, M. Fannes, , **49**, 072104, (2008) M. Mosonyi, , **50**, 032104, (2009) M. Mosonyi, N. Datta: ; **50**, 072104 (2009) M. Mosonyi, F. Hiai: ; , **57**, 2474–2487, (2011) M. Mosonyi, T. Ogawa: ; , (2013) Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr, Serge Fehr, Marco Tomamichel: ; **54**, 122203, (2013) H. Nagaoka: ; , World Scientific, (2005) H. Nagaoka: ; M.A. Nielsen, I.L. Chuang: ; , Cambridge, UK, (2000) J. Nötzel: ; , (2013) T. Ogawa, H. Nagaoka: ; , vol. 45, no. 7, pp. 2486-2489, (1999) T. Ogawa, H. Nagaoka: ; , 2428–2433 (2000). D. Petz: ; **23**, 57–65, (1986) R. Renner: [*Security of Quantum Key Distribution*]{}, PhD dissertation, Swiss Federal Institute of Technology Zurich, Diss. ETH No. 16242, (2005). A. Rényi: ; , Vol. I, pp. 547–561, Univ. California Press, Berkeley, California, (1961) B. Schumacher: ; **51**, 2738, (1995) B. Schumacher, M. Westmoreland: ; , vol. 56, no. 1, pp. 131-138, (1997) R. Sibson: ; **14**, 149–161, (1969) M. Tomamichel, R. Colbeck, R. Renner: ; **55**, 5840–5847, (2009) M. Tomamichel: ; , (2006) H. Umegaki: ; **14**, 59–85, (1962) Mark M. Wilde, Andreas Winter, Dong Yang: ; , (2013) A. Winter: ; , vol. 45, no. 7, pp. 2481–2485, (1999)

--- abstract: 'We calculate, in the continuum limit of quenched lattice QCD, the form factor that enters in the decay rate of the semileptonic decay $B\rightarrow D \ell \nu$. Making use of the step scaling method (SSM), previously introduced to handle two scale problems in lattice QCD, and of flavour twisted boundary conditions we extract $G^{B\rightarrow D}(w)$ at finite momentum transfer and at the physical values of the heavy quark masses. Our results can be used in order to extract the CKM matrix element $V_{cb}$ by the experimental decay rate without model dependent extrapolations.' author: - 'G.M. de Divitiis$^{a,b}$, E. Molinaro$^{c}$, R. Petronzio$^{a,b}$, N. Tantalo$^{b,d}$' title: 'Quenched lattice calculation of the $B\rightarrow D \ell \nu$ decay rate' --- Introduction ============ The central goal of flavour physics is the determination of the Cabibbo–Kobayashi–Maskawa [@Cabibbo:1963yz; @Kobayashi:1973fv] quark mixing matrix. A set of redundant and precise measurements can also provide informations about possible new physics [@Bona:2006ah]. In turn, precise measurements need an adequate theoretical determination of hadron matrix elements of the weak currents. Non perturbative tools and in particular lattice QCD can eventually provide the required precision. In this letter we address the determination at finite momentum transfer of the form factor that enters in the semileptonic heavy meson decay rate. In the infinite quark mass limit it is parametrized by the Isgur-Wise function [@Isgur:1989ed]. In particular, we calculate the matrix elements entering the determination of $V_{cb}$. The calculation that we present misses the effects of dynamical fermions and is not the final one. Nevertheless, it accomplishes for the first time the calculation at finite momentum transfer and at the physical values of the heavy quark masses, allowing to compare experimental data without additional model dependent extrapolations. We use a method [@Guagnelli:2002jd] already applied successfully to the determination of heavy quark masses and decay constants [@deDivitiis:2003wy; @deDivitiis:2003iy; @Guazzini:2006bn], called the step scaling method (SSM), and special boundary conditions, called flavour twisted [@deDivitiis:2004kq], that shift by an arbitrary amount the discretized set of lattice momenta (see also [@Sachrajda:2004mi; @Flynn:2005in]). The step scaling method allows to reconcile large quark masses with adequate lattice resolution and large physical volumes; the flavour twisted boundary conditions allow to perform a calculation at non zero momentum transfer with good accuracy. We present the main results of our computation and stress their phenomenological implications; a more detailed discussion on technical aspects will be presented elsewhere [@comingsoon]. Form factors and decay rate =========================== The semileptonic decay of a pseudoscalar meson into another pseudoscalar meson is mediated by the vector part of the weak $V-A$ current. The corresponding matrix element can be parametrized in terms of two form factors. Among possible parametrizations we chose the following one $$\begin{aligned} \frac{\langle \mathcal{M}_f\vert \ V^\mu \ \vert \mathcal{M}_i\rangle}{\sqrt{M_i M_f}}= (v_i+v_f)^\mu \ h_+ + (v_i-v_f)^\mu \ h_- \nonumber\end{aligned}$$ where $M_{i,f}$ and $v_{i,f}=p_{i,f}/M_{i,f}$ are the meson masses and $4$-velocities. The form factors depend upon the masses of the initial and final particles and upon $w\equiv v_f\cdot v_i$ $$\begin{aligned} &&h_\pm \equiv h_\pm^{i\rightarrow f}(w)\equiv h_\pm(w,M_i,M_f) \nonumber \\ \nonumber \\ &&1\le w \le (M_i^2+M_f^2)/2M_iM_f \nonumber\end{aligned}$$ In the case where $M_i$ is the $B$ meson mass and $M_f$ is the $D$ meson mass the maximum value of $w$ is around $1.6$. In the infinite mass limit the form factors become $$\begin{aligned} &&h_+^{i\rightarrow f}(w)=\xi(w) \nonumber \\ &&\qquad \qquad \qquad \qquad \qquad M_f,M_i\rightarrow\infty \nonumber \\ &&h_-^{i\rightarrow f}(w)=0 \nonumber\end{aligned}$$ where $\xi(w)$ is the universal Isgur-Wise function; the conservation of the vector current implies $\xi(1)=1$. In what follows we will find deviations from such a limit that will allow a precise discussion on the onset of the HQET regime [@comingsoon]. The differential decay rate for the process $B\rightarrow D\ell\nu$, in the case of massless leptons, is given by $$\begin{aligned} &&\frac{d\Gamma^{B\rightarrow D\ell\nu}}{dw}= \nonumber \\ \nonumber \\ &&\vert V_{cb}\vert^2 \frac{G_F^2}{48\pi^3}(M_{B}+M_{D})^2M_{D}^3(w^2-1)^{3/2} \left[ G^{B\rightarrow D}(w)\right]^2 \nonumber\end{aligned}$$ The phase space factor $(w^2-1)^{3/2}$ in the decay rate makes its experimental determination harder as $w$ approaches $1$ and the value at zero recoil is obtained from an extrapolation. The function $G^{B\rightarrow D}(w)$, $$G^{i\rightarrow f}(w)=h^{i\rightarrow f}_+(w)\ -\ \frac{M_f-M_i}{M_f+M_i}\ h^{i\rightarrow f}_-(w)$$ is needed in order to extract $V_{cb}$ by the measurement of the decay rate. Previous lattice calculations by the Fermilab collaboration [@Hashimoto:1999yp; @Okamoto:2004xg] quote the value of $G^{B\rightarrow D}(w)$ only at zero recoil where it can be extracted with good statistical accuracy by using the so called “double ratio” technique. In the following we calculate $G^{B\rightarrow D}(w)$ in the range $1\le w \le 1.2$ that includes values of $w$ where experimental data are available. Lattice Observables {#sec:notations} =================== We have carried out the calculation within the $O(a)$ improved Schrödinger Functional formalism [@Luscher:1992an; @Sint:1993un] with $T=2L$ and vanishing background fields. In order to fix the notations, we introduce the following lattice operators $$\begin{aligned} &&O_{rs}=\frac{a^6}{L^3}\sum_{{\bf y},{\bf z}}{\bar{\zeta}_r({\bf y})\gamma_5\zeta_s({\bf z})} \nonumber \\ &&O_{rs}^\prime=\frac{a^6}{L^3}\sum_{{\bf y},{\bf z}}{\bar{\zeta}^\prime_r({\bf y})\gamma_5 \zeta_s^\prime({\bf z})} \nonumber \\ \nonumber \\ \nonumber \\ &&A^0_{rs}(x)=\bar{\psi}_r(x)\gamma_5\gamma^0\psi_s(x), \quad P_{rs}(x)=\bar{\psi}_r(x)\gamma_5\psi_s(x) \nonumber \\ \nonumber \\ &&V^\mu_{rs}(x)=\bar{\psi}_r(x)\gamma^\mu\psi_s(x), \quad T^{\mu\nu}_{rs}(x)=\bar{\psi}_r(x)\gamma^\mu\gamma^\nu\psi_s(x) \nonumber \\ \nonumber \\ \nonumber \\ &&\mathcal{A}^0_{rs}(x)=A^0_{rs}(x)+ac_A\frac{\partial_0+\partial_0^*}{2}P_{rs}(x) \nonumber \\ \nonumber \\ &&\mathcal{V}^\mu_{rs}(x)=V^\mu_{rs}(x)+ac_V\frac{\partial_\nu+\partial_\nu^*}{2}T^{\mu\nu}_{rs}(x) \nonumber\end{aligned}$$ where $r$ and $s$ are flavour indexes while $\zeta$ and $\zeta^\prime$ are boundary fields at $x_0=0$ and $x_0=T$ respectively. The improvement coefficients $c_A$ and $c_V$ have been taken from refs. [@Luscher:1996ug; @Guagnelli:1997db; @Sint:1997jx]. We have calculated the following correlation functions $$\begin{aligned} &&\mathcal{F}_{i\rightarrow f}^\mu(x_0;{\bf p_i},{\bf p_f})= \frac{a^3}{2}\sum_{\bf x}{ \langle O_{li} \ \mathcal{V}^\mu_{if}(x)\ O_{fl}^\prime \rangle } \nonumber \\ \nonumber \\ &&f^{\mathcal{A}}_f(x_0,{\bf p_f})=-\sum_{{\bf x}}{ \langle O_{lf} \mathcal{A}^0_{fl}(x)\rangle } \nonumber\end{aligned}$$ where $i$ and $f$ refer to the heavy flavours while $l$ to the light one. The external momenta have been set by using flavour twisted b.c. for the heavy flavours; in particular we have used $$\begin{aligned} \psi_{i,f}(x+\hat{1}L)=e^{i\theta_{i,f}}\psi_{i,f}(x) \nonumber\end{aligned}$$ leading to $$\begin{aligned} p_1=\frac{\theta_{i,f}}{L}+\frac{2\pi k_1}{L},\qquad k_1\in \mathbb{N} \nonumber\end{aligned}$$ and ordinary periodic b.c. in the other spatial directions and for the light quarks. We work in the Lorentz frame in which the parent particle is at rest (${\bf p_i=0}$). In this frame $w$ is obtained from the ratio between the energy and the mass of the final particle $w=E_f/M_f$. $\beta$ $T\times L^3$ $N_{cnfg}$ -------- --------- ----------------- ------------ $L_0A$ 7.300 $48\times 24^3$ 277 $L_0B$ 7.151 $40\times 20^3$ 224 $L_0C$ 6.963 $32\times 16^3$ 403 $L_0a$ 6.737 $24\times 12^3$ 608 $L_0b$ 6.420 $16\times 8^3$ 800 $L_1A$ 6.737 $48\times 24^3$ 260 $L_1B$ 6.420 $32\times 16^3$ 350 $L_1a$ 6.420 $32\times 16^3$ 360 $L_1b$ 5.960 $16\times 8^3$ 480 $L_2A$ 6.420 $48\times 24^3$ 250 $L_2B$ 5.960 $24\times 12^3$ 592 : \[tab:sims\] Table of lattice simulations. By assuming ground state dominance and by relying on the conservation of the vector current, the matrix elements of $V^\mu$ can be extracted by considering the ratio $$\begin{aligned} \langle V^\mu \rangle_{D1}^{i\rightarrow f}\equiv \langle \mathcal{M}_f\vert \ V^\mu \ \vert \mathcal{M}_i\rangle_{D1}= \nonumber \\ \nonumber \\ 2\sqrt{M_i E_f}\frac{\mathcal{F}_{i\rightarrow f}^\mu(T/2;{\bf 0},{\bf p_f})}{ \sqrt{ \mathcal{F}_{i\rightarrow i}^0(T/2;{\bf 0},{\bf 0}) \mathcal{F}_{f\rightarrow f}^0(T/2;{\bf p_f},{\bf p_f}) }} \nonumber \\ \nonumber \\ \label{eq:def1}\end{aligned}$$ An alternative definition of the matrix elements ($D2$), which reduces to the previous one ($D1$) in the limits of infinite volume and zero lattice spacing, can be obtained by considering $$\begin{aligned} \langle V^\mu \rangle_{D2}^{i\rightarrow f}\equiv \langle \mathcal{M}_f\vert \ V^\mu \ \vert \mathcal{M}_i\rangle_{D2}= \nonumber \\ \nonumber \\ 2\frac{\sqrt{M_i} E_f f_A^f(T/2,{\bf 0})}{\sqrt{M_f}f_A^f(T/2,{\bf p_f})} \frac{\mathcal{F}_{i\rightarrow f}^\mu(T/2;{\bf 0},{\bf p_f})}{ \sqrt{ \mathcal{F}_{i\rightarrow i}^0(T/2;{\bf 0},{\bf 0}) \mathcal{F}_{f\rightarrow f}^0(T/2;{\bf 0},{\bf 0}) }} \nonumber \\ \nonumber \\ \label{eq:def2}\end{aligned}$$ In eqs. (\[eq:def1\]) and (\[eq:def2\]) the renormalization factors $Z_V$ and $Z_A$ cancel in the ratios together with factors containing the improvement coefficients $b_V$ and $b_A$. By introducing the following ratio $$\begin{aligned} x_f&=&\frac{\mathcal{F}_{f\rightarrow f}^1(T/2;{\bf 0},{\bf p_f})} {\mathcal{F}_{f\rightarrow f}^0(T/2;{\bf 0},{\bf p_f})} \nonumber \\ \nonumber \\ &=& \frac{\langle \mathcal{M}_f\vert \ V^1 \ \vert \mathcal{M}_f\rangle} {\langle \mathcal{M}_f\vert \ V^0 \ \vert \mathcal{M}_f\rangle} = \frac{\sqrt{w^2-1}}{w+1} \nonumber\end{aligned}$$ we can define $w$, as well as $E_f$ and $M_f$, entirely in terms of three point correlation functions. This definition of $w$ is noisier than the one that can be obtained in terms of ratios of two point correlation functions; however it leads to exact vector current conservation when $M_f=M_i$ and reduces the final statistical error on the form factors. The two definitions of the matrix elements lead to two definitions of $G^{i\rightarrow f}$ $$\begin{aligned} G^{i\rightarrow f} &=& \frac{\sqrt{r}\langle V^0 \rangle^{i\rightarrow f}}{M_i(1+r)}\left\{ 1\; + \;\frac{wr-1}{r\sqrt{w^2-1}} \; \frac{\langle V^1 \rangle^{i\rightarrow f}} {\langle V^0 \rangle^{i\rightarrow f}}\right\} \nonumber \\ \nonumber \\ r&=&\frac{M_f}{M_i} \label{eq:fdefme}\end{aligned}$$ The last equation is not defined at $w=1$; this is due to the second term in parenthesis that we extrapolate at zero recoil before calculating $G^{i\rightarrow f}(w=1)$. the step scaling method ======================= ![\[fig:sigmas\] Step scaling functions $\sigma^{i\rightarrow D}(w;L_0,L_1)$, lower plot, and $\sigma^{i\rightarrow D}(w;L_1,L_2)$, upper plot, as functions of the inverse of the RGI heavy quark mass $m_i$ of the parent meson measured in GeV. The black vertical lines represent the physical values of $m_c$ and $m_b$. The data correspond to the definition $D1$.](pics/ssf_F_D_both){width="48.00000%"} The SSM has been introduced to cope with two-scale problems in lattice QCD. In the calculation of heavy-light meson properties the two scales are the mass of the heavy quarks ($b$,$c$) and the mass of the light quarks ($u$,$d$,$s$). Here we consider the form factor $G^{i\rightarrow f}$ as a function of $w$, the volume, $L^3$, and identify heavy meson states by the corresponding RGI quark masses that in the infinite volume limit lead to the physical meson spectrum [@deDivitiis:2003iy]; the RGI quark masses are measured by the lattice version of the PCAC relation and are not affected by finite volume effects (see [@comingsoon] for further details). First we compute the observable $G^{B\rightarrow D}(w;L_0)$ on a small volume, $L_0$, which is chosen to accommodate the dynamics of the $b$-quark. As in our previous work we fixed $L_0=0.4$ fm. Then we evaluate a first effect of finite volume by evolving the volume from $L_0$ to $L_1=0.8$ fm and computing the ratio $$\sigma^{i\rightarrow D}(w;L_0,L_1)=\frac{G^{i\rightarrow D}(w;L_1)}{G^{i\rightarrow D}(w;L_0)}$$ The crucial point is that the step scaling functions are calculated by simulating heavy quark masses $m_i$ smaller than the $b$-quark mass. The physical value $\sigma^{B\rightarrow D}(w;L)$ is obtained by a smooth extrapolation in $1/m_i$ that relies on the HQET expectations and upon the general idea that finite volume effects, measured by the $\sigma$’s, are almost insensitive to the high energy scale. The final result is obtained by further evolving the volume from $L_1$ to $L_2=1.2$ fm, according to $$\begin{aligned} &&G^{B\rightarrow D}(w;L_2)= \nonumber \\ \nonumber \\ &&=G^{B\rightarrow D}(w;L_0)\ \sigma^{B\rightarrow D}(w;L_0,L_1)\ \sigma^{B\rightarrow D}(w;L_1,L_2) \nonumber\end{aligned}$$ Physical values require also usual continuum and chiral extrapolations. ![\[fig:chiral\] Light quark mass dependence of $G^{B\rightarrow D}(w;L_0)$ in the continuum limit. The different sets of points correspond to different values of $m_l$ ranging from about $m_s$ to about $m_s/4$. The data correspond to the definition $D1$.](pics/F_D_04_ml){width="48.00000%"} ![\[fig:2defs\] Comparison of the two definitions of $G^{B\rightarrow D}(w;L)$ at $L_0=0.4$ fm (upper plot) and at $L_2=1.2$ fm (lower plot).](pics/F_D_2defs){width="48.00000%"} results ======= In figure \[fig:sigmas\] we can test the validity of the SSM. The step scaling functions are almost insensitive to the heavy quark mass $m_i$ of the parent meson for values larger than $m_c$. Moreover, finite volume effects are already very small at $L=0.8$ fm, in particular at zero recoil. We present results already extrapolated to the chiral and continuum limits. We have simulated three different lattices for $L_0$ and two for the other volumes (see table \[tab:sims\]) observing small $O(a)$ effects on the form factor and on the step scaling functions. Further details on the continuum extrapolations will be given in ref. [@comingsoon]. Concerning the chiral behaviour, our results do not show any sizable dependence upon the light quark mass, as shown in figure \[fig:chiral\] for $G^{B\rightarrow D}(w;L_0)$. The same feature is observed for all the combinations of heavy quark masses and on all the volumes that we have simulated. Nevertheless we make a linear extrapolation to reach the chiral limit; the resulting error largely accounts for the systematics due to these extrapolations. As discussed in sec. \[sec:notations\], we used two definitions of the form factor that, at finite volume, differ by the finite volume effects. A check of the convergence of our SSM can be obtained by comparing these two definitions on the smallest volume (figure \[fig:2defs\] upper plot) and on the largest one (figure \[fig:2defs\] lower plot). We see that, while the small volume results differ, the final ones converge to a common value giving us confidence of a correct accounting of finite volume effects. [cccc]{} $w$ & $G^{B\rightarrow D}(w)$ & $N_f$ & reference\ \ 1.00 & 1.026(17) & 0 & this work\ 1.03 & 1.001(19) & 0 & this work\ 1.05 & 0.987(15) & 0 & this work\ 1.10 & 0.943(11) & 0 & this work\ 1.20 & 0.853(21) & 0 & this work\ \ 1.00 & 1.058(20) & 0 & [@Hashimoto:1999yp]\ 1.00 & 1.074(24) & 2+1 & [@Okamoto:2004xg]\ concluding remarks ================== In table \[tab:gw\] we quote our final results obtained by averaging over the two definitions and by combining in quadrature statistical errors with the systematic ones due to the small residual dispersion between $D1$ and $D2$. As a comparison we show previous lattice results obtained by the Fermilab collaboration at zero recoil. The existence of predictions up to $w\simeq 1.2$ and physical $b$ and $c$ quark masses allows a direct comparison with experimental data, as shown in figure \[fig:vcb\]. The comparison has been done by extracting the value of $V_{cb}$ by the ratio of the experimental and lattice data at $w=1.2$; as an indication, we get $V_{cb}=3.84(9)(42)\times10^{-2}$, where the first error is from our lattice result, $G^{B\rightarrow D}(w=1.2)=0.853(21)$, and the second from the experimental decay rate, $|V_{cb}|G^{B\rightarrow D}(w=1.2)=0.0327(35)$, as deduced from the plots of refs. [@Belle; @CLEO; @Albertus:2005vd]. The extension of this calculation to the unquenched case does not present problems of principle. The recursive matching process can be extended to the sea quark masses that, alternatively, can be kept to their physical values if the Schrödinger Functional formalism is used. Moreover, flavour twisted boundary conditions can be used for heavy valence quarks also in the $N_f=3$ unquenched theory. The real case will further differ by the heavy flavour determinants that can be computed perturbatively. ![\[fig:vcb\] Comparison of $|V_{cb}|\ G^{B\rightarrow D}(w)$ with available experimental data. The plot has been done by normalizing our lattice data with the value of $V_{cb}$ extracted at $w=1.2$.](pics/vcb_12){width="48.00000%"} The simulations required to carry on this project have been performed on the INFN apeNEXT machines at Rome “La Sapienza”. We warmly thank A. Lonardo, D. Rossetti and P. Vicini for technical advice. [99]{} N. Cabibbo, Phys. Rev. Lett.  [**10**]{} (1963) 531. M. Kobayashi and T. Maskawa, Prog. Theor. Phys.  [**49**]{} (1973) 652. M. Bona [*et al.*]{} \[UTfit Collaboration\], JHEP [**0610**]{} (2006) 081 \[arXiv:hep-ph/0606167\]. N. Isgur and M. B. Wise, Phys. Lett.  B [**237**]{} (1990) 527. M. Guagnelli, F. Palombi, R. Petronzio and N. Tantalo, Phys. Lett. B [**546**]{} (2002) 237 \[arXiv:hep-lat/0206023\]. G. M. de Divitiis, M. Guagnelli, F. Palombi, R. Petronzio and N. Tantalo, Nucl. Phys.  B [**672**]{}, 372 (2003) \[arXiv:hep-lat/0307005\]. G. M. de Divitiis, M. Guagnelli, R. Petronzio, N. Tantalo and F. Palombi, Nucl. Phys.  B [**675**]{}, 309 (2003) \[arXiv:hep-lat/0305018\]. D. Guazzini, R. Sommer and N. Tantalo, PoS [**LAT2006**]{} (2006) 084 \[arXiv:hep-lat/0609065\]. G. M. de Divitiis, R. Petronzio and N. Tantalo, Phys. Lett.  B [**595**]{}, 408 (2004) \[arXiv:hep-lat/0405002\]. C. T. Sachrajda and G. Villadoro, Phys. Lett.  B [**609**]{}, 73 (2005) \[arXiv:hep-lat/0411033\]. J. M. Flynn, A. Juttner and C. T. Sachrajda \[UKQCD Collaboration\], Phys. Lett.  B [**632**]{}, 313 (2006) \[arXiv:hep-lat/0506016\]. G. M. de Divitiis, R. Petronzio and N. Tantalo, “Quenched lattice calculation of semileptonic heavy-light meson form factors” \[arXiv:0707.0587\]. S. Hashimoto, A. X. El-Khadra, A. S. Kronfeld, P. B. Mackenzie, S. M. Ryan and J. N. Simone, Phys. Rev.  D [**61**]{} (2000) 014502 \[arXiv:hep-ph/9906376\]. M. Okamoto [*et al.*]{}, Nucl. Phys. Proc. Suppl.  [**140**]{} (2005) 461 \[arXiv:hep-lat/0409116\]. M. Luscher, R. Narayanan, P. Weisz and U. Wolff, Nucl. Phys.  B [**384**]{} (1992) 168 \[arXiv:hep-lat/9207009\]. S. Sint, Nucl. Phys.  B [**421**]{}, 135 (1994) \[arXiv:hep-lat/9312079\]. M. Luscher, S. Sint, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys.  B [**491**]{} (1997) 323 \[arXiv:hep-lat/9609035\]. M. Guagnelli and R. Sommer, Nucl. Phys. Proc. Suppl.  [**63**]{}, 886 (1998) \[arXiv:hep-lat/9709088\]. S. Sint and P. Weisz, Nucl. Phys.  B [**502**]{}, 251 (1997) \[arXiv:hep-lat/9704001\]. J. E. Bartelt [*et al.*]{} \[CLEO Collaboration\], Phys. Rev. Lett.  [**82**]{} (1999) 3746 \[arXiv:hep-ex/9811042\]. K. Abe [*et al.*]{} \[Belle Collaboration\], Phys. Lett. B [**526**]{} (2002) 258 \[arXiv:hep-ex/0111082\]. C. Albertus, E. Hernandez, J. Nieves and J. M. Verde-Velasco, Phys. Rev.  D [**71**]{} (2005) 113006 \[arXiv:hep-ph/0502219\].

--- abstract: 'Let $X$ be a finite collection of sets. We count the number of ways a disjoint union of $n-1$ subsets in $X$ is a set in $X$, and estimate this number from above by $|X|^{c(n)}$ where $$c(n)=\left(1-\frac{(n-1)\ln (n-1)}{n\ln n} \right)^{-1}.$$ This extends the recent result of Kane–Tao, corresponding to the case $n=3$ where $c(3)\approx 1.725$, to an arbitrary finite number of disjoint $n-1$ partitions.' author: - | Paata Ivanisvili[^1]\ Department of Mathematics, Kent State University\ 1300 Lefton Esplanade, Kent OH 44242\ pivanisv@kent.edu date: | \ Mathematics Subject Classifications: 11B30 title: '**Convolution estimates and the number of disjoint partitions**' --- Introduction ============ Let $\{0,1\}^{m}$ be the Hamming cube of dimension $m \geq 1$. Set $1^{m}:=(1,1,\ldots, 1)$ to be the corner of $\{0,1\}^{m}$. Take a finite number of functions $f_{1}, \ldots, f_{n} : \{0,1\}^{m} \to \mathbb{R}$, and define the convolution at the corner $1^{m}$ as $$\begin{aligned} f_{1}*f_{2}*\ldots*f_{n}(1^{m}):=\sum_{x_{j} \in \{0,1\}^{m}\, :\, x_{1}+\cdots+x_{n}=1^{m}} f_{1}(x_{1})\cdots f_{n}(x_{n}). \end{aligned}$$ Given $f :\{0,1\}^{m} \to \mathbb{R}$ define its $L^{p}$ norm ($p\geq 1$) in a standard way $$\begin{aligned} \|f\|_{p} := \left( \sum_{x \in \{0,1\}^{m}} |f(x)|^{p}\right)^{1/p}.\end{aligned}$$ For $n\in \mathbb{N}$ we set $$\begin{aligned} p_{n} := \frac{\ln \frac{n^{n}}{(n-1)^{n-1}}}{\ln n}.\end{aligned}$$ Our main result is the following theorem \[taot\] For any $n,m \geq 1$, and any $f_{1}, \ldots, f_{n} :\{0,1\}^{m} \to \mathbb{R}$ we have $$\begin{aligned} \label{maininequality} f_{1}*f_{2}*\ldots*f_{n}(1^{m})\leq \prod_{j=1}^{n} \|f_{j}\|_{p_{n}}.\end{aligned}$$ Moreover, for each fixed $n$ exponent $p_{n}$ is the best possible in the sense that it cannot be replaced by any larger number. As an immediate application we obtain the following corollary (see Section \[kak\] below). \[taoc\] Let $X$ be a finite collection of sets. Then $$\begin{aligned} \label{bolof} \left| \left\{ (A_{1}, \ldots, A_{n-1}, A) \in \underbrace{X\times \cdots \times X}_{n}\; :\ A = \bigsqcup_{j=1}^{n-1} A_{j} \right\} \right|\leq |X|^{\frac{n}{p_{n}}},\end{aligned}$$ where $\bigsqcup$ denotes the disjoint union, and $|X|$ denotes cardinality of the set. The corollary extends a recent result of Kane–Tao [@KT], corresponding to the case $n=3$ where $\frac{3}{p_{3}} \approx 1.725$, to an arbitrary finite number $n \geq 3$ disjoint partitions. The proof of the theorem ======================== Following [@KT] the proof goes by induction on the dimension of the cube $\{0,1\}^{m}$. The case $m=1$, which is the most difficult, is the main contribution of the current paper. Basis: $m=1$ ------------ In this case, set $f_{j}(0)=u_{j}$ and $f_{j}(1)=v_{j}$ for $j=1,\ldots, n$. Then the inequality (\[maininequality\]) takes the form $$\begin{aligned} \label{i1} \sum_{j=1}^{n} u_{j} \prod_{\substack{i=1 \\ i\neq j }}^{n}v_{i} \leq \prod_{j=1}^{n} \left(|u_{j}|^{p_{n}}+|v_{j}|^{p_{n}} \right)^{1/p_{n}}.\end{aligned}$$ We do encourage the reader first to try to prove (\[i1\]) in the case $n=3$, or visit [@KT], to see what is the obstacle. For example, when $n=3$ equality in (\[i1\]) is attained at several points. Besides, direct differentiation of (\[i1\]) reveals many “bad” critical points at which finding the values of (\[i1\]) would require numerical computations [@KT]. The number of critical points together with equality cases increases as $n$ becomes larger, therefore one is forced to come up with a different idea. We will overcome this obstacle by looking at (\[i1\]) in dual coordinates. Without loss of generality we can assume that $u_{j}$ and $v_{j}$ are nonnegative for $j=1,\ldots, n$. Moreover, we can assume that $v_{j}\neq 0$ for all $j$ otherwise the inequality (\[i1\]) is trivial. Let us divide (\[i1\]) by $\prod_{j=1}^{n} v_{j}$. Denoting $x_{j}:=(u_{j}/v_{j})^{p_{n}}$ we see that it is enough to prove the following lemma. For any $n\geq 2$ and all $x_{1}, \ldots, x_{n} \geq 0$ we have $$\begin{aligned} \label{mine} \left(\sum_{i=1}^{n} x^{1/p_{n}}_{i}\right)^{p_{n}} \leq \prod_{i=1}^{n} (1+x_{i}),\end{aligned}$$ where $p_{n} = \frac{\ln \frac{n^{n}}{(n-1)^{n-1}}}{\ln n}$ For $n=2$ the lemma is trivial. By induction on $n$, monotonicity of the map $$\begin{aligned} p \to \left(\sum_{i=1}^{n} x^{1/p}_{i}\right)^{p},\end{aligned}$$ and the fact that $p_{n}$ is decreasing, we can assume that all $x_{i}$ are strictly positive. For convenience we set $p:=p_{n}$. Introducing new variables we rewrite (\[mine\]) as follows $$\begin{aligned} p \ln \left( \sum x_{i} \right) \leq \sum \ln (1+x_{i}^{p}).\end{aligned}$$ Concavity of the function $\ln (x)$ provides us with a simple representation of the logarithmic function $$\begin{aligned} \ln(x) = \min_{b \in \mathbb{R}} (b+e^{-b}x-1).\end{aligned}$$ Therefore we are left to show that for all $x_{i} >0$ and all $b_{i} \in \mathbb{R}$ we have $$\begin{aligned} B(x,b):= \sum (b_{i}+(1+x_{i}^{p})e^{-b_{i}}-1)-p \ln \left( \sum x_{i} \right) \geq 0,\end{aligned}$$ where $x=(x_{1}, \ldots, x_{n})$ and $b = (b_{1}, \ldots, b_{n})$. Notice that given a vector $b \in \mathbb{R}^{n}$, the infimum of $B(x,b)$ in $x$ cannot be reached at infinity because of the slow growth of the logarithmic function. Therefore, we look at critical points of $B$ in $x$ $$\begin{aligned} x^{*}_{k} = \frac{e^{\frac{b_{k}}{p-1}}}{\left(\sum_{i} e^{\frac{b_{i}}{p-1}} \right)^{\frac{1}{p}}} \quad \text{for} \quad k=1, \ldots, n. \end{aligned}$$ Notice that $\sum x^{*}_{i} = \left( \sum_{i} e^{\frac{b_{i}}{p-1}} \right)^{\frac{p-1}{p}}$. Therefore $$\begin{aligned} B(x^{*}, b) = \sum_{k} (b_{k} + e^{-b_{k}})+1-n-(p-1)\ln \left(\sum_{k} e^{\frac{b_{k}}{p-1}} \right).\end{aligned}$$ Setting $r:=p-1>0$, and introducing new variables again we are left to show that $$\begin{aligned} f(y):=1-n+\sum \ln y^{r}_{i} + \sum \frac{1}{y_{i}^{r}} - r \ln \left(\sum y_{i} \right) \geq 0\end{aligned}$$ for all $y_{i}>0$. It is straightforward to check that $f(y) \geq 0$ on the diagonal, i.e., when $y_{1}=y_{2}=\ldots=y_{n}$. In general, we notice that critical points of $f(y)$ satisfy the equation $$\begin{aligned} \label{crit} \frac{1}{y_{i}} - \frac{1}{y_{i}^{r+1}} = \frac{1}{y_{j}}- \frac{1}{y_{j}^{r+1}}=\frac{1}{\sum y_{k}}.\end{aligned}$$ Equation (\[crit\]) gives the identity $\sum y_{i}^{-r}=n-1$, and so at critical points (\[crit\]) we are only left to show $$\begin{aligned} \label{lll} \sum \ln y_{i} - \ln \left( \sum y_{i} \right)\geq 0. \end{aligned}$$ Since the mapping $$\begin{aligned} s \to \frac{1}{s} - \frac{1}{s^{r+1}}, \quad s>0,\end{aligned}$$ is increasing on $(0, (1+r)^{1/r})$ and decreasing on the remaining part of the ray, we can assume without loss of generality that $k$ numbers of $x_{i}$ equal to $u\geq (1+r)^{1/r}$, and the remaining $n-k$ numbers of $x_{i}$ equal to $v\leq (1+r)^{1/r}$. Moreover, we can assume that $0<k<n$ otherwise the statement is already proved. From (\[crit\]), we have $$\begin{aligned} \label{eq1} \frac{1}{u}-\frac{1}{u^{r+1}} = \frac{1}{v} - \frac{1}{v^{r+1}} = \frac{1}{k u + (n-k)v}.\end{aligned}$$ From the equality of the first and the third expressions in (\[eq1\]) it follows that $$\begin{aligned} \label{expv} v = \frac{u^{r+1}(1-k)+ku}{(u^{r}-1)(n-k)}.\end{aligned}$$ In order $v$ to be positive we assume that the numerator of (\[expv\]) is non negative. If we plug the expression for $v$ from (\[expv\]) into the first equality of (\[eq1\]) then after some simplifications we obtain the following equation in the variable $z:=u^{r} \geq 1+r$ $$\begin{aligned} \label{yrb} \frac{(z-1)^{r} (n-k)^{r+1}}{(z(1-k)+k)^{r}} = (n-1)z-k.\end{aligned}$$ It follows from (\[eq1\]) that $(ku+(n-k)v)^{r} = \left(\frac{z}{z-1}\right)^{r} z$, and so using (\[yrb\]) we obtain $$\begin{aligned} v^{r} = \frac{z (n-k)}{(n-1)z-k}.\end{aligned}$$ Therefore at critical points (\[lll\]) simplifies as follows $$\begin{aligned} \label{last} (n-1) \ln z + (n-k) \ln \frac{n-k}{(n-1)z-k} - r \ln \frac{z}{z-1} \geq 0.\end{aligned}$$ Now it is pretty straightforward to show that (\[last\]) is non negative even under the assumption $z \geq 1+r$ for $r = p-1 = \frac{(n-1) \ln \frac{n}{n-1}}{\ln n}$. Indeed, notice that $z > \frac{n}{n-1}$, and the map $$\begin{aligned} k \to (n-k)\ln \frac{n-k}{(n-1)z-k} \end{aligned}$$ is increasing on $[1,n-1]$. Therefore it is enough to check nonnegativity of (\[last\]) when $k=1$, in which case the inequality follows again using $z > \frac{n}{n-1}$, and the fact that the map $$\begin{aligned} z \to \frac{\ln \left(1+\frac{1}{z(n-1)-1} \right)}{\ln \left(1+\frac{1}{z-1} \right)}\end{aligned}$$ is increasing for $z \geq \frac{n}{n-1}$. Choice $x_{1}=\ldots=x_{n}=\frac{1}{n-1}$ gives equality in , and this confirms the fact that $p_{n}$ is the best possible in Theorem . Inductive step -------------- Inductive step is the same as in [@KT] without any modifications. This is a standard argument for obtaining estimates on the Hamming cube (see for example [@IV]). In order to make the paper self contained we decided to repeat the argument. Suppose (\[maininequality\]) is true on the Hamming cube of dimension $m$. Without loss of generality assume $f_{j} \geq 0$, and set $g_{j} :=f_{j}^{p}$ for all $j$. Define $$\begin{aligned} B_{n}(y_{1}, \ldots, y_{n}) := y_{1}^{1/p_{n}}\cdots y_{n}^{1/p_{n}}.\end{aligned}$$ For $x_{j} \in \{0,1\}^{m+1}$, let $x_{j} = (\bar{x}_{j}, x'_{j})$ where $\bar{x}_{j}$ is the vector consisting of the first $m$ coordinates of $x_{j}$, and number $x'_{j}$ denotes the last $m+1$ coordinate of $x_{j}$. Set $$\begin{aligned} \tilde{g}_{j}(x'_{j}):=\sum_{\bar{x}_{j} \in \{0,1\}^{m}} g_{j}(\bar{x}_{j}, x'_{j}) \quad j=1,\ldots, n. \end{aligned}$$ We have $$\begin{aligned} &\sum_{x_{j} \in \{0,1\}^{m+1}\; :\; x_{1}+\cdots+x_{n}=1^{m+1}.}B(g_{1}(x_{1}),\ldots, g_{n}(x_{n}))=\\ &\sum_{x'_{j}\in\{0,1\}\; :\; x'_{1}+\cdots+x'_{n}=1.}\; \sum_{\bar{x}_{j} \in \{0,1\}^{m}\; :\; \bar{x}_{1}+\cdots+\bar{x}_{n}=1^{m}.}B(g_{1}(x_{1}),\ldots, g_{n}(x_{n}))\stackrel{\mathrm{induction}}{\leq}\\ & \sum_{x'_{j}\in\{0,1\}\; :\; x'_{1}+\cdots+x'_{n}=1.} B(\tilde{g}_{1}(x'_{1}), \ldots, \tilde{g}_{n}(x'_{n}))\stackrel{\mathrm{basis}}{\leq} \\ &B\left( \sum_{x_{1} \in \{0,1\}^{m+1}} g_{1}(x_{1}), \ldots, \sum_{x_{n} \in \{0,1\}^{m+1}} g_{n}(x_{n})\right).\end{aligned}$$ The proof of Corollary \[taoc\] {#kak} ------------------------------- Without loss of generality we may assume that all the sets $A$ in $X$ are subsets of $\{1,\ldots, m\}$ with some natural $m \geq 1$ (see [@KT]). For $j=1,\ldots, n$ define functions $$f_{j} : \{0,1\}^{m} \to \{0,1\}$$ as follows: $$\begin{aligned} f_{1}(a_{1}, \ldots, a_{m})=\ldots=f_{n-1}(a_{1}, \ldots, a_{m})=1\end{aligned}$$ if the set $\{1\leq i \leq m\; : \; a_{i}=1\}$ lies in $X$, and $f_{j}=0$ otherwise. Finally we define $$\begin{aligned} f_{n}(a_{1}, \ldots, a_{m})=1\end{aligned}$$ if the set $\{1\leq i \leq m\; : \; a_{i}=0\}$ lies in $X$, and $f_{n}=0$ otherwise. Notice that in this case inequality (\[maininequality\]) becomes (\[bolof\]). Acknowledgements {#acknowledgements .unnumbered} ---------------- I would like to thank an anonymous referee, and Benjamin Jaye for helpful comments. [10]{} D. Kane, T. Tao, A bound on Partitioning Clusters. , Volume 24, Issue 2 (2017), Paper \#P2.31, Pages 13. P. Ivanisvili, A. Volberg. Poincaré inequality 3/2 on the Hamming cube. [[`arXiv:1608.04021`](http://arxiv.org/abs/1608.04021)]{} (2016). [^1]: This paper is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

--- abstract: | By making use of the Lewis-Riesenfeld invariant theory, the solution of the Schrödinger equation for the time-dependent linear potential corresponding to the quadratic-form Lewis-Riesenfeld invariant $I_{\rm q}(t)$ is obtained in the present paper. It is emphasized that in order to obtain the general solutions of the time-dependent Schrödinger equation, one should first find the complete set of Lewis-Riesenfeld invariants. For the present quantum system with a time-dependent linear potential, the linear $I_{\rm l}(t)$ and quadratic $I_{\rm q}(t)$ (where the latter $I_{\rm q}(t)$ cannot be written as the squared of the former $I_{\rm l}(t)$, [*i.e.*]{}, the relation $I_{\rm q}(t)= cI_{\rm l}^{2}(t)$ does not hold true always) will form a complete set of Lewis-Riesenfeld invariants. It is also shown that the solution obtained by Bekkar [*et al.*]{} more recently is the one corresponding to the linear $I_{\rm l}(t)$, one of the invariants that form the complete set. In addition, we discuss some related topics regarding the comment \[Phys. Rev. A [**68**]{}, 016101 (2003)\] of Bekkar [*et al.*]{} on Guedes’s work \[Phys. Rev. A [**63**]{}, 034102 (2001)\] and Guedes’s corresponding reply \[Phys. Rev. A [**68**]{}, 016102 (2003)\].\ \ [*PACS:*]{} 03.65.Fd, 03.65.Ge\ [*Keywords:*]{} exact solutions, Lewis-Riesenfeld invariant formulation, unitary transformation address: | $^{1}$ Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical Instrumentation,\ Zhejiang University, Hangzhou SpringJade 310027, P.R. China\ $^{2}$ Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, P.R. China author: - 'Jian-Qi Shen $^{1,}$$^{2}$ [^1]' title: 'Solutions of the Schrödinger equation for the time-dependent linear potential[^2]' --- Introduction ============ Recently, Guedes used the Lewis-Riesenfeld invariant formulation[@Lewis] and solved the one-dimensional Schrödinger equation with a time-dependent linear potential[@Guedes]. More recently, Bekkar [*et al.*]{} pointed out that[@Bekkar] the result obtained by Guedes is merely the particular solution (that corresponds to the null eigenvalue of the linear Lewis-Riesenfeld invariant) rather than a general one. In the comment[@Bekkar], Bekkar [*et al.*]{} stated that they correctly used the invariant method[@Lewis] and gave the general solutions of the time-dependent Schrödinger equation with a time-dependent linear potential[@Bekkar]. However, in the present paper, I will show that although the solutions of Bekkar [*et al.*]{} is more general than that of Guedes[@Guedes], what they finally achieved in their comment[@Bekkar] is still [*not*]{} the [*general*]{} solutions, either. On the contrary, I think that their result [@Bekkar] also belongs to the particular one. The reason for this may be as follows: according to the Lewis-Riesenfeld invariant method[@Lewis], the solutions of the time-dependent Schrödinger equation can be constructed in terms of the eigenstates of the Lewis-Riesenfeld (L-R) invariants. It is known that both the squared of a L-R invariant (denoted by $I(t)$) and the product of two L-R invariants are also the invariants, which agree with the Liouville-Von Neumann equation $\frac{\partial }{\partial t}I(t)+\frac{1}{i}\left[ I(t),H(t)\right] =0$, and that if $I_{a}$ and $I_{b}$ are the two L-R invariants of a certain time-dependent quantum system and $|\psi(t)\rangle$ is the solution of the time-dependent Schrödinger equation (corresponding to one of the invariants, say, $I_{a}$), then $I_{b}|\psi(t)\rangle$ is another solution of this quantum system. So, in an attempt to obtain the [*general*]{} solutions of a time-dependent system, one should first analyze the complete set of all L-R invariants of the system under consideration. Historically, in order to obtain the complete set of invariants, Gao [*et al.*]{} suggested the concept of basic invariants which can generate the complete set of invariants[@Gao], as stated in Ref.[@Gao], the basic invariants can be called invariant generators. As far as Bekkar [*et al.*]{}’s result[@Bekkar] is concerned, the obtained solutions are the ones corresponding only to the linear invariant ([*i.e.*]{}, $I_{\rm l }(t)=A(t)p+B(t)q+C(t)$) that is simply one of the L-R invariants, which form a complete set. It is apparently seen that the quadratic form, $I_{\rm q}(t)=D(t)p^{2}+E(t)(pq+qp)+F(t)q^{2}+A'(t)p+B'(t)q+C'(t)$, is also the one that can satisfy the Liouville-Von Neumann equation, since it is readily verified that the Lie algebraic generators of $I_{\rm q}(t)$ form a Lie algebra, which possesses the following commutators $$\begin{aligned} \left[q^{2}, p^{2}\right]=2i(pq+qp), \quad \left[pq+qp, q^{2}\right]=-4iq^{2}, \quad \left[pq+qp, p^{2}\right]=4ip^{2}, \nonumber \\ \left[q, p^{2}\right]=2ip, \quad \left[p, q^{2}\right]=-2iq, \quad \left[q, pq+qp\right]=2iq, \quad \left[p, pq+qp\right]=-2ip. \label{algebra}\end{aligned}$$ However, for the cubic-form invariant, it is easily seen that there exists no such closed Lie algebra. This point holds true also for the algebraic generators in any high-order L-R invariants $I_{\rm l}^{n}$. So, it is concluded that for the driven oscillator, only the linear $I_{\rm l}(t)$ and quadratic $I_{\rm q}(t)$ will form a complete set of L-R invariants. Note that here $I_{\rm q}(t)$ should not be the squared of $I_{\rm l}(t)$, [*i.e.*]{}, $I_{\rm q}(t)\neq cI_{\rm l}^{2}(t)$, where $c$ is an arbitrary c-number. It is emphasized here that Bekkar [*et al.*]{}’s solution is the one constructed only in terms of the eigenstates of the linear invariant $I_{\rm l}(t)$. Even though only for the linear invariant $I_{\rm l}(t)$ Bekkar [*et al.*]{}’s result[@Bekkar] can truly be viewed as the complete set of solutions, it still cannot be considered general one of the Schrödinger equation, since the latter should contain those corresponding to the quadratic invariant $I_{\rm q}(t)$. In brief, Bekkar [*et al.*]{}’s solution and my solution, which will be found in what follows, together constitute the complete set of solutions of the Schrödinger equation involving a time-dependent linear potential. On the complete set of L-R invariants ===================================== According to the L-R invariant theory[@Lewis], if the eigenstate of the linear invariant $I_{\rm l}(t)$ corresponding to $\lambda_{n}$, [*i.e.*]{}, one of the eigenvalues of $I_{\rm l}(t)$, is $|\lambda_{n}, t\rangle$, then the solution of the Schrödinger equation can be written in the form $$|\Psi(t)\rangle_{\rm Schr}=\sum_{n}c_{n}\exp \left[\frac{1}{i}\phi_{n}(t)\right]|\lambda_{n}, t\rangle \label{2}$$ with $c_{n}$’s and $\phi_{n}(t)$’s being the time-independent coefficients and time-dependent phases[@Lewis; @Gao], respectively. This, therefore, means that the solutions of the time-dependent Schrödinger equation can be constructed in terms of the complete set of eigenvector basis, $\{|\lambda_{n}, t\rangle\}$, of $I_{\rm l}(t)$. Moreover, one can readily verify that the squared, $I_{\rm l}^{2}$, of the linear invariant is the one satisfying the Liouville-Von Neumann equation, and that $I_{\rm l}(t)|\Psi(t)\rangle_{\rm Schr}$ is also a solution (but not another new general one) of the same time-dependent Schrödinger equation, since it is readily verified that $I_{\rm l}(t)|\Psi(t)\rangle_{\rm Schr}$ can also be the linear combination of the eigenstate basis set $\{|\lambda_{n}, t\rangle\}$ of $I_{\rm l}(t)$, [*i.e.*]{}, $$I_{\rm l}(t)|\Psi(t)\rangle_{\rm Schr}=\sum_{n}b_{n}\exp \left[\frac{1}{i}\phi_{n}(t)\right]|\lambda_{n}, t\rangle, \label{1}$$ where the time-independent coefficients $b_{n}$’s are taken $b_{n}=\lambda_{n}c_{n}$, which is obtained via the comparison of the expression (\[1\]) with (\[2\]). Thus, the above discussion shows that the linear invariant $I_{\rm l}$ and its squared $I_{\rm l}^{2}$ have the same eigenstate basis set and therefore $I_{\rm l}$ and $I_{\rm l}^{2}$ cannot form a complete set of L-R invariants. In contrast, if for any c-number $c$, the quadratic $I_{\rm q}$ cannot be written as the squared of linear $I_{\rm l}$ with various integral constants $A_{0}$, $B_{0}$ and $C_{0}$ (for the definition of $A_{0}$, $B_{0}$ and $C_{0}$, see, for example, in Ref.[@Bekkar]), namely, the relation $I_{\rm q}=cI_{\rm l}^{2}$ is always not true, then $\{I_{\rm l}, I_{\rm q}\}$ is the complete set of L-R invariants, which enables us to obtain the general solutions (complete set of solutions) of the time-dependent Schrödinger equation. Perhaps someone will ask such question as, “Does there really exist such quadratic $I_{\rm q}$ that can always not be written in the form $cI_{\rm l}^{2}$?” or “Maybe any $I_{\rm q}$ that satisfies the Liouville-Von Neumann equation can surely be written as the squared of certain $I_{\rm l}$. Really?” Now I will discuss these questions. Consider a given quadratic invariant $I_{\rm q}$ that is written $I_{\rm q}(t)=D(t)p^{2}+E(t)(pq+qp)+F(t)q^{2}+A'(t)p+B'(t)q+C'(t)$ whose time-dependent parameters are determined by the Liouville-Von Neumann equation, and a certain linear invariant $I_{\rm l }(t)=A(t)p+B(t)q+C(t)$, the squared of which is $I_{\rm l}^{2}=A^{2}p^{2}+AB(pq+qp)+B^{2}q^{2}+2C(Ap+Bq+\frac{C}{2})$. Since the functions $A$, $B$ and $C$ can also be determined by the Liouville-Von Neumann equation, the only retained parts left to us to determine is the integral constants $A_{0}$, $B_{0}$ and $C_{0}$. Choose the appropriate integral constants in $A$, $B$ and $C$, and let $I_{\rm q}$ be the squared of $I_{\rm l}$ (should such case exist), and then we have $$\begin{aligned} D=cA^{2}, \quad E=cAB, \quad F=cB^{2}, \nonumber \\ A'=2cAC, \quad B'=2cBC, \quad C'=cC^{2}. \label{eqq6}\end{aligned}$$ If a given $I_{\rm q}$ can really be written as the squared of $I_{\rm l}$, the above six equations are just used to determine the c-number $c$ and the suitable integral constants $A_{0}$, $B_{0}$ and $C_{0}$ in the functions $A$, $B$ and $C$. It is seen that there are only four numbers expected to be determined, and that, in contrast, we have six equations. So, it is possible that there exist potential parameters $c$ and $A_{0}$, $B_{0}$, $C_{0}$ which will not agree with Eqs.(\[eqq6\]) always for a given parameter set $\{D, E, F, A', B', C'\}$, or, for a given parameter set $\{D, E, F, A', B', C'\}$ there are always no such parameters $c$ and $A_{0}$, $B_{0}$, $C_{0}$ which satisfy Eqs.(\[eqq6\]). The existence of $I_{\rm q}$ that cannot be written as the squared of any $I_{\rm l}$ is thus demonstrated. So, in the above we indicate that such two invariants $I_{\rm l}$ and $I_{\rm q}$ (which are independent) form a complete set of L-R invariants. Unitary transformation associated with L-R invariants ===================================================== Now I will solve the time-dependent Schrödinger equation, of which the time-dependent Hamiltonian[@Guedes] is given $$H(t)=\frac{p^{2}}{2m}+f(t)q,$$ by making use of the Lewis-Riesenfeld invariant theory[@Lewis]. The time-dependent L-R invariant used here takes the form $$I_{\rm q}(t)=D(t)p^{2}+E(t)(pq+qp)+F(t)q^{2}+A(t)p+B(t)q+C(t). \label{eq11}$$ With the help of the Liouville-Von Neumann equation, one can arrive at $$\begin{aligned} \dot{D}+\frac{2E}{m}=0, \quad \dot{E}+\frac{F}{m}=0, \quad \dot{F}=0, \nonumber \\ \dot{A}+\frac{B}{m}-2Df=0, \quad \dot{B}-2Ef=0, \quad \dot{C}-fA=0\end{aligned}$$ with dot denoting the derivative with respect to time $t$. The above six equations (referred to as the auxiliary equations[@Gao]) can be used to determine all the time-dependent parameters $A(t)$, $B(t)$, $C(t)$ and $D(t)$, $E(t)$, $F(t)$. In accordance with the L-R theory, solving the eigenstates of the invariant (\[eq11\]) will enable us to obtain the solutions of the time-dependent Schrödinger equation. But, unfortunately, it is not easy for us to immediately solve the eigenvalue equation of the time-dependent invariant (\[eq11\]), for the invariant (\[eq11\]) involves the time-dependent parameters. So, in the following we will use the invariant-related unitary transformation formulation[@Gao], under which the [*time-dependent*]{} invariant in (\[eq11\]) can be transformed into a [*time-independent*]{} one $I_{V}$, and if the eigenstates of $I_{V}$ can be obtained conveniently, the eigenstates of $I_{\rm q}(t)$ can then be easily achieved. Here we will employ two time-dependent unitary transformation operators $$V_{1}(t)=\exp [\eta(t)q+\beta(t)p], \quad V_{2}(t)=\exp [\alpha(t)p^{2}+\rho(t)q^{2}] \label{eq21}$$ to get a [*time-independent*]{} $I_{V}$. The time-dependent parameters $\eta$, $\beta$, $\alpha$ and $\rho$ in (\[eq21\]) are purely imaginary functions, which will be determined in the following subsections. Since the canonical variables (operators) $q$ and $p$ form a non-semisimple Lie algebra, here the first step is to transform $I_{\rm q}(t)$ into $I_{1}(t)$, [*i.e.*]{}, $I_{1}(t)=V_{1}^{\dagger}(t)I_{\rm q}(t)V_{1}(t)$, which no longer involves the canonical variables $q$ and $p$, and the retained Lie algebraic generators in $I_{1}(t)$ are only $p^{2}$, $pq+qp$, $q^{2}$. Note that these three generators also form a Lie algebra (see the commutators (\[algebra\])) . The second step is to obtain the time-independent $I_{V}$, which will be gained via the calculation of $I_{V}=V_{2}^{\dagger}(t)I_{1}(t)V_{2}(t)$. In this step, the obtained $I_{V}$ has no other generators (and time-dependent c-numbers) than $p^{2}$ and $q^{2}$, namely, $I_{V}$ may be written in the form $I_{V}=\varsigma(p^{2}+q^{2})$ with $\varsigma$ being a certain parameter independent of time. It is well known that the eigenvalue equation of $I_{V}$ is of the form $I_{V}|n, q\rangle=(2n+1)\varsigma|n, q\rangle$, where $|n, q\rangle$ stands for the familiar harmonic-oscillator wavefunction. Hence, the eigenstates of the time-dependent L-R invariant $I_{\rm q}(t)$ in (\[eq11\]) can be achieved and the final result is $V_{1}(t)V_{2}(t)|n, q\rangle$ with the eigenvalue being $(2n+1)\varsigma$. The calculation of $I_{1}(t)=V_{1}^{\dagger}(t)I_{\rm q}(t)V_{1}(t)$ -------------------------------------------------------------------- By the aid of the Glauber formula, one can arrive at $$\begin{aligned} I_{1}(t)&=&Dp^{2}+E(pq+qp)+Fq^{2}+[A+2i(E\beta-D\eta)]p+[B+2i(F\beta-E\eta)]q \nonumber \\ &+&C-[-i(B\beta-A\eta)+D\eta^{2}+F\beta^{2}-2E\beta\eta].\end{aligned}$$ If the two relations $$A+2i(E\beta-D\eta)=0, \quad B+2i(F\beta-E\eta)=0 \label{eq22}$$ are satisfied, then we can obtain[^3] $$I_{1}(t)=D(t)p^{2}+E(t)(pq+qp)+F(t)q^{2}.$$ It follows from (\[eq22\]) that the time-dependent parameters in the unitary transformation $V_{1}(t)$ are expressed by $$\eta=\frac{EB-FA}{2i(E^{2}-DF)}, \quad \beta=\frac{DB-EA}{2i(E^{2}-DF)}.$$ The calculation of $I_{V}=V_{2}^{\dagger}(t)I_{1}(t)V_{2}(t)$ ------------------------------------------------------------- By using the Glauber formula, one can arrive at $$I_{V}\equiv V^{\dagger}_{2}(t)I_{1}(t)V_{2}(t)={\mathcal D}p^{2}+{\mathcal E}(pq+qp)+{\mathcal F}q^{2},$$ where ${\mathcal D}$, ${\mathcal E}$ and ${\mathcal F}$ are of the form $$\begin{aligned} {\mathcal D}&=&D+\frac{4iE\alpha}{(16\rho\alpha)^{\frac{1}{2}}}\sinh (16\rho\alpha)^{\frac{1}{2}} +\frac{-8(F\alpha-D\rho)\alpha}{16\rho\alpha}\left[\cosh (16\rho\alpha)^{\frac{1}{2}}-1\right], \nonumber \\ {\mathcal E}&=&\frac{2i(F\alpha-D\rho)}{(16\rho\alpha)^{\frac{1}{2}}}\sinh (16\rho\alpha)^{\frac{1}{2}}+E\cosh (16\rho\alpha)^{\frac{1}{2}}, \nonumber \\ {\mathcal F}&=&F+\frac{-4iE\rho}{(16\rho\alpha)^{\frac{1}{2}}}\sinh (16\rho\alpha)^{\frac{1}{2}} +\frac{8(F\alpha-D\rho)\rho}{16\rho\alpha}\left[\cosh (16\rho\alpha)^{\frac{1}{2}}-1\right], \label{eqqq}\end{aligned}$$ respectively. It follows that if the following two equations are satisfied, $$E=\zeta\sinh (16\rho\alpha)^{\frac{1}{2}}, \quad \frac{2i(F\alpha-D\rho)}{(16\rho\alpha)^{\frac{1}{2}}}=-\zeta\cosh (16\rho\alpha)^{\frac{1}{2}}, \label{eq23}$$ then the coefficients of $pq+qp$ in $I_{V}$ is vanishing. In order that we can analyze the above equations (\[eq23\]) conveniently, the time-dependent parameters $\alpha$, $\rho$ (which are expected to be determined) and $F$, $D$ are respectively parameterized to be $$\alpha=\frac{u\theta}{4}, \quad \rho=\frac{v\theta}{4}, \quad F=h\cosh (\sqrt{uv}\theta), \quad D=g\cosh (\sqrt{uv}\theta). \label{eq24}$$ Substitution of the expressions (\[eq24\]) into (\[eq23\]) yields $$E=\zeta \sinh (\sqrt{uv}\theta), \quad \frac{i(hu-gv)}{2\sqrt{uv}}=-\zeta, \label{eq25}$$ which can determine $\zeta$ and $\theta$ (expressed in terms of $E$, $h$, $g$ and $u$, $v$). It is noted that if the functions $u$ and $v$ are finally determined, then the time-dependent parameters $\alpha$ and $\rho$ in the unitary transformation operator $V_{2}(t)$ (\[eq21\]) can be obtained. In what follows we will determine $u$ and $v$ via setting ${\mathcal D}={\mathcal F}=\varsigma$ with $\varsigma$ being constant ([*i.e.*]{}, time-independent). Insertion of (\[eq24\]) into (\[eqqq\]) will yield $$D+\frac{hu-gv}{2v}[\cosh (\sqrt{uv}\theta)-1]=\varsigma, \quad F-\frac{hu-gv}{2u}[\cosh (\sqrt{uv}\theta)-1]=\varsigma. \label{eq26}$$ Eq.(\[eq26\]) can determine the functions $u$ and $v$, although the problem is very complicated. Here it should be noted that $\theta$ which has been determined by (\[eq25\]) is also the function of $u$ and $v$. Thus, in principle, we can obtain the time-dependent functions $\alpha$ and $\rho$ in the second unitary transformation operator $V_{2}(t)=\exp [\alpha(t)p^{2}+\rho(t)q^{2}]$. Now under the unitary transformation $V_{1}(t)V_{2}(t)$ the time-dependent invariant $I_{\rm q}(t)$ is changed into a time-independent one, [*i.e.*]{}, $$I_{V}\equiv\left[V_{1}(t)V_{2}(t)\right]^{\dagger}I_{\rm q}(t)\left[V_{1}(t)V_{2}(t)\right]=\varsigma(p^{2}+q^{2})$$ whose eigenvalue is $(2n+1)\varsigma$ and the corresponding eigenstate is $|n, q\rangle$ that is the familiar stationary harmonic-oscillator wavefunction, and the eigenvalue equation of the time-dependent invariant $I_{\rm q}(t)$ is thus given as follows $$I_{\rm q}(t)V_{1}(t)V_{2}(t)|n, q\rangle=(2n+1)\varsigma V_{1}(t)V_{2}(t)|n, q\rangle.$$ The solutions of the time-dependent Schrödinger equation -------------------------------------------------------- According to the L-R invariant theory, the particular solution $\left| n, t\right\rangle _{\rm Schr}$ of the time-dependent Schrödinger equation is different from the eigenfunction of the invariant $I_{\rm q}(t)$ only by a phase factor $\exp \left[\frac{1}{i}\phi _{n}(t)\right]$, the time-dependent phase of which is written as $$\phi _{n}(t)=\int_{0}^{t}\langle n, q|\left [ V_{1}(t')V_{2}(t')\right]^{\dagger}\left[H(t')-i\partial/\partial t' \right]\left[V_{1}(t')V_{2}(t')\right]|n, q\rangle {\rm d}t'.$$ This phase $\phi _{n}(t)$ can be calculated with the help of the Glauber formula and the Baker-Campbell-Hausdorff formula[@Wei; @EPJD]. The particular solution $\left| n, t\right\rangle _{\rm Schr}$ of the time-dependent Schrödinger equation corresponding to the invariant eigenvalue $(2n+1)\varsigma$ is thus of the form $$\left| n, t\right\rangle _{\rm Schr}=\exp \left[\frac{1}{i}\phi _{n}(t)\right]V_{1}(t)V_{2}(t)|n, q\rangle.$$ Hence the general solution of the Schrödinger equation can be written in the form $$|\Psi(q, t)\rangle_{\rm Schr}=\sum_{n}c_{n}\left| n, t\right\rangle _{\rm Schr},$$ where the time-independent c-number $c_{n}$’s are determined by the initial conditions, [*i.e.*]{}, $c_{n}= _{\rm Schr}\langle n, t=0|\Psi(q, t=0)\rangle_{\rm Schr}$.\ \ In the above we thus found the general solutions of the Schrödinger equation for the time-dependent linear potential, which corresponds only to the quadratic-form invariant (\[eq11\]). It is concluded here that the solutions obtained above does not form a complete set of solutions of this time-dependent Schrödinger equation, and that Bekkar [*et al.*]{}’s solution and my solution presented here will constitute together such complete set of solutions of the Schrödinger equation. Discussions and Conclusions =========================== \(i) In the present paper we show that since Bekkar [*et al.*]{}’s solution[@Bekkar] has not yet contain those corresponding to the quadratic invariant, it is not the true general solution of the Schrödinger equation for the time-dependent linear potential. Instead, it is the solution corresponding only to the linear L-R invariant. The obtained solution here is the one that corresponds to the invariant (\[eq11\]), which is of the quadratic form. Since the linear and quadratic invariants form a complete set of L-R invariants, Bekkar [*et al.*]{}’s solution[@Bekkar] and my solution presented here constitute such complete set of solutions of the Schrödinger equation involving a time-dependent linear potential.\ \ (ii) It is well known that in quantum optics there are three kinds of photonic quantum states, [*i.e.*]{}, Fock state, coherent state and squeezed state. From my point of view, the calculation of the variations of creation and annihilation operators ($a^{\dagger}$, $a$) of photons under the translation ([*e.g.*]{}, $V_{1}$ of (\[eq21\])) and squeezing transformation ([*e.g.*]{}, $V_{2}$ of (\[eq21\])) operators shows that the variations of $a^{\dagger}$ and $a$ are exactly analogous to that of space-time coordinate variations under the translation, Lorentz rotation (boosts) and dilatation (scale) transformation[@Fulton] and thus these three quantum states (coherent, squeezed and Fock states) correspond to the above three conformal transformations, respectively. I think that this connection between them is of physical interest and deserves further consideration.\ \ (iii) Guedes recently stated that in order to obtain the general solutions of the Schrödinger equation one must follow the L-R invariant theory [*step by step*]{}[@Guedes2]. I don’t approve of this point of view, however. Personally speaking, in fact, the L-R method has only one step, namely, the particular solution of the time-dependent Schrödinger equation is different from the eigenfunction of the invariant only by a time-dependent phase factor. In Ref.[@Bekkar] and [@Shenarxiv], although we follow the L-R method step by step, what we obtained still cannot be viewed as the general solutions of the Schrödinger equation. For this reason, I think that “step by step” is not the essence of getting the general solutions of Schrödinger equation. Instead, the key point for the present subject is that one should first find the complete set of all L-R invariants of the time-dependent quantum systems under consideration. For some systems in the Hamiltonian there may exist no such closed Lie algebra as (\[algebra\]), the complete set of exact solutions can be found by working in a sub-Hilbert-space corresponding to a particular eigenvalue of one of the invariants, namely, only in the sub-algebra (quasi-algebra) corresponding to a particular eigenvalue of this invariant will such time-dependent quantum systems (which have no closed Lie algebra) be solvable[@japan]. For the time-dependent quantum systems, there are no other eigenvalue equations of Hamiltonian than that of the L-R invariants with time-dependent eigenvalues. The complete set of invariants, instead of the time-dependent Hamiltonian, can describe completely the time-dependent quantum systems. For this reason, it is essential to find the complete set of invariants for the time-dependent Hamiltonian of a given quantum system.\ \ (iv) In the Ref.[@Guedes], the author says that to the best of his knowledge there was no publication reporting the solution of the Schrödinger equation for the system described by $H(t)=\frac{p^{2}}{2m}+f(t)q $ without considering approximate and/or numerical calculations. I think that this is, however, not the true case. In the literature, at least in the early of 1990’s, Gao [*et al.*]{} had reported their investigation of the driven generalized time-dependent harmonic oscillator which is described by the following Hamiltonian $H(t)=\frac{1}{2}[X(t)q^{2}+Y(t)(pq+qp)+Z(t)p^{2}]+F(t)q$[@Gao]. It is believed that my solution presented here is only the special case of what they obtained[@Gao].\ \ **Acknowledgements** This project was supported partially by the National Natural Science Foundation of China under the project No. $90101024$. H.R. Lewis, Jr. and W.B. Riesenfeld, J. Math. Phys.(N.Y) [**10**]{}, 1458 (1969). I. Guedes, Phys. Rev. A [**63**]{}, 034102 (2001). H. Bekkar, F. Benamira, and M. Maamache, Phys. Rev. A [**68**]{}, 016101 (2003) X.C. Gao, J.B. Xu, and T.Z. Qian, Phys. Rev. A [**44**]{}, 7016 (1991). J.Q. Shen, arXiv: math-ph/0301026 (2003). J. Wei and E. Norman, J. Math. Phys.(N.Y) [**4**]{}, 575 (1963). J.Q. Shen, H.Y. Zhu, and P. Chen, Euro. Phys. J. D [**23**]{}, 305 (2003). I. Guedes, Phys. Rev. A [**68**]{}, 016102 (2003). T. Fulton, F. Rohrlich, L. Witten, Rev. Mod. Phys. [**34**]{}, 442 (1962). J.Q. Shen, H.Y. Zhu, and H. Mao, J. Phys. Soc. Jpn. [**[71]{}**]{}, 1440 (2002). [^1]: Electronic address: jqshen@coer.zju.edu.cn [^2]: I think that this paper will be a supplement to the recent comment \[Phys. Rev. A [**68**]{}, 016101 (2003)\] of Bekkar [*et al.*]{} on Guedes’s work \[Phys. Rev. A [**63**]{}, 034102 (2001)\] and Guedes’s reply to Bekkar [*et al.*]{}’s comment. It will be submitted nowhere else for publication, just uploaded at the e-print archives. [^3]: In general, for the case of three-generator Hamiltonian (the generators of which form a non-semisimple algebra), the time-dependent c-number $C(t)-[-i(B\beta-A\eta)+D\eta^{2}+F\beta^{2}-2E\beta\eta]$ in $I_{1}(t)$ are vanishing. See, for example, in Ref.[@Shenarxiv], which is a special case of the present problem.

--- abstract: 'We propose an approach of open-ended evolution via the simulation of swarm dynamics. In nature, swarms possess remarkable properties, which allow many organisms, from swarming bacteria to ants and flocking birds, to form higher-order structures that enhance their behavior as a group. Swarm simulations highlight three important factors to create novelty and diversity: (a) communication generates combinatorial cooperative dynamics, (b) concurrency allows for separation of timescales, and (c) complexity and size increases push the system towards transitions in innovation. We illustrate these three components in a model computing the continuous evolution of a swarm of agents. The results, divided in three distinct applications, show how emergent structures are capable of filtering information through the bottleneck of their memory, to produce meaningful novelty and diversity within their simulated environment.' author: - Olaf Witkowski - Takashi Ikegami bibliography: - 'witkowski.bib' title: | How to Make Swarms Open-Ended?\ Evolving Collective Intelligence Through a Constricted Exploration of Adjacent Possibles --- OEE === Life has been evolving on our planet over billions of years, undergoing several major transitions which transformed the way it stored, processed and transmitted information. All these transitions, from multicellularity to the formation of eusocial systems and the development of complex brains, seem to lead to the idea that the evolution of living systems is *open-ended*. In other words, life appears to be capable of increasing its complexity indefinitely Another formulation of open-endedness, echoed by Standish [@standish2003open] and Soros [@soros2014identifying], is that open-endedness depends fundamentally on the continual production of novelty. Life keeps uncovering new inventions, in a process which never seems to stop. Since the 1950’s, open-ended evolution (OEE) has been a central topic of research for artificial life approaches to the fundamental principles of life. Soon after, John Von Neumann [@von1966theory] has contributed to the issue as well, with his early model of self-reproducing automata. Since 2015, a series of workshops have been taking place at Artificial Life conferences [@taylor2016open], the last of which[^1] was a launchpad for the present special issue. In general, an evolutionary system is considered to be open-ended when it is able to endlessly generate diverse novel entities of growing complexity. Engineering open-ended systems in the lab is not easy, and the main obstacle is that the designed evolutionary systems are subject to a thermodynamic drift making them collapse into equilibrium states. Once local optima are reached, they do not produce novelty anymore, which bounds their complexity and diversity. Innovation seems to emerge from collective intelligence, a phenomenon which refers to groups or networks of agents that develop together the ability to enhance the group’s cognitive capacity or creativity. This is reminiscent of the ongoing innovative process of science, which does not have any other fixed objective than the production of new knowledge, but makes discoveries mostly through accidents. Stuart Kauffman advocated for the idea of the adjacent possible, claiming that a biosphere can be viewed as a secular or long-term trend and it can maximize the rate of exploration of the adjacent possible of an existing organization [@kauffman2000investigations; @kauffman2003adjacent]. Ikegami et al. (2017) [@ikegami2017life] builds on that idea to explain how, in terms of evolutionary transitions [@MaynardSmithSzathmary1997], a new stage (e.g. multicellular oranism) of evolution may be produced without any information being passed on from the previous stage (e.g. from single cells). Rather, structural properties are assembled, producing a stepping stone to the next level of innovation. These structural properties of a collective group can be compared to a bottleneck that acts as a filter on several levels of the system, implementing computation that is not present in any of its parts. Part of this idea is analog to Tishby and Polani (1999) [@tishby1999information], where the information is squeezed through a bottleneck, to extract relevant or meaningful information from an environment. The resulting “filtered” information through the bottleneck, retains only the features most relevant to general concepts. In this paper, we present three “C” factors that we deem important for novelty and diversity. We then introduce a swarm model to study these factors, applied in three different studies. We conclude with a discussion on open-endedness at large, framing the three factors in terms of the emergence of collective intelligence in swarm simulations. Conditions for OEE ================== The OEE literature has proposed various conditions that are supposed to lead to the successful production of OEE in a system. Number of studies have attempted to formalize necessary conditions for OEE [@holland1999echoing; @conrad1970evolution]. Taking a recent example, Soros and Stanley (2014) [@soros2014identifying] suggest four conditions at the scale of single reproducing individuals in the system, which should each fulfill some nontrivial minimal criterion, be able to create novelty, act autonomously, and dispose of access to unbounded memory. Such papers have typically been proposing their own model, to demonstrate the importance of the hypothesized conditions for the emergence of OEE within it. However, most evolutionary algorithms seem to either converge very quickly to a solution, or get stuck in a confined area of the search space. Either way, they don’t seem to be able to intrinsically generate the amounts of complexity and novelty we find in nature, even at a scale. Although this failure of simulated evolution to match open-ended properties found in natural evolution may be explained by shorter timescales, in principle one would have expected decades of efforts, and increasingly larger resources poured into research in evolutionary computation, to have unlocked more of its potential to create novelty. However, even the latest technologies don’t seem to keep their inventivity. In general, promising models [@lenski2003evolutionary; @lehman2011evolving; @goodfellow2014generative; @greenbaum2016digital; @silver2017mastering] that manage to demonstrate at least a few phases transitions or creative leaps – not necessarily with evolutionary computation – seem to have one common denominator of containing several structural bottlenecks which filter relevant information through them, as a catalyst of creativity. What seems to be missing to achieve general OEE? We choose to emphasize three “C” candidates which we see as worth pursuing – Communication, Concurrency, and Complexity: (a) Communication: constricted information flows between parts of the systems allow for synergy and cooperation effects. (b) Concurrency: the creation of separate space and time scales requires concurrent, nondeterministic, asynchronous models. (c) Complexity: mere system growth can boost novelty and diversity. These are the three C-factors on which we choose to concentrate, in this paper. Next, will expand on each of them a little further, before proposing how to apply them in concrete models. We will now expand on these three points, before presenting concrete examples, with Study 1, 2 and 3. Communication {#communication .unnumbered} ------------- This first point addresses synergies and coordination between components of the systems, using information transfers. One well known example of OEE is combinatorial creativity in human language, where syntactical rules are capable of producing infinite well-formed structures using recursion, thus making the number of potential sentences unbounded [@hauser2002faculty]. Although these may seem slightly dated remarks, at the advent of language studies based on artificial life systems [@kirby2002natural], it is promising to focus on the cultural layer of dynamics, that lives on top of the main layer of entities. For example, in the case of web services (social tagging networks), we can analyze how combinatioral complexity is effective in evoking OEE. In cellular automata, one may want to study the interactions between gliders or other patterns. In artificial chemistry, one may want to look at information flows between types of molecules or replicators. In agent-based modeling, perhaps establishments of protocols between agents or groups of agents can become a factor to focus on. Communication naturally adds relevant computing filters on unexploited information flows, effectively increasing the bandwidth of useful information flows within the system per clock cycle, communication offers a layer for metadynamics at a different timescale from the first-order dynamics. This induces a separation of timescales, thus doubling the system’s capacity to implement learning mechanisms. Designing information to circulate between a sub-entities of the system forces the creation of more structural bottlenecks. We propose information exchanges as a central mechanism promoting open-ended evolution. From information flows in groups of individuals, a system can boost its own production of creativity to achieve indefinite complexity. Examples are detailed in Study 1 and 2, below. Concurrency {#concurrency .unnumbered} ----------- In many situations, a system cannot scale up to larger space-time scales as it is. We need to add some ingredient to make it work in the larger scales. One such remedy is to give it asynchronous updates. Removing the global clock is needed to make larger systems function consistently without constantly checking local consistency. On the other hand, we know that cellular automata (CA) tend to lose their complexity by adopting asynchronous dynamics. Yet asynchrony is an original natural phenomena difficult to bring to artificial systems. According to Dave Ackley [@ackley2015artificial], models should be indefinitely scalable, ruling out deterministic, synchronous models (such as simple cellular automata), and suggesting nondeterministic, asynchronous ones. Bersini et al. (1994) [@bersini1994asynchrony] proposed that asynchronous rather than synchronous updating may be key factor in inducing stability in simulations. Although they were examining a variant of cellular automata, their results, based on an analysis of the Lyapunov exponent, indicated the responsibility of asynchrony for sensitivity of the update function. Ackley and Ackley (2015) [@ackley2015artificial] propose to use asynchrony. The concurrency is also closely related to the ability a system has to evolve separate timescales. Although highly contingent on other properties of a simulation, the capacity to develop heterogeneous timescales often constitutes a barrier to producing intrinsic novelty. Researchers used to separate lifetime learning and evolutionary learning, as two distinct mechanisms [@nolfi1990learning]. However, the effects of accumulating and filtering information into and out of one system’s memory occurs at a much more continuous range of different timescales. In nature, from phenotypic plasticity, to maternal effects, to sexual selection, or to gene flow, many events have their time scales intricately interlaced. We will address this in particular in Study 1 and 2. Complexity {#complexity .unnumbered} ---------- We have no grounds to argue that nature is its own only possible realization, since there would be no satisfactory explanation for that. One important feature of nature though, is its complexity, which can translate into both system size and landscape complexity. In the simplest of all cases, complexity can be reached merely with large population sizes. Ikegami et al. (2017) [@ikegami2017life] proposed that large groups of individuals, given the right set of structural characteristics, may be the main driver for emergence. They discussed this hypothesis in relation to large-scale boid swarm simulations [@reynolds1987flocks], in which the nucleation, organization and collapse dynamics were found to be more diverse in larger flocks than in smaller flocks. Collective behaviors can be qualitatively different by increasing the number of agents, i.e. a colony or group size. In the actual observation, e.g., the individual bees change their behaviors depending on the colony size. Also the fish change their performance of sensing the environmental gradient depending on the school size. In previous works [@mototake2015simulation], we simulated half a million birds flock using a boids model [@reynolds1987flocks; @toner1998flocks] and found that qualitatively different behavior emerges when the total number of individuals exceeds a few thousands or so. Flocks of different sizes and different shapes interact to diminish some flocks but to generate new ones. Different types of fluctuation become dominant in different size of flocks. A correlation of the local density fluctuation becomes dominant in the larger size flocks and that of the velocity fluctuation dominates in the smaller size flocks. An example of that is offered in Study 3. Stretching the argument on size, environmental complexity is definitely necessary to a certain extent to create complex behaviors. Only with richer environments, encompassing complex distributions of energy resources and ways for systems to survive, can emerging individuals explore a rich set of strategies and increasingly increasingly complex solutions. As mentioned earlier, evolutionary landscapes have become an important concept in biology to analyze the dynamics at play in an ecosystem. The picture to have is the one of a unit of selection (e.g. a gene, among many other options) being represented by a point in a multidimensional search space. That space is typically given as many dimensions as there are degrees of freedom for the entity to vary and evolve in the space (e.g. combination of nucleotide sequences). The search space is mapped onto an additional dimension, which is usually the reproductive success, or fitness. The shape of the fitness across all degrees of freedom of the system have a strong impact on the dynamics that it can achieve. Malan et al. (2013) [@malan2013survey] identifies eleven characteristics of fitness, which make them more or less difficult to solve. These characteristics include the degree of variable inter-dependency, noise, fitness distribution, fitness distribution in search space, modality, information to guide search and deception, global landscape structure, ruggedness, neutrality, symmetry, and searchability. In evolutionary systems, richer environments, benefiting of a complex distribution of energy sources and ways for systems to survive, give rise to richer sets of pathways. The larger the search spaces, the more complex fitness functions are potentially evolved. Another way is to make the environment a more complex function of time, which the agents will need to learn in order to extract more energy from it. In Study 1 and 2, we present results suggesting that simulations should be ensured to provide sufficient system complexity in terms of the environment of agents. Simulation time and memory, though not mentioned yet, are important components to consider. Computationally, the whole course of evolution on Earth is like a single run of a single algorithm that invented all of nature, and seems like it will never end. One obvious difference is the size of the systems, which might be the missing element to get ever-greater emergent complexity and novelty through very long time. However, we do not insist on this component in this paper, as we consider trivial that a system with too low computational power will not be able to achieve OEE to any extent. Similarly, there is point to be made about endo-OEE, producing novelty from within, against exo-OEE, which makes use of input from outside the system. Picbreeder [@secretan2008picbreeder], for example, explicitly requires external human input to function, which makes it a debatable generator of OEE. Nevertheless, OEE is not about new information, but rather inventions achieved by the system. In that respect, swarms are a promising model: without increasing ensemble size, they let us focus on how coordination patterns self-organize, generating intrinsic novelty. To give another example, even increasing the number of neurons in a neural network still requires neurons to differentiate themselves, and create coordinated networks before they get to foster innovative ideas. We exemplify the importance of size and complexity in Study 1 and 3, while discussing how to make simulations parallelizable, to save considerable amounts of time and memory by distributing them over many machines. Concurrent evolutionary neural boids model ========================================== The model we choose to present puts together the abovementioned series of features, as a means to promote the system’s open-endedness. We give some details here, and will go over the details of several applications of it in the next section. The evolutionary system is an agent-based simulation, based off Reynolds’ boids model [@reynolds1987flocks]. The boids model was based on simple rules computed locally, allowing to simulate flocks of agents moving through artificial environments. As with most artificial life simulations, boids showed emergent behavior, that is, the complexity of boids arises from the interaction of individual agents adhering to a set of simple rules of separation (steer to avoid crowding local flockmates), alignment (steer towards the average heading of local flockmates, and cohesion (steer to move toward the center of mass of local flockmates). In our model, like in Reynolds’ model, the population of agents moves around in a continuous three-dimensional space, with periodic boundary conditions (Figure \[fig:worldsim\]). However, instead of using fixed rules to control the boids’ motion, we allow agents to evolve their own controllers through concurrent evolutionary computation. Each agent, instead of responding to simple rules, is controlled by its own neural network. The parameters of the neural network are encoded in a genotype vector, which determines the individual’s sensorimotor choices at each moment in time. This corresponds to standard evolutionary robotics methodologies [@nolfi2000evolutionary], although we introduce the following variant. The genotype is evolved through the course of the simulation, via a continuous variant of an evolutionary algorithm [@witkowski2014a], that is, agents with high level of fitness are allowed at any point to replicate with mutation in the middle of the running simulation. This model also builds up on prior work on the effect of self-organized inter-agent communication and cooperative behavior on the performance of agents to solve tasks [@prokopenko2006evolving; @olson2013predator]. Previous research has shown the difficulty of using communication channels [@rasmusen1994games; @mitri2013using] but showed cooperative value of information transfers [@witkowski2016emergence]. This will be complemented by the results from previous information-theoretic analyses of learning systems, which managed to shed light on the role of information flows in learning [@tishby2015deep; @lin2017does; @tishby2011information]. Agents are given a certain energy, that also acts as their fitness. This will be specific to the study cases. Each agent comes with a set of 12 different sensors. The neural network (represented on Figure \[fig:ann\]) takes the information from those sensors as inputs, in order to decide the agent’s actions at every time step. The possible actions amount to the agent’s motion, and in the specific variant shown here, a Prisoner’s Dilemma action (cooperate or defect), as well as two output signals. The architecture is composed of a 12 input, 10 hidden, 5 output, and 10 context neurons connected to the hidden layer (see Figure \[fig:ann\]). The agents’ motion is controlled by $M_1$ and $M_2$, outputting two Euler rotation angles: $\psi$ for pitch (i.e. elevation) and $\theta$ for yaw (i.e. heading), with floating point values between $0$ and $\pi$. Even though the agents’ speed is fixed, the rotation angles still allow the agent to control its average speed (for example, if $\psi$ is constant and $theta$ equals zero, the agents will continuously loop on a circular trajectory, which results in an almost-zero average speed over 100 steps). The outputs $S_\text{out}^{(1)}$ and $S_\text{out}^{(2)}$ control the signals emitted on two distinct channels, which are propagated through the environment to the agents within a neighboring radius set to $50$. The choice for two channels was made to allow for signals of higher complexity, and possibly more interesting dynamics than greenbeard studies [@gardner2010]. The received signals are summed separately for each direction (front, back, right, left, up, down), and weighted by the squared inverse of the emitters distance. This way, agents further away have much less impact on the sensors than closer ones do. Every agent is able to receive signals on the two emission channels, from 6 different directions, totalling $12$ different values sensed per time step. For example, the input $S_\text{in}^{(6,1)}$ corresponds to the signals reaching the agent from the neighbors below. ![ [**Architecture of the agent’s controller.**]{} The network is composed of 12 input neurons, 10 hidden neurons, 10 context neurons and 5 output neurons.[]{data-label="fig:ann"}](witkowski-fig2-ann.png){width=".7\textwidth"} The evolution is performed continuously over the population. Agents with negative or zero energy are removed, while agents with energy above a threshold are forced to reproduce, within the limits of one infant per time step. The reproduction cost is low enough, considering the threshold, to not put the life of the agent at risk. Study cases =========== We go over the application of this model in three selected examples of studies. Each of them highlights a specific property for OEE. The first model shows how agents can form patterns to accelerate their search for energy, distributed over an n-dimensional space, collaborating via local signaling with their neighbors. The second study shows the invention of dynamical group strategies in a spatial prisoner’s dilemma, allowing for specific cooperation effects. The third example shows the impact of growth on the emergence of noise-canceling effects. OEE via collective search based on communication ------------------------------------------------ Since Reynold’s boids, coordinated motion has often been reproduced in number of artificial models, but the conditions leading to its emergence are still subject to research, with candidates ranging from obstacle avoidance to virtual leaders. The relation of spatial coordination and group cooperation has long been studied in game theory and evolutionary biology. We here apply our model of agents exchanging signals and moving in a three-dimensional environment, to a task of dynamical search for free energy in space [@witkowski2014a; @witkowski2016emergence]. Each agent’s movements are controlled by artificial networks, evolved through generations of an asynchronous selection algorithm. At the term of the evolution, the agents are able to communicate to produce cooperative, coordinated behavior. Individuals develop swarming using only their ability to listen to each other’s signals. The agents are selected based on their performance at finding invisible resources in space giving them fitness. The agents are shown to use the information exchanged between them via signaling to form temporary leader-follower relations allowing them to flock together. The swarmers outperform the non-swarmers at finding the resource, thus reaching a neutral evolutionary space which leads to a genetic drift. This work constructs an adaptive system to evolve swarming based only on individual sensory information and local communication with close neighbors. This addresses directly the problem of group coordination without central control, without being aware of the position direct neighbors, nor any use of the substrate where to deposit information (stigmergy) [@hauert2009evolved]. The approach has also the advantage of yielding original and efficient swarming strategies. A detailed behavioral analysis is then performed on the fittest swarm to gain insight as to the behavior of the individual agents. The results show that agents progressively evolve the ability to flock through communication to perform a foraging task. We observe a dynamical swarming behavior, including coupling/decoupling phases between agents, allowed by the only interaction at their disposal, that is signaling. Eventually, agents come to react to their neighbors’ signals, which is the only information they can use to improve their foraging. This can lead them to either head towards or move away from each other. While moving away from each other has no special effect, moving towards each other, on the contrary, leads to swarming. Flocking with each other may lead agents to slow down their pace, which for some of them may keep them closer to a food resource. This creates a beneficial feedback loop, since the fitness brought to the agents will allow them to reproduce faster, and eventually multiply this type of behavior within the total population. The algorithm converges to build a heterogeneous population, as shown on Figure \[fig:treehorizontal\]. The phylogeny is represented horizontally in order to compare it to the average number of neighbors throughout the simulation. The neighborhood becomes denser around iteration $400k$, showing a higher portion of swarming agents. This leads to a firstly strong selection of the agents able to swarm together over the other individuals, a selection that is soon relaxed due to the signaling pattern being largely spread, resulting in a heterogeneous population, as we can see on the upper plot, with numerous branches towards the end of the simulation. ![*Top:* average number of neighbors during a single run. *Bottom:* agents phylogeny for the same run. The roots are on the left, and each bifurcation represents a newborn agent.[]{data-label="fig:treehorizontal"}](witkowski-fig3-phylogeny.jpeg){width=".9\textwidth"} In this scenario, agents do not need extremely complex learning to swarm and eventually get more easily to the resource, but rather rely on dynamics emerging from their communication system to create inertia and remain close to goal areas. ![ Two principal components of a PCA on the genotypes of all agents of a typical run, over one million iterations. Each circle represents one agent’s genotype, the diameter representing the average number of neighbors around the agent over its lifetime, and the color showing its time of death ranging from bright green (at time step $0$, early in the simulation) to red (at time step $10^6$, when the simulation approaches one million iterations). []{data-label="fig:pca"}](witkowski-fig4-pca.png){width=".85\textwidth"} The simulated population displays strong heterogeneity due to the asynchronous reproduction schema, which can be seen in the phylogenetic tree (Figure \[fig:treehorizontal\]). The principal component analysis plotted on Figure \[fig:pca\] shows a large cluster (left side) in addition to a series of smaller ones (right side). The genotypes in the early stages of the simulation belong to the right clusters, but get to the left cluster later on, reaching a higher number of neighbors. The plot shows a diverse set of late clusters, which translates to numerous distinct behaviors in the late stage of the simulation. Such heterogeneity may suppress swarming but the evolved signaling helps the population to form and keep swarming. The simulations do not exhibit strong selection pressures to adopt specific behavior apart from the use of the signaling. Without high homogeneity in the population, the signaling alone allows for interaction dynamics sufficient to form swarms, which proves in turn to be beneficial to get extra fitness. These results represent an improvement on previous models using hard-coded rules to simulate swarming behavior, as they are evolved from very simple conditions. Our model also does not rely on any explicit information from leaders, like previously used in part of the literature [@cucker2008flocking; @su2009flocking]. It does not impose any explicit leader-follower relationship beforehand, letting simply the leader-follower dynamics emerge and self-organize. In spite of being theoretical, the swarming model presented in this paper offers a simple, general approach to the emergence of swarming behavior once approached via the boids rules. This simulation improves on previous work because agents naturally switch leadership and followership by exchanging information over a very limited channel of communication. Finally, our results also show the advantage of swarming for resource finding. It’s only through swarming, enabled by signaling behavior, that agents are able to reach and remain around the goal areas. In terms of cooperation, this model exemplifies a case of multilevel selection theory [@wilson1994reintroducing; @traulsen2006evolution], which models the layers of competition and evolution, within an ecological system. Our system shows the emergence of different levels which function cohesively to maximize reproductive success. The fitness value of the group level dynamics outweighs the competitive costs, resulting in individuals constantly innovating in ways they are cooperating in a non-trivial way, to create behaviors which are not centrally coded for. This study shows swarming dynamics emerging from a communication system between agents, immersed in a simulated environment with spatial distribution of energy resource. The concurrent evolution scheme, running at the same time as the simulation itself, led to decentralized leader-follower interactions, which allowed for collective motion patterns, which in turn significicantly improved the groups’ fitnesses. This model encodes the stochastic evolution of a controller that maps sensory input onto motor output, to optimize the performance on a task, as framed broadly by Nolfi and Floreano (2000) [@nolfi2000evolutionary]. We capture the fight against a difficult wall [@schmickl2016sooner], which simulations typically fail at because they suddenly hit a so-called “wall of complexity”: trivial tasks are solved easily, but it’s hard to jump to solving difficult ones. If we take the no-free-lunch argument from Wolpert and Macready (1997) [@wolpert1997no] that no optimisation algorithm is at all times superior to others, it is natural that the more specific the algorithm, the more it is likely to fail with new problems. Our results suggest that novelty can be produced by the asynchronous evolution of a heterogeneous community of agents, which through their mixture of strategies, may achieve open-ended, uninformed learning. The heterogeneity present in the model also offers an extension to the advantages of particle swarm optimization (PSO) [@eberhart1995new]. While PSO only offers one unique objective function to optimize, each agent in the swarm effectively runs its own function, which are combined into a swarm behavior. Although these results suggest open-endedness, it is worth noting we do not bring a proof that the phenomenon is truly open-ended, which may require the emergence of ever-complexifying communication, or an uninterrupted sequence of evolutionary innovations. The information flows were a focus of the original work [@witkowski2016emergence]. From these flows, one can notice three main bottlenecks. The evolutionary computation contains a bottleneck effect, as a result from the selection based on the agents’ performance on the task. Another bottleneck can be found between the sensory inputs of each agent, and its motor outputs, as the neural controller acts as a filter for the information. The agents’ signaling also naturally contains a bottleneck effect, as the information transmitted from agent to agent is constrained by the physical communication bandwidth. The combination of these three bottlenecks allows for relevant information to be filtered into the swarm, which is able learn certain behaviors (see also next section). OEE via cooperative flocking ---------------------------- The evolution of cooperation is studied in game theory, and stretches have been made to include spatial dimensions. This problem is often tackled by using simple models, such as considering interactions to be a game of Prisoner’s Dilemma (PD). We examined a variation of the model with a distinct fitness function in a separate study, based this time on the agents playing a spatial version of the Prisoner’s Dilemma [@witkowski2014b]. We study the impact of the movement control on optimal strategies, and show that cooperators rapidly join into static clusters, creating favorable niches for fast replications. It is also noted that, while remaining inside those clusters, cooperators still keep moving faster than defectors. The system dynamics are analyzed further to explain the stability of this behavior. This work presents, in an even more explicit fashion than the previous study, a model aimed at showing emergent levels of selection for cooperative behavior [@wilson1994reintroducing]. At every time step, agents are playing a N-player version of the prisoner’s dilemma with their surrounding, meaning that they make a single decision that affects all agents around them. They get reward and/or punishment based on the number of cooperator around them. Their decision is one of the outputs of their neural network. Effectively, the payoff matrix we used is an extension of Chiong and Kirley (2012) [@chiong2012], where we added the distance to take into account the spatial continuity. Based on the outcome of the match, agents can choose a new direction, which is similar to leaving the group in the walk away strategy [@aktipis2004], the main difference being that, in our case, it is also possible for groups to split. It is also similar in another aspect: there is a cost to leaving a group, as a lone agent may need time to meet others. At the beginning of each run, the environment is seeded with random agents. Since all weights in their neural network are set at random, roughly half of the agents initially choose to cooperate while the other half choose to defect. This leads to a fast extinction of cooperators, until approximately 50000 time steps), until a group emerges strong enough to survive. The second phase follows, in which cooperators are quickly increasing in number due to the autocatalytic nature of this strategy. A third step happens eventually, where defectors invade the cluster, followed either by the survival of the cluster due to cooperators running away or a reboot of the cycle. In case of survival, oscillations in the proportion of cooperators can be observed. However, this phenomenon is averaged away over multiple runs, since period and phase of the oscillations are not correlated from one experiment to the other. Were a defector to appear near a cluster of cooperators, the cluster would react by “reproducing away”. However, the chances to be overtaken by the defectors is much higher than in the dynamic case. From this three-dimensional model of agents playing the Prisoner’s Dilemma, the first result is that cooperators, when they are present, quickly evolve to form clusters as they represent a favorable pattern. The clustering behavior can be interpreted as a degenerated version of the simulations presented above, since the cooperating agents present the same capacities of information exchange as that model. We note that this solution is evolved through a longer time scale, as it is not always viable locally, depending on the distribution and behavioral thresholds of defectors. While the clustering itself can be expected, it is interesting to observe that their overall movement rate is still higher than defectors. This is even more surprising considering that those clusters do not seem to move fast. Instead, analysis shows that cooperators are moving quickly inside the cluster, which may be a way to adapt to an aggressive environment. In addition, comparison with the static case showed that movement made the emergence of cooperators harder, but more stable in the long run. Since it is harder for defectors to overtake a cluster of cooperators, our systems often show a soft bistability, meaning that they will eventually switch from one state to the other. It is even possible to observe a sort of symbiosis, where cooperators are generating more energy than necessary, which is in turn used by peripheral defectors. In this case, replacement rates allow cooperators to stay ahead, keeping this small ecosystem stable. This cohesion among cooperators seems to be enhanced by signaling, even though signals might attract defectors. Additional investigation on the transfer entropy, for instance, could be a promising next step. Another result is found in the choice of actions, generated by the neural networks without consideration of the past actions. We notice the emergence of a dynamical memory effect, that otherwise requires to be encoded in each agent, here emerging from the agents’ motion in space. Since the Prisoner’s Dilemma game has become a common model used in evolutionary biology to study the outcomes depending on the costs that characterize an ecosystem, this model, with a fitness based on the results of such game, showed the emergence of spatial coordination based on a the exchange of signals between agents. The signals remained very simple, and the environment was fixed in time. This model’s evolutionary computation reached solutions composed of different parts, including soft bistable strategies, different radiuses of clusters, as well as the use of dynamical patterns to improve their fitness. The solutions were also distributed over different timescales. The communication between agents also allowed them to converge on these behaviors more quickly. These elements refer back to our 3-C arguments earlier, for the discovery of novel solutions to a simple PD game. Lastly, we note that many different neural architectures may coexist as only a part of the neural architecture is used to implement flocking. This neural heterogeneity is something we’d like to insist in the context of OEE. Additionally, communication is important to filter out the neural architecture heterogeneity, which potentially holds the heterogeneity in a community (i.e. agents can stay in a community if they can communicate to each other). The communication may therefore indirectly help preserving the heterogeneity. OEE via large scale swarms -------------------------- Studying flocking models can also lead to the emergence of OEE, by focusing on emergent phenomena as macroscopic layers of patterns and structures that appear as a result of cooperative phenomena between autonomously behaving elements. A group of elements creates a self-organizing structure, which governs the individual micro rules and creates a new macro structure. Therefore, consecutive micro–macro recurrent self-organization is defined as an emergent phenomenon. Here, we describe the contribution of the same swarming simulation, scaled up, showcasing the effect of size on emergence of open-endedness [@drozd2016critical]. Before that, we start by presenting a degenerated version of that model, which shows large-scale dynamics in the less computationally costly case of agents that don’t preserve any internal state other than position and velocity [@ikegami2017life]. Starting with this simpler stateless model, we observe a noticeable change when the total number of boids increases from 2048 to 524,288, while the density is kept constant (Figure \[fig:largeboids\]). In order to compute large swarming behaviour, we parallelized the computational steps using the general-purpose computing on graphics processing units (GPGPU) method. The next step was to extend it to stateful agents. ![Visualization of swarming behavior, simulated by a large scale stateless boids model [@ikegami2017life]. The total number of boids in each panel is (a) 2048, (b) 16 384, (c) 131 072 and (d) 524 288, respectively. Some flocks are composed of a very large number of boids with narrow filament patterns. The initial velocity of each boid is set at random, and the density of the total number of boids is kept constant at 16,384 (number per cubic unit). The minimum and the maximum speed are set at 0.001 and 0.005 (unit per step), respectively. []{data-label="fig:largeboids"}](witkowski-fig6-largeboids.jpg){width=".7\columnwidth"} We explore the effect of reaching a critical mass, and how it impact the efficiency of the swarm’s foraging behavior. In particular, we study the problem of maintaining the swarm’s resilience to noisy signals from the environment. To do so, we look at stateful boids, i.e. moving agents controlled by neural network controllers, which we evolve through time in order to explore further the emergence of swarming, like in the previous two sections. However, we now ground our model in a more realistic setting where information about the resource location made partly accessible to the agents, but only through a highly noisy channel. The swarming is shown to critically improve the efficiency of group foraging, by allowing agents to reach resource areas much more easily by correct individual mistakes in group dynamics. As high levels of noise may make the emergence of collective behavior depend on a critical mass of agents, it is crucial to reach in simulation sufficient computing power to allow for the evolution of the whole set of dynamics. Because this type of simulations based on neural controllers and information exchanges between agents is computationally intensive, it is critical to optimize the implementation in order to be able to analyze critical masses of individuals. In this work, we address implementation challenges, by showing how to apply techniques from astrophysics known as treecodes to compute the signal propagation, and efficiently parallelize for multi-core architectures. The results confirm that signal-driven swarming improves foraging performance. The agents overcome their noisy individual channels by forming dynamic swarms. The measured fitness is found to depend on the population size, which suggests that large scale swarms may behave qualitatively differently. The minimalist study presented in this paper together with crucial computational optimizations opens the way to future research on the emergence of signal-based swarming as an efficient collective strategy for uninformed search. Future work will focus on further information analysis of the swarming phenomenon and how swarm sizes can affect foraging efficiency. In this model, we specifically focus on the addition of noise to the food detection sense that the agents possess, and hypothesize that it can be overcome by the emergence of a collective behavior involving sufficiently large groups of agents. Many systems, from atomic piles to swarms, seem to work towards preserving a precarious balance right at their critical point [@bak2013nature]. An atomic pile is said to be “critical” when a chain reaction of nuclear fission becomes self-sustaining. A minimal amount of fissionable material has to be compacted together to keep the dynamics from fading away. The notion of critical mass as a crucial factor in collective behavior has been studied in various areas of application [@marwell1993critical; @oliver2001whatever]. Similarly, the size of the formed groups of agents may be crucial, in order to reach a critical mass in swarms, enough to overcome very noisy environments. Part of the focus will therefore be on the optimization of the computer simulation itself, as large-scale swarms may qualitatively differ in behavior from regular-sized ones. The model extends the original setup described before, which proposed an asynchronous simulation evolving a swarming behavior based on signaling between individuals. However, unlike the original model, where the individuals don’t perceive directly either the food patches or the other agents around them, here we give a sense of vision to every agent, allowing them to detect nearby resources. However, we add a high level of noise to make this information highly imperfect. We used an agent-based simulation to show how signal-driven swarming, emerging in an evolutionary simulation such as in Witkowski and Ikegami (2014) [@witkowski2014a], allow agents to overcome noisy information channels an improve their performance in a resource finding task. Our first contribution is the very introduction of noise, demonstrating that the algorithm performs well against noises filling up channels of information almost up to their full capacity, in the inputs of agents. The individuals, by means of a swarming behavior helped by basic signaling, manage to globally filter out the noise present in the information from their sensory inputs, to reach the food sites. We proposed a hierarchical method based on the Barnes-Hut simulation in computational physics and its parallel implementation. We achieved a performance improvement of a few orders of magnitude over the previous implementation [@witkowski2014a]. This implementation is crucial to achieve the simulation of a sufficient number of agents to test for large-scale swarms (i.e. involving a very large number of individuals), which have been suggested to generate qualitatively different dynamics. The optimization of the fitness acquired by phenotypes using efficient patterns of behavior (motion and signaling), which themselves are encoded in the weights of agents’ neural networks. The real optimization therefore occurs at the higher level of the darwinian-like process in the genotypic search space. Efficient genotypes are selected by the asynchronous genetic algorithm throughout a simulation run. We observed that signaling improves the foraging of agents (see Figure \[fig:boidsefficiency\] for plots from Drozd et al. [@drozd2016critical] of efficiency or fitness against simulation time), the average resource retrieved per agent per iteration as a measure of the population’s fitness. Without noise, the agents using signaling are less efficient than their silent counterpart, which we found is not due to the cost of signaling, but rather because of the excess of noise brought by the signal inputs. The difference remains very small between signaling and non-signaling agents. ![Agents’ efficiency plot with and without signal, from a stateful (neural-network-controlled) boids model in the original work [@drozd2016critical], with mean (central line) and standard deviation range (area plot) over 10 runs. The plots correspond to noiseless (top), constant noise 20 (middle), and constant noise 40 (bottom), respectively.[]{data-label="fig:boidsefficiency"}](witkowski-fig7-efficiency.jpeg){width=".7\columnwidth"} We find however that from a certain noise level, the cost to signal is fully compensated by the benefits of signaling, as it helps the foraging of agents. The average fitness becomes even higher as we increase the noise level, which suggest that the signaling behavior increases in efficiency for high levels of noise, allowing the agents to overcome imperfect information by forming swarms. We also observe scale effects in the influence of the signal propagation on the average fitness of the population. For a smaller population, only middle values of signal propagation seem to bring about fitter behaviors, whereas this is not the case for larger sizes of population. On the contrary, larger populations are most efficient for lower levels of signal propagation. This may suggest a phase transition in the agents’ behavior for large populations, eventually in the way the swarming itself helps foraging. Understanding criticality seems strongly related to a broad, fundamental theory for the physics of life as it could be, which still lacks a clear description of how it can arise and maintain itself in complex systems. The effects of criticality have recently been investigated futher by one of the authors, using a similar setup [@khajehabdollahi2018critical]. The results showed exploratory dynamics at criticality in the evolution of foraging swarms, and the tension between local and global scale adaptation. Through this work, by increasing the number of simulated boids that maintain their own states, we may introduce more than the mere number. By allowing for many information exchanges between computing agents, the simulation can effectively take leaps of creativity. In Stanley and Lehman’s 2015 book [@stanley2015greatness], objective functions are presented as a distraction, as novelty and diversity might not be achieved by hard-coding the arrival point. Here, in contrast, we have many evolvable objective functions cooperate in reaching a solution, as a stepping stone to reach the search for novelty. By letting the swarm grow, we see the emergence of collective intelligence, which corresponds to the invention of signal-based error correction. By exchanging signals, the agents are able to correct the error induced by the noise we injected in the simulation. Like for the large scale boids simulation, the invention happens after a critical mass of agents is reached, suggesting similar dynamics with stateful agents. Discussion ========== OEE is the everlasting innovative processes found in human technologies and biological evolutions, and we barely observe open-endedness outside these examples. Yet, some artificial systems demonstrate close-to-OEE phenomena, which we have discussed earlier in this paper. Achieving real OEE remains an open challenge, and at this point all works in the literature works fall short of that objective. Although that may be the case with the swarm models presented above, it was one of our goals to emphasize the importance of maintaining the evolvability of a system. In an adversarial game theoretical setup for example, reaching an ESS or a local attractor may keep the system from inventing new solutions. In this sense, explicitly stopping the system from learning too much may allow the system to avoid being stuck in such attractor, and possibly keep innovating forever. In this paper, we propose collective intelligence as a driving force towards open-ended evolution, suggesting that collective groups can develop the ability to be more innovative. Instead of aiming at optimizing one fixed objective function, a collective swarm of agents works with as many competing objectives as there are agents in the swarm. Through information exchanges between a certain number of agents, these objectives, embodied in the agent’s behaviors, can collaborate to implement a search for novelty. All agents contribute to the search in behavioral space, as one whole organization, by exploring the adjacent possible. Each novel discovery in the system, or emergent level of organization, can be reached from an adjacent state where the system was previously. The way one moves from one point to the next, which should retain information accumulated in the past, is constrained by the structure of the swarm, in a bottleneck effect. We discuss several instances of bottlenecks in this paper. One is a task or environmental condition which each agent must overcome. In the case of foraging environment, organizing swarming turned out to be a critical step. So it became a major transition from non-swarming to swarming agents. Swarming behavior was obtained by organizing a hill side function by the neural controller. After the swarming behavior has been achieved, other properties (e.g. individual pattern) start to evolve. So for the task, swarming was a necessary behavior to organize and was a bottleneck for the entire evolutionary process. In other words, OEE emerges by setting up a right environmental condition. In case of a game-theoretic situation, such as Study 2, the communication system among all agents constituted a bottleneck to achieve mutual cooperation. With the emergence of niche construction, the door opens for regulating mechanisms such as cooperation, reciprocal altruism or social punishment, to get implemented. In this example, the OEE in terms of the invention of cooperation mechanisms, can only evolve as a secondary structure once the swarming structure is already established. In the case of large swarm models, the bottleneck is twofold. One is scale itself and also its CPU resource. We discussed the evolution of swarm by increasing its size, showing there is a critical size where the different kinds of fluctuation dominates in larger flocks (i.e. heading direction fluctuation to the density fluctuation). If such a transition occurs at the larger size simulation (we expect it can happen in each 3-4 order difference in sizes), we say that OEE is caused by increasing the size. In addition to this point, 3D swarm models require a huge computational power and we need to elaborate programming for a large scale systems. In Study 3, each boid has a list of neighbors and it is updated periodically. This speeds up the calculation of the distance from the one to its neighboring boids. In study 1, each bird can listen to the sound sent unidirectionally from the other boids, so that we don’t have to calculate the exact distance. Real birds will never measure the distance to the other individuals. So measuring the distance is an unnecessary bottleneck due to the computational model. Here the OEE is the new computational techniques to overcome this computational bottlenecks. The computation of a swarm displays a bottleneck effect, in the sense that the emergent properties of the swarm and its embodiment in a simulated environment may constrain the way the information (communication, lineage information) flows within the system, and the way relevant information (strategies, motion patterns) is progressively retained[^2] through time in its structure. Nevertheless, the simplicity of such information flows may be limiting, more complex information transfer protocols may need to emerge from bottlenecks to bootstrap OEE. For open-endedness, bottlenecks are crucial to, perhaps counterintuitively, act against learning. We observe examples of such bottlenecks in systems like Picbreeder [@secretan2008picbreeder], where one must find a way to avoid the system from assuming that the current apparent goal is the ultimate goal, as this would preclude further innovations. Picbreeder-like systems present similarities with our signal-based swarms, as they have communication between many agents filter information to let innovations come about. As suggested in the beginning of the paper, bottlenecks can be caused by different components: an explicit communication system, a concurrent evolutionary system, and a greater complexity. These three components are highlighted in the studies described above, and we propose them as the principal ones to create novelty and heterogeneity in solutions. First, the communication between agents is shown to catalyze swarming and cooperation strategies. In previous work of turn-taking interaction between two agents installed with neural networks [@iizuka2003adaptive], we noticed that performing democratic turn-taking offers novel styles of motion evolutionarily. Accordingly, here, the local interactions between agents in a flock allow for the swarm to take particular shapes (Study 1), invent an explicit cooperative protocol (Study 2), and implement a noise-canceling policy (Study 3). To reach OEE, perhaps more than mere signaling, higher complexity levels of language may need to emerge. Second, the concurrent evolution algorithm essentially selects for meaningful information in behavioral space, by squeezing noisy behavior through a selective bottleneck. However, instead of using one unique objective function, the selection is distributed asynchronously in space and time. Differential timescales also helps accelerating the learning, which should happen as fast as possible, while retaining the way to generate the best found patterns discovered in the past. Lastly, once past the selection bottleneck, heterogeneity seems to increase considerably in genotypic space. Third, in terms of complexity, given the population size is large enough, with a consequently large number of degrees of freedom, we notice the swarm dynamics significantly change, in various ways. The flock’s surface curvature may vary for large or small flocks, as well as the attraction and repulsion induced by the exchange of different signals. The motor responses may be amplified, since the input signals may significantly increase, given a higher density of neighboring agents, as seen in Study 1. Similarly, smaller flocks may display a more ordered behavior, with the trade-off however of being more sensitive to noise, since the critical mass is not reached to implement noise-canceling effects, as demonstrated in Study 3. Larger flocks can also be a source of individual behavioral differentiation, when a higher order of organization emerges. The key is not the size nor the amount of new information, but rather the system promoting the invention of new coordination patterns within itself. We have shown how collective intelligence has the ability to augment the creation of new and diverse solutions in a swarm, when given limited channels of communication, a concurrent evolution bottleneck and a large number of constrainted degrees of freedom. It come as an inspiration for scientists: a good way to build an open-ended system, able to indefinitely discover new inventions, seems not to reside in centralized computation, but rather in distributed systems, composed of large collectives of communicating agents. Acknowledgements ================ The authors would like to thank their collaborators who contributed partially to this work: Nathanael Aubert-Kato, Aleksandr Drozd, Yasuhiro Hashimoto, Norihiro Maruyama, Yoh-ichi Mototake and Mizuki Oka. [^1]: at ALIFE 2018, in Tokyo [^2]: A swarm can be shown to act as a collective memory, either explicitly/statefully [@witkowski2016emergence] or dynamically/statelessly [@couzin2002collective].

--- abstract: | We show that a strong P-Cygni feature seen in the far-UV spectra of some very hot (${\mbox{\,$T_{eff}$}}\gtrsim 85$ kK) central stars of planetary nebulae (CSPN), which has been previously identified as [$\lambda$]{}977, actually originates from  [$\lambda$]{}973. Using stellar atmospheres models, we reproduce this feature seen in the spectra of two \[WR\]-PG 1159 type CSPN, Abell 78 and NGC 2371, and in one PG 1159 CSPN, K 1-16. In the latter case, our analysis suggests an enhanced neon abundance. Strong neon features in CSPN spectra are important because an overabundance of this element is indicative of processed material that has been dredged up to the surface from the inter-shell region in the “born-again” scenario, an explanation of hydrogen-deficient CSPN. Our modeling indicates the  [$\lambda$]{}973 wind feature may be used to discern enhanced neon abundances for stars showing an unsaturated P-Cygni profile, such as some PG 1159 stars. We explore the potential of this strong feature as a wind diagnostic in stellar atmospheres analyses for evolved objects. For the \[WR\]-PG 1159 objects, the line is present as a P-Cygni line for ${\mbox{\,$T_{eff}$}}\gtrsim 85$ kK, and becomes strong for $100 \lesssim {\mbox{\,$T_{eff}$}}\lesssim 155$ kK when the neon abundance is solar, and can be significantly strong beyond this range for higher neon abundances. When unsaturated, [*i.e.*]{}, for very high  and/or very low mass-loss rates, it is sensitive to ${\mbox{\,$\dot{M}$}}$ and very sensitive to the neon abundance. The  classification is consistent with recent identification of this line seen in absorption in many PG 1159 spectra. author: - 'J.E. Herald, L. Bianchi' - 'D.J. Hillier' title: 'DISCOVERY OF NeVII IN THE WINDS OF HOT EVOLVED STARS[^1]' --- INTRODUCTION {#sec:intro} ============ Central stars of planetary nebulae (CSPN) represent an evolutionary phase the majority of low/intermediate mass stars will experience. A small subset of CSPN have been termed “PG 1159-\[WR\]” stars, as they represent objects transitioning from the \[WR\] to the PG 1159 class. The former are objects moving along the constant-luminosity branch of the H-R diagram which have optical spectra rich in strong emission line features, similar to those of Wolf-Rayet (WR) stars, which represent a late evolutionary stage of massive stars (the “\[WR\]” designation is meant to distinguish the two). The majority of \[WR\] CSPN show prominent carbon features and are termed “\[WC\]”, while a handful show strong oxygen lines (“\[WO\]”). This difference is believed to reflect a difference in the ionization of the winds rather than in the elemental abundances [@crowther:02]. Unlike massive WR stars, CSPN of the nitrogen-rich \[WN\] subtype are very rare, the two candidates being LMC-N66 in the LMC [@pena:04] and PM5 in the Galaxy [@morgan:03]. The PG 1159 class marks the entry point onto the white-dwarf cooling sequence, and these stars display mainly absorption line profiles in the optical, as their stellar wind has almost all but faded. Both classes are examples of hydrogen-deficient CSPN, which presumably make up 10-20% of the CSPN population ([@demarco:02; @koesterke:98b] and references therein), and are believed to represent subsequent evolutionary stages based on their similar parameters and abundances. An explanation for the origin of such objects is the “born-again” scenario (see [@iben:95; @herwig:99] and references therein). In this scenario, helium shell flashes produce processed material between the H- and He-burning shells (the intershell region). This material is enriched in He from CNO hydrogen burning, but through 3$\alpha$ process burning it also becomes enriched in C, O, Ne, and deficient in Fe (see, [*e.g.*]{}, [@werner:04], and references therein). After the star initially moves off the asymptotic giant-branch (AGB), it experiences a late helium-shell flash, causing the star to enter a second (or “born-again”) AGB phase. Flash induced mixing dredges the processed intershell material to the surface, resulting in H-deficient surface abundances. When the star enters its second post-AGB phase, its spectrum can develop strong wind features causing it to resemble that of (massive) WR stars, perhaps because the chemically enriched surface material increases the efficiency of radiative momentum transfer to the wind. Eventually, as the wind fades, the object moves onto the white dwarf cooling sequence. As this happens, observable wind spectral features may only be present in the far-UV and UV regions. @herald:04b (hereafter, HB04) modeled the far-UV and UV spectra of four Galactic CSPN, including Abell 78 (A78 hereafter), considered the proto-typical transition star and candidate for the born-again scenario. @crowther:98 classified it as a PG 1159-\[WO1\] star based on its high / ratio. HB04 also presented an analysis of NGC 2371, which the authors argue is of similar nature, although possessing a wind of even higher ionization. In those analyses, HB04 were unable to reproduce the prominent P-Cygni feature seen in the spectra of both stars at $\sim 975$ Å, identified in the spectra of A78 as  [$\lambda$]{}977 in the past [@koesterke:98b]. Prominent  [$\lambda$]{}977 P-Cygni profiles do occur in the far-UV spectra of CSPN of cooler temperatures (${\mbox{\,$T_{eff}$}}\lesssim 80$ kK, see, [*e.g.*]{}, [@herald:04a]), as well as in massive hot stars. However, effective temperatures of the transition objects (such as A78) are found to be very high ([*i.e.*]{}, $\gtrsim 90$ kK) from both stellar atmospheres codes ([*e.g.*]{}, [@werner:03], HB04) and nebular line analyses ([*e.g.*]{}, [@kaler:93; @grewing:90]). HB04 noted that although this feature has been assumed to be from , a strong transition from an ion of relatively low ionization potential (48 eV) in such highly-ionized winds was strongly questionable. HB04 investigated and excluded the possibility that the line originated from a highly ionized iron species. @werner:04 reported the identification of a narrow absorption feature at [$\lambda$]{}973 in the spectra of several PG 1159 stars as . In some cases, this line is superimposed on a broad P-Cygni feature, which they identified as  [$\lambda$]{}977. Given that the high ionization potential of  (207 eV) is more consistent with the conditions expected in objects of such high temperatures, we were motivated to include neon in the model atmospheres of A78 and NGC 2371 presented in HB04. Neon had not been included in any previous modeling. We present the results of this analysis, and report that neon can adequately account for this hitherto unexplained wind feature. Additionally, we show that  is also responsible for the broad P-Cygni feature seen in the spectra of the PG 1159 star K 1-16. We also investigate, with a grid of models, the usefulness of this line as a wind diagnostic for very hot CSPN. As the neon abundance is also of interest with respect to massive stars with winds, this work may have application to the study of the hottest Wolf-Rayet stars as well. This paper is arranged as follows: the observations are described in § \[sec:obs\]. The models are described in § \[sec:modeling\]. Our results are discussed in § \[sec:discussion\] and our conclusions in § \[sec:conclusions\]. OBSERVATIONS AND REDUCTION {#sec:obs} ========================== The data sets utilized in this paper are summarized in Table \[tab:obs\]. For NGC 2371 and K 1-16, we have used far-UV data from *Far-Ultraviolet Spectroscopic Explorer* (FUSE), and for A78, from the *Berkeley Extreme and Far-UV Spectrometer* (BEFS). For NGC 2371, we have also made use of a UV *International Ultraviolet Explorer* (IUE) spectrum. The data characteristics, and the reduction of the FUSE data, are described in HB04. The data were acquired from the MAST archive. For K 1-16, the FUSE data were reduced in a similar manner as described in HB04, except with the latest version of the FUSE pipeline (CALFUSE v2.4). The count-rate plots show that the star was apparently out of the aperture during part of the observation, and data taken during that period was omitted from the reduction process. The radial velocities of NGC 2371 and A78 are $+20.6$ and $+17$ , respectively [@acker:92]. All observed spectra presented in this paper have been velocity-shifted to the rest-frame of the star based on these values. The far-UV spectra of our sample are shown in Fig. \[fig:fuv\_all\], along with our models (described in § \[sec:modeling\]). They are mainly dominated by two strong P-Cygni features -  [$\lambda\lambda$]{}1032,38 and  [$\lambda$]{}973. Both features are saturated in the spectra of NGC 2371 and A78, while they are unsaturated in that of K 1-16. The numerous absorption lines seen are due to the Lyman and Werner bands of molecular hydrogen (), which resides in both the interstellar and circumstellar medium (discussed in HB04). Close inspection of the  P-Cygni profile reveals that there does appear to be some absorption due to  [$\lambda$]{}977 in each case, as well as emission in the case of NGC 2371. This apparently arises from absorption by cooler carbon material in the circumstellar environment, perhaps similar to the “carbon curtain” @bianchi:87b invoked to explain the similar  [$\lambda\lambda$]{}1334.5,1335.7 features seen in the spectra of NGC 40 (that CSPN has a temperature of ${\mbox{\,$T_{eff}$}}= 90$ kK, as estimated from the UV spectrum, too hot for  to be present in the stellar atmosphere). MODELING {#sec:modeling} ======== To analyze the spectra of our sample, we have computed non-LTE line-blanketed models which solve the radiative transfer equation in an extended, spherically-symmetric expanding atmosphere. The models are identical to those described in HB04, except neon is now included in the model atmospheres. The reader is referred to that work for a more detailed description of the models, here we give only a summary. The intense radiation fields and low wind densities of CSPN invalidate the assumptions of local thermodynamic equilibrium, and their extended atmospheres necessitate a spherical geometry for solving the radiative transfer equation. To model these winds, we have used the CMFGEN code [@hillier:98; @hillier:99b; @hillier:03]. The detailed workings of the code are explained in the references therein. To summarize, the code solves for the non-LTE populations in the co-moving frame of reference. The fundamental photospheric/wind parameters include , , , the elemental abundances and the velocity law (including ). The *stellar radius* () is taken to be the inner boundary of the model atmosphere (corresponding to a Rosseland optical depth of $\sim20$). The temperature at different depths is determined by the *stellar temperature* , related to the luminosity and radius by $L = 4\pi{\mbox{\,$R_{*}$}}^2\sigma{\mbox{\,$T_{*}$}}^4$, whereas the *effective temperature* () is similarly defined but at a radius corresponding to a Rosseland optical depth of 2/3. The luminosity is conserved at all depths, so $L = 4\pi{\mbox{\,$R_{2/3}$}}^2\sigma{\mbox{\,$T_{eff}$}}^4$. We assume what is essentially a standard velocity law $v(r) = {\mbox{\,v$_{\infty}$}}(1-r_0/r)^\beta$ where $r_0$ is roughly equal to , and $\beta =1$. For the model ions, CMFGEN utilizes the concept of “superlevels”, whereby levels of similar energies are grouped together and treated as a single level in the rate equations [@hillier:98]. Ions and the number of levels and superlevels included in the model calculations are listed in Table \[tab:ion\_tab\]. The atomic data references are given in HB04, except for neon (discussed in § \[sec:neon\]). The parameters of the models presented here are given in Table \[tab:mod\_param\_dist\]. Abundances {#sec:abund} ---------- Throughout this work, the nomenclature $X_i$ represents the mass fraction of element $i$, “” denotes the solar abundance, with the values for “solar” taken from @grevesse:98 (their solar abundance of neon is 1.74 by mass). As explained in HB04, an abundance pattern of ,,= 0.54, 0.36, 0.08 was adopted to model these hydrogen deficient objects. The nitrogen abundance was taken to be = 0.01, and solar values were adopted for the other elements, except for iron. HB04 and @werner:03 found a sub-solar iron abundance was required to match observations of A78, and our models of that star have ${\mbox{\,$X_{Fe}$}}= 0.03{\mbox{\,$\rm{X_{\odot}}$}}$. Neon {#sec:neon} ---- The prominent far-UV  feature arises from the $2p^1P^o - 2p^2 \:^1D$ transition. As discussed by @werner:04, there is some uncertainty in the corresponding wavelength. We adopt 973.33 Å, the value found in the Chianti database [@young:03] and which was was measured by @lindeberg:72. The corresponding lower and upper level energies are 214952.0 and 317692.0 $\rm{cm^{-1}}$, respectively. The neon atomic data was primarily taken from the Opacity Project [@seaton:87; @opacity:95; @opacity:97] and the Atomic Spectra Database at the NIST Physical Laboratory. For , energy level data have been taken from NIST with the exception of the $2p^2 \:^1D$ level, for which we have used the value from the Chianti database. Individual sources of atomic data (photo-ionization and cross-sections) include the following: @luo:89a (), @tully:90 (), and @peach:88 (). RESULTS {#sec:discussion} ======= The goal of this work was to test whether the inclusion of neon in the models of HB04 could account for the strong P-Cygni feature appearing at $\sim975$ Å in the spectra of two transition stars (NGC 2371 and A78), which previously has lacked a plausible explanation (HB04). The previous common identification with  [$\lambda$]{}977 was questioned by HB04, as the presence of this ion would imply a much lower , inconsistent with other spectral diagnostics. @koesterke:98b speculated that for A78, neglected iron lines might sufficiently cool the outer layers of the (otherwise hot) wind to allow for the formation of . However, HB04 computed models which included highly ionized iron, and excluded this explanation for the observed feature. As we discuss below, we find that this P-Cygni line originates from in both stars. We also show that this is the case for a PG 1159 star, K 1-16. Additionally, we explore the usefulness of this feature as a diagnostic for stellar parameters, which is very important given the scarcity of diagnostic lines at high effective temperatures (discussed by HB04). NGC 2371 & A78 {#sec:transition} -------------- We initially re-calculated the NGC 2371 and A78 best-fit models of HB04 (Table \[tab:mod\_param\_dist\]) including neon at solar abundance. The resulting models reproduced the [$\lambda$]{}973 P-Cygni feature at a strength comparable to the observations, (although a bit weak in both cases), showing that  is indeed responsible for this line. We also computed models with higher neon abundances, which is a predicted consequence of the “born-again” scenario (see § \[sec:diag\]). In Fig. \[fig:fuv\_all\], we show the ${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$ models, which are nearly indistinguishable from the solar abundance models (see § \[sec:diag\]). We have applied the effects of  and  absorption to the model spectra as described in HB04. The feature is weaker in the observations of A78 than in those of NGC 2371, and its blue P-Cygni edge is more severely affected by absorption from , making assessments of the quality of its model fit more uncertain. We note here that the parameters derived by HB04 were determined from a variety of diagnostic lines, and the listed uncertainties take into account all the different adjustments needed to fit them all, not just the ones shown here. In addition, the inclusion of neon in the calculation does not change the ionization significantly for other abundant ions in the wind (as can be seen in Fig. \[fig:fuv\_all\], where the HB04 models are also plotted), thus there was no need for a revision of the stellar parameters. K 1-16 {#sec:k116} ------ Based on the similar parameters ( and abundances) of NGC 2371 and A78 to those of PG 1159 stars discussed by @werner:04, we suspected that the broad P-Cygni profile identified therein as  [$\lambda$]{}977 in the spectra of a few of their objects originated from as well. @werner:04 do classify a  line from their (static) models, but only attribute this identification to a narrow absorption feature, while identifying the broad P-Cygni wind feature as . We decided to test this hypothesis for the case of the PG 1159 star K 1-16. @koesterke:98b determined the following parameters for K 1-16 from a hydrostatic analysis: ${\mbox{\,$T_{*}$}}= 140$ kK, $\log{L/{\mbox{\,$\rm{L_{\odot}}$}}} = 3.6$ (which imply ${\mbox{\,$R_{*}$}}= 0.11$ ), ${\mbox{\,$\log{g}$}}=6.1$, , , = 0.38, 0.56, 0.06, and the following parameters from a wind-line analysis: ${\mbox{\,v$_{\infty}$}}= 4000$  and $\log{{\mbox{\,$\dot{M}$}}} = -8.1$  from the  resonance lines. Although the mass-loss rate is lower, the other parameters are close to those NGC 2371, so we first took the parameters of our NGC 2371 model shown in Fig. \[fig:fuv\_all\], and scaled them to K 1-16’s radius of ${\mbox{\,$R_{*}$}}= 0.11$  as determined by @koesterke:98b. Further scaling of the model flux is needed to match the observed flux levels of K 1-16, and this scaling is equivalent to the star lying at a distance of 2.05 kpc. The only distance estimates to this star are statistical (based on nebular relations), ranging from 1.0 to 2.5 kpc ([@cahn:92; @maciel:84], respectively). The FUSE spectrum of K 1-16 (Fig. \[fig:fuv\_all\]) shows a unsaturated  P-Cygni profile of comparable strength to the  feature (which is also unsaturated), in contrast to that of NGC 2371, where the profiles are saturated and that of  is stronger than that of . We therefore decreased the mass-loss rate of the scaled model with solar neon abundance until the feature was fit adequately (the resulting parameters of the best-fit model for K 1-16 are listed in Table \[tab:mod\_param\_dist\]). The resulting model’s  feature is unsaturated and is weak compared to the observations, as shown in Fig. \[fig:k116\_XNe\]. We also computed an enriched model with ${\mbox{\,$X_{Ne}$}}=10$  (also shown). As the figure illustrates, the feature (unsaturated in this case) is now very sensitive to the abundance (unlike the cases of NGC 2371 and A78), and the profile of the Ne-enriched model is now too strong. Fig. \[fig:fuv\_all\] shows both the  and  lines of the Ne-enriched K 1-16 model. Our models undoubtedly show that , not , accounts for the broad P-Cygni profile seen in the PG 1159 stars as well. Furthermore, our models demonstrate how this feature can be used to detect a supersolar neon abundance in the case of an unsaturated [$\lambda$]{}973 profile. Although these results suggest a supersolar abundance for K 1-16, a more complete photospheric/wind-line analysis should be performed, given that other parameters influence the strength of this line (§ \[sec:diag\]). To account for the interstellar  absorption, we have used the parameters utilized by @kruk:98 ($\log{N({\mbox{\rm{\ion{H}{1}}}})} = 20.48$ cm$^{-2}$, $b=20$ ). We modeled the  absorption using $\log{N({\mbox{\rm{H}$_2$}})} = 16.0$ cm$^{-2}$, which produces fits adequate for our purpose (Figs. \[fig:fuv\_all\] and \[fig:k116\_XNe\]). We have assumed  and  gas temperatures of 80 K, and used the same methods described in HB04 to calculate the absorption profiles. We found a slightly higher reddening value than that of @kruk:98 to produce better results (${\mbox{\,$E_{\rm{B-V}}$}}= 0.025$ vs. 0.02 mag). We note that the far-UV spectrum of K 1-16 shows very little absorption from  compared to other CSPN ([*e.g.*]{}, [@herald:02; @herald:04a; @herald:04b]), presenting a very “clean” example of a far-UV CSPN spectrum.  [$\lambda$]{}973 as a DIAGNOSTIC {#sec:diag} --------------------------------- To investigate the potential of this feature as a diagnostic of stellar parameters, we computed exploratory models varying either the neon abundance, the mass-loss rate or the temperature of the A78 and NGC 2371 models (while keeping the other parameters the same) to study the sensitivity of this line to each parameter. Evolutionary calculations of stars experiencing the “born-again” scenario ([*e.g.*]{}, see [@herwig:01]) predict a neon abundance of $\sim2$% by mass in the intershell region, produced via $\:^{14}\rm{N}(\alpha,\gamma)\:^{18}\rm{F}(\rm{e}^{+}\nu)\:^{18}\rm{O}(\alpha,\gamma)\:^{22}\rm{Ne}$. This material later gets “dredged up” to the surface, resulting in a surface abundance enhancement of up to 20 times the solar value. We have thus calculated models with super-solar neon abundances (with ${\mbox{\,$X_{Ne}$}}= 10$ , and 50 ) to gauge the effects on the [$\lambda$]{}973 feature (shown in Fig. \[fig:diag\]). As expected, because the feature is nearly saturated in both cases, the line shows only a weak dependence on . The enriched models do result in a better fit than the solar abundance models. However, given the sensitivity of this line to ${\mbox{\,$T_{eff}$}}$ (discussed below) and the uncertainty in this parameter (see HB04), we cannot make a definitive statement about the neon abundance of these transition objects based solely on this wind line. On the other hand, the strength of the  line depends dramatically on the neon abundance in parameter regimes where it is not saturated, [*e.g.*]{}, for very high  or very low mass-loss rates, as discussed below and shown in Figs. \[fig:k116\_XNe\]-\[fig:k116\_mdot\]. To test the sensitivity of the [$\lambda$]{}973 feature to , we have computed a range of models varying the mass-loss rates of our models (with solar neon abundance) while keeping the other parameters fixed for each. We find virtually no change while the [$\lambda$]{}973 profile remains saturated, until the change in  induces a significant change in the ionization of the wind. This is shown for the model parameters of K 1-16 in Fig. \[fig:k116\_mdot\], where the  profile is essentially unchanged for $5{\mbox{\,$\rm x 10^{-8}$}} < {\mbox{\,$\dot{M}$}}< 1{\mbox{\,$\rm x 10^{-7}$}}$ , and then weakens dramatically as  lowered to $1{\mbox{\,$\rm x 10^{-8}$}}$ . However, if the atmosphere is Ne-enriched, this limit could be significantly lower, as illustrated in the $5{\mbox{\,$\rm x 10^{-9}$}}$  (${\mbox{\,$T_{eff}$}}=135$ kK) case. The ionization structures of neon for the A78 and NGC 2371 models (with ${\mbox{\,$X_{Ne}$}}=1{\mbox{\,$\rm{X_{\odot}}$}}$) are shown in Fig. \[fig:ion\]. In the cooler A78 model,  is only dominant deep in the wind, with being dominant in the outer layers. We have explored the temperature sensitivity of the feature by adjusting the luminosity of our default models while keeping the other parameters ([*i.e.*]{},  and ) the same. The results are shown in Fig. \[fig:diag\]. For the A78 model parameters, the  [$\lambda$]{}973 wind feature weakens as the temperature is decreased, becoming insignificant for ${\mbox{\,$T_{eff}$}}\lesssim 85$ kK. For the NGC 2371 model parameters, it weakens significantly as the temperature is lowered from $\simeq 130$ kK to $\simeq 110$ kK. The profile is fairly constant for $130 \lesssim {\mbox{\,$T_{eff}$}}\lesssim 145$ kK, and then starts to weaken as the temperature is increased, as  ceases to be dominant in the outer wind (around ${\mbox{\,$T_{eff}$}}\simeq 150$ kK), and becomes a pure absorption line for ${\mbox{\,$T_{eff}$}}\gtrsim 170$ kK. However, these thresholds are dependent on the neon abundance, as illustrated by the ${\mbox{\,$T_{eff}$}}= 165$ kK models. For that temperature, the solar neon abundance model shows only a weak, unsaturated P-Cygni profile, but in the ${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$ model, the line increases dramatically, becoming saturated again. This shows the potential of this line to exist in strength over a wide range of effective temperatures, as well as being an Ne-abundance diagnostic. The presence of  as a P-Cygni profile sets a lower limit to  ($\sim 80$ kK), quite independent of the neon abundance. Although our modeling indicates the  [$\lambda$]{}973 feature may not be useful in diagnosing a super-solar neon abundance in the case of NGC 2371 (because it is saturated) and A78 (because its blue edge is obscured by  absorption), it does show other neon transitions in the far-UV and UV which do not produce significant spectral features for a solar neon abundance, but do for enriched neon abundances (see examples in Fig. \[fig:neon\_lines\]). The strongest examples are the [$\lambda$]{}2213.13 and [$\lambda$]{}2229.05 transitions from the $3d \:^2D - 3p \:^2P^o$  triplet which become evident in models of both objects when ${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$. For this abundance,  $3p \:^2P^o - 3s \:^2S$ transitions are also seen at [$\lambda$]{}2042.38 and [$\lambda$]{}2055.94 in the A78 models, and the  $3s^1S - 3p^1 P^o$ transition ([$\lambda$]{}3643.6) in the models of NGC 2371 (this line has been used by [@werner:94] to deduce enhanced neon abundances in a few PG 1159 stars, including K 1-16). Also seen in the ${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$ models of NGC 2371 is the  $3p^3P^o - 3d \:^3D$ multiplet, which @werner:04 observed at positions shifted about 6 Å blueward of the wavelengths listed in the NIST database (the strongest observed component occurs at [$\lambda$]{}3894). For ${\mbox{\,$X_{Ne}$}}= 50{\mbox{\,$\rm{X_{\odot}}$}}$, in the model of NGC 2371,  $2p^2 \:^4P - 2p^2 \:^2P^o$ features are seen from 993 Å to 1011 Å, as well as blend of -  transitions at $\sim2300$ Å. Although the resolution and/or quality of the available IUE data in this range are not sufficient to rigorously analyze these lines, we note that the observations seem to favor a higher neon abundance, based on a significantly strong feature at 2230 Å in the IUE observations of NGC 2371 that is only matched by the ${\mbox{\,$X_{Ne}$}}= 50{\mbox{\,$\rm{X_{\odot}}$}}$ model. We note that introducing neon at solar abundance in the model atmospheres of these objects does not have a significant impact on the ionization structure of other relevant ions in the wind. For A78, at ${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$, the ionization structures of other elements changes slightly, but not enough to result in spectral differences. Very high neon abundances (${\mbox{\,$X_{Ne}$}}= 50{\mbox{\,$\rm{X_{\odot}}$}}$) result in a less ionized wind, with, for example, the  [$\lambda$]{}1371 feature strengthening as this ion becomes more dominant in the outer parts of the wind. The neon ionization structure is most dramatically affected - with becoming the dominant ion in the outer parts of the wind. For NGC 2371, introducing neon at solar abundance reduces the ionization of the wind slightly, with the effect becoming more significant at ${\mbox{\,$X_{Ne}$}}\ge 10{\mbox{\,$\rm{X_{\odot}}$}}$ when it leads to stronger  and  features at UV wavelengths. Thus, high neon abundances can significantly influence the atmospheric structure, and fitting a spectrum using the same set of non-neon diagnostics with a model with an enriched neon abundance generally seems to require a higher luminosity. CONCLUSIONS {#sec:conclusions} =========== We have shown that the strong P-Cygni wind feature seen around 975 Å (hitherto unidentified or mistakenly identified as [$\lambda$]{}977) in the far-UV spectra of very hot (${\mbox{\,$T_{eff}$}}\gtrsim 100$ kK) CSPN can be reproduced by models which include neon in the stellar atmosphere calculations. We have demonstrated this identification in the case of A78, a transitional \[WO\]-PG 1159 star, and in a similar object with winds of even higher ionization, NGC 2371. Through a comparison of our models with the far-UV spectrum of the PG 1159-type CSPN K 1-16, we have also demonstrated that the broad wind feature seen at this wavelength in some PG 1159 objects originates not from  (as indicated in [@werner:04]), but from  as well. Our grid of models show that  [$\lambda$]{}973.33 is a very strong wind feature detectable at solar abundance levels, in contrast to photospheric optical neon features [@werner:04], in CSPN of high stellar temperatures (${\mbox{\,$T_{eff}$}}\gtrsim 85$ kK). For the parameters of NGC 2371 ($\log{{\mbox{\,$\dot{M}$}}} = -7.1$ ) and A78 ($\log{{\mbox{\,$\dot{M}$}}} = -7.3$ ), the strength of the feature peaks for $130 \lesssim {\mbox{\,$T_{eff}$}}\lesssim 145$ kK, and weakens dramatically for ${\mbox{\,$T_{eff}$}}\gtrsim 160$ (these cutoffs depend on the value of  and the neon abundance). For an enhanced neon abundance (${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$), the feature remains strong even in models of very high temperatures (${\mbox{\,$T_{eff}$}}\gtrsim 165$ kK) or very low mass-loss rates (${\mbox{\,$\dot{M}$}}\simeq 1{\mbox{\,$\rm x 10^{-8}$}}$ ), while the lower  limits remains approximately the same. We note here that the far-UV spectra of these objects show the  feature being weaker that the  line, while in some PG 1159 stars ([*e.g.*]{}, K 1-16 and Longmore 4), they are of comparable strength. Since PG 1159 stars represent a more advanced evolutionary stage when the star is getting hotter and the wind is fading,  [$\lambda$]{}973 may be the last wind feature to disappear if the atmosphere is enriched in neon. In hydrogen-deficient objects, an enhanced neon abundance lends credence to evolutionary models which have the star experiencing a late helium shell flash, and predict a neon enrichment of about 20 times the solar value. When saturated ([*e.g.*]{}, in the case in NGC 2371), the feature is insensitive to , and only weakly sensitive to the neon abundance. Although models of these objects with enriched neon abundances do result in better fits for our transition objects, the sensitivity of the feature to ${\mbox{\,$T_{eff}$}}$ prevents us making a quantitative statement regarding abundances based on this feature alone. Other far-UV/UV lines from  (at 2042, 2056, 2213, and 2229 Å) and  (at 3644 Å) which only appear in Ne-enriched models, could in principle be used for this purpose, but we lack observations in this range of sufficient quality/resolution to make a quantitative assessment. In the case of K 1-16,  [$\lambda$]{}973 is unsaturated, and our models require an enhanced neon abundance to fit it simultaneously with the  [$\lambda\lambda$]{}1032,38 profile. This result is in line with those of @werner:94, who derived a neon abundance of 20 times the solar value for this object from analysis of the  [$\lambda$]{}3644 line. The neon overabundance is further evidence that this PG 1159 object has experienced the “born-again” scenario.  [$\lambda$]{}973 has diagnostic applications not only to late post-AGB objects, but also for evolved massive stars. Evolutionary models predict the surface neon abundance to vary dramatically as Wolf-Rayet stars evolve (see, [*e.g.*]{}, [@meynet:05]). For cooler WR stars (such as those of the WN-type), the neon abundance can be estimated from low-ionization features in the infrared and ultraviolet ([*e.g.*]{}, \[\] 12.8$\mu$m, \[\] 15.5 $\mu$m,  [$\lambda$]{}2553). The  [$\lambda$]{}973 feature may provide a strong neon diagnostic for hotter, more evolved WR stars. For example, the WO star Sanduleak 2 has ${\mbox{\,$T_{*}$}}\simeq 150$ kK [@crowther:00], and appears to have a feature at that wavelength in a FUSE archive spectrum. Neon enhancements produced in massive stars may explain the discrepancy in the $\:^{22}$Ne/$\:^{20}$Ne ratio between the solar system and Galactic cosmic ray sources (see, [*e.g.*]{}, [@meynet:01]). We are grateful to the anonymous referee for a careful reading of the manuscript and their constructive comments. We are indebted to the members of the Opacity Project and Iron Project and to Bob Kurucz for their continuing efforts to compute accurate atomic data, without which this project would not have been feasible. The SIMBAD database was used for literature searches. This work has been funded by NASA grants NAG 5-9219 (NRA-99-01-LTSA-029) and NAG-13679. The BEFS and IUE data were obtained from the Multimission Archive (MAST) at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. [33]{} natexlab\#1[\#1]{} Acker, A., Marcout, J., Ochsenbein, F., Stenholm, B., & Tylenda, R. 1992, Strasbourg-ESO Catalogue of Galactic Planetary nebulae (ESO Publication) Bianchi, L. & Grewing, M. 1987, , 181, 85 Cahn, J. H., Kaler, J. B., & Stanghellini, L. 1992, , 94, 399 Crowther, P. A. 2000, , 356, 191 Crowther, P. A., De Marco, O., & Barlow, M. J. 1998, , 296, 367 Crowther, P. A., Dessart, L., Hillier, D. J., Abbott, J. B., & Fullerton, A. W. 2002, , 392, 653 De Marco, O. & Soker, N. 2002, , 114, 602 Grevesse, N. & Sauval, A. J. 1998, , 85, 161 Grewing, M. & Neri, R. 1990, , 236, 223 Herald, J. E. & Bianchi, L. 2002, , 580, 434 —. 2004, , 611, 294 —. 2004, , 609, 378 Herwig, F. 2001, , 275, 15 Herwig, F., Blöcker, T., Langer, N., & Driebe, T. 1999, , 349, 5 Hillier, D. J., Lanz, T., Heap, S. R., Hubeny, I., Smith, L. J., Evans, C. J., Lennon, D. J., & Bouret, J. C. 2003, , 588, 1039 Hillier, D. J. & Miller, D. L. 1998, , 496, 407 —. 1999, , 519, 354 Iben, I. & McDonald, J. 1995, in LNP, Vol. 443, White Dwarfs, ed. D. Koester & K. Werner (Heidelberg: Springer), 48 Kaler, J. B., Stanghellini, L., & Shaw, R. A. 1993, , 279, 529 Koesterke, L. & Werner, K. 1998, , 500, 55 Kruk, J. W. & Werner, K. 1998, , 502, 858 Lindeberg, S. 1972, Uppsala Univ. Inst. Phys. Report UUIP-759, 1 Luo, D. & Pradhan, A. K. 1989, , 22, 3377 Maciel, W. J. 1984, , 55, 253 , G., [Arnould]{}, M., [Paulus]{}, G., & [Maeder]{}, A. 2001, Space Science Reviews, 99, 73 , G. & [Maeder]{}, A. 2005, , 429, 581 Morgan, D. H., Parker, Q. A., & Cohen, M. 2003, , 346, 719 . 1995, The Opacity Project, Vol. 1 (Bristol: Institute of Physics Publications) —. 1997, The Opacity Project, Vol. 2 (Bristol: Institute of Physics Publications) Peach, G., Saraph, H. E., & Seaton, M. J. 1988, , 21, 3669 Peña, M., Hamann, W.-R., Ruiz, M. T., Peimbert, A., & Peimbert, M. 2004, , 419, 583 Seaton, M. J. 1987, , 20, 6363 Tully, J. A., Seaton, M. J., & Berrington, K. A. 1990, , 23, 3811 Werner, K., Dreizler, S., Koesterke, L., & Kruk, J. W. 2003, in IAU Symp., Vol. 209, Planetary Nebulae: Their Evolution and Role in the Universe, ed. M. Dopita & S. Kwok (San Fransisco: ASP), 239 Werner, K. & Rauch, T. 1994, , 284, 5 Werner, K., Rauch, T., Reiff, E., Kruk, J. W., & Napiwotzki, R. 2004, , 427, 685 Young, P. R., Del Zanna, G., Landi, E., [et al.]{} 2003, , 144, 135 [^1]: Based on observations made with the NASA-CNES-CSA Far Ultraviolet Spectroscopic Explorer and data from the MAST archive. FUSE is operated for NASA by the Johns Hopkins University under NASA contract NAS5-32985.

0.7cm UT-09-14\ IPMU-09-0060 1.35cm [**Inverse Problem of Cosmic-Ray Electron/Positron\ from Dark Matter** ]{} 1.2cm Koichi Hamaguchi$^{1,2}$, Kouhei Nakaji$^1$ and Eita Nakamura$^1$ 0.4cm [*$^1$ Department of Physics, University of Tokyo, Tokyo 113-0033, Japan\ $^2$ Institute for the Physics and Mathematics of the Universe, University of Tokyo, Chiba 277-8568, Japan* ]{} 1.5cm Introduction ============ The existence of Dark Matter (DM) is by now well established [@DM], yet its identity is a complete mystery; it has no explanation in the framework of the Standard Model of particle physics. Recently, several exciting data have been reported on cosmic-ray electrons and positrons, which may be indirect signatures of decaying/annihilating DM in the present universe. The PAMELA collaboration reported that the ratio of positron and electron fluxes increases at energies of $\sim$ 10–100 GeV [@Adriani:2008zr], which shows an excess above the expectations from the secondly positron sources. In addition, the ATIC collaboration reported an excess of the total electron + positron flux at energies between 300 and 800 GeV [@:2008zzr]. (cf. the PPB-BETS observation [@Torii:2008xu].) More recently, the Fermi-LAT collaboration has just released precise, high-statistics data on cosmic-ray electron + positron spectrum from 20 GeV to 1 TeV [@Fermi], and the HESS collaboration has also reported new data [@HESS]. Although these two new data sets show no evidence of the peak reported by ATIC, their spectra still indicate an excess above conventional models for the background spectrum [@Fermi], and the decaying/annihilating DM remains an interesting possibility [@post-Fermi]. In the previous studies of DM interpretations of the cosmic-ray electrons/positrons, the analyses have been done by (i) assuming a certain source spectrum from annihilating/decaying DM, (ii) solving the propagation and (iii) comparing the predicted electron and positron fluxes with the observation. In this letter, we discuss the possibility of solving its inverse problem, namely, try to reconstruct the source spectrum from the observational data. For that purpose, we adopt an analytical approach, since the inverse problem would be very challenging in the numerical approach. We show that the inverse problem can indeed be solved analytically under certain assumptions and approximations, and provide analytic formulae to reconstruct the source spectrum of the electron/positron from the observed flux. As an illustration, we apply the obtained formulae to the electron and positron flux above $\sim 100$ GeV for the just released Fermi data [@Fermi], together with the new HESS data [@HESS]. It is found that the reconstructed spectrum for high energy range is almost independent of the propagation models and whether the DM is decaying or annihilating. We also estimate the errors of the reconstructed source spectrum. The obtained result implies that electrons/positrons at the source have a broad spectrum ranging from ${\cal O}(1)$ TeV to ${\cal O}(100)$ GeV, with a broad but clear peak around $E\sim 400$ GeV and another peak at $E\sim 1$ TeV. Inverse problem of cosmic-ray electron and\ positron from dark matter =========================================== Let us first summarize the procedure to calculate the $e^\pm$ fluxes. The electron/positron number density per unit kinetic energy $f_e(E,\vec{r},t)$ evolves as [@Strong:2007nh] $$\begin{aligned} \frac{\partial}{\partial t}f_e(E,\vec{r},t) &=& \nabla\cdot [K(E,\vec{r})\nabla f_e(E,\vec{r},t)] + \frac{\partial}{\partial E}[b(E,\vec{r}) f_e(E,\vec{r},t)] +Q(E,\vec{r},t)\,,\end{aligned}$$ where $K(E,\vec{r})$ is the diffusion coefficient, $b(E,\vec{r})$ is the rate of energy loss, and $Q(E,\vec{r},t)$ is the source term of the electrons/positrons. The effects of convection, reacceleration, and the annihilation in the Galactic disk are neglected. We only consider the electrons and positrons from dark matter decay/annihilation, which we assume time-independent and spherical; $$\begin{aligned} Q(E,\vec{r}) = q(|\vec{r}|) \frac{dN_e(E)}{dE}\,,\end{aligned}$$ where $dN_e/dE$ is the energy spectrum of the electron and positron from one DM decay/annihilation, and $q(|\vec{r}|)$ is given by $$\begin{aligned} q(|\vec{r}|) &=& \frac{1}{m_X\tau_X}\rho(|\vec{r}|)\quad{\rm for~decaying~DM}, \\ q(|\vec{r}|) &=& \frac{{ \left\langle {\sigma v} \right\rangle }}{2m_X^2}\rho(|\vec{r}|)^2\quad{\rm for~annihilating~DM},\end{aligned}$$ where $m_X$ and $\tau_X$ are the mass and the lifetime of the DM, and ${ \left\langle {\sigma v} \right\rangle }$ is the average DM annihilation cross section. We adopt a stationary two-zone diffusion model with cylindrical boundary conditions, with half-height $L$ and a radius $R$, and spatially constant diffusion coefficient $K(E)$ and the energy loss rate $b(E)$ throughout the diffusion zone. The diffusion equation is then $$\begin{aligned} K(E)\nabla^2 f_e(E,\vec{r}) + \frac{\partial}{\partial E}[b(E)f_e(E,\vec{r})] + q(|\vec{r}|) \frac{dN_e(E)}{dE} = 0\,, \label{eq:diffusion}\end{aligned}$$ with the boundary condition $f_e(E,\vec{r})=0$ for $r=\sqrt{x^2+y^2}=R$, $-L\le z\le L$ and $0\le r \le R$, $z=\pm L$. As shown in Appendix A, its solution at the Solar System, $r=r_\odot\simeq 8.5$ kpc, $z=0$ is given by $$\begin{aligned} f_e(E) = f_e(E,\vec{r}_\odot) =\frac{1}{b(E)}\int^{E_{\rm max}}_EdE' \frac{dN_e(E')}{dE'} g\left(L(E')-L(E)\right), \label{eq:dN/dE}\end{aligned}$$ where $$\begin{aligned} g(x)&=&\sum_{n,m=1}^\infty J_0 \left(j_n \frac{r_\odot}{R}\right) \sin\left(\frac{m\pi}{2}\right) q_{nm}e^{-d_{nm}x}, \label{eq:gx} \\ q_{nm}&=& \frac{2}{J_1^2(j_n) \pi} \int^1_0 d\hat{r}\, \hat{r} \int^\pi_{-\pi} d\hat{z}\, J_0(j_n \hat{r})\sin\left(\frac{m}{2}(\pi-\hat{z})\right) q\left(\sqrt{(R\hat{r})^2 + \left(\frac{L\hat{z}}{\pi}\right)^2}\right), \label{eq:qnm} \\ d_{nm} &=& \frac{j_n^2}{R^2}+\frac{m^2 \pi^2}{4L^2}, \label{eq:dnm} \\ L(E) &=& \int^EdE'\frac{K(E')}{b(E')}, \label{eq:L}\end{aligned}$$ where $j_n$ are the successive zeros of $J_0$. The electron/positron flux is given by $\Phi_e(E) = (c/4\pi)f_e(E)$. Our main purpose is to solve the inverse problem, namely, to reconstruct the source spectrum $dN_e(E)/dE$ for a given $f_e(E)$. As shown in Appendix \[inv\_sol\], this inverse problem can indeed be solved, and the solution is given by $$\begin{aligned} \label{solution_Fourier} \frac{dN_e(E)}{dE} &=& \frac{dL(E)}{dE} \int_{-\infty}^{\infty} \frac{dk}{2\pi} e^{-ikL(E)} \frac{\int_{-\infty}^{\infty}dw e^{-ikw}\tilde{A}(w)}{\int_{0}^{\infty}dz e^{-ikz}g(z)},\end{aligned}$$ where $$\begin{aligned} A(E)=f_e(E)b(E)\end{aligned}$$ and $$\tilde{A}(x=-L(E))=A(E),$$ or alternatively, $$\begin{aligned} \label{solution_Laplace} \frac{dN_e(E)}{dE}=-\frac{1}{g(0)}\left[\frac{dA(E)}{dE}+ \frac{dL(E)}{dE}\int_E^{E_{\rm max}}dE'\,\frac{dA(E')}{dE'}\Gamma\left(L(E')-L(E)\right)\right],\end{aligned}$$ where the function $\Gamma(x)$ is determined from $g(x)$. See Appendix \[inv\_sol\] for details. Several comments are in order. First of all, from Eq. (\[solution\_Laplace\]) one can see that the source flux at an energy $E_{\rm src}$ depends only on the observed flux at $E_{\rm obs}\geq E_{\rm src}$. This may be counterintuitive, because the solution to the diffusion equation (\[eq:dN/dE\]) tells us that the observed flux at $E=E_{\rm obs}$ is determined by the source flux at $E_{\rm src}\geq E_{\rm obs}$. It can be understood by considering the reconstruction of the source spectrum from the highest energy and gradually to the lower energy. Secondly, as we will see in the explicit examples, for high energy range the above formulae can be approximated by the first term of Eq. (\[solution\_Laplace\]); $$\begin{aligned} \frac{dN_e(E)}{dE}\simeq -\frac{1}{g(0)} \frac{dA(E)}{dE}\,. \label{eq:simple}\end{aligned}$$ This is because at higher energy the electrons lose their energy quickly and hence only local electrons contribute. Technically, at higher energy $g(x)$ can be approximated as $g(x)\simeq g(0)$. The approximated formula Eq. (\[eq:simple\]) is then directly obtained from Eq. (\[eq:dN/dE\]). Note that $g(0)$ is given by the local DM density $\rho_\odot\simeq 0.30$ GeV/cm$^3$; $$\begin{aligned} g(0) &=& q(r=r_\odot, z =0) =\left\{ \begin{array}{ll} \displaystyle{\rho_\odot\frac{1}{m_X\tau_X}} & {\rm for~decaying~DM}, \\ & \\ \displaystyle{\rho_\odot^2\frac{{ \left\langle {\sigma v} \right\rangle }}{2 m_X^2}} & {\rm for~annihilating~DM}. \end{array} \right.\end{aligned}$$ Examples ======== In this section, we show some examples. The energy loss rate is taken as $b(E) = E^2/E_0\tau_E$, with $E_0=1$ GeV and $\tau_E=10^{16}$ sec, and the propagation models are parametrized by $K(E) = K_0(E/E_0)^\delta$ which leads to $$\begin{aligned} L(E) = -\frac{\tau_E K_0}{1-\delta}\left(\frac{E_0}{E}\right)^{1-\delta}.\end{aligned}$$ Models $R$ \[kpc\] $L$ \[kpc\] $\delta$ $K_0$ \[kpc$^2/$Myr\] -------- ------------- ------------- ---------- ----------------------- M2 20 1 0.55 0.00595 MED 20 4 0.70 0.0112 M1 20 15 0.46 0.0765 : The diffusion model parameters consistent with the observed B/C ratio [@Delahaye:2007fr]. We consider the three benchmark models from Ref. [@Delahaye:2007fr], M2, MED, and M1, which are summarized in Table 1. The parameters $K_0$ and $\delta$ are chosen so that the observed B/C ratio is reproduced. Reconstruction with the Fermi data ---------------------------------- ![The Fermi data [@Fermi] and a fitting curve, together with three different background spectra.[]{data-label="fig:Fermidata"}](Fermi_flux.eps "fig:"){width="10cm"}\ As the first example, we consider the Fermi data of the the electron + positron flux [@Fermi], at energy above 100 GeV. As the astrophysical background, we take a simple power low $\Phi_e^{\rm bg}\propto E^{-\alpha}$, and vary the index as $\alpha = 3.2\pm 0.1$ to represent the effect of the background uncertainties.[^1] The normalization is determined by fitting the data for $E< 100$ GeV, assuming that the electron + positron flux is dominated by the background in this energy range.[^2] For illustration, we fit the observed data by a polynomial as $E^3 \Phi_e(E) = \sum_{n=0}^{n_{\rm max}} c_n E^n$ with $n_{\rm max} = 5$, which is shown in Fig. \[fig:Fermidata\] together with three different background spectra ($\alpha = 3.1$, 3.2, and 3.3). ![The source spectra reconstructed from the Fermi data in Fig. \[fig:Fermidata\], for different background spectrum indices $\alpha = 3.1$ (top left), 3.2 (top right), and 3.3 (bottom). In each case, the results of the analytic formula \[(\[solution\_Fourier\]) or (\[solution\_Laplace\])\] are shown for the decaying DM with the three propagation models. The results for the simple approximation formula (\[eq:simple\]) are also shown, which almost overlap with the M1 model lines in this figure.[]{data-label="fig:FermiSource"}](Fermi_3_1_spec.eps "fig:"){width="7cm"} ![The source spectra reconstructed from the Fermi data in Fig. \[fig:Fermidata\], for different background spectrum indices $\alpha = 3.1$ (top left), 3.2 (top right), and 3.3 (bottom). In each case, the results of the analytic formula \[(\[solution\_Fourier\]) or (\[solution\_Laplace\])\] are shown for the decaying DM with the three propagation models. The results for the simple approximation formula (\[eq:simple\]) are also shown, which almost overlap with the M1 model lines in this figure.[]{data-label="fig:FermiSource"}](Fermi_3_2_spec.eps "fig:"){width="7cm"}\ ![The source spectra reconstructed from the Fermi data in Fig. \[fig:Fermidata\], for different background spectrum indices $\alpha = 3.1$ (top left), 3.2 (top right), and 3.3 (bottom). In each case, the results of the analytic formula \[(\[solution\_Fourier\]) or (\[solution\_Laplace\])\] are shown for the decaying DM with the three propagation models. The results for the simple approximation formula (\[eq:simple\]) are also shown, which almost overlap with the M1 model lines in this figure.[]{data-label="fig:FermiSource"}](Fermi_3_3_spec.eps "fig:"){width="7cm"} In Fig. \[fig:FermiSource\], the source spectra reconstructed by the analytic formula \[(\[solution\_Fourier\]) or (\[solution\_Laplace\])\] are shown for the decaying DM, with the three background spectra and the three propagation models. Here and hereafter, we assume the isothermal DM distribution $\rho(|\vec{r}|) = \rho_\odot(r_c^2 + r_\odot^2)/(r_c^2 + |\vec{r}|^2)$ with $r_c=3.5$ kpc. We also assume that there is no excess above the energy at which the fitting curve crosses the background. Interestingly, in all cases there is a clear but broad peak around 400 GeV. There is another increase of the spectrum for $E{ \mathop{}_{\textstyle \sim}^{\textstyle >} }800$ GeV, which is due to the last two data points (see Fig. \[fig:Fermidata\]). As can be seen from the figure, the reconstructed source spectrum is almost independent of the propagation models, except for the energy range $E{ \mathop{}_{\textstyle \sim}^{\textstyle <} }200$ GeV for the M2 model with background index $\alpha { \mathop{}_{\textstyle \sim}^{\textstyle >} }3.2$. In Fig. \[fig:FermiSource\], we also show the approximated formula (\[eq:simple\]). As discussed in the previous section, the simple approximation formula (\[eq:simple\]) well reproduces the results of the full formula. In some cases, the reconstructed source spectrum becomes unphysical (negative) in the energy range of $E{ \mathop{}_{\textstyle \sim}^{\textstyle <} }200$ GeV and $700{ \mathop{}_{\textstyle \sim}^{\textstyle <} }E{ \mathop{}_{\textstyle \sim}^{\textstyle <} }800$ GeV. This is because of the relatively hard spectrum of the observed flux in this energy range. (The simple formula (\[eq:simple\]) implies that an observed flux much harder than $E^{-2}$ is difficult to obtain from the DM decay/annihilation.) In order to see the dependence on the DM distribution (or whether DM is decaying or annihilating), we show in Fig. \[fig:FermiComparison\] the case of annihilating DM compared with the decaying DM, for MED propagation with the background spectrum $\alpha=3.2$. As expected, the reconstructed spectrum is almost independent of whether the DM is decaying or annihilating. This is also understood from the approximated formula (\[eq:simple\]). ![The comparison of the source spectra for the decaying and annihilating DM, reconstructed from the Fermi data in Fig. \[fig:Fermidata\] with a background spectrum index $\alpha = 3.2$ and MED propagation. The results for the simple approximation formula (\[eq:simple\]) are also shown.[]{data-label="fig:FermiComparison"}](Fermi_3_2_ann_spec.eps){width="7cm"} Reconstruction with the HESS data --------------------------------- ![The HESS data and a fitting curve [@HESS], together with the three different background spectra.[]{data-label="fig:HESSdata"}](HESS_flux.eps "fig:"){width="10cm"}\ ![The source spectra reconstructed from the HESS data in Fig. \[fig:HESSdata\], for different background spectrum indices $\alpha = 3.1$ (top left), 3.2 (top right), and 3.3 (bottom). In each case, the results of the analytic formula \[(\[solution\_Fourier\]) or (\[solution\_Laplace\])\] are shown for the decaying DM with the MED propagation models, together with the simple approximation formula (\[eq:simple\]).[]{data-label="fig:HESS_source"}](HESS_3_1_spec.eps "fig:"){width="7cm"} ![The source spectra reconstructed from the HESS data in Fig. \[fig:HESSdata\], for different background spectrum indices $\alpha = 3.1$ (top left), 3.2 (top right), and 3.3 (bottom). In each case, the results of the analytic formula \[(\[solution\_Fourier\]) or (\[solution\_Laplace\])\] are shown for the decaying DM with the MED propagation models, together with the simple approximation formula (\[eq:simple\]).[]{data-label="fig:HESS_source"}](HESS_3_2_spec.eps "fig:"){width="7cm"}\ ![The source spectra reconstructed from the HESS data in Fig. \[fig:HESSdata\], for different background spectrum indices $\alpha = 3.1$ (top left), 3.2 (top right), and 3.3 (bottom). In each case, the results of the analytic formula \[(\[solution\_Fourier\]) or (\[solution\_Laplace\])\] are shown for the decaying DM with the MED propagation models, together with the simple approximation formula (\[eq:simple\]).[]{data-label="fig:HESS_source"}](HESS_3_3_spec.eps "fig:"){width="7cm"} Next, we apply the formula to the newly published HESS data [@HESS], shown in Fig. \[fig:HESSdata\]. As for the fitting function, we adopt the broken power law described in [@HESS]. We assume the same background spectra $E^{-(3.2\pm 0.1)}$ as before. The resultant source spectrum is shown in Fig. \[fig:HESS\_source\] for decaying DM, isothermal DM profile and MED propagation model. One can see that the breaking of the power at $E\sim 1$ TeV results in a peak at that energy in the reconstructed source spectrum. Estimation of the uncertainties ------------------------------- ![The reconstructed spectrum for the Fermi and HESS data, with errors corresponding to the statistical errors of their data [@Fermi; @HESS], with background spectrum indices $\alpha = 3.1$ (top left), 3.2 (top right), and 3.3 (bottom). See text for details.[]{data-label="fig:withError"}](witherror_3_1.eps "fig:"){width="7cm"} ![The reconstructed spectrum for the Fermi and HESS data, with errors corresponding to the statistical errors of their data [@Fermi; @HESS], with background spectrum indices $\alpha = 3.1$ (top left), 3.2 (top right), and 3.3 (bottom). See text for details.[]{data-label="fig:withError"}](witherror_3_2.eps "fig:"){width="7cm"}\ ![The reconstructed spectrum for the Fermi and HESS data, with errors corresponding to the statistical errors of their data [@Fermi; @HESS], with background spectrum indices $\alpha = 3.1$ (top left), 3.2 (top right), and 3.3 (bottom). See text for details.[]{data-label="fig:withError"}](witherror_3_3.eps "fig:"){width="7cm"} So far, we have neglected the errors in the observed flux. Here, we estimate the effects of the uncertainties in the observed flux. As we have seen in the previous two subsections, the simple formula (\[eq:simple\]) is a good approximation in the high energy range $E{ \mathop{}_{\textstyle \sim}^{\textstyle >} }200$ GeV. Therefore, we use it to estimate the errors of the reconstructed spectrum. The derivative $dA(E)/dE$ and the corresponding error $\delta[dA(E)/dE]$ at $E=E_i$ are estimated by fitting the sequential three data points at $E=(E_{i-1}, E_i, E_{i+1})$ with a quadratic function. The resultant source spectrum is shown in Fig. \[fig:withError\] for the Fermi and HESS data, with the three background spectra. Here, we only adopt the statistical errors of the Fermi and HESS data [@Fermi; @HESS]. The broad peaks around 400 GeV and around 1 TeV remain even after taking into account the errors in the observed flux. Although it is difficult to precisely reconstruct the source spectrum because of the limited data as well as the lack of the knowledge of the astrophysical background, Fig. \[fig:withError\] implies that the electrons/positrons from DM have a broad spectrum, ranging from ${\cal O}(1)$ TeV to ${\cal O}(100)$ GeV. For instance, a direct two-body decay of DM into electron(s), or a three-body decay into electron(s) with a smooth matrix element, would have harder spectrum and does not fit the reconstructed spectrum well. The obtained result seems to suggest that the source spectrum has a larger soft component, like the one from cascade decays. Discussion ========== In this letter, we have discussed the possibility of solving the inverse problem of the cosmic-ray electron and positron from the dark matter annihilation/decay. Some simple analytic formulae are shown, with which the source spectrum of the electron and positron can be reconstructed from the observed flux. It is shown that the reconstructed source spectrum at an energy $E_{\rm src}$ depends only on the observed flux above that energy, $E_{\rm obs}\ge E_{\rm src}$. We also illustrated our approach by applying the obtained formula to the electron + positron flux above 100 GeV for the just released Fermi data [@Fermi] and the HESS data [@HESS]. Assuming simple power-law backgrounds, the reconstructed source spectrum indicates two peaks, a broad one at around 400 GeV and another one at around 1 TeV. It is shown that the reconstructed spectrum at high energy is almost independent of the propagation models, and whether it is decaying or annihilating. The effect of the errors in the observed flux is also discussed. Within the uncertainties, the obtained result implies that the electrons/positrons at the source have a broad spectrum ranging from ${\cal O}(1)$ TeV to ${\cal O}(100)$ GeV, with a large soft component, like the one from cascade decays. It is difficult to precisely reconstruct the source spectrum at the present stage, because of the limited data as well as the lack of the knowledge of the astrophysical background. Future measurements, such as the PAMELA data at higher energy range, will allow better understandings. There is also a proposed experiment CALET [@Torii:2006qb], which can measure the electron + positron flux up to 10 TeV with a significant statistics. (cf. [@Chen:2008fx]). We expect that our approach will be a useful tool in shedding light on the mystery of DM. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Satoshi Shirai and Tsutomu Yanagida for helpful discussions and comments. This work was supported by World Premier International Center Initiative (WPI Program), MEXT, Japan. The work of EN is supported in part by JSPS Research Fellowships for Young Scientists. Solution to the diffusion equation ================================== The diffusion equation (\[eq:diffusion\]) can be solved in the following way (cf. Ref. [@Hisano:2005ec].) Using the cylindrical coordinate, $f_e(E,\vec{r})=f_e(E,r,z)$, where $r=\sqrt{x^2+y^2}$ and $$\begin{aligned} \nabla^2=\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r} +\frac{\partial^2}{\partial z^2},\end{aligned}$$ the diffusion equation (\[eq:diffusion\]) becomes $$\begin{aligned} K(E)\left[\frac{\partial^2}{\partial r^2}+\frac{1}{r} \frac{\partial}{\partial r}+\frac{\partial^2}{\partial z^2}\right]f_e(E,r,z) +\frac{\partial}{\partial E}[b(E) f_e(E,r,z)] +q(r,z)\frac{dN_e(E)}{dE} = 0.\end{aligned}$$ We expand $f_e$ as $$\begin{aligned} f_e(E,r,z)&=&\sum_{n,m=1}^\infty f_{nm}(E) J_0\left(j_n\frac{r}{R}\right) \sin\Big(\frac{m\pi}{2L}(L-z)\Big), \\ f_{nm}(E) &=& \frac{2}{J_1^2(j_n)\pi}\int_0^1d\hat{r}\,\hat{r} \int_{-\pi}^\pi d\hat{z}\, J_0(j_n\hat{r})\sin\left(\frac{m}{2}(\pi{-}\hat{z})\right) f_e\left(E,R\hat{r},\frac{L}{\pi}\hat{z}\right),\end{aligned}$$ where $J_0$ and $J_1$ are the zeroth and first order Bessel functions of the first kind, respectively, and $j_n$ are the successive zeros of $J_0$. In this expansion, the boundary condition $f_e(E,r,z)=0$ for $r=R$ and $z=\pm L$ is automatically satisfied. Conversely, any (sufficiently good) function which satisfies the boundary condition can be expanded as above. The orthogonal relations are $$\begin{aligned} && \int_0^1dx\,xJ_0(j_n x)J_0(j_{n'}x) = \frac{1}{2}J_1^2(j_n)\delta_{nn'}, \\ && \int_{-\pi}^\pi dx\,{\rm sin}\left(\frac{m}{2}(\pi{-}x)\right) {\rm sin}\left(\frac{m'}{2}(\pi{-}x)\right)=\pi\delta_{mm'}\quad(m,m'\in\mathbb{Z}_{>0}).\end{aligned}$$ Using the differential equation $$\begin{aligned} \left[\frac{d^2}{dx^2}+\frac{1}{x}\frac{d}{dx}+j_n^2\right]J_0(j_n x)=0,\end{aligned}$$ one obtains $$\begin{aligned} \label{eq_for_fnm} \left[-d_{nm}K(E)+\frac{\partial b(E)}{\partial E}+b(E) \frac{\partial}{\partial E}\right]f_{nm}(E)+q_{nm} \frac{dN_e(E)}{dE}=0,\end{aligned}$$ where $d_{nm}$ and $q_{nm}$ are given by Eqs. (\[eq:dnm\]) and (\[eq:qnm\]). Imposing the boundary condition for $f_{nm}(E)$ as \[fmn\_eq\] f\_[nm]{}(E\_[max]{})=0E\_[max]{}=[max]{}{E|Q\_[nm]{}(E)0}, the solution to Eq. (\[eq\_for\_fnm\]) is given by f\_[nm]{}(E)=\_E\^[E\_[max]{}]{}dE’ q\_[nm]{} [exp]{}, where $L(E)$ is given by Eq. (\[eq:L\]). This leads to Eq. (\[eq:dN/dE\]). Solution to the inverse problem {#inv_sol} =============================== Here we show two ways of reconstructing the source spectrum $dN_e(E)/dE$. By a change of variables, the equation (\[eq:dN/dE\]) is rewritten as $$\begin{aligned} \label{int_eq} \tilde{A}(x)&=& - \int_0^x dy \frac{d\tilde{N}_e(y)}{dy} g(x-y),\end{aligned}$$ where $x=-L(E)$, $y=-L(E')$, $\tilde{N}_e(y={-}L(E'))=N_e(E')$, $A(E)=f_e(E)b(E)$, and $\tilde{A}(x={-}L(E))=A(E)$. We assumed $L(E_{\rm max})=0$. Here, $\tilde{A}(x)$ and $g(x)$ are known functions once we fix the propagation model and have the observational data. Our inverse problem is now reduced to the problem of solving for the function $d\tilde{N}_e(x)/dx$, given the functions $\tilde{A}(x)$ and $g(x)$. The integral equation (\[int\_eq\]) is of the form known as the Volterra’s integral equation. It is known that the solution of the equation exists and is unique. We see that the right-hand side of Eq. (\[int\_eq\]) is a convolution of two functions, so the Fourier transform or the Laplace transform seems to be useful tools. In the following, we solve the equation in two different ways using the Fourier and Laplace transforms. Fourier transform {#fourier-transform .unnumbered} ----------------- Inserting two step functions in the integral, we get $$\begin{aligned} \tilde{A}(x)&=& - \int_{-\infty}^{\infty} dy \frac{d\tilde{N}_e(y)}{dy} \theta(y) \ g(x-y) \theta(x-y).\end{aligned}$$ Operating $\int_{-\infty}^{\infty} dx e^{-ikx} $ on each side of the equation and changing the variable as $z=x-y$ in the right-hand side, we obtain $$\begin{aligned} \int_{-\infty}^{\infty} dx e^{-ikx} \tilde{A}(x) &=& - \int_{-\infty}^{\infty} dy e^{-iky} \frac{d\tilde{N}_e(y)}{dy} \theta(y) \ \int_{-\infty}^{\infty} dz e^{-ikz} g(z) \theta(z).\end{aligned}$$ After dividing each side by $\int_{-\infty}^{\infty} dz e^{-ikz} g(z) \theta(z) $ and operate $ \int_{-\infty}^{\infty} \frac{dk}{2\pi} e^{ikx} $ , we finally get the formula we desire as $$\begin{aligned} \frac{d\tilde{N}_e(x)}{dx} \theta(x) &=& - \int_{-\infty}^{\infty} \frac{dk}{2\pi} e^{ikx} \frac{\int_{-\infty}^{\infty}dw e^{-ikw}\tilde{A}(w)}{\int_{-\infty}^{\infty}dz e^{-ikz}g(z) \theta(z) },\end{aligned}$$ or $$\begin{aligned} \frac{dN_e(E)}{dE} \theta\left(-L(E)\right) &=& \frac{dL(E)}{dE} \int_{-\infty}^{\infty} \frac{dk}{2\pi} e^{-ikL(E)} \frac{\int_{-\infty}^{\infty}dw e^{-ikw}\tilde{A}(w)}{\int_{0}^{\infty}dz e^{-ikz}g(z)}.\end{aligned}$$ Laplace transform {#laplace-transform .unnumbered} ----------------- Eq. (\[int\_eq\]) can be solved in an alternative way. We first differentiate the both side of the equation with respect to $x$ and then divide by $g(0)$. Then we obtain $$\label{Poisson} F(x)=\varphi(x)-\int_0^xdy\,K(x-y)\varphi(y),$$ where $$\begin{aligned} \varphi(x)&=\frac{d\tilde{N}_e(x)}{dx}, \\ F(x)&=-\frac{1}{g(0)}\frac{d\tilde{A}(x)}{dx},\end{aligned}$$ and $$K(x)=-\frac{1}{g(0)}\frac{dg(x)}{dx}.$$ The Laplace transform of Eq. (\[Poisson\]) is $${\cal L}F(\xi)={\cal L}\varphi(\xi)-{\cal L}K(\xi){\cal L}\varphi(\xi),$$ where we denote the Laplace transform of a function $f(x)$ by ${\cal L}f(\xi)$: $${\cal L}f(\xi)=\int_0^\infty dx\,e^{-\xi x}f(x).$$ This equation can be solved as $$\varphi(x)=F(x)+\int_0^xdy\,\Gamma(x-y)F(y),$$ where $$\Gamma(x)={\cal L}^{-1}\left[\frac{{\cal L}K}{1-{\cal L}K}\right](x)$$ and ${\cal L}^{-1}$ denotes the inverse Laplace transform. In the original variable, the solution can be written as $$\frac{dN_e(E)}{dE}=-\frac{1}{g(0)}\left[\frac{dA(E)}{dE}+ \frac{dL(E)}{dE}\int_E^{E_{\rm max}}dE'\,\frac{dA(E')}{dE'}\Gamma\left(L(E')-L(E)\right)\right].$$ [99]{} See, for reviews, G. Bertone, D. Hooper and J. Silk, Phys. Rept.  [**405**]{}, 279 (2005) \[arXiv:hep-ph/0404175\].\ C. Amsler [*et al.*]{} \[Particle Data Group\], Phys. Lett.  B [**667**]{} (2008) 1. O. Adriani [*et al.*]{} \[PAMELA Collaboration\], Nature [**458**]{} (2009) 607 \[arXiv:0810.4995 \[astro-ph\]\]. J. Chang [*et al.*]{}, Nature [**456**]{} (2008) 362. S. Torii [*et al.*]{} \[PPB-BETS Collaboration\], arXiv:0809.0760 \[astro-ph\]. Fermi LAT Collaboration, arXiv:0905.0025 \[astro-ph.HE\]. F. Aharonian [*et al.*]{} \[H.E.S.S. Collaboration\], Phys. Rev. Lett.  [**101**]{} (2008) 261104 \[arXiv:0811.3894 \[astro-ph\]\];\ F. Aharonian [*et al.*]{} \[H.E.S.S. Collaboration\], arXiv:0905.0105 \[astro-ph.HE\]. L. Bergstrom, J. Edsjo and G. Zaharijas, arXiv:0905.0333 \[astro-ph.HE\];\ S. Shirai, F. Takahashi and T. T. Yanagida, arXiv:0905.0388 \[hep-ph\];\ P. Meade, M. Papucci, A. Strumia and T. Volansky, arXiv:0905.0480 \[hep-ph\];\ D. Grasso [*et al.*]{}, arXiv:0905.0636 \[astro-ph.HE\];\ C. H. Chen, C. Q. Geng and D. V. Zhuridov, arXiv:0905.0652 \[hep-ph\].\ X. J. Bi, R. Brandenberger, P. Gondolo, T. Li, Q. Yuan and X. Zhang, arXiv:0905.1253 \[hep-ph\].\ K. Kohri, J. McDonald and N. Sahu, arXiv:0905.1312 \[hep-ph\]. See for example: A. W. Strong, I. V. Moskalenko and V. S. Ptuskin, Ann. Rev. Nucl. Part. Sci.  [**57**]{} (2007) 285 \[arXiv:astro-ph/0701517\]. T. Delahaye, R. Lineros, F. Donato, N. Fornengo and P. Salati, Phys. Rev.  D [**77**]{} (2008) 063527 \[arXiv:0712.2312 \[astro-ph\]\]. E. A. Baltz and J. Edsjo, Phys. Rev.  D [**59**]{} (1999) 023511 \[arXiv:astro-ph/9808243\]. I. V. Moskalenko and A. W. Strong, Astrophys. J.  [**493**]{} (1998) 694 \[arXiv:astro-ph/9710124\]. J. Hisano, S. Matsumoto, O. Saito and M. Senami, Phys. Rev.  D [**73**]{} (2006) 055004 \[arXiv:hep-ph/0511118\]. S. Torii \[CALET Collaboration\], Nucl. Phys. Proc. Suppl.  [**150**]{} (2006) 345; J. Phys. Conf. Ser. [**120**]{}, (2008) 062020. C. R. Chen, K. Hamaguchi, M. M. Nojiri, F. Takahashi and S. Torii, arXiv:0812.4200 \[astro-ph\]. [^1]: For instance, for the parametrization of the background spectrum in Ref. [@Baltz:1998xv], based on the simulations of Ref. [@Moskalenko:1997gh], the high energy electron + positron spectrum is well approximated by a single power $\Phi_{e}^{\rm bg} = \Phi_{e^-}^{\rm bg,prim} + \Phi_{e^-}^{\rm bg,sec} + \Phi_{e^+}^{\rm bg,sec}\simeq \Phi_{e^-}^{\rm bg,prim} \propto E^{-3.25}$. [^2]: The PAMELA data [@Adriani:2008zr] shows that the positron fraction is less than ${\cal O}(10)$ % for $E{ \mathop{}_{\textstyle \sim}^{\textstyle <} }100$ GeV. Assuming that the excess is mainly from the DM decay/annihilation, which generates the same amount of electron and positron, at least about 80 % of the total flux is the background in this energy range.

--- abstract: | We study the Art Gallery Problem for face guards in polyhedral environments. The problem can be informally stated as: *how many (not necessarily convex) windows should we place on the external walls of a dark building, in order to completely illuminate its interior?* We consider both *closed* and *open* face guards (i.e., faces with or without their boundary), and we study several classes of polyhedra, including *orthogonal* polyhedra, *4-oriented* polyhedra, and *2-reflex orthostacks*. We give upper and lower bounds on the minimum number of faces required to guard the interior of a given polyhedron in each of these classes, in terms of the total number of its faces, $f$. In several cases our bounds are tight: $\lfloor f/6\rfloor$ *open* face guards for orthogonal polyhedra and 2-reflex orthostacks, and $\lfloor f/4\rfloor$ *open* face guards for $4$-oriented polyhedra. Additionally, for *closed* face guards in 2-reflex orthostacks, we give a lower bound of $\lfloor (f+3)/9\rfloor$ and an upper bound of $\lfloor (f+1)/7\rfloor$. Then we show that it is [[**NP**]{}]{}-hard to approximate the minimum number of (closed or open) face guards within a factor of $\Omega(\log f)$, even for polyhedra that are orthogonal and simply connected. We also obtain the same hardness results for *polyhedral terrains*. Along the way we discuss some applications, arguing that face guards are *not* a reasonable model for guards *patrolling* on the surface of a polyhedron. author: - 'Giovanni Viglietta[^1]' title: 'Face-Guarding Polyhedra' --- Introduction {#s1} ============ #### Previous work. Art Gallery Problems have been studied in computational geometry for decades: given an *enclosure*, place a (preferably small) set of *guards* such that every location in the enclosure is seen by some guard. Most of the early research on the Art Gallery Problem focused on guarding 2-dimensional polygons with either point guards or segment guards [@art; @shermer; @urrutia2000]. Gradually, some of the attention started shifting to 3-dimensional settings, as well. Several authors have considered edge guards in 3-dimensional polyhedra, either in relation to the classical Art Gallery Problem or to its variations [@viglietta4; @edgenew; @wireless; @viglietta2; @thesis]. Recently, Souvaine et al. [@faceguards] introduced the model with *face guards* in 3-dimensional polyhedra. Ideally, each guard is free to roam over an entire face of a polyhedron, including the face’s boundary. Let $g(\mathcal P)$ be the minimum number of face guards needed for a polyhedron $\mathcal P$, and let $g(f)$ be the maximum of $g(\mathcal P)$ over all polyhedra $\mathcal P$ with exactly $f$ faces. For general polyhedra, Souvaine et al. showed that $\lfloor f/5\rfloor \leqslant g(f) \leqslant \lfloor f/2\rfloor$ and, for the special case of orthogonal polyhedra (i.e., polyhedra whose faces are orthogonal to the coordinate axes), they showed that $\lfloor f/7\rfloor \leqslant g(f) \leqslant \lfloor f/6\rfloor$. They also suggested several open problems, such as studying *open* face guards (i.e., face guards whose boundary is omitted), and the computational complexity of minimizing the number of face guards. Subsequently, face guards have been studied to some extent also in the case of polyhedral terrains. In [@terrain2; @terrain3] a tight bound is obtained, and in [@terrain1] it is proven that minimizing face guards in triangulated terrains is [[**NP**]{}]{}-hard. However, since these results apply to terrains, they have no direct implications on the problem of face-guarding polyhedral enclosures. #### Our contribution. In this paper we solve some of the problems left open in [@faceguards], and we also expand our research in some new directions. A preliminary version of this paper has appeared at CCCG 2013 [@mycccg]. In Section \[s2\] we discuss the face guard model, arguing that a face guard fails to meaningfully represent a guard “patrolling” on a face of a polyhedron. Essentially, there are cases in which the path that such a patrolling guard ought to follow is so complex (in terms of the number of turns, if it is a polygonal chain) that a much simpler path, striving from the face, would guard not only the region visible from that face, but the entire polyhedron. However, face guards are still a good model for illumination-related problems, such as placing (possibly non-convex) windows in a dark building. In Section \[s3\] we obtain some new bounds on $g(f)$, for both closed and open face guards. First we generalize the upper bounds given in [@faceguards] by showing that, for $c$-oriented polyhedra (i.e., whose faces have $c$ distinct orientations), $g(f)\leqslant \lfloor f/2 - f/c\rfloor$. We also provide some new lower bound constructions, which meet our upper bounds in two notable cases: orthogonal polyhedra with open face guards ($g(f)=\lfloor f/6\rfloor$), and $4$-oriented polyhedra with open face guards ($g(f)=\lfloor f/4\rfloor$). Then we go on to study a special class of orthogonal polyhedra, namely *2-reflex orthostacks*. The following table summarizes our new results, as well as those that were already known. Each entry contains a lower and an upper bound on $g(f)$, or a single tight bound. When applicable, a reference is given to the paper in which each result was first obtained. Observe that, for open face guards in triangulated terrains, $f$ guards are easily seen to be necessary in the worst case. Indeed, if the terrain is a convex “dome” (i.e., if no edges are reflex), then every face requires an open face guard. In the case of closed face guards in triangulated terrains, we remark that the bound given in [@terrain3] is expressed in terms of the number of vertices. Therefore we rewrote it in terms of $f$, using Euler’s formula. **Open face guards** **Closed face guards** --------------------------- ----------------------------------------------------------------- ------------------------------------------------------------------------------------------------------- **2-reflex orthostacks** $g(f)=\lfloor f/6\rfloor$ $\lfloor (f+3)/9\rfloor\leqslant g(f)\leqslant\lfloor (f+1)/7\rfloor$ **Orthogonal polyhedra** $g(f)=\lfloor f/6\rfloor$ $\lfloor f/7\rfloor\leqslant_{\cite{faceguards}} g(f)\leqslant_{\cite{faceguards}}\lfloor f/6\rfloor$ **4-oriented polyhedra** $g(f)=\lfloor f/4\rfloor$ $\lfloor f/5\rfloor\leqslant g(f)\leqslant\lfloor f/4\rfloor$ **General polyhedra** $\lfloor f/4\rfloor\leqslant g(f)\leqslant\lfloor f/2\rfloor-1$ $\lfloor f/5\rfloor\leqslant_{\cite{faceguards}} g(f)\leqslant\lfloor f/2\rfloor-1$ **Triangulated terrains** $g(f)=f$ $g(f)=_{\cite{terrain3}}\lfloor (f+3)/6\rfloor$ In Section \[s4\] we provide an approximation-preserving reduction from [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}to the problem of minimizing the number of (closed or open) face guards in simply connected orthogonal polyhedra. It follows that the minimum number of face guards is [[**NP**]{}]{}-hard to approximate within a factor of $\Omega(\log f)$. We also obtain the same result for (non-triangulated) terrains. This adds to the result of [@terrain1], which states that minimizing closed face guards is [[**NP**]{}]{}-hard in triangulated terrains. We also briefly discuss the membership in [[**NP**]{}]{}of the minimization problem, pointing out some difficulties in applying previously known techniques. We leave as an open problem the task to tighten all the bounds in the table above, as well as to prove or disprove that minimizing face guards is in [[**NP**]{}]{}. We conjecture that all the lower bounds are tight, and that the minimization problem does belong to [[**NP**]{}]{}. Model and motivations {#s2} ===================== #### Definitions. A *polyhedron* is a connected subset of ${\mathbb{R}}^3$, union of finitely many closed tetrahedra embedded in ${\mathbb{R}}^3$, whose boundary is a (possibly non-connected) orientable 2-manifold. Since a polyhedron’s boundary is piecewise linear, the notion of *face* of a polyhedron is well defined as a maximal planar subset of its boundary with connected and non-empty relative interior. Thus a face is a plane polygon, possibly with holes, and possibly with some degeneracies, such as hole boundaries touching each other at a single vertex. Any vertex of a face is also considered a *vertex* of the polyhedron. *Edges* are defined as minimal non-degenerate straight line segments shared by two distinct faces and connecting two vertices of the polyhedron. Since a polyhedron’s boundary is an orientable 2-manifold, the relative interior of an edge lies on the boundary of exactly two faces, thus determining an internal dihedral angle (with respect to the polyhedron). An edge is *reflex* if its internal dihedral angle is reflex, i.e., strictly greater than $180^\circ$. Given a polyhedron, we say that a point $x$ is *visible* to a point $y$ if no point in the straight line segment $xy$ lies in the exterior of the polyhedron. For any point $x$, we denote by $\mathcal V(x)$ the *visible region* of $x$, i.e., the set of points that are visible to $x$. In general, for any set $S\subset \mathbb R^3$, we let $\mathcal V(S)=\bigcup_{x\in S}\mathcal V(x)$. A set is said to *guard* a polyhedron if its visible region coincides with the entire polyhedron (including its boundary). The Art Gallery Problem for face guards in polyhedra consists in finding a (preferably small) set of faces whose union guards a given polyhedron. If such faces include their relative boundary, they are called *closed* face guards; if their boundary is omitted, they are called *open* face guards. A polyhedron is *$c$-oriented* if there exist $c$ unit vectors such that each face is orthogonal to one of the vectors. If these unit vectors form an orthonormal basis of $\mathbb R^3$, the polyhedron is said to be *orthogonal*. Hence, a cube is orthogonal, a tetrahedron and a regular octahedron are both 4-oriented, etc. We will refer informally to the $z$ axis as the *vertical* axis. Specifically, the positive $z$ direction will be *up*, and the opposite direction will be *down*. Hence, a direction parallel to the $xy$ plane will be said to be *horizontal*. The positive $x$ direction will be *right*, the negative $x$ direction will be *left*, and so on. #### Motivations. There is a straightforward analogy between *guarding* problems and *illumination* problems: placing guards in a polyhedron corresponds to placing light sources in a dark building, in order to illuminate it completely. For instance, a point guard would model a *light bulb* and a segment guard could be a *fluorescent tube*. Because face guards are 2-dimensional and lie on the boundary of the polyhedron, we may think of them as *windows*. A window may have any shape, but should be flat, and hence it should lie on a single face. It follows that, if our purpose is to illuminate as big a region as possible, we may assume without loss of generality that a window always coincides with some face. Face guards were introduced in [@faceguards] to represent guards *roaming* over a face. This is in accordance with the traditional usage of segment guards as a model for guards that *patrol* on a line [@art]. While this is perfectly sound in the case of segment guards, face guards pose additional problems, as explained next. We begin by observing that, even in 2-dimensional polygons, there may be edge guards that cannot be locally “replaced” by finitely many point guards. Figure \[fig1:a\] shows an example: if a subset $G$ of the top edge $\ell$ is such that $\mathcal V(G)=\mathcal V(\ell)$, then the right endpoint of $\ell$ must be a limit point of $G$. We can exploit this fact to construct the class of polyhedra sketched in Figure \[fig2\]. First we cut long parallel *dents* on opposite faces of a cuboid, as in Figure \[fig1:b\], in such a way that the resulting polyhedron looks like an extruded “iteration” of the polygon in Figure \[fig1:a\]. Then we stab this construction with a row of *girders* running orthogonally with respect to the dents, as Figure \[fig2:a\] illustrates. Suppose that a guard has to patrol the top face of this construction, eventually seeing every point that is visible from that face. The situation is represented in Figure \[fig2:b\], where the light-shaded region is the top face, and the dashed lines mark the underlying girders. By the above observation and by the presence of the girders, each thick vertical segment must be approached by the patrolling guard from the interior of the face. Suppose that the polyhedron has $n$ dents and $n$ girders. Then, the number of its vertices, edges, or faces is $\Theta(n)$. Now, if the guard moves along a polygonal chain lying on the top face, such a chain must have at least a vertex on each thick segment, which amounts to $\Omega(n^2)$ vertices. Similarly, if the face guard has to be substituted with segment guards lying on it, quadratically many guards are needed. On the other hand, it is easy to show that a path of linear complexity is sufficient to guard any polyhedron, provided that its boundary is connected: triangulate every face (thus adding linearly many new “edges”) and traverse the resulting 1-skeleton in depth-first order starting from any vertex, thus covering all edges. Because the set of edges is a guarding set for any polyhedron [@thesis Observation 3.10], the claim follows. In general, if the boundary is not connected (i.e., the polyhedron has some “cavities”), then we may repeat the same construction for every connected component. This defeats the purpose of having faces model guards patrolling on segments, as it makes little sense for a face of “unit weight” to represent quadratically many guards. Analogously, a roaming guard represented by a face may have to follow a path that is overly complex compared to the guarding problem’s optimal solution. Even if we are allowed to replace a face guard with guards patrolling any segment in the polyhedron (i.e, not necessarily constrained to live on that face), a linear number of them may be required. Indeed, consider a cuboid with very small height, and arrange $n$ thin and long *chimneys* on its top, in such a way that no straight line intersects more than two chimneys. The complexity of the polyhedron is $\Theta(n)$, and a face guard lying on the bottom face must be replaced by $\Omega(n)$ segment guards. On the other hand, we know that a linear number of segment guards is enough not only to “dominate” a single face, but to entirely guard any polyhedron. Summarizing, a face guard appropriately models an entity that is naturally constrained to live on a single face, like a flat window, and unlike a team of patrolling guards. In the case of a single roaming guard, the model is insensitive to the complexity of the guard’s path. Bounds on face guard numbers {#s3} ============================ $c$-oriented polyhedra ---------------------- Here we parameterize polyhedra according to the orientations of their faces, and we give upper and lower bounds on the number of face guards required to guard them. #### Upper bounds. By generalizing the approach used in [@faceguards Lemmas 2.1, 3.1], we provide an upper bound on face guard numbers, which becomes tight for open face guards in orthogonal polyhedra and open face guards in 4-oriented polyhedra. We emphasize that our upper bound holds for both closed and open face guards, and for polyhedra of any genus and number of cavities. \[t3:face\] Any $c$-oriented polyhedron with $f$ faces is guardable by $$\left\lfloor\frac f 2 - \frac f c\right\rfloor$$ closed or open face guards. Let $\mathcal P$ be a polyhedron whose faces are orthogonal to $c\geqslant 3$ distinct vectors. Let $f_i$ be the number of faces orthogonal to the $i$-th vector $v_i$. We may assume that $i<j$ implies $f_i \geqslant f_j$. Then, $$f_1+f_2 \geqslant \frac {2f} c.$$ Let us stipulate that the direction of the cross product $v_1 \times v_2$ is *vertical*. Thus, there are at most $$f-\frac {2f} c$$ non-vertical faces. Some of these are facing up, the others are facing down. Without loss of generality, at most half of them are facing down, and we assign a face guard to each of them. Therefore, at most $$\left\lfloor\frac f 2 - \frac f c\right\rfloor$$ face guards have been assigned. Let $x$ be any point in $\mathcal P$. If $x$ belongs to a face with a guard, $x$ is guarded. Otherwise, consider an infinite circular cone $\mathcal C(x)$ with apex $x$ and axis directed upward. Let $\mathcal G$ be the intersection of $\mathcal V(x)$, $\mathcal C(x)$, and the boundary of $\mathcal P$. Intuitively, $\mathcal G$ is the part of the boundary of the polyhedron that would be illuminated by a spotlight placed at $x$ and pointed upward. We will show that this area contains points belonging to some guard, which make $x$ guarded. If the aperture of $\mathcal C(x)$ is small enough, the relative interior of $\mathcal G$ belongs entirely to faces containing guards and to at most two vertical faces containing $x$. Because these vertical faces obstruct at most one dihedral angle from $x$’s view, the portion of $\mathcal G$ not belonging to them has non-empty interior. If we remove from this portion the (finitely many) edges of $\mathcal P$, we still have a non-empty region. By construction, this region belongs to the interiors of faces containing a guard; hence $x$ is guarded. Our guarding strategy becomes less efficient as $c$ grows. In general, if no two faces are parallel (i.e., $c=f$), we get an upper bound of $\left \lfloor f/2\right\rfloor - 1$ face guards, which improves on the one in [@faceguards] by just one unit. #### Lower bounds. In [@faceguards], Souvaine et al. construct a class of orthogonal polyhedra with $f$ faces that need $\lfloor f/7\rfloor$ *closed* face guards. In Figure \[fig3\] we give an alternative construction, with the additional property of having a 3-regular 1-skeleton and having no vertical reflex edges. Indeed, each small L-shaped polyhedron that is attached to the big cuboid adds seven faces to the construction, of which at least one must be selected. ![Orthogonal polyhedron that needs $\lfloor f/7\rfloor$ closed face guards[]{data-label="fig3"}](faceclosed.pdf){width=".5\linewidth"} For *open* face guards, we have a different construction, shown in Figure \[fig4\]. There are six faces for each of the large flat cuboids, and no open face can guard the center of two different flat cuboids. Therefore, one guard is needed for each of the flat cuboids, and this amounts to $\lfloor f/6\rfloor$ open face guards. Plugging $c=3$ in Theorem \[t3:face\] reveals that our lower bound is also tight. To guard an orthogonal polyhedron having $f$ faces, $\lfloor f/6\rfloor$ open face guards are always sufficient and occasionally necessary. $\square$ Moving on to *closed* face guards in 4-oriented polyhedra, we propose the construction in Figure \[fig5\]. Each closed face sees the tip of at most one of the $k$ tetrahedral *spikes*, hence $k$ guards are needed. Because there are $5k+2$ faces in total, a lower bound of $\lfloor f/5\rfloor$ closed face guards follows. ![4-oriented polyhedron that needs $\lfloor f/5\rfloor$ closed face guards[]{data-label="fig5"}](lower1.pdf){width=".75\linewidth"} For *open* face guards in 4-oriented polyhedra, we modify the previous example by carefully placing additional spikes on the other side of the construction, as Figure \[fig6\] illustrates. Once again, since each open face sees the tip of at most one of the $k$ spikes and there are $4k+2$ faces in total, a lower bound of $\lfloor f/4\rfloor$ open face guards follows. ![4-oriented polyhedron that needs $\lfloor f/4\rfloor$ open face guards[]{data-label="fig6"}](lower3.pdf){width=".75\linewidth"} This bound is also tight, as easily seen by plugging $c=4$ in Theorem \[t3:face\]. To guard a 4-oriented polyhedron having $f$ faces, $\lfloor f/4\rfloor$ open face guards are always sufficient and occasionally necessary. $\square$ 2-reflex orthostacks -------------------- In our pursuit to lower the $\lfloor f/6\rfloor$ upper bound on closed face guards in orthogonal polyhedra, in order to match it with the $\lfloor f/7\rfloor$ lower bound, we study a special class of orthogonal polyhedra. Recall that the lower bound example in Figure \[fig3\] has no vertical reflex edges, but only reflex edges in two horizontal directions. These orthogonal polyhedra are called *2-reflex*, and were first introduced by the author in [@thesis], and studied in conjunction with edge guards. We further restrict our analysis to 2-reflex polyhedra that are also *orthostacks*, as defined in [@orthostacks]. An orthostack is an orthogonal polyhedron whose horizontal cross sections are simply connected. Therefore, a 2-reflex orthostack can be naturally viewed as a pile of cuboidal *bricks* of various sizes, stacked on top of each other. Our motivation for studying 2-reflex orthostacks is that they constitute the most obvious building block for 2-reflex polyhedra. In turn, 2-reflex polyhedra are a natural class of polyhedra of intermediate complexity between orthogonal prisms and orthogonal polyhedra. While 2-reflex orthostacks form a very basic class of polyhedra, they already pose some challenges, and we perceive them as a necessary and critical step toward solving the general face-guarding problem for orthogonal polyhedra. #### Lower bounds. Observe that the polyhedron in Figure \[fig4\] is already a 2-reflex orthostack. Along with Theorem \[t3:face\], this yields a tight bound of $\lfloor f/6\rfloor$ *open* face guards in 2-reflex orthostacks. On the other hand, the one in Figure \[fig3\], despite being 2-reflex, is not an orthostack. However, the example in Figure \[fig4\] can be used once again to obtain a good lower bound on *closed* face guards, as well. ![2-reflex orthostack that needs $\lfloor (f+3)/9\rfloor$ closed face guards[]{data-label="figlowerortho"}](orthostack.pdf){width=".5\linewidth"} As Figure \[figlowerortho\] indicates, each triplet of consecutive bricks can be guarded by a single closed face guard. On the other hand, no closed face guard can see the centers of four bricks. If there are $k$ bricks in total, then there are $f=3k+3$ faces, and $g=\lceil k/3\rceil=\lfloor (k+2)/3\rfloor$ open face guards are needed. By substituting for $k$, we get $g=\lfloor (f+3)/9\rfloor$. #### Upper bounds. We have already given a tight bound for open face guards, so let us consider closed face guards now. Note that two adjacent bricks of a 2-reflex orthostack share a horizontal rectangle, which we call the *contact rectangle* between the two bricks. In general, each contact rectangle is coplanar with at least one horizontal face of the orthostack. If a contact rectangle is coplanar with exactly one face of the orthostack, such a contact rectangle is said to be *canonical*. Observe that any non-canonical contact rectangle between two bricks, having $k>1$ coplanar faces, can always be converted into $k$ canonical ones, by suitably adding new bricks between the initial two. More specifically, “extruding” the initial contact rectangle into a new brick, as shown in Figure \[figstretch\], allows to separate the horizontal faces facing up from those facing down. Subsequently, if two coplanar up-facing (respectively, down-facing) faces are still present, one of them can be “lifted” (respectively, “lowered”) via the addition of another brick. ![Converting a generic contact rectangle into two canonical ones[]{data-label="figstretch"}](stretchb.pdf){width=".75\linewidth"} Note that this transformation does not change the number of faces of the polyhedron. Also, after the transformation, some visibilities between points may be lost, but none is gained. Therefore, if the resulting polyhedron is guardable by $g$ closed face guards, then the original polyhedron is guardable by the $g$ “corresponding” closed face guards. Hence, in the following, we will restrict our analysis to 2-reflex orthostacks with canonical contact rectangles only. Depending to the relative positions of two bricks, the contact rectangle between them can be of one of four different *types*, which are illustrated in Figure \[fig2stack\]. Each type is characterized by how many (reflex) edges are shared between the two adjacent bricks: a contact rectangle of type $i$ has exactly $i$ edges of the orthostack on its perimeter, with $1\leqslant i\leqslant 4$. It is straightforward to see that there are no other possible configurations for a canonical contact rectangle. \ When two bricks are attached on top of each other and a canonical contact rectangle is formed, some of their faces disappear or are merged together, and therefore the total number of faces decreases. The quantity by which it decreases is called the *deficit* of the contact rectangle, and it only depends on its type. In Figure \[fig2stack\], the number of faces of each orthostack consisting of two bricks is denoted by $f$, and the deficit of the corresponding contact rectangle is denoted by $\Delta$. Observe that $f+\Delta=12$, and that the deficit of a contact rectangle of type $i$ is $\Delta=5-i$. It follows that the number of faces of a 2-reflex othostack with $k$ bricks depends only on the types of its $k-1$ contact rectangles, and this number is $6k$ minus the sum of the deficits of the contact rectangles. This is also equal to $k+5$ plus the sum of the types of the contact rectangles. Let two adjacent bricks be given, sharing a canonical contact rectangle of type $i$, with $1\leqslant i\leqslant 4$. Note that the vertical projection of one of the two bricks is strictly contained in the vertical projection of the other brick. If the vertical projection of the lower (respectively, upper) brick is contained in the vertical projection of the upper (respectively, lower) brick, we denote the configuration by the symbol ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ (respectively, ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$). Similarly, we indicate a stack of arbitrarily many bricks by a sequence of labeled $\mathlarger\bigsqcup$ and $\mathlarger\bigsqcap$ symbols, and we call this sequence the *signature* of the orthostack. For instance, the signature of the orthostack in Figure \[figlowerortho\] is $${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cr2} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cr2} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cr2} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cr2} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}.$$ The symbol ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ is shorthand for “${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ or ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$”. Similarly, ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\crj} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ stands for $$\mbox{``${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\crj} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ or ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\crj} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ or ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\crj} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ or ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\crj} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$'',}$$ and so on. As it turns out, to obtain our upper bound on closed face guards, we can almost entirely abstract from the actual shapes of 2-reflex orthostacks, and just reason about their signatures. Any 2-reflex orthostack whose signature is of the form ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\crj} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$, with $j\neq 4$, is guardable by one vertical closed face guard. If the top contact rectangle is not of type 4, there is a vertical face that is shared by the two top bricks, as Figure \[fig7:b\] exemplifies. This face also touches the bottom brick, and so it guards the entire orthostack. Any 2-reflex orthostack with signature of the form ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\crj} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ is guardable by one vertical closed face guard. It is sufficient to choose any vertical face of the middle brick. As Figure \[fig7:c\] suggests, this face sees also the top and bottom bricks. Any 2-reflex orthostack with signature of the form ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\crj} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ is guardable by one horizontal closed face guard that is not the topmost nor the bottommost face of the orthostack. Let us consider the two horizontal faces that border the middle brick: we show that one of them guards the whole polyhedron. If the vertical projection of the top brick is entirely contained in the vertical projection of the bottom brick, as Figure \[fig7:d\] shows, we pick the top face of the middle brick. Otherwise, the vertical projection of the top brick intersects the bottom face of the middle brick. Hence we may pick this face, as it guards all three bricks. From the three previous propositions, we straightforwardly obtain the following. \[l:stack1\] Any 2-reflex orthostack made of three bricks, whose signature is neither of the form ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cr4} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ nor ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cr4} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$, is guardable by one closed face guard that is not the topmost nor the bottommost horizontal face of the orthostack. \[l:stack2\] Any 2-reflex orthostack with signature ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ is guardable by one vertical closed face guard. Figure \[figeeee\] shows the construction of such an orthostack, brick by brick. When only the bottom brick is present, the guard can be chosen among its four vertical faces. Each time a new brick is added on top of the old ones, exactly three vertical faces are extended to form three sides of the new brick. As a consequence, after three bricks have been added on top of the first one, there is still at least one vertical face that stretches from the very bottom to the very top of the construction. This face guards the entire orthostack. Any 2-reflex orthostack with $f$ faces is guardable by $$\left\lfloor \frac{f+1} 7\right\rfloor$$ closed face guards. We will prove a slightly stronger statement: the $\lfloor (f+1)/7\rfloor$ face guards can be chosen in such a way that none of them lies on the topmost horizontal face of the orthostack. We prove this claim by induction on the number of bricks. Given a 2-reflex orthostack $\mathcal P$ with $k\geqslant 0$ bricks, suppose that the claim holds for all 2-reflex orthostacks with fewer bricks, and let us prove that it holds for $\mathcal P$, as well. The cases $k=0$ and $k=1$ are trivial: if $k=0$, then $f=0$, and zero guards are sufficient; if $k=1$, then $f=6$, and $\mathcal P$ is guarded by any vertical face. For the case $k=2$, the possible configurations are represented in Figure \[fig2stack\], which shows that $8\leqslant f\leqslant 11$, and $\mathcal P$ is easily guardable by a single vertical face. If $k=3$, Lemma \[l:stack1\] guarantees that one vertical face guard is sufficient, unless the signature of $\mathcal P$ is of the form ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcap}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcap\cr4} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$ or ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cr4} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigsqcup}\fi \mathpalette\mov@rlay{\mathlarger\bigsqcup\cri} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$. But in this case $f\geqslant 13$, so we are allowed to place two face guards, and it is easy to find two vertical faces that guard all the three bricks. Now suppose that $k\geqslant 4$, and let $1\leqslant j\leqslant k$. Let $\mathcal P'$ be the orthostack formed by the $j$ topmost bricks of $\mathcal P$, having $f'$ faces, and let $\mathcal P''$ be the rest of the orthostack, made of $k-j$ bricks and having $f''$ faces. The total number of faces of $\mathcal P$ is $f=f'+f''-\Delta$, where $\Delta$ is the deficit of the contact rectangle between $\mathcal P'$ and $\mathcal P''$ (if $j=k$, we may take $\Delta=0$). By inductive hypothesis, we can guard $\mathcal P''$ with at most $\lfloor (f''+1)/7\rfloor$ closed face guards, none of which lies on the topmost horizontal face. Suppose that $\mathcal P'$ can be guarded with at most $\lfloor (f'-\Delta)/7\rfloor$ closed face guards, none of which lies on the topmost or the bottommost horizontal face. In total, we would have at most $$\left\lfloor \frac{f''+1} 7\right\rfloor + \left\lfloor \frac{f'-\Delta} 7\right\rfloor \leqslant \left\lfloor \frac{f''+1+f'-\Delta} 7\right\rfloor = \left\lfloor \frac{f+1} 7\right\rfloor$$ closed face guards. Moreover, because none of these face guards would be coplanar with the contact rectangle between $\mathcal P'$ and $\mathcal P''$, they would be naturally mapped into face guards of $\mathcal P$. Indeed, the horizontal guards maintain the same shape and size after the merge, while some vertical guards may be merged with other faces, and thus enlarged, which is not an issue because this only makes them guard a bigger area. Together, these faces would guard all of $\mathcal P$, none of them would lie on its topmost horizontal face, and therefore our main claim on $\mathcal P$ would be proven. Let us show that, in every case, it is always possible to choose $j$ in such a way that the desired conditions on $\mathcal P'$ are met, allowing our previous reasoning to go through. In most cases, choosing $j=2$ is enough, as detailed next. Let $a$ and $b$ be the types of the two topmost contact rectangles, and let $\Delta'$ be the deficit of the topmost contact rectangle. So, if $\mathcal P'$ consists of two bricks, $f'=12-\Delta'$. Now, if $\lfloor (f'-\Delta)/7\rfloor\geqslant1$, we are allowed to place at least one guard in $\mathcal P'$, and it is easy to see that one vertical closed face guard is always sufficient (cf. Figure \[fig2stack\]). Hence we want $f'-\Delta\geqslant 7$ to hold, which is equivalent to $\Delta+\Delta'\leqslant 5$, that is, $a+b\geqslant 5$. The only cases left are those in which $a+b\leqslant 4$. Namely, these are the cases in which the signature of the three topmost bricks is one of the following (or one of their reverses): ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$, ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr2} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$, ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr3} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$, ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr2} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr2} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$. In the last three cases, we choose $j=3$. Indeed, in these cases $f'$ is either 11 or 12, and $\lfloor (f'-\Delta)/7\rfloor=1$. By Lemma \[l:stack1\], we can guard $\mathcal P'$ with one closed face guard that has the desired properties. Finally, let us assume that the signature of the three topmost bricks of $\mathcal P$ is ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$. Choosing $j=3$ works as above, unless the third contact rectangle of $\mathcal P$ is again of type 1 (indeed, if the type is at least 2, we have $f'=10$, $\Delta\leqslant 3$, and therefore $\lfloor (f'-\Delta)/7\rfloor\geqslant 1$). In this last case, we choose $j=4$ (recall that $k\geqslant 4$), so that the signature of $\mathcal P'$ is ${{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}{{ \mathop{\ifx\mathop\mathop\vphantom{\mathlarger\bigbox}\fi \mathpalette\mov@rlay{\mathlarger\bigbox\cr1} } \ifx\mathop\mathop\expandafter\displaylimits\fi}}$. We have $f'=12$, $0\leqslant \Delta\leqslant 4$ ($\Delta=0$ holds if $k=4$), and $\lfloor (f'-\Delta)/7\rfloor=1$. By Lemma \[l:stack2\], $\mathcal P'$ can be guarded by a single vertical guard, and our theorem follows. Minimizing face guards {#s4} ====================== Hardness of approximation ------------------------- In [@faceguards], Souvaine et al. ask for the complexity of minimizing face guards in a given polyhedron. We show that this problem is at least as hard as [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}, and we infer that approximating the minimum number of (closed or open) face guards within a factor of $\Omega(\log f)$ is [[**NP**]{}]{}-hard. This remains true even if we restrict the problem to the class of simply connected orthogonal polyhedra. We also show that the same hardness of approximation result holds for non-triangulated terrains. Recall that, in [@terrain1], Iwamoto et al. proved that minimizing closed face guards in triangulated terrains is [[**NP**]{}]{}-hard. Thus, we improve on their result in the case of non-triangulated terrains, while also extending it to open face guards. #### Orthogonal polyhedra. We give a linear approximation-preserving reduction from [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}, in the sense of [@ausiello Definition 8.4]. \[t3:hard\] [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}is L-reducible to the problem of minimizing (closed or open) face guards in a simply connected orthogonal polyhedron. Let an instance of [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}be given, i.e., a *universe* $U=\{1,\cdots,n\}$, and a collection $S\subseteq \mathcal P(U)$ of $m\geqslant 1$ subsets of $U$. We will construct a simply connected orthogonal polyhedron with $f\in O(mn)$ faces that can be guarded by $k$ (closed or open) faces if and only if $U$ is the union of $k-1$ elements of $S$. Figure \[figset2\] shows our construction for $U=\{1,2,3,4\}$ and $S=\{\{2,4\},\{1,3\},\{2\}\}$. Figure \[figset1\] illustrates the side view of a generic case in which $m=4$. ![[[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}reduction for orthogonal polyhedra, 3D view[]{data-label="figset2"}](faceset2.pdf){width=".75\linewidth"} ![[[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}reduction for orthogonal polyhedra, side view[]{data-label="figset1"}](faceset1.pdf){width=".75\linewidth"} Each of the thin cuboids on the far left is called a *fissure*, and represents an element of $U$. Facing the fissures there is a row of $m$ *mountains* of increasing height, separated by *valleys* of increasing depth. The $m$ vertical walls that are facing the fissures (drawn as thick lines in Figure \[figset1\]) are called *set faces*, and each of them represents an element of $S$. For each $S_i\in S$, we dig a narrow rectangular *dent* in the $i$-th set face in front of the $j$-th fissure, if and only if $j\notin S_i$. Each dent reaches the bottom of its set face, and almost reaches the top, so that it does not separate the set face into two distinct faces. Moreover, every dent (except those in the rightmost set face) is so deep that it connects two neighboring valleys. In Figure \[figset1\], dents are depicted as darker regions; in Figure \[figset2\], the dashed lines mark the areas where dents are *not* placed. We want to fix the width of the fissures in such a way that only a restricted number of faces can see their bottom. Specifically, consider $n$ *distinguished points*, located in the middle of the lower-left edges of the fissures (indicated by the thick dot in Figure \[figset1\]). The $j$-th distinguished point definitely sees some portions of the $i$-th set face, provided that $j\in S_i$. If this is the case, and $i<m$, it also sees portions of two other faces (one horizontal, one vertical) surrounding the same valley. Moreover, if $j\notin S_m$, the $j$-th distinguished point also sees the bottom of a dent in the rightmost set face. We want no face to be able to see any distinguished point, except the faces listed above (plus of course the faces belonging to fissures or surrounding their openings). To this end, assuming that the dents have unit width, we set the width of the fissures to be slightly less than $1/4$. Indeed, referring to Figure \[figset1\], the width of the visible region of a distinguished point, as it reaches the far right of the construction, is strictly less than $$\frac{(m)+(2m+1)}{m}\cdot\frac 14\ =\ \left(3+\frac 1m\right)\cdot\frac 14\ \leqslant\ 4\cdot\frac 14\ =\ 1,$$ because $m\geqslant 1$. Finally, a small *niche* is added in the lower part of the construction. Its purpose is to enforce the selection of a “dedicated” face guard, as no face can see both a distinguished point and the bottom of the niche. Let a guarding set for our polyhedron be given, consisting of $k$ face guards. We will show how to compute in polynomial time a solution of size at most $k-1$ for the given [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}instance, provided that it is solvable at all. We first discard every face guard that is not guarding any distinguished point. Because at least one such face must guard the niche, we are left with at most $k-1$ guards. Then, if any of the remaining face guards borders the $i$-th valley, with $i<m$, we replace it with the $i$-th set face. Indeed, it is easy to observe that such a set face can see the same distinguished points, plus possibly some more. By construction, all the remaining guards can see exactly one distinguished point (they are either faces belonging to some fissure, or surrounding its opening, or rightmost faces of the rightmost dents). We replace each of these face guards with any set face that guards the same distinguished point (which exists, otherwise the [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}instance would be unsolvable). As a result, we have at most $k-1$ set faces guarding all the distinguished points. These immediately determine a solution of equal size to the given [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}instance. Conversely, if the [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}instance has a solution of size $k$, it is easy to see that our polyhedron has a guarding set of $k+1$ guards: all the set faces corresponding to the [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}’s solution, plus the bottom face. \[cor:hard\] Given a simply connected orthogonal polyhedron with $f$ faces, it is [[**NP**]{}]{}-hard to approximate the minimum number of (closed or open) face guards within a factor of $\Omega(\log f)$. The polyhedra constructed by the L-reduction of Theorem \[t3:hard\] have $f\in O(mn)$ faces. It was proved in [@set] that [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}is [[**NP**]{}]{}-hard to approximate within a ratio of $\Omega(\log n)$ and, by inspecting the reduction employed, it is apparent that all the hard [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}instances generated are such that $m\in O(n^c)$, for some constant $c\geqslant 1$. As a consequence, we may assume that $\Omega(\log n)=\Omega(\log n^{c+1})\subseteq \Omega(\log(mn))\subseteq \Omega(\log f)$. Since the minimum is [[**NP**]{}]{}-hard to approximate within some factor belonging to $\Omega(\log n)$, and the same factor also belongs to $\Omega(\log f)$, our claim follows. An analogous of Theorem \[t3:hard\] and of Corollary \[cor:hard\] also holds, with the same proof, for the related problem of guarding the *boundary* of a polyhedron by face guards, as opposed to the whole interior. #### Non-triangulated terrains. We show that the above reduction can be adapted to work with non-triangulated terrains. A *terrain* is a piecewise-linear surface embedded in ${\mathbb{R}}^3$, such that any vertical line intersects the surface in exactly one point. Therefore, a terrain is an unbounded 2-manifold that partitions ${\mathbb{R}}^3$ in an *upper region* and a *lower region*, each of which homeomorphic to a half-space. Faces, vertices and edges of terrains are defined in the same way as for polyhedra (cf. Section \[s2\]). We stipulate that a terrain has exactly one unbounded face, and therefore no unbounded edges. Visibility is defined in the upper region only: two points belonging to the upper region are visible if the line segment connecting them does not intersect the lower region. Therefore, the Art Gallery Problem for face guards in terrains asks for a set of faces that collectively see the whole upper region of a given terrain. The problem of guarding terrains is connected to the problem of guarding polyhedra, in that terrains may be intuitively viewed as a class of “upward-unbounded polyhedra”. A terrain whose bounded faces are triangles is called a *triangulated* terrain. These special terrains are studied in [@terrain2; @terrain1; @terrain3], where it is shown that computing the minimum number of closed face guards in a given triangulated terrain is [[**NP**]{}]{}-hard. Here we strengthen this result by showing that such a minimum is even [[**NP**]{}]{}-hard to approximate within a logarithmic factor, for both open and closed face guards, provided that terrains are not necessarily triangulated. \[t4:hard\] Given a (not necessarily triangulated) terrain with $f$ faces, it is [[**NP**]{}]{}-hard to approximate the minimum number of (closed or open) face guards within a factor of $\Omega(\log f)$. We show that [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}is L-reducible to the problem of minimizing (closed or open) face guards in a non-triangulated terrain, by suitably modifying the construction given in Theorem \[t3:hard\]. Then, our claim follows as in Corollary \[cor:hard\]. Our new construction is sketched in Figures \[figter1\] and \[figter2\], again for $U=\{1,2,3,4\}$ and $S=\{\{2,4\},\{1,3\},\{2\}\}$. The faces that look vertical in Figure \[figter1\] are actually very steep slopes, as the side view of Figure \[figter2\] suggests. ![[[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}reduction for non-triangulated terrains, 3D view[]{data-label="figter1"}](terrain1.pdf){width=".75\linewidth"} We have $n$ very thin *fissures* and $m$ *mountains* of increasing height. The proportions are chosen in such a way that no face in the terrain can see inside two distinct fissures all the way to the far corner (i.e., the *distinguished point* in Figure \[figter2\]), except the *set faces* on the mountains. In particular, the construction is so “long” that the wall opposite to the fissures can see no fissure, as its visual is obstructed by the mountains (refer to the dashed line in Figure \[figter2\]). Moreover, each mountain, due to its *dents*, can see exactly the fissures that correspond to the subset of $U$ that the mountain itself represents. Observe that, once again, $f\in O(mn)$. ![[[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}reduction for non-triangulated terrains, side view[]{data-label="figter2"}](terrain2.pdf){width=".85\linewidth"} Now, regardless of whether face guards are closed or open, all the remaining parts of the terrain can be guarded by a number of face guards that is bounded by a small constant $c$. Indeed, if the mountains are thin enough, all the dents can be collectively guarded by the two large side walls of the terrain (i.e., the light-shaded polygons in Figure \[figter2\]). Because we may assume that the “hard” [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}instances have arbitrarily large optimal solutions, $c$ becomes negligible in the computation of the approximation ratio, and our theorem follows. Computing visible regions ------------------------- The next natural question is whether the minimum number of face guards can be computed in [[**NP**]{}]{}, and possibly approximated within a factor of $\Theta(\log f)$ in polynomial time. Usually, when finitely many possible guard locations are allowed (such as with vertex guards and edge guards), this is established by showing that the visible region of any guard can be computed efficiently, as well as the intersection of two visible regions, etc. As a result, the environment is partitioned into polynomially many regions such that, for every region $R$ and every guard $g$, either $R\subseteq \mathcal V(g)$ or $R\cap \mathcal V(g)=\varnothing$. This immediately leads to a reduction to [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}, which yields an approximation algorithm with logarithmic ratio, via a well-known greedy heuristic [@ghosh]. With face guards (and also with edge guards in polyhedra) the situation is complicated by the fact that the visible region of a guard may not be a polyhedron, but in general its boundary is a piecewise quadric surface. For example, consider the orthogonal polyhedron in Figure \[figvis\]. It is easy to see that the visible region of the bottom face (and also the visible region of edge $a$) is the whole polyhedron, except for a small region bordered by the thick dashed lines. ![The visible region of the bottom face is bounded by a hyperboloid of one sheet.[]{data-label="figvis"}](facevis.pdf){width=".65\linewidth"} The surface separating the visible and invisible regions consists of a right trapezoid plus a bundle of mutually skew segments whose extensions pass through the edges $a$, $b$, and $c$. These edges lie on three lines having equations $$y^2+z^2=0,$$ $$x^2+(z-1)^2=0,$$ $$(x-1)^2+(y-1)^2=0,$$ respectively. A straightforward computation reveals that the bundle of lines passing through these three lines has equation $$xy-xz+yz-y=0,$$ which defines a hyperboloid of one sheet. In general, the boundary of the visible area of a face (or an edge) is determined by lines passing through pairs or triplets of edges of the polyhedron. If three edges are all parallel to a common plane, the surface they determine is a hyperbolic paraboloid (degenerating into a plane if two of the edges are parallel to each other), otherwise they determine a hyperboloid of one sheet, as in the above example. There exists an extensive literature of purely algebraic methods to compute intersections of quadric surfaces (see for instance [@quadric]), but the parameterizations involved may yield coefficients containing radicals. At this stage in our understanding, it is not clear whether any of these methods can be effectively applied to reduce the minimization problem of face-guarding polyhedra (or even edge-guarding polyhedra) to [[<span style="font-variant:small-caps;">Set Cover</span>]{}]{}. Acknowledgments {#acknowledgments .unnumbered} --------------- The author wishes to thank the anonymous reviewers for precious suggestions on how to improve the readability of this paper. [99]{} N. Alon, D. Moshkovitz, and S. Safra. Algorithmic construction of sets for $k$-restrictions. [*ACM Transactions on Algorithms*]{}, vol. 2, pp. 153–177, 2006. C. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi. [*Complexity and approximation: Combinatorial optimization problems and their approximability properties*]{}. Springer, 2003. N. Benbernou, E. D. Demaine, M. L. Demaine, A. Kurdia, J. O’Rourke, G. T. Toussaint, J. Urrutia, and G. Viglietta. Edge-guarding orthogonal polyhedra. In [*Proceedings of the 23rd Canadian Conference on Computational Geometry*]{}, pp. 461–466, 2011. T. Biedl, E. D. Demaine, M. L. Demaine, A. Lubiw, M. H. Overmars, J. O’Rourke, S. Robbins, S. H. Whitesides. Unfolding some classes of orthogonal polyhedra. In [*Proceedings of the 10th Canadian Conference on Computational Geometry*]{}, pp. 70–71, 1998. J. Cano, C. D. Tóth, and J. Urrutia. Edge guards for polyhedra in 3-space. In [*Proceedings of the 24th Canadian Conference on Computational Geometry*]{}, pp. 155–160, 2012. T. Christ and M. Hoffmann. Wireless localization within orthogonal polyhedra. In [*Proceedings of the 23rd Canadian Conference on Computational Geometry*]{}, pp. 467–472, 2011. L. Dupont, D. Lazard, S. Lazard, and S. Petitjean. A new algorithm for the robust intersection of two general quadrics. In [*Proceedings of the 19th Annual ACM Symposium on Computational Geometry*]{}, pp. 246–255, 2003. S. K. Ghosh. Approximation algorithms for art gallery problems. In [*Proceedings of the Canadian Information Processing Society Congress*]{}, pp. 429–434, 1987. C. Iwamoto, J. Kishi, and K. Morita. Lower bound of face guards of polyhedral terrains. [*Journal of Information Processing*]{}, vol. 20, pp. 435–437, 2012. C. Iwamoto, Y. Kitagaki, and K. Morita. Finding the minimum number of face guards is NP-hard. [*IEICE Transactions on Information and Systems*]{}, vol. E95-D, pp. 2716–2719, 2012. C. Iwamoto and T. Kuranobu. Improved lower and upper bounds of face guards of polyhedral terrains. [*IEICE Transactions on Information and Systems (Japanese Edition)*]{}, vol. J95-D, pp. 1869–1872, 2012. J. O’Rourke. [*Art gallery theorems and algorithms*]{}. Oxford University Press, New York, 1987. T. Shermer. Recent results in art galleries. In [*Proceedings of the IEEE*]{}, vol. 80, pp. 1384–1399, 1992. D. L. Souvaine, R. Veroy, and A. Winslow. Face guards for art galleries. In [*Proceedings of the XIV Spanish Meeting on Computational Geometry*]{}, pp. 39–42, 2011. J. Urrutia. Art gallery and illumination problems. In J.-R. Sack and J. Urrutia, editors, [*Handbook of Computational Geometry*]{}, pp. 973–1027, North-Holland, 2000. G. Viglietta. Searching polyhedra by rotating half-planes. [*International Journal of Computational Geometry and Applications*]{}, vol. 22, pp. 243–275, 2012. G. Viglietta. [*Guarding and searching polyhedra*]{}. Ph.D. Thesis, University of Pisa, 2012. G. Viglietta. Face-guarding polyhedra. In [*Proceedings of the 25th Canadian Conference on Computational Geometry*]{}, pp. 277–282, 2013. [^1]: School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa ON, Canada, [viglietta@gmail.com]{}.

--- abstract: 'We investigate the corrections of the littlest Higgs(LH) model and the $SU(3)$ simple group model to single top production at the CERN Large Hardon Collider(LHC). We find that the new gauge bosons $W_{H}^{\pm}$ predicted by the LH model can generate significant contributions to single top production via the s-channel process. The correction terms for the tree-level $Wqq''$ couplings coming from the $SU(3)$ simple group model can give large contributions to the cross sections of the t-channel single top production process. We expect that the effects of the LH model and the $SU(3)$ simple group model on single top production can be detected at the LHC experiments.' author: - | Chong-Xing Yue, Li Zhou, Shuo Yang\ [Department of Physics, Liaoning Normal University, Dalian 116029, China [^1]]{}\ title: | Little Higgs models and single top production\ at the LHC --- 1. Introduction {#introduction .unnumbered} =============== The top quark, with a mass of the order of the electroweak scale $m_{t}=172.7\pm2.9GeV$\[1\] is the heaviest particle yet discovered and might be the first place in which the new physics effects could be appeared. The properties of the top quark could reveal information regarding flavor physics, electroweak symmetry breaking(EWSB) mechanism, as well as new physics beyond the standard model(SM)\[2\]. Hadron colliders, such as the Tevatron and the CERN Large Hadron Collider(LHC), can be seen as top quark factories. One of the primary goals for the Tevatron and the LHC is to determine the top quark properties and see whether any hint of non-SM effects may be visible. Thus, studying the top quark production at hadron colliders is of great interest. It can help the collider experiments to probe EWSB mechanism and test the new physics beyond the SM. In the context of the SM, top quark can be produced singly via electroweak interactions involving the $Wtb$ vertex. There are three production processes which are distinguished by the virtuality $Q^{2}$ of the electroweak gauge boson $W(Q^{2}=-p^{2}$, where $p$ is the four-momentum of the gauge boson $W)$\[2\]. The first process is the so-called $W$-gluon fusion, or t-channel process, which is the dominant process involving a space-like $W$ boson($p^{2}<0$) both at the Tevatron and the LHC. If a $b$ quark distribution function is introduced into the calculation, the leading order process for the $W$-gluon fusion channel is the t-channel process $q+b\rightarrow q'+t$ including $\overline{q}'+b\rightarrow \overline{q}+t$\[3\]. The second process is the s-channel process $q+\overline{q}'\rightarrow t+\overline{b}$ mediated by a time-like $W$ boson ($p^{2}>(m_{t}+m_{b})^{2}$). Single top quark can also be produced in association with a real $W$ boson($p^{2}\approx M_{W}^{2}$). The cross section for the $tW$ associated production process is negligible at the Tevatron, but of considerable size at the LHC, where the production cross section is larger than that of the s-channel process. At the leading order, the production cross sections for all of three processes are proportional to the Cabibbo-Kobayashi-Maskawa(CKM) matrix element $|V_{tb}|^{2}$. Thus, measuring the cross section of single top production generally provides a direct probe of $|V_{tb}|$, the effective $Wtb$ vertex, further the strength and handedness of the top charged-current couplings. This fact has already motivated large number of dedicated experimental and theoretical studies. Since the cross section of single top production is smaller than that of the $t\overline{t}$ production and the final state signals suffer from large background, the observation of the single top events is even more challenging than $t\overline{t}$. It is expected that increased luminosity and improved methods of analysis will eventually achieve detection of single top events. So far, there are not single top events to be observed. The cross sections of single top production for the s- and t-channels might be observed at the Tevatron Run II with a small data sample of only a few $fb^{-1}$. However, the LHC can precisely measure single top production, the CKM matrix element $V_{tb}$ could be measured down to less than one percent error at the ATLAS detector\[4\]. The three processes for single top production can be affected by new physics beyond the SM in two ways. One way proceeds via the modification of the SM couplings between the known particles, such as $Wtb$ and $Wqq'(q, q'= u, d, c, s)$ couplings, and other way involves the effects of new particles that couple to the top quark. Certainly, these two classifications can be seen to overlap in the limit, in which the extra particles are heavy and decouple from the low energy description. The SM couplings between the ordinary particles take well defined and calculable values in the SM, any deviation from these values would indicate the presence of new physics. Thus, single top production at hadron colliders might be sensitive to certain effects of new physics and studying the non-SM effects on single top production is very interesting and needed. To address EWSB and the hierarchy problem in the SM, many alternative new physics models, such as supersymmetry, extra dimensions, topcolor, and the recently little Higgs models, have been proposed over the past three decades. Of particular interest to us is the little Higgs models\[5\]. In this kind of models, the Higgs particle is a pseudo-Goldstone boson of a global symmetry which is spontaneously broken at some high scales. EWSB is induced by radiative corrections leading to a Coleman-Weinberg type of potential. Quadratic divergence cancellations of radiative corrections to the Higgs boson mass are due to contributions from new particles with the same spin as the SM particles. Some of these new particles can generate characteristic signatures at the present and future collider experiments\[6,7\]. The aim of this paper is to study the effects of the little Higgs models on single top production and see whether the corrections of the little Higgs models to the cross section of single top production can be detected at the LHC. The rest of this paper is organized as follows. In the next section, we shall briefly summarize some coupling expressions in the little Higgs models, which are related to single top production. The contributions of the correction terms for the tree-level $Wtb$ and $Wqq'$ couplings to single top production at the LHC are calculated in section 3. In section 4, we discuss the corrections of the new charged gauge bosons, such as $W_{H}^{\pm}$ and $X^{-}$, predicted by the little Higgs models, to single top production at the LHC. Our conclusions and discussions are given in section 5. 2. The relevant couplings {#the-relevant-couplings .unnumbered} ========================= There are several variations of the little Higgs models, which differ in the assumed higher symmetry and in the representations of the scalar multiplets. According the structure of the extended electroweak gauge group, the little Higgs models can be generally divided into two classes\[6,8\]: product group models, in which the SM $SU(2)_{L}$ is embedded in a product gauge group, and simple group models, in which it is embedded in a larger simple group. The littlest Higgs model(LH)\[9\] and the $SU(3)$ simple group model\[8,10\] are the simplest examples of the product group models and the simple group models, respectively. To predigestion our calculation, we will discuss single top production at the LHC in the context of these two simplest models. The LH model\[9\] consists of a nonlinear $\sigma$ model with a global $SU(5)$ symmetry and a locally gauged symmetry $[SU(2)\times U(1)]^{2}$. The global $SU(5)$ symmetry is broken down to its subgroup $SO(5)$ at a scale $f\sim\Lambda_{s}/4\pi\sim TeV$, which results in 14 Goldstone bosons(GB’s). Four of these GB’s are eaten by the gauge bosons($W_{H}^{\pm}, Z_{H}, B_{H}$), resulting from the breaking of $[SU(2)\times U(1)]^{2}$, giving them mass. The Higgs boson remains as a light pseudo-Goldstone boson, and other GB’s give mass to the SM gauge bosons and form a Higgs field triplet. The gauge and Yukawa couplings radiative generate a Higgs potential and trigger EWSB. In the LH model, the couplings constants of the SM gauge boson $W$ and the new gauge boson $W_{H}$ to ordinary particles, which are related to our calculation, can be written as\[11\]: $$\begin{aligned} g_{L}^{Wtb}&=&\frac{ie}{\sqrt{2}S_{W}}[1-\frac{v^{2}}{2f^{2}}(x_{L}^{2}+c^{2}(c^{2}-s^{2}))] ,\hspace{12mm}g_{R}^{Wtb}=0;\\ g_{L}^{Wqq'}&=&\frac{ie}{\sqrt{2}S_{W}}[1-\frac{v^{2}}{2f^{2}}c^{2}(c^{2}-s^{2})],\hspace{23mm}g_{R}^{Wqq'}=0;\\ g_{L}^{W_{H}tb}&=&g_{L}^{W_{H}qq'}=\frac{ie}{\sqrt{2}S_{W}}\frac{c}{s},\hspace{38mm}g_{R}^{W_{H}tb}=g_{R}^{W_{H}qq'}=0.\end{aligned}$$ Where $\nu\approx246GeV$ is the electroweak scale and $S_{W}=\sin\theta_{W}$, $\theta_{W}$ is the Weinberg angle. $c(s=\sqrt{1-c^{2}})$ is the mixing parameter between $SU(2)_{1}$ and $SU(2)_{2}$ gauge bosons and the mixing parameter $x_{L}=\lambda_{1}^{2}/(\lambda_{1}^{2}+\lambda_{2}^{2})$ comes from the mixing between the SM top quark $t$ and the vector-like top quark $T$, in which $\lambda_{1}$ and $\lambda_{2}$ are the Yukawa coupling parameters. The $SU(2)$ doublet quarks $(q,q')$ represent $(u,d)$ or $(c,s)$. In above equations, we have assumed $V_{tb}\approx V_{ud}\approx V_{cs}\approx1$. The $SU(3)$ simple group model\[8,10\] consists of two $\sigma$ model with a global symmetry $[SU(3)\times U(1)]^{2}$ and a gauge symmetry $SU(3)\times U(1)_{X}$. The global symmetry is spontaneously broken down to its subgroup $[SU(2)\times U(1)]^{2}$ by two vacuum condensates $<\Phi_{1,2}>=(0,0,f_{1,2})$, where $f_{1}\sim f_{2}\sim1TeV$. At the same time, the gauge symmetry is broken down to the SM gauge group $SU(2)\times U(1)$ and the global symmetry is broken explicitly down to its diagonal subgroup $SU(3)\times U(1)$ by the gauge interactions. This breaking scenario gives rise to an $SU(2)_{L}$ doublet gauge bosons $(Y^{0},X^{-})$ and a new neutral gauge boson $Z'$. Due to the gauged $SU(3)$ symmetry in the $SU(3)$ simple group model, all of the SM fermion representations have to be extended to transform as fundamental (or antifundamental) representations of $SU(3)$, which demands the existence of new heavy fermions in all three generations. The fermion sector of the $SU(3)$ simple group model can be constructed in two ways: universal and anomaly free, which might induce the different signatures at the high energy collider experiments. However, the coupling forms of the gauge bosons $W$ and $X$ to the SM quarks can be unitive written as\[6\]: $$\begin{aligned} g_{L}^{Wtb}&=&\frac{ie}{\sqrt{2}S_{W}}(1-\frac{1}{2}\delta_{t}^{2}),\hspace{24mm}g_{R}^{Wtb}=0;\\ g_{L}^{Wqq'}&=&\frac{ie}{\sqrt{2}S_{W}}(1-\frac{1}{2}\delta_{\nu}^{2}),\hspace{23mm}g_{R}^{Wqq'}=0;\\ g_{L}^{Xtb}&=&\frac{ie}{\sqrt{2}S_{W}}\delta_{t},\hspace{40mm}g_{R}^{Xtb}=0;\\ g_{L}^{Xqq'}&=&\frac{ie}{\sqrt{2}S_{W}}\delta_{\nu},\hspace{38mm}g_{R}^{Xqq'}=0\end{aligned}$$ with $$\delta_{t}=\frac{\nu}{\sqrt{2}f}t_{\beta}\frac{x_{\lambda}^{2}-1}{x_{\lambda}^{2}+t_{\beta}^{2}}, \hspace{30mm}\delta_{\nu}=-\frac{\nu}{2f t_{\beta}}.$$ Where $f=\sqrt{f_{1}^{2}+f_{2}^{2}}$, $t_{\beta}=\tan\beta=f_{2}/f_{1}$, and $x_{\lambda}=\lambda_{1}/\lambda_{2}$. Using these Feynmen rules, we will estimate the contributions of the LH model and the $SU(3)$ simple group model to single top production at the LHC in the following sections. 3. The contributions of the correction terms to single top production {#the-contributions-of-the-correction-terms-to-single-top-production .unnumbered} ====================================================================== For the t-channel process $q+b\rightarrow q'+t$, at the leading order, there is only one diagram with $W$ exchange in the t-channel. In the context of the little Higgs models, the corresponding scattering amplitude can be written as : $$M_{i}^{t}=\frac{2\pi\alpha_{e}(1+\delta g_{Li}^{Wtb})(1+\delta g_{Li}^{Wqq'})}{S_{W}^{2}(\hat{t}-m_{W}^{2})}[\overline{u}(q')\gamma^{\mu}P_{L}u(q)] [\overline{u}(t)\gamma_{\mu}P_{L}u(b)],$$ where $\hat{t}=(P_{b}-P_{t})^{2}$, $P_{L}=(1-\gamma^{5})/2$ is the left-handed prosection operator. $i=$1 and 2 represent the LH model and the $SU(3)$ simple group model, respectively. $\delta g_{Li}^{Wtb}$ and $\delta g_{Li}^{Wqq'}$ are the correction terms for the $Wtb$ and $Wqq'$ couplings induced by these two little Higgs models, which have been given in Eqs.(1,2,4,5). In the context of the little Higgs models, the scattering amplitude of the s-channel process $q+\overline{q}'\rightarrow t+\overline{b}$ can be written as: $$\begin{aligned} M_{i}^{s}=\frac{2\pi\alpha_{e}(1+\delta g_{Li}^{Wtb})(1+\delta g_{Li}^{Wqq'})}{S_{W}^{2}(\hat{s}-m_{W}^{2})}[\overline{\nu}(\overline{q}')\gamma^{\mu}P_{L}u(q)] [\overline{u}(t)\gamma_{\mu}P_{L}\nu(\overline{b})],\end{aligned}$$ where $\hat{s}=(P_{q}+P_{\overline{q}'})^{2}$ and $\sqrt{\hat{s}}$ is the center-of-mass energy of the subprocess $q+\overline{q}'\rightarrow t+\overline{b}$. At the leading order, the production of single top quark in association with a $W$ boson is given via the processes mediated by the s-channel $b$ quark exchange and the u-channel top quark exchange. In the little Higgs models, the tree-level coupling of the gluon to a pair of fermions is same as that in the SM, thus the scattering amplitude of this process can be written as: $$M^{tW}_{i}=\frac{eg_{s}(1+\delta g_{Li}^{Wtb})}{\sqrt{2}S_{W}}\overline{u}(t)[\frac{\not\varepsilon_{2} P_{L}(\not P_{g}+\not P_{b}+m_{b})\not\varepsilon_{1}}{\hat{s}'-m_{b}^{2}}+\frac{\not\varepsilon_{1} (\not P_{t}-\not P_{g}+m_{t})\not\varepsilon_{2}P_{L}}{\hat{u}-m_{t}^{2}}]u(b),$$ where $\hat{s}'=(P_{g}+P_{b})^{2}=(P_{W}+P_{t})^{2}$, $\hat{u}=(P_{t}-P_{g})^{2}=(P_{b}-P_{W})^{2}$. After calculating the cross sections $\hat{\sigma_{i}}(\hat{s})$ for the t-channel, s-channel, and $tW$ associated production processes, the total cross section $\sigma_{i}(S)$ for each process of single top production at the LHC can be obtained by convoluting $ \hat{\sigma}_{ijl}(\hat{s})[\hat{\sigma}_{i}(\hat{s})=\sum\limits_{j,l} \hat{\sigma_{ijl}}(\hat{s})]$ with the parton distribution functions(PDF’s): $$\sigma_{i}(S)=\sum_{j,l}\int_{0}^{1}dx_{1}\int_{0}^{1}dx_{2}f_{j}(x_{j},\mu_{f}^{2}) f_{l}(x_{l},\mu_{f}^{2})\hat{\sigma_{jl}}(\hat{s}),$$ where $j$ and $l$ are the possible combination of incoming gluon, quark, antiquark. $f(x,\mu_{f}^{2})$ is the PDF evaluated at the factorization scale $\mu_{f}$. Through out this paper, we neglect all quark masses with the exception of $m_{t}$, use CTEQ6L PDF with $\mu_{f}=m_{t}$\[12\], and take the center-of-mass energy $\sqrt{S}=14TeV$ for the process $pp\rightarrow t+X$ at the LHC. To obtain numerical results, we need to specify the relevant SM parameters. These parameters are $m_{t}$=172.7GeV\[1\], $\alpha(m_{Z})$=1/128.8, $\alpha_{s}$=0.118, $S_{W}^{2}$=0.2315, and $m_{W}$=80.425\ GeV\[13\]. Except for these SM input parameters, the contributions of the LH model and the $SU(3)$ simple group model to single top quark production are dependent on the free parameters ($f$, $x_{L}$, $c$) and ($f$, $x_{\lambda}$, $t_{\beta}$), respectively. Considering the constraints of the electroweak precision data on these free parameters, we will assume $f\geq1TeV$, $0.4\leq x_{L}\leq 0.6$ and $0<c\leq 0.5$ for the LH model\[14\] and $f\geq 1TeV$, $x_{\lambda}>1$, and $t_{\beta}>1$ for the $SU(3)$ simple group model\[6,8,10\], in our numerical estimation. The relative corrections of the LH model and the $SU(3)$ simple group model to the cross section $\sigma_{i}$ of single top production at the LHC are shown in Fig.1 and Fig.2, respectively. In these figures, we have taken $\Delta\sigma_{i}=\sigma_{i}-\sigma^{SM}_{i}$, $f=1.0TeV$ and three values of the mixing parameters $x_{L}$ and $x_{\lambda}$. From these figures, we can see that the contributions of the $SU(3)$ simple group model to single top production are larger than those of the LH model. For the LH model, the absolute values of the relative correction $\Delta\sigma_{i}/\sigma_{i}^{SM}$ are smaller than $2\%$ in most of the parameter space preferred by the electroweak precision data. The $SU(3)$ simple group model has negative contributions to single top production at the LHC. For $f=1TeV$, $x_{\lambda}\geq3$, and $1\leq t_{\beta}\leq5$, the absolute values of the relative correction $\Delta\sigma_{i}/\sigma^{SM}_{i}$ for the t-channel, s-channel, and $tW$ associated production processes are in the ranges of $4.3\%\sim10.8\%$, $3.1\%\sim7.5\%$, and $2\%\sim10.6\%$, respectively. In the context of the SM, the production cross sections at hadron colliders for the t-channel and s-channel single top production processes have been calculated at the next leading order(NLO)\[15,2\]. The values of $\sigma_{t}^{SM}(t)[\sigma_{t}^{SM}(\overline{t})]$ and $\sigma_{s}^{SM}(t)[\sigma_{s}^{SM}(\overline{t})]$ at the LHC are given as $(156\pm8)pb[(91\pm5)pb]$ and $(6.6\pm0.6)pb[(4.1\pm0.4)pb]$, respectively. A NLO calculation of the $tW$ associated production cross section at the LHC has been recently given in Ref.\[16\]. The large backgrounds of the signature from single top production come from W$+$jets and $t\overline{t}$ production. Despite the relatively large expected rate and $D0$ has developed several advanced multivariate techniques to discriminate single top production from backgrounds\[17\], single top production has not been discovered yet. For all of three processes for single top production, the production cross section of the t-channel process can be mostly precise measured at the LHC, which is expected to be measured to $2\%$ accuracy\[4\]. Thus, at least we can say that, in most of the parameter space, the effects of the $SU(3)$ simple group model on the t-channel single top production process might be detected at the LHC. In general, the contributions of the little Higgs models to observables are proportional to the factor $1/f^{2}$. To see the $f$ dependence of the corrections of the $SU(3)$ simple group model to single top production, we plot the relative correction $\Delta\sigma_{i}/\sigma_{i}^{SM}$ as a function of the scale parameter $f$ for $t_{\beta}=3$ and $x_{\lambda}=3$ in Fig.3, in which the solid line, dotted line, and dashed line represent the s-channel, t-channel, and $ tW $ associated production processes, respectively. One can see from Fig.3 that the value of the relative correction $\Delta\sigma_{i}/\sigma_{i}^{SM}$ gets close to zero as $f$ increasing. Thus, the contributions of the $SU(3)$ simple group model to single top production decouple for large value of the scale parameter $f$. However, for $t_{\beta}>3$, $x_{\lambda}>3$, and $1TeV<f\leq2.5TeV$, the absolute value of the relative correction $\Delta\sigma_{t}/\sigma_{t}^{SM}$ is larger than $2\%$, which might be detected at the LHC. 4. The contributions of the new gauge bosons $W_{H}$ and $X$ to single top production {#the-contributions-of-the-new-gauge-bosons-w_h-and-x-to-single-top-production .unnumbered} ===================================================================================== Some of the new particles, such as new charged gauge boson $W'$ and scalar boson $\Phi$, can couple the top quark to one of the lighter SM particles and thus can generate contributions to single top production at tree-level or at one-loop. The one-loop contributions are generally too small to be observed at hadron colliders\[18\]. From Eqs.(3,6,7), we can see that the new charged gauge bosons $W_{H}^{\pm}$ and $X^{-}$ have contributions to the t-channel and s-channel processes for single top production. However, since these new gauge bosons must have space-like momentum in the t-channel process $q+b\rightarrow q'+t$, their contributions to the production cross section of the t-channel process are suppressed by the factor $1/M_{W_{H}}^{2}(M_{X}^{2})$\[19\]. The masses $M_{W_{H}}$ and $M_{X}$ are at the order of TeV. Thus, the contributions of these heavy gauge bosons to the t-channel process are very small, which can be neglected. For the s-channel process $q+\overline{q}'\rightarrow t+\overline{b}$, the new charged gauge boson $W'$ might generate significant contributions to its production cross section because of the possibility of $W'$ resonant production\[19,20\]. So, in this section, we will only consider that the contributions of the new gauge bosons $W^{-}_{H}$ and $X^{-}$ to the s-channel process $\overline{q}+q'\rightarrow \overline{t}+b$. Certainly, our numerical results are easily transferred to those of the new charged gauge boson $W^{+}_{H}$ for the process $q+\overline{q}'\rightarrow t+\overline{b}$ by replacing $\overline{b}$ as $b$ and $t$ as $\overline{t}$. The center-of-mass energy $\sqrt{S}$ of the LHC is large enough to produce the heavy gauge bosons $W_{H}^{-}$ or $X^{-}$ on shell, thus these heavy gauge bosons might produce significant contributions to the s-channel process $\overline{q}+q'\rightarrow \overline{t}+b$. The corresponding scattering amplitude including the SM gauge boson $W$ can be written as: $$M_{i}=\frac{2\pi\alpha_{e}}{S_{W}^{2}}[\frac{1}{\hat{s}-m_{W}^{2}}+ \frac{AB}{\hat{s}-M_{i}^{2}+iM_{i}\Gamma_{i}}] [\overline{u}(\overline{q})\gamma^{\mu}P_{L}\nu(q')] [\overline{u}(b)\gamma_{\mu}P_{L}\nu(\overline{t})],$$ where $i$ represents the gauge boson $W_{H}^{-}$ or $X^{-}$. For the gauge boson $W_{H}^{-}$, $A=B=c/s$, and for the gauge boson $X^{-}$, $A=\delta_{t}$ and $B=\delta_{\nu}$. The expression of the total decay width $\Gamma_{W_{H}}$ has been given in Ref.\[21\]. If the decay of the gauge boson $X^{-}$ to one SM fermion and one TeV-scale fermion partner is kinematically forbidden, then it can decay to pairs of SM fermions through their mixing with the TeV-scale fermion partners, which is independent of the fermion embedding\[6\]. For the gauge boson $X^{-}$, the possible decay modes are $\overline{t}b$, $\overline{u}d$, $\overline{c}s$, and $l\nu_{l}$, in which $l$ presents all three generation leptons $e$, $\mu$, and $\tau$. The total decay width $\Gamma_{X^{-}}$ can be written as\[6\]: $$\Gamma_{X^{-}}=\frac{\alpha M_{X}}{4S_{W}^{2}}(\delta_{t}^{2}+5\delta_{\nu}^{2}).$$ The new gauge bosons predicted by the little Higgs models get their masses from the $f$ condensate, which breaks the extended gauge symmetry. At the leading order, the masses of the new charged gauge bosons $W_{H}^{\pm}$ and $X^{-}$ can be written as\[5,6,7\]: $$\begin{aligned} M_{W_{H}}&=&\frac{gf}{2sc}\approx 0.65f\cdot\frac{c}{s},\\ M_{X}&=&\frac{gf}{\sqrt{2}}\approx 0.46f.\end{aligned}$$ For the LH model, if we assume that the free parameters $f$ and $c$ are in the ranges of $1\sim3TeV$ and $0\sim0.5$, then we have $M_{W_{H}}\geq1.12TeV$. While the mass of the gauge boson $X^{-}$ predicted by the $SU(3)$ simple group model is larger than $920GeV$ even for $f\geq2TeV$. As numerical estimation, we will simple assume $1TeV\leq M_{W_{H}}\leq3TeV$ and $1TeV\leq M_{X}\leq3TeV$. In Fig.4 and Fig.5, we plot the relative correction parameters $R_{W}=\Delta\sigma_{s}^{W}/\sigma_{s}^{SM}$ and $R_{X}=\Delta\sigma_{s}^{X}/\sigma_{s}^{SM}$ as functions of the gauge boson masses $M_{W_{H}}$ and $M_{X}$ for $t_{\beta}=3$ and three values of the mixing parameters $c$ and $x_{\lambda}$, respectively. In these figures, we have assumed $\Delta\sigma_{s}^{W}=\sigma_{s}(W+W_{H})-\sigma_{s}(W)$ and $\Delta\sigma_{s}^{X}=\sigma_{s}(W+X)-\sigma_{s}(W)$. From these figures, we can see that the value of $R_{W}$ is significantly larger than that of $R_{X}$. This is because, compared to the gauge boson $W_{H}^{-}$, the contributions of the gauge boson $X^{-}$ to the s-channel process $\overline{q}+q'\rightarrow \overline{t}+b$ are suppressed by the factor $\nu^{2}/f^{2}$. For $0.3\leq c\leq 0.6$ and $1TeV \leq M_{W_{H}}\leq 2TeV$, the value of the relative correction parameter $R_{W}$ is in the range of $ 1.5\%\leq R_{W}\leq 90\% $. Even for $M_{W_{H}}\geq2.0TeV(f \sim 2TeV)$, the value of the relative correction parameter $R_{W}$ can reach $6.3\%$. Thus, the effects of the new gauge boson $W_{H}$ to the s-channel process for single top production might be detected at the LHC. 5. Conclusions and discussions {#conclusions-and-discussions .unnumbered} ============================== The electroweak production of single top quark at hadron colliders is an important prediction of the SM which proceeds through three distinct subprocesses. These subprocesses are classified by the virtuality of the electroweak gauge boson $W$ involved: t-channel($p^{2}<0$), s-channel($p^{2}>0$), and associated $tW$($p^{2}=m_{W}^{2}$) production. Each process has rather distinct event kinematics, and thus are potentially observable separately from each other\[2\]. All of these processes are sensitive to modification of the $Wtb$ coupling and the s-channel process is rather sensitive to some heavy charged particles. Thus, studying single top production at the LHC can help to text the SM and further to probe new physics beyond the SM. To solve the so-called hierarchy or fine-tuning problem of the SM, the little Higgs theory was proposed as a kind of models to EWSB accomplished by a naturally light Higgs boson. All of the little Higgs models predict the existence of the new heavy gauge bosons and generate corrections to the SM tree-level $Wqq'$ couplings. Thus, the little Higgs models have effects on single top production at hadron colliders. Little Higgs models can be generally divided in two classes: product group models and simple group models. The LH model and the $SU(3)$ simple group model are the simplest examples of the two class models, respectively. In this paper, we have investigated single top production at the LHC in the context of the LH model and the $SU(3)$ simple group model. We find that these two simplest little Higgs models generate contributions to single top production at hadron colliders via two ways: correcting the SM tree-level $Wqq'$ couplings and new charged gauge boson exchange. For the LH model, the contributions mainly come from the s-channel $W_{H}$ exchange. For $0.3\leq c\leq 0.6$ and $1TeV \leq M_{W_{H}}\leq 2TeV$, the relative correction of the new gauge boson $W_{H}$ to the production cross section of the s-channel process at the LHC with $\sqrt{S}$=14TeV is in the range of $ 1.5\%\leq R_{W}\leq 90\% $. For the $SU(3)$ simple group model, the contributions of the new gauge boson $X$ to the s-channel single top production process is very small. However, in most of the parameter space, the correction terms to the tree-level $Wtb$ and $Wqq'$ couplings can generate significant corrections to all production cross sections of the three processes for single top production at the LHC. It is well known that precise electroweak data provide strong constraints on any extensions of the SM. Most of the little Higgs models are severed constrained by the precise electroweak data, with the exception of the littlest Higgs model with T parity, in which a low scale parameter $f$ is allowed. However variations in the model can give rise to very different constraints. For example, for the LH model, if the SM fermions are charged under $U(1)_{1}\times U(1)_{2}$, the constraints become relaxed. The scale parameter $f=1\sim2TeV$ is allowed for the mixing parameter $c$, $c'$, and $x_{L}$ in the ranges of $0\sim0.5$, $0.62\sim0.73$, and $0.3\sim0.6$, respectively\[6,14\]. In this case, the mass of the new charged gauge boson $W_{H}$ is allowed in the range of $1TeV\sim3TeV$. Thus, as numerical estimation, we have simply assumed the scale parameter $f\geq1TeV$. Certainly, the effects of the little Higgs models on single top production decrease as $f$ increasing, as shown in Fig.3. However, our numerical results shown that, even for $f\geq2TeV$, the relative correction of the $SU(3)$ simple group model to the cross section for the t-channel single top production process can reach $-9\%$. Even we assume that the mass of the new charged gauge boson $W_{H}$ predicted by the LH model is larger than $2TeV$, it can also make the cross section of the s-channel single top production process enhance about $6\%$. So we expect that the effects of the $SU(3)$ simple group model on the t-channel process for single top production and the contributions of the new charged gauge bosons $W_{H}^{\pm}$ predicted the LH model to the s-channel process can be detected at the LHC experiments. [**Acknowledgments**]{} This work was supported in part by Program for New Century Excellent Talents in University(NCET-04-0290), the National Natural Science Foundation of China under the Grants No.10475037. [99]{} CDF Collaboration, D0 Collaboration, and Tevatron Electroweak Working Group, [*Combination of CDF and D0 results on the top-quark mass*]{}, [*hep-ex*]{}/[**0507091**]{} M. Beneke et al., [*Top quark physics*]{}, [ *hep-ph*]{}/[**0003033**]{}; D. Chakraborty, J. Konigsberg and D. Rainwater, [*Ann. Rev. Nucl. Part. Sci.*]{} [**53**]{}(2003)301; W. Wagner, [*Rept. Prog. Phys.*]{} [**68**]{}(2005)2409. Z. Sullivan, [*Phys. Rev. D*]{}[**70**]{}(2004)114012; J. Campbell, R. K. Ellis and F. Tramontano, [*Phys. Rev. D*]{}[**70**]{}(2004)094012; Chuan-Ren Chen, F. Larios, C. -P. Yuan, [*Phys. Lett. B*]{}[**631**]{}(2005)126. J. Parsons, [*Top Quark Physics at the LHC I*]{}, talk presented at the Top Quark Symposium, April [**7-8, 2005,**]{} Michigan Centor for Theoretical Physics, University of Michiyan, Ann Arbor. For recent review see: M.Schmaltz and D.Tucker-Smith, [*Ann. Rev. Nucl. Parti. Sci.*]{}[ **55**]{}(2005)229. T. Han, H. E. Logan, and L. T. Wang, [*JHEP*]{} [**0601**]{}(2006)099. M. Perelstein, [*hep-ph*]{}/[**0512128**]{}. D. E. Kaplan and M. Schmaltz, [*JHEP*]{} [**0310**]{}(2003)039. N. Arkani-Hamed, A. G. Cohen, E. Katz, A. E. Nelson, [*JHEP*]{} [**0207**]{}(2002)034. M. Schmaltz, [*JHEP*]{} [**0408**]{}(2004)056. T. Han, H. E. Logan, B. McElrath and L. T. Wang, [*Phys. Rev. D*]{}[**67**]{}(2003)095004. J. Pumplin et al., [*JHEP*]{} [**0207**]{}(2002)012; D. Stump et al., [*JHEP*]{} [**0310**]{}(2003)046. S. Eidelman et al\[Particle Data Gruop\], [*Phys. Lett. B*]{}[ **592**]{}(2004)1. R. Casalbuoni, A. Deandrea, M. Oertel, [**JHEP 0402**]{}(2004)032; Mu-Chun Chen and S. Dawson, [*Phys. Rev. D*]{}[**70**]{}(2004)015003; Chong-Xing Yue and Wei Wang, [*Nucl. Phys. B*]{}[ **683**]{}(2004)48; W. Kilian and J. Reuter, [*Phys. Rev. D*]{}[ **70**]{}(2004)015004; A. Deandrea, [*hep-ph*]{}/[**0405120**]{}; J. L. Hewett, F. J. Petriello and T. G. Rizzo, [*JHEP*]{} [ **0310**]{}(2003)062. C. Csaki, J.Hubisz, G. D. Kribs, P. Meade, and J. Terning, [*Phys. Rev. D*]{}[**67**]{}(2003)115002; C. Csaki et al., [*Phys. Rev. D*]{}[**68**]{}(2003)035009. T. Gregoire, D. R. Smith and J. G. Wacker, [*Phys. Rev. D*]{}[ **69**]{}(2004)115008; Mu-Chun Chen, [*Mod. Phys. Lett. A*]{}[**21**]{}(2006)621. B. W. Harris et al., [*Phys. Rev. D*]{}[**66**]{}(2002)054024. J. Campbell and F. Tramontano, [*Nucl. Phys. B*]{}[**726**]{}(2005)109; M. Beccaria, F. M. Renard, C. Verzegnassi, [*Phys. Rev. D*]{}[**71**]{}(2005)033005; M. Beccaria, G. Macorini, F. M. Renard and C. Verzegnassi, [*Phys. Rev. D*]{}[**73**]{}(2006)093001. V. M. Abazov et al.\[D0 Collaboration\], [*Phys. Lett. B*]{}[**622**]{}(2005)265. C.-S. Li, R. Oakes, J.-M. Yang and H.-Y. Zhou, [*Phys. Rev. D*]{}[ **57**]{}(1998)2009; S. Bar-Shalom, D. Atwood, and A. Soni, [*Phys. Rev. D*]{}[**57**]{}(1998)1495. Tim Tait and C.-P. Yuan, [*Phys. Rev. D*]{}[ **63**]{}(2000)014018. Z. Sullivan, [*Phys. Rev. D*]{}[**66**]{}(2002)075011. H. E. Logan, [*hep-ph*]{}/[**0307340**]{}; Chong-Xing Yue, Feng Zhang, and Wei Wang, [*Chin. Phys. Lett.*]{} [ **22**]{}(2005)1083. [^1]: E-mail:cxyue@lnnu.edu.cn

--- abstract: 'Stochastic Gradient Descent (SGD) plays a central role in modern machine learning. While there is extensive work on providing error upper bound for SGD, not much is known about SGD error lower bound. In this paper, we study the convergence of constant step-size SGD. We provide error lower bound of SGD for potentially non-convex objective functions with Lipschitz gradients. To our knowledge, this is the first analysis for SGD error lower bound without the strong convexity assumption. We use experiments to illustrate our theoretical results.' author: - Zhiyan Ding - Yiding Chen - Qin Li - Xiaojin Zhu bibliography: - 'SGD.bib' title: 'Error Lower Bounds of Constant Step-size Stochastic Gradient Descent' --- Introduction ============ Stochastic Gradient Descent (SGD) is one of the most popular optimization algorithms in modern machine learning  [@bottou2010large; @bottou2018optimization; @zhang2004solving; @badrinarayanan2017segnet; @zinkevich2010parallelized]. SGD enjoys computational efficiency and is easy to implement. In terms of theoretical guarantee, there are extensive studies on analyzing the error upper bound for SGD.  [@zhang2004solving] studies SGD on regularized forms of linear prediction methods, where an error upper bound is given. A non-asymptotic convergence analysis is given in [@moulines2011non]. [@nguyen2018sgd] studies the convergence of SGD with decaying step-size SGD. [@rakhlin2011making] gives convergence analysis for averaged SGD. [@nguyen2018tight] discusses the tightness of existing theoretic upper bound and provides a particular objective function whose SGD error almost touches this upper bound. [@gower2019sgd] gives general convergence analysis under expected smooth assumption. However, there is much less work on the SGD error lower bound. [@dieuleveut2017bridging] provides the lower bound for averaged SGD with strongly convex objective functions. [@jentzen2019lower] quantifies the convergence speed of SGD and gives error lower bound but they are considering SGD with decaying step-size. Their discussion is limited to quadratic objective functions and lack explicit formula of coefficient. When compared to prior work, our work is more general in the sense that we consider a family of objective function with Lipschitz gradient (both **convex** and **non-convex** functions are included). The error lower bound of SGD is of interest for several reasons. On one hand, the error lower bound sharpens our understanding of SGD. [@zhang2004solving] givens an error upper bound for constant step-size SGD when applied to a linear regression learning task. But there exists a term that does not vanish as iteration goes to infinity. This term depends on the step-size $\eta$ and will approach $0$ as $\eta \rightarrow 0$. Our error lower bound accounts for the existence of such a term. On the other hand, SGD error lower bound might be an interesting topic to the adversarial machine learning community. For training data poisoning attacks where an attacker changes training data before learning, a learner using constant step-size SGD will be partially immune to the attack in the sense that the attacker cannot precisely control the resulting model. Our contributions include the following: 1. We give asymptotic and non-asymptotic analysis for SGD error lower and upper bounds when the objective function is strongly convex and has Lipschitz gradients (Theorem \[Lemma:iteration\], \[ThCB1\]); 2. We give asymptotic analysis for SGD error lower bound when the objective function is potentially non-convex and has Lipschitz gradients (Theorem \[ThCB2\]). We also give a loose but uniform constant lower bound (Theorem \[constantlowerbound\]). Preliminaries ============= Stochastic gradient descent (SGD) is an iterative method for optimizing an objective function with the following structure $$\label{OMP} \min_{x\in\mathbb{R}^d} f(\theta):=\frac{1}{J}\sum^J_{j=1}f_j(\theta).$$ where $f, f_j:\mathbb{R}^d\rightarrow\mathbb{R}$ for $j=1,\dots,J$. In machine learning applications, $f$ is the total loss function whereas each $f_j$ represents the loss due to the $j$-th training sample. $\theta$ is a vector of trainable parameters and $J$ is the training sample size, which is typically very large. Throughout the paper we call $\theta^\dagger$ the optimal solution. Under the assumption that all objective functions are differentiable, we have $$\label{udagger} \nabla f(\theta^\dagger) = 0\,.$$ At each iteration, SGD randomly picks one of $\{f_j\}$ and performs gradient descent along the gradient provided by the chosen $f_j$. More explicitly, to obtain the $k+1$-th step solution, one has the following formula: $$\label{Istep} \theta_{k+1}=\theta_k-\eta \nabla f_{\gamma_k}(\theta_k)\,,$$ where $\eta>0$ is the step-size, and $\gamma_k$ is an $i.i.d.$ random variable from the uniform distribution on $\{1,2,\dots,J\}$. In this paper we focus on SGD with constant step-size. Comparing to Gradient Descent (GD), which requires gradients of the entire list of objective functions, SGD requires fewer number of gradients, and thus is computationally much cheaper per iteration. However, since stochasticity is involved, the convergence analysis is significantly more complicated. In particular, since $\nabla f_j (\theta^\dagger)$ may not be zero (despite $\nabla f(\theta^\dagger)=0$), SGD cannot terminate itself even at the global minimum. In the strongly convex case, we show that instead of converging to the minimum point, SGD forms a “fuzzy error ball” around it. We assume the gradients of $f$ and each $f_j$ to be Lipschitz: \[ass:smooth\] $\lambda$-smooth assumption: there exist constants $\{\lambda_{\max,j}\}^{J}_{j=0}>0$ such that $$\label{eqn:assum_Lip} \begin{aligned} |\nabla f(\theta_1)-\nabla f(\theta_2)| &\leq \lambda_{\max,0}|\theta_1-\theta_2| \\ |\nabla f_j(\theta_1)-\nabla f_j(\theta_2)| &\leq \lambda_{\max,j}|\theta_1-\theta_2|\;(\forall j)\,. \end{aligned}$$ In this paper, we use $| \cdot |$ to denote the $2$-norm of a vector. Our analysis is divided into two situations: **Lipschitz** (Assumption \[ass:smooth\]) and **strongly convex** (Assumptions \[ass:convex\]); **Lipschitz** and potentially **non-convex**. For the first situation, we will assume: \[ass:convex\] Strongly convex assumption: there exist a constant $\lambda_{\min}>0$ such that $$\label{eqn:assum_conv} \left\langle \nabla f(\theta_1)-\nabla f(\theta_2), \theta_1-\theta_2\right\rangle\geq \lambda_{\min}|\theta_1-\theta_2|^2\,,\\$$ Under the strong convexity assumption the solution $\theta^\dagger$ satisfying  is the unique solution to . We introduce the following notations: $$D_0=\left(\frac{1}{J}\sum^J_{j=1}|\nabla f_j(\theta^\dagger)|^2\right)^{1/2}\,,\quad\Lambda=\left(\frac{1}{J}\sum^J_{j=1}\lambda^2_{\max,j}\right)^{1/2}\,.$$ We now show two examples that satisfy both Assumption \[ass:smooth\] and \[ass:convex\]: **Linear Regression** Let $\{(\mathbf{x}_j, y_j)\}_{j = 1}^J$ be a training set, where $\mathbf{x}_j \in \mathbb{R}^{d \times 1}$ and $y_j \in \mathbb{R}$. We assume $J >> d$ and $X = (\mathbf{x}_1, \ldots, \mathbf{x}_J) \in \mathbb{R}^{d\times J}$ has full row rank. We consider a linear regression problem. Let $f_j(\theta) = 1/2(\theta^\top\mathbf{x}_j - y_j)^2$. Then $f(\theta) = 1/(2J) \left|X^\top \theta - \mathbf{y}\right|^2$, where $\mathbf{y} = (y_1, \ldots, y_J)^\top \in \mathbb{R}^{J \times 1}$. In this case,  has a closed-form solution and $\theta^\dagger$ is given by $(XX^\top)^{-1}X\mathbf{y}$. $\nabla f_j(\theta) = \mathbf{x}_j (\mathbf{x}_j^\top \theta - y_j)$, $\nabla f(\theta) = X(X^\top \theta - \mathbf{y})/J$, $\nabla^2 f_j(\theta) = \mathbf{x}_j \mathbf{x}_j^\top$, $\nabla^2 f(\theta) = XX^\top /J$. Thus we can choose $\lambda_{\max,0} = \sigma_{\max} (XX^\top)/J$, $\lambda_{\max,j} = \mathbf{x}_j^\top \mathbf{x}_j$ and $\lambda_{\min} = \sigma_{\min}(XX^\top)/J$. We use $\sigma_{\min}(\cdot)$ and $\sigma_{\max}(\cdot)$ to denote the smallest and largest eigenvalues. **Logistic Regression with $L_2$ Regularization** Let $\{(\mathbf{x}_j, y_j)\}_{j = 1}^J$ be a training set, where $\mathbf{x}_j \in \mathbb{R}^{d \times 1}$ and $y_j \in \{0,1\}$. The objective function of Logistic Regression with regularization (of weight 1) is $$\theta^\top \theta/2-1/J\sum_{j=1}^J(y_j \log(S(\theta^\top \mathbf{x}_j)) + (1-y_j) \log(1- S(\theta^\top \mathbf{x}_j))),$$ where $S(\cdot)$ is the Sigmoid function defined by $S(x) = 1/(1+e^{-x})$. In this case, $$f_j(\theta) = -(y_j \log(S(\theta^\top \mathbf{x}_j)) + (1-y_j) \log(1- S(\theta^\top \mathbf{x}_j))) + \theta^\top \theta/2,$$ Then $$\nabla f_j(\theta) = (S(\theta^\top \mathbf{x}_j) - y_j) \mathbf{x}_j + \theta$$ By $$\begin{aligned} |\nabla f_j(\theta_1) - \nabla f_j(\theta_2)| &\le |S(\theta_1^\top \mathbf{x}_j) - S(\theta_2^\top \mathbf{x}_j)| |\mathbf{x}_j| + |\theta_1 - \theta_2| \\ &\le |\theta_1^\top \mathbf{x}_j - \theta_2^\top \mathbf{x}_j| |\mathbf{x}_j| + |\theta_1 - \theta_2| \\ & \le (|\mathbf{x}_j|^2+1) |\theta_1 - \theta_2|,\end{aligned}$$ and $$\nabla^2 f_j(\theta) = S(\theta^\top \mathbf{x}_j) (1 - S(\theta^\top \mathbf{x}_j))\mathbf{x}_j \mathbf{x}_j ^\top + I,$$ we can choose $$\lambda_{\max,j} = (|\mathbf{x}_j|^2 + 1), \lambda_{\max,0} = 1+1/J\sum_{j=1}^J|\mathbf{x}_j|^2, \lambda_{\min} = 1$$ Main Results ============ We present our main theory in this section. It is our goal to show that SGD is bounded away from the minimum. The quantity of interest is: $$\label{Def:WK} {\mathcal{R}}_k=\mathbb{E}\left|\theta_k-\theta^\dagger\right|^2\,.$$ It is the expected squared distance between the $k$-iteration solution and the target $\theta^\dagger$. The expectation takes into account the randomness in SGD. The analysis builds upon two steps: in Step 1, we derive the upper and lower bound of the iterative formula that updates ${\mathcal{R}}_{k+1}$ from ${\mathcal{R}}_k$. This step heavily depends on rewriting the SGD formula. We present the result in Subsection \[Sec:recuresive\]. In Step 2, we analyze the long time behavior of the iterative formula. The behavior depends on the strong convexity of the cost function, and thus we separate the discussion for the strongly convex and non-convex case, and present them in Subsection \[Sec:convex\] and \[Sec:nonconvex\], respectively. Key Recursive Inequalities {#Sec:recuresive} -------------------------- The derivation of the updating formula for ${\mathcal{R}}_k$ comes directly from SGD , which we rewrite into: $$\label{eqn:lstep_V} \theta_{k+1}=\theta_k-\eta\nabla f(\theta_k)+\eta V\left(\theta_k\right)\,,$$ where the variation $$V\left(\theta\right)=\nabla f(\theta)-\nabla f_{\gamma_k}(\theta)\,.$$ Due to the randomness involved in the selection of $\gamma_k$, $V$ is also a random variable with explicitly expressible mean and variance: $$\begin{aligned} &\mathbb{E}\left(V\left(\theta\right)\right)=\overrightarrow{0}\,,\\ &\mathrm{Var}\left(V\left(\theta\right)\right)=\frac{1}{J}\sum^{J}_{j=1}\left(\nabla f(\theta)-\nabla f_j(\theta)\right)\left(\nabla f(\theta)-\nabla f_j(\theta)\right)^\top\,. \end{aligned}$$ Building upon these, we have the following theorem on the recursive formula. \[Lemma:iteration\] The following upper and lower bounds hold true if SGD uses fixed step-size $\eta<\frac{1}{\lambda_{\max,0}}$: - If the objective functions are $\lambda$-smooth (satisfy Assumption \[ass:smooth\]), then $$\label{infinequality} {\mathcal{R}}_{k+1}-{\mathcal{R}}_k\geq \left[-2\eta\lambda_{\max,0}\right]{\mathcal{R}}_k-2\eta^2\Lambda D_0{\mathcal{R}}^{1/2}_{k}+\eta^2D^2_0\,.$$ - In addition, if the objective functions are also strongly convex (satisfy both Assumption \[ass:smooth\] and \[ass:convex\]), then: $$\begin{aligned} \label{supinequality} {\mathcal{R}}_{k+1}-{\mathcal{R}}_k\leq & \left[-2\eta\lambda_{\min}+\eta^2\left(\lambda^2_{\max,0}+\Lambda^2-\lambda^2_{\min}\right)\right]{\mathcal{R}}_k +2\eta^2\Lambda D_0{\mathcal{R}}^{1/2}_{k}+\eta^2D^2_0\,.\end{aligned}$$ There are several comments: - Equation  gives the lower bound for ${\mathcal{R}}_{k+1}$ while  gives the upper bound. The lower bound relies only on the $\lambda$-smooth assumption while the upper bound furthermore requires strong convexity. This is in line with the standard GD analysis. - For very small $\eta$, the $\eta^2$ terms in  and  become negligible, and the linear term on ${\mathcal{R}}_k$ dominates the estimates. - In the recursive formula for the classical GD analysis the new-step error linearly depends on the old-step error. We do have an extra square root term $\sqrt{{\mathcal{R}}_k}$. This is exactly because the randomness adds a direction different from the gradient descent direction. To handle this new direction, the Hölder inequality is utilized and the power of ${\mathcal{R}}_k$ is thus sacrificed. - In the second statement of the theorem, strongly convex property is imposed. This may not be necessary, see [@gower2019sgd]. However, it is our main goal to understand the lower but not the upper bound, and thus we do not pursue a tighter condition in this paper. - The statement of the theorem still holds true for varying time step, assuming that $\eta_k$ satisfies the condition at every step: $\eta_k<\frac{1}{\lambda_{\text{max},0}}$ for all $k$. - For $J = 1$, SGD degenerates to GD and $D_0 = 0$. recover the classical analysis for GD. We start from , subtract $\theta^\dagger$ and take expectation of the $L_2$ norm and get: $$\label{Secondcal} \begin{aligned} {\mathcal{R}}_{k+1} &= \mathbb{E}\left|\theta_{k+1}-\theta^\dagger\right|^2=\mathbb{E}\left|\theta_k-\theta^\dagger-\eta\nabla f(\theta_k)\right|^2+\eta^2\mathbb{E}\left|V(\theta_k)\right|^2\\ &=\mathbb{E}\left|\theta_k-\theta^\dagger-\eta\left(\nabla f(\theta_k)-\nabla f(\theta^\dagger)\right)\right|^2+\eta^2\mathbb{E}\left|V(\theta_k)\right|^2\\ &= \mathbb{E}\left|\theta_k-\theta^\dagger-\eta\left(\nabla f(\theta_k)-\nabla f(\theta^\dagger)\right)\right|^2+\frac{1}{J}\sum^J_{j=1}\mathbb{E}|\nabla f_j(\theta_k)|^2-\mathbb{E}|\nabla f(\theta_k)|^2\,. \end{aligned}$$ In the first equation we used the fact that $\mathbb{E}\left(V\left(\theta\right)\right)=0$, and in the second we use the fact that $\nabla f(\theta^\dagger)=0$. Lastly we expand $\mathbb{E}|V(\theta_k)|^2$ using the definition of $V$. To obtain  and  amounts to give upper and lower bounds for the three terms on the right hand side. To lower bound the first term, we note that using the triangle inequality: $$\label{eqn:term1_lower} \begin{aligned} \mathbb{E}\left|\theta_k-\theta^\dagger-\eta\left(\nabla f(\theta_k)-\nabla f(\theta^\dagger)\right)\right|^2\geq&\mathbb{E}\left(\left|\theta_k-\theta^\dagger\right|-\eta\left|\nabla f(\theta_k)-\nabla f(\theta^\dagger)\right|\right)^2\\ \geq&\mathbb{E}\left(\left(1-\eta\lambda_{\max,0}\right)\left|\theta_k-\theta^\dagger\right|\right)^2\\ \geq&\left(1-\eta\lambda_{\max,0}\right)^2\mathbb{E}\left|\theta_k-\theta^\dagger\right|^2 = (1-\eta\lambda_{\max,0})^2{\mathcal{R}}_k\,, \end{aligned}$$ where the second inequality comes from the $\lambda$-smooth assumption and the requirement on $\eta$. To upper bound the first term, we note that: $$\begin{aligned} &\left|\theta_k-\theta^\dagger-\eta\left(\nabla f(\theta_k)-\nabla f(\theta^\dagger)\right)\right|^2\\ =&\langle \theta_k-\theta^\dagger\,, \theta_k-\theta^\dagger\rangle - 2\eta \langle \theta_k-\theta^\dagger\,, \nabla f(\theta_k)-\nabla f(\theta^\dagger)\rangle +\eta^2\langle \nabla f(\theta_k)-\nabla f(\theta^\dagger)\,,\nabla f(\theta_k)-\nabla f(\theta^\dagger)\rangle\,, \end{aligned}$$ which gives $$\label{eqn:term1_upper} \begin{aligned} \mathbb{E}\left|\theta_k-\theta^\dagger-\eta\left(\nabla f(\theta_k)-\nabla f(\theta^\dagger)\right)\right|^2\leq&\left(1-2\eta\lambda_{\min}+\eta^2\lambda^2_{\max,0}\right)\mathbb{E}\left|\theta_k-\theta^\dagger\right|^2=\left(1-2\eta\lambda_{\min}+\eta^2\lambda^2_{\max,0}\right){\mathcal{R}}_k\,. \end{aligned}$$ by the strong convexity condition. To estimate the third term in , we note that on one hand, by using the $\lambda$-smoothness assumption \[ass:smooth\], we have: $$\label{eqn:term2_2} \begin{aligned} \mathbb{E}|\nabla f(\theta_k)|^2&=\mathbb{E}|\nabla f(\theta_k)-\nabla f(\theta^\dagger)|^2\leq \lambda^2_{\max,0}\mathbb{E}\left|\theta_k-\theta^\dagger\right|^2 = \lambda^2_{\max,0}{\mathcal{R}}_k\,, \end{aligned}$$ and on the other, by using the strong convexity property , we have: $$\label{eqn:term2_3} \begin{aligned} \mathbb{E}|\nabla f(\theta_k)|^2&=\mathbb{E}|\nabla f(\theta_k)-\nabla f(\theta^\dagger)|^2\geq \lambda^2_{\min}\mathbb{E}\left|\theta_k-\theta^\dagger\right|^2= \lambda^2_{\min}{\mathcal{R}}_k\,. \end{aligned}$$ To estimate the second term in , we insert $\theta^\dagger$ to have: $$\label{eqn:term2_1_upper} \begin{aligned} &\frac{1}{J}\sum^J_{j=1}\mathbb{E}|\nabla f_j(\theta_k)|^2 = \frac{1}{J}\sum^J_{j=1}\mathbb{E}|\nabla f_j(\theta_k)-\nabla f_j(\theta^\dagger)+\nabla f_j(\theta^\dagger)|^2\\ =&\frac{1}{J}\sum^J_{j=1}\mathbb{E}|\nabla f_j(\theta^\dagger)|^2+\frac{1}{J}\sum^J_{j=1}\mathbb{E}|\nabla f_j(\theta_k)-\nabla f_j(\theta^\dagger)|^2+\frac{2}{J}\sum^J_{j=1}\mathbb{E}\left\langle \nabla f_j(\theta^\dagger),\nabla f_j(\theta_k)-\nabla f_j(\theta^\dagger)\right\rangle\,. \end{aligned}$$ Noting that $\frac{1}{J}\sum_{j=1}^J\mathbb{E}|\nabla f_j(\theta^\dagger)|^2 = D^2_0$, $$0\leq \frac{1}{J}\sum^J_{j=1}\mathbb{E}|\nabla f_j(\theta_k)-\nabla f_j(\theta^\dagger)|^2\leq \Lambda^2\mathbb{E}|\theta_k-\theta^\dagger|^2$$ and that $$\begin{aligned} \frac{1}{J}\sum^J_{j=1}\mathbb{E}\left\langle \nabla f_j(\theta^\dagger),\nabla f_j(\theta_k)-\nabla f_j(\theta^\dagger)\right\rangle\leq& \frac{1}{J}\sum^J_{j=1}\left|\nabla f_j(\theta^\dagger)\right|\mathbb{E}\left|\nabla f_j(\theta_k)-\nabla f_j(\theta^\dagger)\right|\\ \leq& \frac{1}{J}\sum^J_{j=1}|\nabla f_j(\theta^\dagger)|\mathbb{E}\left(\lambda_{\max,j}\left|\theta_k-\theta^\dagger\right|\right)\\ \leq& \frac{1}{J}\sum^J_{j=1}|\nabla f_j(\theta^\dagger)|\lambda_{\max,j} \left(\mathbb{E}|\theta_k-\theta^\dagger|^2\right)^{1/2}\\ \leq& \left(\frac{1}{J}\sum_{j=1}^J|\nabla f_j(\theta^\dagger)|^2\right)^{1/2}\left(\frac{1}{J}\sum_{j=1}^J\lambda^2_{\max,j}\right)^{1/2} \sqrt{{\mathcal{R}}_k}\\ =&D_0{\Lambda}\sqrt{{\mathcal{R}}_k}\,. \end{aligned}$$ The second to last inequality relies on the $\lambda$-smooth assumption and the last one comes from Cauchy-Schwartz inequality. Putting together these estimates leads to the upper and lower bound of the second term in : $$\label{eqn:term2} D^2_0- 2\Lambda D_0\sqrt{{\mathcal{R}}_k}\leq \frac{1}{J}\sum^J_{j=1}\mathbb{E}|\nabla f_j(\theta_k)|^2\leq D^2_0 + \Lambda^2{\mathcal{R}}_k + 2\Lambda D_0\sqrt{{\mathcal{R}}_k}\,.$$ To finish the proof, we combine , and  for the lower bound: $${\mathcal{R}}_{k+1}\geq \left[1-2\eta\lambda_{\max,0}\right]{\mathcal{R}}_k-2\eta^2\Lambda D_0{\mathcal{R}}^{1/2}_{k}+\eta^2D^2_0\,.$$ With the strongly convex assumption \[ass:convex\], combining ,  and , we have: $$\begin{aligned} {\mathcal{R}}_{k+1}\leq&\left[1-2\eta\lambda_{\min}+\eta^2\left(\lambda^2_{\max,0}+\Lambda^2-\lambda^2_{\min}\right)\right]{\mathcal{R}}_k+2\eta^2\Lambda D_0{\mathcal{R}}^{1/2}_{k}+\eta^2D^2_0\,, \end{aligned}$$ concluding the proof. Strongly convex Objective Function {#Sec:convex} ---------------------------------- If the objective function is strongly convex, it is a well-known result that SGD converges to the optimal solution with decaying step-size ([@nguyen2018sgd; @gower2019sgd; @zhang2004solving; @nguyen2018tight]). With fixed step-size, however, it is widely believed that $\theta_k$ oscillates around the true solution with a noise determined by the step-size. We give a tight trajectory of ${\mathcal{R}}_k$ in the following theorem based on the recursive formula above. \[ThCB1\] Suppose the objective functions in  are $\lambda_{max,0}$-smooth and strongly convex (satisfying  and ), then using SGD with step-size: $$\label{lambdacondition} \eta<\frac{\lambda_{\min}}{\Lambda^2+\lambda^2_{\max,0}}\,,$$ we have constants $C_{0-4}$ depending on $f$, $f_i$ and ${\mathcal{R}}_0$ only so that: 1. **Non-asymptotic rate for all $k$** $$\label{supiteration} \begin{aligned} &{\mathcal{R}}_{k}\leq \left[1-2\eta\lambda_{\min}+\eta^2\mathsf{\Phi}^2\right]^k{\mathcal{R}}_0+C_2\eta\,,\\ &{\mathcal{R}}_{k}\geq \left[1-2\eta\lambda_{\max}\right]^k{\mathcal{R}}_0+C_3\eta\, \end{aligned}$$ 2. **Non-asymptotic rate for large $k$:** Let $K_0=\left\lceil\frac{\log(\eta)}{\log(1-\eta\lambda_{\min})}\right\rceil$, then we have $${\mathcal{R}}_{K_0}\leq C_4\eta\,,$$ and for $k\geq K_0$: $$\label{supiteration2} \begin{aligned} {\mathcal{R}}_{k} \leq &\left[1-2\eta\lambda_{\min}+\eta^2\mathsf{\Phi}^2\right]^{k-K_0}{\mathcal{R}}_{K_0}+\frac{\eta D^2_0+2C^{1/2}_4\eta^{3/2}\Lambda D_0}{2\lambda_{\min}-\eta\mathsf{\Phi}^2}\,,\\ {\mathcal{R}}_k \geq &\left[1-2\eta\lambda_{\max,0}\right]^{k-K_0}{\mathcal{R}}_{K_0}+\frac{\eta D^2_0-2C^{1/2}_4\eta^{3/2}\Lambda D_0}{2\lambda_{\max,0}}\,. \end{aligned}$$ 3. **Asymptotic rate for large $k$:** $$\begin{aligned} &\limsup_{k\rightarrow\infty}{\mathcal{R}}_{k}\leq \frac{\eta D^2_0+2C^{1/2}_4\eta^{3/2}\Lambda D_0}{2\lambda_{\min}-\eta\mathsf{\Phi}^2}\,,\label{supb}\\ &\liminf_{k\rightarrow\infty}{\mathcal{R}}_{k}\geq \max\left\{\frac{\eta D^2_0-2C^{1/2}_4\eta^{3/2}\Lambda D_0}{2\lambda_{\max,0}},0\right\}\,.\label{infb}\end{aligned}$$ In the estimate, $$\mathsf{\Phi}=\sqrt{\Lambda^2+\lambda^2_{\max,0}-\lambda^2_{\min}}\,,$$ and the constants $C_{0-4}$ can be made explicit: $$\begin{aligned} & C_0(\eta) =\frac{\eta\Lambda D_0+\sqrt{\eta^2\Lambda^2D^2_0+\eta D^2_0\left(2\lambda_{\min}-\eta\mathsf{\Phi}^2\right)}}{2\lambda_{\min}-\eta\mathsf{\Phi}^2}\,, \\ & C_1(\eta)=\max\left\{{\mathcal{R}}_0\,, C_0^2\right\}\,,\\ &C_2(\eta)=\frac{2\Lambda D_0\left[C_1+\left(D^2_0+1/2\right)\eta^2\right]^{1/2}+D^2_0}{2\lambda_{\min}-\eta\mathsf{\Phi}^2}\,, \\ &C_3(\eta)=\frac{-2\Lambda D_0\left[C_1+\left(D^2_0+1/2\right)\eta^2\right]^{1/2}+D^2_0}{2\lambda_{\max,0}}\,, \\ & C_4(\eta)={\mathcal{R}}_0+C_2(\eta)\,.\end{aligned}$$ Several comments are in order. - If ${\mathcal{R}}_0\sim O(1)$, by definition, except $C_0\sim O(\sqrt{\eta})$, all other constants $C_{1-4}$ are of $O(1)$ on $\eta$. - The asymptotic rate , are not very tight when time step is not small enough. If $\eta$ further satisfies $$\frac{\eta^2\Lambda D_0}{1-2\eta\lambda_{\max,0}}\leq \frac{-2\eta\Lambda D_0+\sqrt{4\eta^2\Lambda^2 D^2_0+8\eta\lambda_{\max,0}D^2_0}}{4\lambda_{\max,0}}\,,$$ we could improve them as $$\begin{aligned} \limsup_{k\rightarrow\infty}{\mathcal{R}}_{k}\leq &\left[\frac{2\eta\Lambda D_0+\sqrt{4\eta^2(\Lambda^2 D^2_0-\mathsf{\Phi}^2D^2_0)+8\eta D^2_0\lambda_{\min}}}{4\lambda_{\min}-2\eta\mathsf{\Phi}^2}\right]^2\,,\label{supbre}\\ \liminf_{k\rightarrow\infty}{\mathcal{R}}_{k}\geq &\left[\frac{-2\eta\Lambda D_0+\sqrt{4\eta^2\Lambda^2 D^2_0+8\eta\lambda_{\max,0}D^2_0}}{4\lambda_{\max,0}}\right]^2\,.\label{infbre}\end{aligned}$$ We proved in the non-convex case . The upper bound can be proved using similar techniques, which we omitted. In Theorem \[ThCB1\], we use estimation of ${\mathcal{R}}^{1/2}_k$. This will give us some higher order (w.r.t $\eta$) error in the trajectory track ,. In real application, if time step $\eta$ is not small enough, we can choose to calculate , directly to track the trajectory of ${\mathcal{R}}_k$. It’s clear from equation  and  the impact of the error introduced due to the random initial guess is eliminated at an exponential rate. As $k$ increases, the second term dominates. It is a term of $\mathcal{O}(\eta)$, with the coefficient mainly determined by $D_0$, that reflects the influence of the stochasticity in the algorithm. This is a term that cannot be eliminated. Furthermore, comparing  with , we see that in the leading order, for small $\eta$, approximately $$\begin{aligned} \limsup_{k\rightarrow\infty}{\mathcal{R}}_{k}\lesssim \frac{\eta D_0}{2\lambda_{\min}}\,,\quad\text{while}\quad \liminf_{k\rightarrow\infty}{\mathcal{R}}_{k}\gtrsim \frac{\eta D_0}{2\lambda_{\max,0}}\,.\end{aligned}$$ This implies in the strongly convex case, if we run SGD forever, the expectation of the error will only be determined by the total loss function $f$ and the step-size $\eta$. Potentially Non-Convex Objective {#Sec:nonconvex} -------------------------------- Without the strongly convex assumption \[ass:convex\], it’s hard to track the trajectory of ${\mathcal{R}}$. The main reason is we only have one direct inequality in Theorem \[Lemma:iteration\]. However, the lower bound of $\liminf$ has a relaxed requirement on strong convexity and the properties of objective function, and thus a quantitative result is still available for the lower bound of $\liminf$. The following theorem gives us the asymptotic lower bound of ${\mathcal{R}}$: \[ThCB2\] Suppose the objective functions are $\lambda_{\max,0}$-smooth (satisfying ), then using SGD with step-size $\eta$ such that: $$\eta<\frac{1}{2\lambda_{\max,0}},$$ there exists $z_0$ so that 1. **Asymptotic rate of $\limsup$:** $$\label{supb2} \limsup_{k\rightarrow\infty}{\mathcal{R}}_{k}\geq z_0^2\,.$$ 2. **Asymptotic rate of $\liminf$:** If $\eta$ is small enough such that $$\label{smallcondition2} \frac{\eta^2\Lambda D_0}{1-2\eta\lambda_{\max,0}}\leq z_0\,,$$ then $$\label{infb2} \liminf_{k\rightarrow\infty}{\mathcal{R}}_{k}\geq z^2_0\,.$$ $z_0$ can be made explicit: $$\label{eqn:C_0C_1} z_0(\eta)=\frac{-2\eta\Lambda D_0+\sqrt{4\eta^2\Lambda^2 D^2_0+8\eta\lambda_{\max,0}D^2_0}}{4\lambda_{\max,0}}.$$ Several comments are in order: - In this theorem, if we further have $\eta<\frac{\lambda_{\max,0}}{2\Lambda^2}$, then we have $$z_0\geq \sqrt{\frac{\eta D^2_0-\eta^{3/2}D_0\sqrt{2D^2_0\Lambda^2/\lambda_{\max,0}}}{2\lambda_{\max,0}}}.$$ Therefore, when $\eta$ is small enough, the leading term of $\liminf$ is the same as the strongly convex case . - Condition  is a very weak requirement on $\eta$. Indeed the left hand is a second order term of $\eta$ while the right hand side is $z_0$ which is about $\mathcal{O}(\eta^{1/2})$. This condition always holds true for small $\eta$. Compare Theorem \[ThCB2\] with Theorem \[ThCB1\], the major difference is we lost the non-asymptotic estimation in the non-convex case. Without further assumption, it’s hard to find upper bounds in each iteration like . This makes controlling negative term $-{\mathcal{R}}^{1/2}_k$ and tracking tight lower bound difficult. However, using structure of the left hand side in , we can prove $z^2_0$ is also a loose lower bound for every iteration as the following theorem states: \[constantlowerbound\] If is true, then ${\mathcal{R}}_k$ satisfies the following properties: 1. For any $k\geq 0$, $${\mathcal{R}}_{k}>\min\left\{{\mathcal{R}}_{0},z^2_0\right\}$$ 2. For any $k\geq 0$, if ${\mathcal{R}}_k<z^2_0$, then $$\label{alwayslarge1} {\mathcal{R}}_{k+1}>{\mathcal{R}}_{k}$$ 3. If there exists $k^*$ such that ${\mathcal{R}}_{k^*}\geq z^2_0$, then we have $$\label{alwayslarge2} {\mathcal{R}}_k\geq z^2_0,\quad \forall k>k^*\,.$$ In Theorem \[constantlowerbound\], equation , show if there is one step whose error, in expectation sense, is small, then the error will grow and if there is one step the error is above a certain threshold, in expectation sense, the error will be constantly above the threshold, leading to no-convergent case. In real applications, most likely ${\mathcal{R}}_0>z^2_0$. By , although we cannot get a tight lower bound of ${\mathcal{R}}_k$ for any $k$, at least we know it will not decrease below the threshold $z^2_0$ in any steps. Experiments =========== We now illustrate our theoretical results with experiments. In Section \[sec:LR\], we use SGD to learn a linear regression model. In Section \[sec:NonConvexExper\], we use SGD to minimize a $4$th order piecewise polynomial on $\mathbb{R}$. SGD for Strongly Convex Quadratic Objective Function \[sec:LR\] --------------------------------------------------------------- In this experiment, we run SGD to learn a linear regression model $y = \theta^\top x + \epsilon$, where $\theta \in \mathbb{R}^{2\times 1}$. We will use this experiment to show the existence of an error lower bound and visualize the “fuzzy ball” around the global minimum. The training data $\{(\mathbf{x}_j, y_j)\}_{j=1}^J \subset \mathbb{R}^{d \times 1} \times \mathbb{R}$ is generated as follows: the size of training set is $J = 30$, the dimension of the feature vector is $d = 2$. Each $\mathbf{x}_j$ is a random vector on the unit sphere generated as follows: first draw $\bm{x}_j \sim N(\bm{0}, I_{2 \times 2})$, then normalize it into $\mathbf{x}_j = \bm{x}_j / |\bm{x}_j|$. We use $\theta^*= (-1.27,-0.49)^\top$ to generate the $y_j$’s. For $j = 1, \ldots, J$, $y_j = \theta^{* \top}\mathbf{x}_j + \epsilon_j$, where $\epsilon_j \sim N(0, 0.1^2)$. $\theta^\dagger = (-1.27,-0.48)^\top$ is the Ordinary Least Square solution of this linear regression problem. When running SGD, we set the step-size to be $\eta = 0.01$ and set the maximum iteration to be $50000$. The initial point is $(0,0)^\top$. For the same data set, we run $1000$ trials. The only difference between these trials is due to the randomness in SGD. The error ${\mathcal{R}}_k$ is an expectation, we estimate ${\mathcal{R}}_k$ by averaging the error $|\theta_k - \theta^\dagger|^2$ over the $1000$ trials. We visualize the $\theta_k$’s for some particular iterations and plot the error curve in Figure \[fig:exper1\]. In the error curve, the lower and upper bounds are obtained by computing  and  in Theorem \[Lemma:iteration\], the asymptotic lower bound $z_0^2$ is the result in Theorem \[ThCB2\]. ![**Upper panels**: upper left panel shows $\theta_k$ for $k = 25, 50, 100, 250, 1000, 5000, 10000, 50000$ in each trial; upper middle panel shows $\theta_{10000}$ in each trial; upper right panel shows $\theta_{50000}$ in each trial. In these panels, the blue circle is the initial point ($\theta_0$), the blue star is the global optimal ($\theta^\dagger$). We can only find $6$ clusters in the upper left panel because the clusters in the $5000, 10000$ and $50000$th iterations overlap. **Lower panel**: the SGD error curve for a strongly convex objective function\[fig:exper1\].](exper1-eps-converted-to.pdf){width="60.00000%"} The upper panels in Figure \[fig:exper1\] shows the distribution of $\theta_k$ in particular iterations. Intuitively, the variance of $\theta_k$ does not decrease to zero as $t$ becomes large even after the SGD has stabilized. The error curve in Figure \[fig:exper1\] shows a positive constant lower bound of the error, which demonstrates that the error of solution ${\mathcal{R}}_k$ will not decrease to zero. Instead, the solution will be bounded away from a small fuzzy ball with a fixed radius in the sense of expected $L_2$ norm. So our results do not mean SGD will never touch the minimum point $\theta^\dagger$. SGD for Non-Convex Objective Function \[sec:NonConvexExper\] ------------------------------------------------------------ In this experiment, we run SGD to minimize a non-convex function. We use this experiments to show the existence of an error lower bound when the objective function is non-convex. For simplicity, we let the objective function be the following function: $f(\theta) = \theta^4 + 2/3 \theta^3 - \theta^2$, $f_1(\theta) = \theta^4 + 2/3 \theta^3 -\theta^2 +\theta$, $f_2(\theta) = \theta^4 + 2/3 \theta^3 -\theta^2 -\theta$ for $\theta \in [-2,2]$. To ensure that $f,f_1,f_2$ has Lipschitz derivatives, we define $f,f_1,f_2$ to be linear functions for $\theta \notin [-2,2]$ such that $f,f_1,f_2,f',f_1',f_2'$ are all continuous on $\mathbb{R}$. See Figure \[fig:fplot\]. It is clear that $f$ has two local minima at $\theta = -1$ and $\theta = 0.5$. $\theta = -1$ is the global minimum, i.e. $\theta^\dagger = -1$. ![$f,f_1,f_2$ for $\theta\in[-3, 3]$\[fig:fplot\]](fplot-eps-converted-to.pdf){width="80.00000%"} When running SGD, we set the step-size to be $1/(4\lambda_{\max,0}) \approx 0.069$ and set the maximum iteration to be $500$. The initial point is either $\theta_0 = 2$ or $\theta_0=-2$. For each of the initial points, we run $100$ trials. For each trial, we plot the value of $\theta$ at each iteration. We show the $\theta$ trajectories of all the $500$ trials and error curves for each initial points in Figures \[fig:err2\_1\] and \[fig:err2\_2\]. In the error curves, the lower bounds are obtained by computing  in Theorem \[Lemma:iteration\], the asymptotic lower bound $z_0^2$ is the result in Theorem \[ThCB2\]. ![$\theta$ trajectories and error curve for SGD starts at $\theta_0 = -2$, $500$ trials.\[fig:err2\_2\]](errorCur2_1-eps-converted-to.pdf){width="100.00000%"} ![$\theta$ trajectories and error curve for SGD starts at $\theta_0 = -2$, $500$ trials.\[fig:err2\_2\]](errorCur2_2-eps-converted-to.pdf){width="100.00000%"} It is not surprising that SGD will get trapped into local minimum for improper initial point. Figure \[fig:err2\_1\] shows that when starting at $\theta_0 = 2$, SGD is very likely to get trapped at $\theta = 0.5$. In this case, our lower bound is rather loose. However, as shown in Figure \[fig:err2\_2\], even when SGD is trapped at $\theta^\dagger$ the global minimum, there is still an error lower bound, which is very similar to the results in Section \[sec:LR\]. Intuitively, when the objective function is non-convex and has local minima, SGD might get trapped in the local minima. If so, it is obvious that there exists an error lower bound. Our theorems show that even if SGD escapes from the local minimum and approaches the global minimum, there is still a positive error lower bound because of the randomness in constant step-size SGD. Conclusion ========== In this paper, we study the error lower bound of constant step-size SGD. We use expected $L_2$ distance from the global minimum to characterize the error. Our theoretical results show that for potentially non-convex objective function with Lipschitz gradient, there always exists a positive error lower bound for SGD. The lower bounds account for a non-decaying term in classic SGD error upper bound analysis. Proof of Theorem 3.2 ==================== We first reformulate the right hand side of the iterative formula $$\label{supinequalityapp} {\mathcal{R}}_{k+1}-{\mathcal{R}}_k\leq \left[-2\eta\lambda_{\min}+\eta^2\mathsf{\Phi}^2\right]{\mathcal{R}}_k+2\eta^2\Lambda D_0{\mathcal{R}}^{1/2}_{k}+\eta^2D^2_0\,.$$ as a quadratic function, with ${\mathcal{R}}_k^{1/2}$ replaced by $x$: $$H(x)=\eta\left(\left[-2\lambda_{\min}+\eta{\mathsf{\Phi}}^2\right]x^2+2\eta\Lambda D_0x+\eta D^2_0\right)\,.$$ Since $\eta<\frac{\lambda_{\min}}{\Lambda^2+\lambda^2_{\max,0}}<\frac{\lambda_{\min}}{\Lambda^2+\lambda^2_{\max,0}-\lambda^2_{\min}}$, the coefficient of the quadratic term is negative and the function has a maximum value. By direct calculation, we obtain $$\max\,H(x)= \eta^2\left(\frac{\eta\Lambda^2}{2\lambda_{\min}-\eta{\mathsf{\Phi}}^2}+D^2_0\right)\leq \left(D^2_0+\frac{1}{2}\right)\eta^2\,,$$ where we used the condition of $\eta$. Also, the function achieves zero at a root: $$x_0=\frac{\eta\Lambda D_0+\sqrt{\eta^2\Lambda^2D^2_0+\eta D^2_0\left(2\lambda_{\min}-\eta{\mathsf{\Phi}}^2\right)}}{2\lambda_{\min}-\eta{\mathsf{\Phi}}^2}\,.$$ Therefore, we obtain $$H(x_0)\leq \left(D^2_0+\frac{1}{2}\right)\eta^2\,,\quad\text{and}\quad C_0=x_0\,.$$ Since $H$ is monotonically decreasing in the region of $[x_0\,,\infty)$, one has $H(C_0)\leq 0$. We now use the iterative formula to discuss two cases: $$\begin{aligned} \text{if}\quad {\mathcal{R}}_k\leq C^2_0:\quad {\mathcal{R}}_{k+1}&\leq {\mathcal{R}}_{k}+H(W^{1/2}_k)\leq {\mathcal{R}}_{k}+\left(D^2_0+\frac{1}{2}\right)\eta^2 \leq C^2_0+\left(D^2_0+\frac{1}{2}\right)\eta^2\,,\\ \text{if}\quad {\mathcal{R}}_{k}>C^2_0:\quad {\mathcal{R}}_{k+1}&\leq {\mathcal{R}}_{k}+H(W^{1/2}_k)\leq {\mathcal{R}}_{k}+H(C_0)\leq {\mathcal{R}}_{k}\,.\end{aligned}$$ This implies, for all $k$: $$\label{crKupp} {\mathcal{R}}_k\leq \max\left\{C^2_0,{\mathcal{R}}_0\right\}+\left(D^2_0+\frac{1}{2}\right)\eta^2= C_1+\left(D^2_0+\frac{1}{2}\right)\eta^2\,.$$ Plug this into , we arrive at: $$\begin{aligned} {\mathcal{R}}_{k+1} \leq & \left[1-2\eta\lambda_{\min}+\eta^2{\mathsf{\Phi}}^2\right]{\mathcal{R}}_k +2\eta^2\Lambda D_0\left(C_1+\left(D^2_0+1/2\right)\eta^2\right)^{1/2}+\eta^2D^2_0\,,\end{aligned}$$ and then by setting $$\begin{aligned} \alpha & =1-2\eta\lambda_{\min}+\eta^2{\mathsf{\Phi}}^2, \\ \beta & =2\eta^2\Lambda D_0\left(C_1+\left(D^2_0+1/2\right)\eta^2\right)^{1/2}+\eta^2D^2_0\,,\end{aligned}$$ we have a simpler version of the updating formula: $$\begin{aligned} {\mathcal{R}}_{k}&\leq \alpha^k{\mathcal{R}}_0+\left(\sum^k_{i=0}\alpha^i\right)\beta\leq \alpha^k{\mathcal{R}}_0+\frac{\beta}{1-\alpha} =\left(1-2\eta\lambda_{\min}+\eta^2{\mathsf{\Phi}}^2\right)^k{\mathcal{R}}_0+C_2\eta\,.\label{nonasyminquality}\end{aligned}$$ Similarly, plug into, $$\label{infinequalityapp} {\mathcal{R}}_{k+1}-{\mathcal{R}}_k\geq \left[-2\eta\lambda_{\max,0}\right]{\mathcal{R}}_k-2\eta^2\Lambda D_0{\mathcal{R}}^{1/2}_{k}+\eta^2D^2_0\,,$$ we also have $$\begin{aligned} {\mathcal{R}}_{k}\geq & \left[1-2\eta\lambda_{\max,0}\right]{\mathcal{R}}_k -2\eta^2\Lambda D_0\left(C_1+\left(D^2_0+1/2\right)\eta^2\right)^{1/2}+\eta^2D^2_0\,,\end{aligned}$$ and $${\mathcal{R}}_{k}\geq \left[1-2\eta\lambda_{\max,0}\right]^k{\mathcal{R}}_0+C_3\eta.$$ Because $\eta<\frac{\lambda_{\min}}{\Lambda^2+\lambda^2_{\max,0}}<\frac{\lambda_{\min}}{\Lambda^2+\lambda^2_{\max,0}-\lambda^2_{\min}}$, we have $$\begin{aligned} \alpha &= 1-2\eta\lambda_{\min}+\eta^2\left(\Lambda^2+\lambda^2_{\max,0}-\lambda^2_{\min}\right)\leq 1-\eta\left(2\lambda_{\min}-\eta\left(\Lambda^2+\lambda^2_{\max,0}-\lambda^2_{\min}\right)\right)\leq 1-\eta\lambda_\text{min}\,,\end{aligned}$$ and setting $K_0=\left\lceil\frac{\log(\eta)}{\log(1-\eta\lambda_{\min})}\right\rceil$ in , we have $$\begin{aligned} {\mathcal{R}}_{K_0}\leq & \alpha^{K_0}{\mathcal{R}}_0+\frac{\beta}{1-\alpha}\\ \leq & \left[1-\eta\lambda_{\min}\right]^{\frac{\log(\eta)}{\log(1-\eta\lambda_{\min})}}{\mathcal{R}}_0 +\frac{2\eta\Lambda D_0\left[C_1+\left(D^2_0+1/2\right)\eta^2\right]^{1/2}+\eta D^2_0}{2\lambda_{\min}-\eta{\mathsf{\Phi}}^2}\\ = &\left({\mathcal{R}}_0+\frac{2\Lambda D_0\left[C_1+\left(D^2_0+1/2\right)\eta^2\right]^{1/2}+D^2_0}{2\lambda_{\min}-\eta{\mathsf{\Phi}}^2}\right)\eta=C_4\eta\,. \end{aligned}$$ What’s more, for $k\geq K_0$, we also have $${\mathcal{R}}_{k}\leq \alpha^{K_0}{\mathcal{R}}_0+\frac{\beta}{1-\alpha}\leq C_4\eta\quad \Rightarrow\quad {\mathcal{R}}^{1/2}_{k}\leq C^{1/2}_4\eta^{1/2}\,.$$ Plug this into the iteration formula and , we obtain for $k\geq K_0$ $${\mathcal{R}}_{k+1}\leq \left[1-2\eta\lambda_{\min}+\eta^2{\mathsf{\Phi}}^2\right]{\mathcal{R}}_k+2C^{1/2}_4\eta^{5/2}\Lambda D_0+\eta^2D^2_0$$ $${\mathcal{R}}_{k+1}\geq \left[1-2\eta\lambda_{\max,0}\right]{\mathcal{R}}_k-2C^{1/2}_4\eta^{5/2}\Lambda D_0+\eta^2D_0,$$ where we also use $$1-2\eta\lambda_{\max,0}>0,\quad \forall \eta<\frac{\lambda_{\min}}{\Lambda^2+\lambda^2_{\max,0}}.$$ Running the same argument using $\alpha$ and $\beta$ as above, we obtain $$\label{supiteration2app} \begin{aligned} {\mathcal{R}}_{k} \leq &\left[1-2\eta\lambda_{\min}+\eta^2\mathsf{\Phi}^2\right]^{k-K_0}{\mathcal{R}}_{K_0}+\frac{\eta D^2_0+2C^{1/2}_4\eta^{3/2}\Lambda D_0}{2\lambda_{\min}-\eta\mathsf{\Phi}^2}\,,\\ {\mathcal{R}}_k \geq &\left[1-2\eta\lambda_{\max,0}\right]^{k-K_0}{\mathcal{R}}_{K_0}+\frac{\eta D^2_0-2C^{1/2}_4\eta^{3/2}\Lambda D_0}{2\lambda_{\max,0}}\,. \end{aligned}$$ Finally, the last two inequalites are direct result by letting $k\rightarrow$ in . Proof of Theorem 3.3,3.4 ======================== First, take $\limsup$ on both sides of , we can obtain $$\begin{aligned} \limsup_{k\rightarrow\infty}{\mathcal{R}}_{k+1}\geq & \left[1-2\eta\lambda_{\max,0}\right]\limsup_{k\rightarrow\infty}{\mathcal{R}}_k -2\eta^2\Lambda D_0\limsup_{k\rightarrow\infty}{\mathcal{R}}^{1/2}_{k}+\eta^2D^2_0\,,\end{aligned}$$ which implies $$\label{eqn:limsup_nonconv} 2\eta\lambda_{\max,0}\left(\limsup_{k\rightarrow\infty}{\mathcal{R}}_k\right)+2\eta^2\Lambda D_0\left(\limsup_{k\rightarrow\infty}{\mathcal{R}}_{k}\right)^{1/2}-\eta^2D^2_0\geq 0\,.$$ Define a quadratic function to be: $$h_1(z)=2\lambda_{\max,0}z^2+2\eta\Lambda D_0z-\eta D^2_0\,.$$ Comparing it with , we see that $$h_1\left(\left(\limsup_{k\rightarrow\infty}{\mathcal{R}}_{k}\right)^{1/2}\right)\geq 0\,.$$ It is also straightforward to verify that $h_1(z_0)= 0$ by plugging in the definition of $z_0$. Considering $h_1$ is a monotonically increasing function in the region of $[z_0,\infty)$, $$\limsup_{k\rightarrow\infty}{\mathcal{R}}_k\geq z_0^2\,.$$ Finally, to prove $$\label{infb2app} \liminf_{k\rightarrow\infty}{\mathcal{R}}_{k}\geq z^2_0\,.$$ and Theorem 3.4, we need to consider two different cases. - In the first scenario, we assume there exists $k^*\geq0$ such that $${\mathcal{R}}_{k^*}\geq z_0^2\,.$$ Let $$h_2(z)=z^2-h_1(z)=\left[1-2\eta\lambda_{\max,0}\right]z^2-2\eta^2\Lambda D_0z+\eta^2D^2_0\,,$$ then naturally: ${\mathcal{R}}_{k^*+1}\geq h_2({\mathcal{R}}^{1/2}_{k^*})$ according to . One can also show that $h_2$ achieves its minimum at $$z^*=\frac{\eta^2\Lambda D_0}{1-2\eta\lambda_{\max,0}}\,,$$ then because of condition on $\eta$, $z^\ast<z_0$, and $h_2$ is a monotonically increasing function in the region $[z_0,\infty]$, and thus $${\mathcal{R}}_{k^*+1}\geq h_2\left({\mathcal{R}}^{1/2}_{k^*}\right)\geq h_2(z_0)=z_0^2\,.$$ which implies $${\mathcal{R}}_{k}\geq z^2_0,\quad \forall k>k^*\quad\Rightarrow\quad \liminf_{k\rightarrow\infty}{\mathcal{R}}_k\geq z_0^2\,.$$ - In the second scenario, we assume for all $k\geq0$, we have $$0\leq {\mathcal{R}}_{k}\leq z_0^2\quad\Rightarrow\quad h_1\left({\mathcal{R}}^{1/2}_{k}\right)\leq h_1(z_0)=0\,.$$ Here we used the fact that $h_1$ is monotonically increasing in the region of $[0,z_0]$. Using , we have, for all $k\geq 0$: $$z_0^2\geq {\mathcal{R}}_{k+1}\geq {\mathcal{R}}_{k}-h_1\left({\mathcal{R}}^{1/2}_{k}\right)\geq {\mathcal{R}}_k-h_1(z_0)={\mathcal{R}}_{k}\,,$$ meaning $\{{\mathcal{R}}_k\}$ is an increasing sequence with an upper bound $z^2_0$. However, we also have, according to $\limsup_{k\rightarrow\infty}{\mathcal{R}}_k\geq z^2_0$, therefore, we finally obtain $$\lim_{k\rightarrow\infty}{\mathcal{R}}_k=z_0^2\,.$$ Combining the discussion of the two scenarios, we prove and Theorem 3.4.

--- abstract: 'We study conformal structure and topology of leaves of singular foliations by Riemann surfaces.' address: - 'Nessim Sibony, Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France. Department of Mathematics, University of Oslo, PO-BOX 1053 Blindern, Oslo.' - 'Erlend Fornæss Wold, Department of Mathematics, University of Oslo, PO-BOX 1053 Blindern, Oslo.' author: - Nessim Sibony - Erlend Fornæss Wold title: Topology and complex structures of leaves of foliations by Riemann surfaces --- [^1] [^2] introduction ============ The question about the topological types of leaves in a lamination has been addressed in several important works. The first striking result is due to Cantwell-Conlon [@CantwellConlon2] and asserts that any open surface is a leaf of a compact nonsingular lamination. On the other hand Ghys [@Ghys2] has considered the following situation. Let $(X,\mathcal L)$ be a compact non-singular lamination by surfaces, and let $\mu$ be a harmonic measure for $X$, as constructed in [@Garnett]. Then $\mu$-almost every leaf is of one of the following six topological types: a plane, a cylinder, a plane with infinitely many handles attached, a cylinder with infinitely many handles attached, a sphere with a Cantor set taken out, or a sphere minus a Cantor set with a handle attached to every end. Ghys uses ergodic theory - Brownian motion with respect to $\mu$. Cantwell-Conlon [@CantwellConlon] obtained the topological analogue of Ghys’ Theorem, *i.e.*, if the lamination is minimal there is a $G_\delta$-dense set of leaves of one of the six types described by Ghys. Here we address the problem of finding conditions for a generic leaf to be a holomorphic disk, and study the conformal structure of leaves in a singular foliation. We are motivated by a conjecture of Anosov: *for a generic polynomial foliation on $\mathbb P^2$, all but countably many leaves are disks* (see e.g. Ilyashenko [@Ilyashenko]). Our main concern is holomorphic foliations on $\mathbb P^k$. Generically, such a foliation does not admit a nontrivial directed image of the complex plane, and in particular, all leaves are hyperbolic, *i.e.*, they are universally covered by the unit disk. This allows us to introduce several functions on $X\setminus E$, that for a point $z\in X\setminus E$ measure how much the leaf passing through $z$ looks like the unit disk, observed from the point $z$. For instance, if we let $k(z,v)$ denote the leafwise Kobayashi metric, and $k_\iota(z,v)$ the leafwise *injective* Kobayashi metric (defined completely analogously to $k$), we set $\rho(z):=k(z,v)/k_\iota(z,v)$. Then $0<\rho\leq 1$, and $\rho(z)=1$ if and only if the leaf passing through $z$ is the disk (see Section \[metrics\] for more details). \[main\] Let $(X,\mathcal L,E)$ be a Brody hyperbolic holomorphic foliation on a compact complex manifold $X$ of dimension $d=2$, where the singular set $E$ is finite. Assume that there is no compact leaf, and that all singularities are hyperbolic. Suppose that there is a sequence $\{z_j\}\subset X\setminus E$ of points such that $\rho(z_j)\rightarrow 1$, and $z_j\rightarrow z\in X\setminus E$. Then there is a nontrivial minimal closed saturated set $Y\subset X$ such that all but countably many leaves in $Y$ are disks. The proof of Theorem \[main\] uses ergodic theory, and will in fact, granted the assumptions, produce a directed positive $\partial\overline\partial$-closed current $T$, such that all but countably many leaves in $\mathrm{Supp}(T)$ are disks. Recall that a foliation is Brody hyperbolic if it does not admit a nontrivial holomorphic image of the complex plane directed by the foliation, possibly passing through the singularities. For a generic foliation on $\mathbb P^k$ of degree $d>1$, there is no compact leaf, all singularities are hyperbolic, all leaves are hyperbolic, and by Brunella [@Brunella], there is no directed closed current. In particular it is Brody hyperbolic. The function $\rho$ is only lower semicontinuous, so the assumption above does not imply $\rho(z)=1$. When a singular point $p\in E$ is hyperbolic, there is always a sequence $z_j\rightarrow p$ such that $\rho(z_j)\rightarrow 1$, see Example \[hypsing\]. It is conjectured that for $X=\mathbb P^2$, generically the foliation $(X,\mathcal L,E)$ is minimal, in which case $Y=X$. Note that for $X=\mathbb P^2$, generically there is a unique minimal saturated set $\tilde Y\subset X$, due to the unique ergodicity theorem proved in [@FS1]. In that case $Y=\tilde Y$ although there is no assumption that $z\in\tilde Y$. A consequence of Theorem \[main\] is that either *all* leaves are far away from resembling the disk, or all but countably many leaves are disks. We have the following dichotomy: \[dichotomy\] Let $(X,\mathcal L,E)$ be a Brody hyperbolic holomorphic foliation on a compact complex manifold $X$ of dimension $d=2$, where the singular set $E$ is finite. Assume that there is no compact leaf, and that all singularities are hyperbolic. Then either - there is a minimal closed saturated set $Y\subset X$ such that all but countably many leaves in $Y$ are disks, or - the limit set $\Lambda_L$ associated to *any* leaf $L$ is equal to $b\triangle$. Recall that if $\Gamma_f$ is the Deck-group associated to a universal covering map $f:\triangle\rightarrow L$, then $\Lambda_f$ is the cluster set of $\{\gamma(0)\}_{\gamma\in\Gamma_f}$. The limit set $\Lambda_L$ of the Riemann surface $L$ is well defined modulo conformal transformations of $\triangle$. In Section 5 we will strengthen this result. We remark that recently, Goncharuk-Kundryashov [@GoncharukKundryashov] have constructed examples of foliations on $\mathbb P^2$ with the line at infinity invariant, such that all leaves have infinitely many handles. In this case, all leaves except the line at infinity are contained in $\mathbb C^2$. Metrics and functions on Riemann surface laminations {#metrics} ==================================================== The Kobayashi Metric -------------------- Let $L$ be a Riemann surface. The Kobayashi pseudo-metric $k(z,v)$ is defined as follows. For $z\in L$ and $v\in T_zL$ we set $$k(z,v)=\inf\{\frac{1}{|\lambda|}:\exists f:\triangle\rightarrow L, f(0)=z, f'(0)=\lambda v\},$$ where $f$ ranges over all holomorphic maps. This metric is non-degenerate if and only if $L$ is hyperbolic, *i.e.*, if $L$ is universally covered by the unit disk $\triangle$. In this case, a universal covering map $f:\triangle\rightarrow L$ is a local isometry with respect to the Poincaré metric, *i.e.*, $$f^*k(\zeta)=\frac{1}{1-|\zeta|^2}|d\zeta|.$$ The Injective Kobayashi Metric ------------------------------ The injective Kobayashi pseudo-metric $k_\iota(z,v)$ is defined as follows. For $z\in L$ and $v\in T_zL$ we set $$k_\iota(z,v)=\inf\{\frac{1}{|\lambda|}:\exists f:\triangle\rightarrow L, f(0)=z, f'(0)=\lambda v\},$$ where $f$ ranges over all *injective* holomorphic maps. The Suita Metric ---------------- Assume for a moment that the Riemann surface $L$ is hyperbolic in the sense of Ahlfors, *i.e.*, that $L$ carries Green’s functions. For a point $z\in L$ we let $G_z(x)$ denote the negative Green’s function with pole at $z$. In a local coordinate system $w$ with $w(z)=0$ we may write $$G_z(w)=\log |w| + h_z(w),$$ with $h_z$ harmonic. In local coordinates $z$ on $L$ the Suita metric $s(z,v)$ is defined by $c_\alpha(z)|dz|=\exp(h_z(0))|dz|$. If $L$ does not support a Green’s function we set $c_\alpha\equiv 0$. Some functions on Riemann surfaces. ----------------------------------- We now define some functions with values in $[0,1]$, and which achieve the value one at a point $z\in L$ if and only if $L$ is the unit disk. We let $f:\triangle\rightarrow L$ be a universal covering map with $f(0)=z$, and we let $\Gamma$ denote the associated Deck-group. $$\rho(z):=\frac{k(z,v)}{k_\iota(z,v)},$$ $$\alpha(z):=\frac{s(z,v)}{k(z,v)},$$ $$\beta(z):=\min\{|\gamma(0)|:\gamma\in\Gamma\}.$$ These functions have concrete geometric interpretations. Note first that by Hurwitz Theorem and a normal family argument there exists an injective holomorphic map $g:\triangle\rightarrow L, g(0)=z,$ that realises $k_\iota$, and a universal covering map $f$ realises $k$. Then $g$ can be factored through $f$, *i.e.*, there exists an injective holomorphic $h:\triangle\rightarrow\triangle, h(0)=0$, such that $f(h(z))=g(z)$. By the chain rule we see that $$\label{intrho} \rho(z)=h'(0).$$ For an interpretation of $\alpha$ we have by Myrberg’s theorem [@Myrberg],[@Tsuji] that $$G_z(f(\zeta))=\sum_{\gamma\in\Gamma}\log|\frac{\zeta-\gamma(0)}{1-\overline{\gamma(0)}\zeta}|.$$ In the coordinate system given by $f$ we have that $k(z,v)=|v|$, and it follows that $$\label{suitacover} \alpha(z)=\Pi_{\gamma\neq\mathrm{id}}|\gamma(0)|.$$ So $\alpha(z)$ is the product of the Möbius lengths of the shortest elements in each homotopy class based at $z$. Finally, $\beta(z)$ is the shortest length occuring in this product, *i.e.*, the length of the shortest non-trivial loop based at $z$. Let $Y$ be a compact Riemann surface, and let $\Omega\subset Y$ be a domain. We define the metrics on the Riemann surface $\Omega$. Set $K=Y\setminus\Omega$, and let $K_1\subset K$ be a compact set with $K\setminus K_1$ closed. Assuming that $K_1$ has logarithmic capacity zero, then $\lim_{z\rightarrow K}\alpha(z)=0$ (the simplest example would be if $K_1$ is an isolated point). Assuming that $\lim_{z\rightarrow K_1}\beta(z)=0$ we will also have $\rho(z),\alpha(z)\rightarrow 0$. Convergence of $\alpha(z)$ is clear since $\alpha<\beta$. To see the convergence of $\rho$ we let $z_j\rightarrow z_0\in K_1$, we let $f_j:\triangle\rightarrow\Omega$ be a universal covering map with $f_j(0)=z$, and we let $g_j:\triangle\rightarrow\Omega$ be injective holomorphic with $h_j(0)=z$ and $\kappa_\iota(z)=|h'(0)|^{-1}$. Let $h_j:\triangle\rightarrow\triangle$ factor $g_j$ through $f_j$, *i.e.*, we have that $g_j=f_j\circ h_j$, so that $|h'_j(0)|=\rho(z_j)$. Since $\beta(z_j)\rightarrow 0$ we see that $f_j$ cannot be injective on disks of radius $r_j$ where $r_j\rightarrow 0$, and so by Lemma \[SCH\] we have that $\rho(z_j)\rightarrow 0$. We have further the following relations between the functions. \[implications\] Let $L$ be a hyperbolic Riemann surface and let $\{z_j\}\subset L$ be a sequence of points. Then $$\alpha(z_j)\rightarrow 1\Rightarrow \beta(z_j)\rightarrow 1\Leftrightarrow\rho(z_j)\rightarrow 1.$$ Moreover, if $g$ denotes any of the three functions, the following holds: For any $\delta>0$ (small) and $R>0$ (large) there exists $\epsilon>0$ such that if $L$ is any Riemann surface, and $z\in L$ with $g(z)\geq 1-\epsilon$, then $g(y)\geq 1-\delta$ for all $y\in L$ with $\mathrm{d_K}(z,y)\leq R$. The first implication is clear since $\alpha<\beta$. For the second right implication, fix a universal covering map $f:\triangle\rightarrow L$ with $f(0)=z$, and fix $\gamma\in\Gamma$ such that $\beta(z)=|\gamma(0)|$. Set $$b=\frac{1}{2}\log(\frac{1+\beta(z)}{1-\beta(z)}),$$ *i.e.*, the Kobayashi length from $0$ to $\gamma(0)$. By the triangle inequality $\Gamma$ cannot identify points in a disk of Kobayashi radius $b/3$, hence $f$ is injective on the disk centred at the origin of radius $$i(\beta(z)):=\frac{(1+\beta(z))^{\frac{1}{3}}-(1-\beta(z))^{\frac{1}{3}}}{(1+\beta(z))^{\frac{1}{3}} + (1-\beta(z))^{\frac{1}{3}}}.$$ So $$\label{estrho} \rho(z)\geq i(\beta(z)).$$ For the last implication, we again fix a universal covering map at a point $z$, and we fix an injective $h:\triangle\rightarrow\triangle$ with $h(0)=0$ such that $\rho(0)=h'(0)$. By the following lemma we see that $$\label{estbeta} \beta(z)\geq (\frac{1-\sqrt{1-\rho(z)^2}}{\rho(z)})^2$$ \[SCH\] Let $h:\triangle\rightarrow\triangle$ be a holomorphic map with $h(0)=0$, and set $\lambda=|h'(0)|$. Then $\triangle_r\subset h(\triangle)$ with $$r=(\frac{1-\sqrt{1-\lambda^2}}{\lambda})^2$$ Assume that $-r\notin h(\triangle), r>0$, and set $\phi(\zeta)=\frac{\zeta+r}{1+r\zeta}$, and then $g(\zeta)=\phi(h(\zeta))$. Then $g(0)=r$. Let $f$ the square root of $g$ such that $f(0)=\sqrt r$, set $\psi(\zeta)=\frac{\zeta-\sqrt r}{1-\sqrt r\zeta}$, and then $q(\zeta)=\psi(f(z))$. Now $g'(0)=(1-r^2)\lambda$, and since $(f\cdot f)'(0)=2f(0)f'(0)$ and $\psi'(\sqrt r)=1/(1-r)$ we get that $$q'(0)=\frac{(1+r)\lambda}{2\sqrt r}\leq 1\Leftrightarrow \lambda r - 2\sqrt{r} + \lambda <0,$$ by Schwarz Lemma. The expression on the right is zero when $$\sqrt{r}=\frac{2\pm\sqrt{4-4\lambda^2}}{2\lambda}.$$ Finally we consider the last claim. If $g$ is equal to $\rho$ or $\beta$, it suffices by and to prove the claim for either of them. For $g=\beta$ this is a simple consequence of the triangle inequality. For $g=\alpha$ we fix $x\in L$ and let $f:\triangle\rightarrow L$ be a universal covering map. Then by $$\alpha(f(\zeta))=\Pi_{\gamma\neq\mathrm{id}}|\frac{\zeta-\gamma(\zeta)}{1-\overline{\gamma(\zeta)}\zeta}|=\Pi_{\gamma\neq\mathrm{id}}\mathrm{d_M}(\zeta,\gamma(\zeta)),$$ where $\mathrm{d_M}$ denotes the Möbius distance on $\triangle$. Letting $\mathrm{d_P}$ denote the Poincaré distance on $\triangle$ this can be rewritten as $$\alpha(f(\zeta)) = \Pi_{\gamma\neq\mathrm{id}}\frac{{\mathrm{e}}^{2\mathrm{d_P}(\zeta,\gamma(\zeta))}-1}{{\mathrm{e}}^{2\mathrm{d_P}(\zeta,\gamma(\zeta))}+1}.$$ If we set $r=\frac{{\mathrm{e}}^{2R}-1}{{\mathrm{e}}^{2R}+1}$ we have that $|\zeta|<r$ for all $\zeta$ with $f(\zeta)=y$ with $\mathrm{d_P}(x,y)<R$. By the triangle inequality we have that $$\mathrm{d_P}(\zeta,\gamma(\zeta))\geq\mathrm{d_P}(0,\gamma(\zeta)) - \mathrm{d_P}(0,\zeta)\geq \mathrm{d_P}(0,\gamma(\zeta))-R,$$ and furthermore $$\mathrm{d_P}(0,\gamma(\zeta))\geq\mathrm{d_P}(0,\gamma(0))-R.$$ It follows that $$\begin{aligned} \alpha(f(\zeta)) & \geq \Pi_{\gamma\neq\mathrm{id}}\frac{{\mathrm{e}}^{2\mathrm{d_P}(0,\gamma(0))}{\mathrm{e}}^{-4R}-1}{{\mathrm{e}}^{2\mathrm{d_P}(0,\gamma(0))}{\mathrm{e}}^{-4R}+1} \\ & = \Pi_{\gamma\neq\mathrm{id}}(1 - \frac{2}{{\mathrm{e}}^{2\mathrm{d_P}(0,\gamma(0))}{\mathrm{e}}^{-4R}+1})\end{aligned}$$ Fix $\tilde\delta>$ such that $\alpha(f(\zeta))>1-\delta$ if $\log\alpha(f(\zeta))>-\tilde\delta$. Now $$\log(\alpha(f(\zeta)))\geq \sum_{\gamma\neq\mathrm{id}}-\frac{4}{{\mathrm{e}}^{2\mathrm{d_P}(0,\gamma(0))}{\mathrm{e}}^{-4R}+1}\geq\sum_{\gamma\neq\mathrm{id}}-\frac{8{\mathrm{e}}^{4R}}{{\mathrm{e}}^{2\mathrm{d_P}(0,\gamma(0))}+1}$$ if $\epsilon>0$ is small enough. On the other hand $$\log\alpha(f(0))=\sum_{\gamma\neq\mathrm{id}}\log(1- \frac{2}{{\mathrm{e}}^{2\mathrm{d_P}(0,\gamma(0))}+1})\leq \sum_{\gamma\neq\mathrm{id}}- \frac{1}{{\mathrm{e}}^{2\mathrm{d_P}(0,\gamma(0))}+1},$$ and so $$\log\alpha(f(\zeta))\geq 8e^{4R}\log\alpha(f(0)),$$ which is greater than $-\tilde\delta$ if $\epsilon>0$ is small enough. The values of the functions $\alpha,\beta$ and $\rho$ at a point $z$ on a Riemann surface $X$, can be regarded as measuring how much $L$ resembles the unit disk observed from the point $z$. Indeed, each of them take values in the unit interval, and $\alpha(z)=1$ for a point $z\in L$ if and only if $L$ is biholomorphic to the unit disk. On the other hand, there are many Riemann surfaces $L$ for which the quantity $$s_\alpha(L):=\sup_{z\in L}\{\alpha(z)\}$$ is equal to one. For instance we have the following. Let $\Gamma$ be a Fuchsian group such that the limit set $\Lambda(\Gamma)$ is different from $b\triangle$, and let $L=\triangle/\Gamma$ denote the underlying Riemann surface. Then $s_\alpha(L)=1$. Fix $\theta$ such that $e^{i\theta}\notin\Lambda(\Gamma)$. By below there exists a constant $C>0$ such that $\alpha(f(re^{i\theta}))\geq C(1-r)$, where $f:\triangle\rightarrow X$ denotes the universal covering map (in fact the constant $C$ depends only on the distance to the limit set). We remark that when $\Lambda(\Gamma)\neq b\triangle$, then $\Gamma$ is of convergence type, or equivalently, $\triangle/\Gamma$ supports a non-trivial bounded subharmonic function. The reason is that if $p\in b\triangle\setminus\Lambda(\Gamma)$, the group $\Gamma$ cannot identify points near $p$, and so it is easy to construct $\Gamma$-invariant subharmonic functions. In the final section we will construct a further example where $\Lambda(\Gamma)=b\triangle$ but still $s_\alpha(L)=1$ where $L=\triangle/\Gamma$.   Riemann surface laminations --------------------------- We will be interested in the metrics and functions defined above in the setting of laminations by Riemann surfaces. Recall that a non-singular lamination $(X,\mathcal L)$ in a complex manifold $M$ is a closed subset $X\subset M$ such that for each point $p\in X$, there are local coordinates $\phi(x)=(z,w)\in\triangle\times\triangle^{n-1}$ near $p$, and a closed subset $T\in\triangle^{n-1}$ such that $\phi(U_p\cap X)$ is a disjoint union of holomorphic graphs $(z,g_t(z))$ with $g_t(0)=t\in T$. Moreover $g_t$ varies continuously with $t$ (the last assumption is unnecessary if $n=2$ in which case $g_t$ is automatically almost Lipschitz). The concept of a lamination generalises to that of an abstract lamination. An abstract lamination by Riemann surfaces $(X,\mathcal L)$ is a locally compact topological space $X$ covered by charts $U_i$ with embeddings $\phi_i:U_i\rightarrow\triangle\times T_i$, and continuous transition mappings $$(z,t)\rightarrow (z'(z,t),t'(t)),$$ with $z'(z,t)$ holomorphic in $z$. Depending on the transversals $T_i$ one can also consider higher transverse regularity. Finally a singular Riemann surface lamination $(X,\mathcal L,E)$ is a compact topological space $X$ with $E\subset X$ a finite set of points, $X\setminus E$ is a non-singular lamination, and for each point $p\in E$ there exists an open neighbourhood $U_p$ of $p$, and a homeomorphism $\phi_p$ from $U_p$ onto a closed set $Y\subset \mathbb B^n$ with $\phi_p(p)=0$, where $Y\setminus\{0\}$ is a non-singular lamination, and $\phi_p$ is holomorphic along leaves. From now on we will consider compact Riemann surface laminations $(X,\mathcal L,E)$. Outside of $E$ we have that $X$ can be equipped with a leafwise hermitian metric $\omega$, to obtain a refined structure $(X,\mathcal L,E,\omega)$. Near a singular point $p\in E$ we will always assume that such a metric is comparable to $\phi_p^*\omega_E$, where $\omega_E$ is the euclidean metric. The main examples we have in mind are laminated sets in compact complex manifolds, in particular in $\mathbb P^2$, and laminations constructed as suspensions or towers of compact Riemann surfaces, see e.g. [@FS2], [@FSW], [@Ghys] and references in there. From now on we assume that all leaves of a lamination are hyperbolic. Then the Kobayashi metrics, Suita metric, and the functions $\rho,\alpha,\beta$ can be defined along the leaves of $(X,\mathcal L,E)$. Recall that the Suita metric is set to be zero on a Riemann surface that does not support a Green’s function. However, a leaf not supporting a Green’s function would give rise to a positive closed current [@PaunSibony]. So by [@Brunella], for a generic foliation on $\mathbb P^k$ of degree $d>1$, the Suita metric is non-degenerate on all leaves. Let $(X,\mathcal L,E)$ be a compact hyperbolic Riemann surface lamination. Assume that there is no non-constant holomorphic map $f:\mathbb C\rightarrow X$ weakly directed by $\mathcal L$, and assume that all singularities are hyperbolic. Then the following holds: - $\rho$ is lower semi-continuous, and continuous on all leaves without holonomy. - $\alpha$ is upper semi-continuous on all leaves without holonomy. - $\beta$ is lower semi-continuous, and continuous on leaves without holonomy. First, lower semi-continuity in (i) follows from Proposition \[productstructure\] since $k(z,v)$ is continuous, and injective holomorphic maps will lift to nearby leaves. Next, assume to get a contradiction that there is a point $z_0$ on a leaf $L_0$ without holonomy at which $\rho$ is not upper semicontinuous, *i.e.*, $k_\iota(z,v)$ is not lower semi-continuous. Then there exists a sequence $z_j\subset L_j$ with $z_j\rightarrow z_0$, and $\lim_{j\rightarrow\infty}k_\iota(z_j,v)<k_\iota(z_0,v)$. If we let $f_j:\triangle\rightarrow L_j$ realise $k_\iota$ at $z_j$ for $j=0,1,2,3,...,$ this means that $\lim_{j\rightarrow\infty} |f_j'(0)|>|f_0'(0)|$ (evaluated in some local coordinates). Since $\{f_j\}$ is a normal family, we may assume that $f_j\rightarrow\tilde f_0$ uniformly on compacts. Then $|\tilde f_0'(0)|>|f_0'(0)|$ and so $\tilde f_0$ cannot be injective. Choose distinct points $a,b\in\triangle$ such that $\tilde f_0(a)=\tilde f_0(b)$, and let $l$ be the straight line segment between $a$ and $b$. Then $\tilde f_0(l)$ is a closed loop in $L_0$, and $\tilde f_j(l)$ would determine a lifting of this loop to an open curve for $j$ large, a contradiction to the fact that $L_0$ is without holonomy, *i.e*, a any compact in $L_0$ has a fundamental neighborhood system with product structure (see [@EpsteinMillettTischler], [@Hector] and Proposition \[productstructure\] below). To show (ii), fix a point $w\in L$ where $L$ is a leaf without holonomy. Since $k(z,v)$ is continuous it suffices to show that $c_\beta$ is upper semi-continuous at $w$. For $\epsilon>0$ choose a smooth domain $Y\subset L$ with $w\in Y$ such that $c_{\alpha,Y}(w)<c_\alpha(w)+\epsilon$. By Proposition \[productstructure\] the foliation has a product structure $Y\times T$ near $Y$ and so for any leaf $L_t$ near $L$ there is a domain $Y_t\subset L_t$ such that $c_{\alpha,Y_t}(w_t)<c_\alpha(w)+2\epsilon$ for $w_t$ close to $w$. Since the Suita metric is decreasing with respect to increasing domains, this gives the upper semi-continuity of $\alpha$. To show (iii), note first that $\beta$ is lower semi-continuous by the continuity of the Kobayashi metric and continuity of the universal covering maps after appropriate normalisation. For the last claim, fix a point $w$ on a leaf $L_t, t\in T$, without holonomy, and let $f:\triangle\rightarrow L_t$ be a universal covering map. Let $\epsilon>0$ be small, and let $Y\subset L_t$ be a smooth domain such that $f(\triangle_{|\gamma(0)|+\epsilon})\subset Y$. By Proposition \[productstructure\] there is a product structure near $Y$ and since the universal covering maps may be chosen to vary continuously, there are sequences of points $a_j\rightarrow 0$ and $b_j\rightarrow\gamma(0)$ such that $f_{t_j}(a_j)=f_{t_j}(b_j)$ when $t_j\rightarrow t$. This means that there are elements $\gamma_j$ in the Deck-groups such that $\gamma_j(a_j)=b_j$, hence $\gamma_j(0)\rightarrow\gamma(0)$. Let $(X,\mathcal L,E)$ be a foliation on $\mathbb P^n$ , and assume that all singularities are hyperbolic. Then there exists a $\delta>0$ such that $\beta(z)\geq\delta$ if $z\in\mathbb P^n\setminus E$, unless $z$ is on a separatrix and is close to $E$. For such a foliations all leaves are hyperbolic, and the Kobayashi metric is continuous (see [@CandelGomez-Mont] and the survey [@FS2]). Near any point $p\in E$, all leaves except finitely many separatrices are simply connected. So in local coordinates where $p=0$, there are $0<\delta_1<\delta_2<<2$ such that if $z\in B_{\delta_1}(0)$, not on a separatrix, then any nontrivial loop based at $z$ will have to leave $B_{\delta_2}(0)$, and so the length of such a curve is bounded away from zero. Suppose $(X,\mathcal L,E)$ is a compact minimal Brody hyperbolic Riemann surface lamination, and assume that all singularities are hyperbolic. Assume that one leaf is a disk. Then a generic leaf is a disk, and for any leaf $L$ there exists a sequence $z_j\in L$ with $z_j\rightarrow z_0\notin E$, and $$\lim_{j\rightarrow\infty}\beta(z_j)=\lim_{j\rightarrow\infty}\rho(z_j)=1.$$ For each $n$ we have that the set $U_n=\{\rho>1-1/n\}$ is open by lower semi-continuity of $\rho$, and by minimality we have that $U_n$ is dense. So $\cap_n U_n$ is a dense $G_\delta$ set on which $\rho\equiv 1$, and so the corresponding leaves are disks. Furthermore, any leaf $L$ will have to cluster onto a disk away from $E$, and lower semi-continuity implies (2.28). We may now also define the functions $s_{\alpha,\beta,\rho}(X)$ on a Riemann surface lamination $X$; we simply take the suprema over all leaves. Suppose $(X,\mathcal L,E)$ is a compact Brody hyperbolic Riemann surface lamination, and assume that all singularities are hyperbolic. Then if $L$ is a dense leaf we have that $s_\rho(X)=s_\rho(L)$ and $s_\beta(X)=s_\beta(L)$. This follows by lower semi-continuity of the functions $\rho$ and $\beta$. Product structures on laminations by complex manifolds ====================================================== In this section we give some basic results about holomorphic maps into holomorphic foliations, and product structures on leaves without holonomy. As observed in [@FS2], for foliations on complex manifolds, these results follows by a construction due to Royden [@Royden]; our emphasis here is on abstract laminations. A complex manifold $M$ is a *Stein manifold* if $M$ admits a strictly plurisubharmonic exhaustion function $\rho$. A compact set $K\subset M$ of a complex manifold $M$ is a *Stein compact* if it has a fundamental system of open Stein neighbourhoods. A pair $(A,B)$ of compact subsets in a complex manifold $X$ is a Cartan pair if the following holds - $A,B, D=A\cup B$ and $C=A\cap B$ are Stein compacta, and - $\overline{A\setminus B}\cap\overline{B\setminus A}=\emptyset$. \[productstructure\] Let $(X,\mathcal L)$ be a lamination by complex manifolds of dimension $d$ on a metric space $X$. Fix a local transversal $T$ and a point $t_0$ in $T$. Let $M$ be a Stein manifold with a strictly plurisubharmonic Morse exhaustion function $u$, and let $M_c=\{u\leq c\}$ be a smooth simply connected sublevel set. Then for any holomorphic immersion $f:M_c\rightarrow L_{t_0}$ with $t_0\in f(M_c)$, there exists an open neighbourhood $T'\subset T$ of $t_0$ and a continuous map $F:M_c\times T'\rightarrow X$ such that - $F_t:M_c\times\{t\}\rightarrow L_{t}$ is holomorphic for all $t$, and - $F_{t_0}=f$. Furthermore, if $Y_0\subset L_{t_0}$ is a strictly psedoconvex domain with boundary and without holonomy and $t_0\in Y_0$, there exists an open neighbourhood $T'\subset T$ of $t_0$ and a continuous map $F:Y_0\times T'\rightarrow X$ such that - $F_t:Y_0\times\{t\}\rightarrow L_{t}$ is a holomorphic embedding for all $t$, and - $F_{t_0}=\iota$, where $\iota$ is the inclusion map. Let $A_k$ be a finite sequence of compact strongly pseudoconvex domains with $A_{k+1}=A_k\cup B_k$, such that $(A_k,B_k)$ is a Cartan pair for each $k$, and $M_c=\cup_k A_k$. If $\{s_j\}, j=1,...,m,$ are the singular values of $u$ less than $c$, and $0<\epsilon<<1$, each sublevel set $\{u\leq s_j-\epsilon\}$ will occur as $A_k$ for some $k$, and in that case $B_k$ will be a topological $p$-cell, $p$ being the Morse index of $u$ at the critical point, such that $A_{k+1}$ is diffeomorphic to $\{u\leq s_j+\epsilon\}$. For any other $A_k$, the set $B_k$ will be a small “bump” on $A_k$, such that $A_k\cap B_k$ is connected, and $A_{k+1}$ is diffeomorphic to $A_{k}$. Furthermore, we ensure that $f(A_1)$ is contained in the flow box $\mathbb B^d\times T_1, T_1=T$, $f(B_k)$ is contained in a flow box $\mathbb B^d\times T_k$ for all $k$, and that $f$ is injective on $A_1$ and on each $B_k$ (see e.g. [@Forstnericbook], 3.10 and 5.10 for details on the existence of such a family of bumps). We will prove, by induction on $k$, that there are transversals $T^k\subset T$ such that (i) and (ii) holds with $M_c$ replaced by $A_k$ and $T'$ replaced by $T^k$. This is clearly the case for $A_1$ and $T^1=T_1$, since we can lift $f$ inside the flowbox. Assume now that (i) and (ii) hold for the pair $(A_k,T^k)$, *i.e.*, we have constructed a map $F_k:A_k\times I^k\rightarrow X$ satisfying (i) and (ii). First we will construct a local lifting $G_{k+1}$ of $f:B_{k}\rightarrow\mathbb B^d\times T_{k+1}$. For each $t\in T^{k}$ the map $G_{k+1}:A_{k}\cap B_{k}\rightarrow X$ will determine which leaf we should lift to in the flow box, and then we lift using the projection in the flow box. This is where simply connectedness is used in the case where attaching $B_k$ corresponds to attaching a 1-cell. The maps $F_k$ and $G_{k+1}$ do not match, but $\gamma_t(\cdot)=G_{k+1}^{-1}\circ F_{k}(\cdot,t)$ converges to the identity on a neighbourhood of $B_{k}$ as $t\rightarrow t_0$. By Theorem 8.7.2 in [@Forstnericbook] there exist an open set $U\supset A_k$ and an open subset $V\supset B_k$ such that for $t$ close enough to $t_0$, there are continuous families of injective holomorphic maps $\alpha_t:U\rightarrow M, \beta_t:V\rightarrow M$ such that $\beta_t\circ\alpha_t^{-1}=\gamma_t$ near $A_k\cap B_k$, and $\alpha_{t_0}=\beta_{t_0}=\mathrm{id}$. Then the maps $F_t\circ\alpha_t$ and $G_t\circ\beta_t$ fit together near $A_k\cap B_k$ to form a map $F_{k+1}(\cdot,t)$ as long as $T$ belongs to a transversal $T^{k+1}$ contained in a small neighbourhood of $t_0$. Finally, the existence of a map $F$ satisfying (iii) and (iv) is proved in exactly the same way, inductively constructing liftings of $A_k$ where $(A_k,B_k)$ is a family of “bumps” on $Y_0$; the absence of holonomy makes sure that the leafs match when attaching $B_k$ corresponds to crossing a singular point of Morse index one. Proof of Theorem \[main\] ========================= We will now prove Theorem \[main\]. \[main2\] Let $(X,\mathcal L,E)$ be a be a Brody hyperbolic holomorphic foliation on a compact complex manifold $X$ of dimension $d=2$, where the singular set $E$ is finite, Assume that there is no compact leaf, and that all singularities are hyperbolic. Suppose that there is a sequence $\{z_j\}\subset X\setminus E$ of points such that $\rho(z_j)\rightarrow 1$, and $z_j\rightarrow z\in X\setminus E$. Then there is a nontrivial closed minimal saturated set $Y\subset X$ such that all but countably many leaves in $Y$ are disks. Note that $E$ might be empty, in which case the assumption $z_j\rightarrow z$ is unnecessary. It is seen from the proof below, that if we add some extra conditions, similar results hold also for $d>2$, and for for compact Riemann surface laminations $(X,\mathcal L,E)$. For a holomorphic foliation $(X,\mathcal L,E)$ we add the condition that there is no positive directed $\partial\overline\partial$-closed current $T$, whose support contains only leaves with holonomy in $(X,\mathcal L,E)$. For a compact Riemann surface lamination $(X,\mathcal L,E)$, we assume in addition that the lamination is minimal, in which case $Y=X$, and replace “all but countably many leaves” by “a residual set of leaves”. Let $f:\triangle\rightarrow X$ be a universal covering map with $f(0)=z$. For $0<r<1$ we define a $(1,1)$-current $G_r$ on $\triangle$ by setting $$\langle G_r,\alpha\rangle := \int\int_\triangle\log^+\frac{r}{|\zeta|}\alpha,$$ and we set $T_r=f_*G_r$. Then $T_r$ is a positive current of bidimension $(1,1)$. By Theorem 5.3 in [@FS3] we have that $\|T_r\|\rightarrow\infty$ as $r\rightarrow 1$, *i.e.*, $$\lim_{r\rightarrow 1}\int\int_\triangle\log^+\frac{r}{|\zeta|}f^*\omega=\infty.$$ By Proposition \[implications\] we may choose a sequence $k(j)$ such that if $f_j:\triangle\rightarrow X$ is a universal covering map of the leaf passing through $z_{k(j)}$ sending $0$ to $z_{k(j)}$, then $\beta(f_j(\zeta))>1-1/j$ for all $|\zeta|<1-1/j$. Furthermore, $k(j)$ may be chosen such that $f_j$ approximates $f$ arbitrarily well on $\triangle_{1-1/j}$, and so we have that $$\lim_{j\rightarrow\infty}\int\int_\triangle\log^+\frac{1-1/j}{|\zeta|}f_j^*\omega=\infty.$$ So if we set $T_j:=f_{j*}G_{1-1/j}$ we have that $\|T_j\|\rightarrow\infty$ as $j\rightarrow\infty$ and that $\beta(z)>1-1/j$ for all $z$ in the support of $T_j$. Let $T$ be any cluster point of $\{T_j/\|T_j\|\}$. Then $T$ has mass one. Since the masses of $\partial\overline\partial T_j$ are uniformly bounded the current $T$ is $\partial\overline\partial$-closed, and so its support is a sub-foliation of $(X,\mathcal L,E)$. The current $T$ admits in a flow-box a decomposition $$\langle T,\omega\rangle=\int (\int_{\triangle_t}h_t\omega) d\mu(t)$$ where $h_t$ is harmonic for each $t$ on the transversal, and since the foliation has no closed leaf, the measure $\mu$ is diffuse. Let $Y\subset X$ be a minimal foliation on $\mathrm{Supp}(T)$. Then, since $Y$ is not a single compact leaf, there are uncountably many leaves in $Y$. Moreover, the leaves with holonomy in $(X,\mathcal L,E)$ form a countable set. This follows from [@EpsteinMillettTischler] since leaves with holonomy correspond to fix points for the holonomy pseudogroup, which is countable since we are in the holomorphic category, and the transversal is one dimensional. Fix a leaf $L$ without holonomy in $Y$ and a point $w\in L$. By the construction there exists a sequence of points $w_j\in X\setminus E$ such that $w_j\rightarrow w$ and $\beta(w_j)\rightarrow 1$ as $j\rightarrow\infty$. By the upper semi-continuity of $\beta$ we have that $\beta(w)=1$. Hence all leaves without holonomy in $Y$ (resp. in Supp($T$)) are disks. Proof of Theorem \[dichotomy\] ============================== We will give in this section a stronger version of Theorem \[dichotomy\], but first we give a proof of Theorem \[dichotomy\] using Theorem \[main\]. *Proof of Theorem \[dichotomy\]:* Assume that there exists a leaf $L$ with a universal covering map $f:\triangle\rightarrow L$ such that the limit set $\Lambda(\Gamma_f)\neq b\triangle$. In that case, we may assume that the segment $\{e^{i\theta}, \theta\in I_s\}, I_s=[-s,s],$ does not intersect $\Lambda(\Gamma_f)$ for some $s>0$. By the proof of Theorem 5.3 in [@FS3] we cannot have that $\lim_{r\rightarrow 1}f(re^{i\theta})\in E$ for all $\theta\in I_s$, and so in particular there is a $\theta_0\in I_s$ and $r_j\rightarrow 1$, such that $\mathrm{dist}(f(r_j e^{i\theta_0}),E)\geq \epsilon>0$, measured by some given metric on $\mathbb P^n$. We will show that $\lim_{j\rightarrow\infty}\alpha(f(r_j e^{i\theta_0}))=1$, in which case Theorem \[dichotomy\] follows by Theorem \[main\] and Proposition \[implications\]. Recall that $$\alpha(f(z))=\Pi_{\gamma\neq\mathrm{id}}|\frac{z-\gamma(z)}{1-\overline{\gamma(z)}z}|.$$ Now, by [@Rao], (3.8), we have for any $\gamma$ that $$\label{estterms} \log|\frac{1-\overline{\gamma(z)}z}{z-\gamma(z)}|\leq\frac{(1-|z|^2)^2(1-|\gamma(0)|^2)}{|\gamma(0)|^2|z-\zeta_1|^2|z-\zeta_2|^2},$$ where $\zeta_1$ and $\zeta_2$ are the two fixed points of $\Gamma_f$. All these fixed points are in $\Lambda(\Gamma_f)$, and so $$\log|\frac{1-\overline{\gamma(re^{i\theta})}re^{i\theta}}{re^{i\theta}-\gamma(re^{i\theta})}|\leq C(1-r^2)^2(1-|\gamma(0)|^2),$$ for all $\theta\in I_s$ and $C>0$ fixed. That the leaf $L$ carries a Green’s function is equivalent to $\Gamma_f$ being of convergence type, *i.e.*, we have that $$\sum_{\gamma\neq\mathrm{id}}1-|\gamma(0)|<\infty,$$ and so $$\label{estlog} -\log\alpha(f(re^{i\theta}))\leq C'(1-r^2) \mbox{ for } \theta\in I_s.$$ $\hfill\square$ Before giving a strengthening of Theorem \[dichotomy\], we give a corollary to it. Let $(X,\mathcal L)$ be a Brody hyperbolic holomorphic foliation on a compact complex manifold $X$ of dimension $d=2$. Assume that no leaf is compact, and that all singular points are hyperbolic. Then if there exists a leaf $L$ of finite genus and with countably many ends, there is a minimal set $Y\subset X$ such that a generic leaf in $Y$ is a disk. By [@HeSchramm] such a leaf is biholomorphic to subset of a compact Riemann surface, all of whose boundary components are smoothly bounded or points. Since there are only countably many boundary components there has to be an isolated component. This component cannot be a point, because there would be arbitrarily Kobayashi-short nontrivial curves, hence there is a smoothly bounded isolated boundary component. This implies that the limit set of the group associated to a universal covering map of $L$ is not everything. Let $(X,\mathcal L,E)$ be a Brody hyperbolic holomorphic foliation on a compact complex manifold $X$ of dimension $d=2$. Assume that there is no compact leaf and that all singularities are hyperbolic. Assume further that there is a leaf $L$ with universal covering map $f:\triangle\rightarrow L$, and a set $F\subset b\triangle$ of positive measure, such that at each point $\zeta\in F$ there is a horocycle $$D_\zeta=D_r((1-r)\zeta), 0<r<1, r=r(\zeta),$$ on which $f$ is injective. Then there is a minimal set $Y\subset X$ such that all but countably many leaves in $Y$ are disks. A consequence of the Theorem is that the structure of some “complicated” leaves, imply that generic leaves are discs. In the final section we will give an example of a Riemann surface $\triangle\overset{f}{\rightarrow}\triangle/\Gamma$ such that $\Lambda(\Gamma)=b\triangle$, but there is a set $F\subset b\triangle$ of full measure, on which there are injective horocycles for $f$. For each $\zeta\in b\triangle$ we set $$\sigma(\zeta):=\sup\{0<r<1:f \mbox{ is injective on } D_{r}((1-r)\zeta)\}.$$ Then $\sigma$ is upper semi-continuous by Hurwitz’ Theorem, and so each set $E_n:=\{\zeta:\sigma(\zeta)\geq 1/n\}$ is measurable and closed. So there is a set $E_N$ of positive measure. We will now construct a sequence of $\partial\overline\partial$-closed currents $T_j$ and a sequence $r_j\rightarrow 1$ such that $\beta\geq r_j$ at all points on all leaves without holonomy in $\mathrm{Supp}(T_j)$. Then if $T$ is any cluster point of $\{T_j\}$, by upper semi-continuity of $\beta$ we have that $\beta\equiv 1$ on all leaves without holonomy in $\mathrm{Supp}(T)$. Let $\zeta\in E_N$. Then for all $n>N$ we have for $z\in D_{1/n}((1-1/n)\zeta)$ that $$\label{betaest} \beta(f(z))\geq\frac{\frac{n(2-1/n)}{N(2-1/N)}-1}{\frac{n(2-1/n)}{N(2-1/N)}+1}=:M(N,n).$$ The Kobayashi distance between $1-1/N$ and $1-1/n$ is $$\frac{1}{2}[\log(\frac{2-1/n}{1/n}) - \log(\frac{2-1/N}{1/N})]=\frac{1}{2}\log\frac{n(2-1/n)}{N(2-1/N)}=m(N,n).$$ We let $s<1$ approach 1, and choose $r=r(s)$ such that $\phi_r(z)=\frac{z+r}{1+rz}$ maps the point $-s$ to $1-1/N$. Then $\phi_r(D_s(0))\subset D_{1/N}(1-1/N)$. If we choose $\tilde s<s$ such that the Kobayashi distance between $\tilde s$ and $s$ is $m(N,n)$ then $\phi_r(-\tilde s)=1-1/n$ and any point in $D_{1/n}(1-1/n)$ is eventually contained in $\phi_r(D_{\tilde s}(0))$ as $s\rightarrow 1$. This shows that the Kobayashi distance from any point in $D_{1/n}(1-1/n)$ to the complement of $D_{1/N}(1-1/N)$ is at least $m(N,n)$. By the injectivity of $f$ on $D_{1/N}(1-1/N)$ this means that $\Gamma$ cannot identify a point $z\in D_{1/n}(1-1/n)$ with a point closer too it than $m(N,n)$. Choosing another universal covering map $\tilde f$ with $\tilde f(0)=z$ this means precisely . Now for each $n>N$ we define $$U_n:=(\cup_{\zeta\in E_N}D_{1-1/n}((1-1/n)\zeta))\cup D_{1-1/n}(0).$$ Then for each $n$ the domain $U_n$ is simply connected, and we let $\varphi_n:\triangle\rightarrow U_n$ be a Riemann map with $\varphi(0)=0$. Set $f_n:=f\circ\varphi_n$ and $G_n=\log\frac{1}{|\varphi_n^{-1}|}$. Now for $0<r<1$ we define a (1,1)-current $G_{r}$ on $\triangle$ by $$\langle G_r,\omega\rangle:=\int_{\triangle}\log^+(\frac{r}{|\zeta|})\omega,$$ where $\omega$ is a $(1,1)$-form, and then $T_{n,r}:=(f_n)_*G_r$. As before we want to show that $\|T_{n,r}\|\rightarrow\infty$ as $r\rightarrow 1$. In that case any cluster point $T_n$ of $\{\frac{T_{n,r}}{\|T_{n,r}\|}\}$ as $r\rightarrow 1$ will be $\partial\overline\partial$-closed, and so its support is a Riemann surface lamination $X_n\subset X$. By the previous lemma and upper semi-continuity of $\beta$ on all leaves without holonomy in $X_n$, we have that $\beta\geq M(N,n)$ on these leaves. Finally we may consider a cluster point $T$ of $\{T_n\}$. So we fix a positive smooth test form $\omega$ of type $(1,1)$ and estimate $$\begin{aligned} \langle T_{n,r},\omega\rangle & =\int_\triangle\log^+(\frac{r}{|\zeta|})f_n^*\omega=\int_\triangle (\varphi_n)_*(\log^+(\frac{r}{|\zeta|})f_n^*\omega)\\ & = \int_{U_n}\log^+(\frac{r}{|\varphi_n^{-1}(z)|})f^*\omega.\end{aligned}$$ To get the unbounded mass we need to show that $$\int_{U_n}\log|\frac{1}{\varphi_n^{-1}(z)}|f^*\omega = \int_{U_n} G_n(z)f^*\omega \sim \int_{U_n}G_n(z)|f'(z)|^2dV = \infty,$$ where $G_n$ is the Green’s function on $U_n$. Now by [@FS2] there exists a constant $c_0>0$ such that $|f'(se^{i\theta})|\leq\frac{c_0}{1-s}$ for all $se^{i\theta}$ and for each compact set $K\subset X\setminus E$ there is a constant $c_K$ such that $|f'(se^{i\theta})|\geq\frac{c_K}{1-s}$ when $f(se^{i\theta})\in K$. Set $$A:=\{e^{i\theta}:\lim_{s\rightarrow 1} f(se^{i\theta})\in E\}.$$ By [@FS3] page 951 we have that $A$ has measure zero, and so there exists a set $\tilde A\subset E_N$ of positive measure and $\epsilon>0$, such that for all $e^{i\theta}\in\tilde A$, we have that $\mathrm{dist}(f(se^{i\theta}),E)>\epsilon$ for infinitely many $s\rightarrow 1$. By Harnack’s inequality there exists a constant $c_1>0$ such that for all $\zeta\in E_N$ we have that $G_n(s\zeta)\geq c_1\cdot (1-s)$. By Fubini’s Theorem it suffices for us to show that $$\label{intest} \int_0^1 (1-s)|f'(se^{i\theta})|^2ds=\infty,$$ for $e^{i\theta}\in\tilde A$. Fix $c_2>0$ such that $|f'(\zeta)|\geq\frac{c_2}{1-|\zeta|}$ for all $\zeta$ with $\mathrm{dist}(f(\zeta),E)\geq\epsilon/2$. Then if $\mathrm{dist}(f(se^{i\theta}),E)\geq\epsilon/2$ for $s\in [a,b]$ we have that $$\label{interval} \int_a^b (1-s)|f'(se^{i\theta})|^2ds\geq \int_a^b\frac{c_2}{1-s}ds.$$ Fix $e^{i\theta}\in\tilde A$. If $\mathrm{dist}(f(se^{i\theta}),E)\geq\epsilon/2$ for all $s$ close enough to one, it is clear that is infinite. So we have to consider the case where $\mathrm{dist}(f(se^{i\theta}),E)<\epsilon/2$ for infinitely many $s$. Then we may choose a sequence $a_1<b_1<a_2<b_2<\cdot\cdot\cdot<a_j<b_j<\cdot\cdot$, such that $\mathrm{dist}(f(a_je^{i\theta}),E)<\epsilon/2$ and $\mathrm{dist}(f(b_je^{i\theta}),E)\geq\epsilon$. So $$\sum_{j=1}^\infty\int_{a_j}^{b_j}\frac{1}{1-s}ds=\infty.$$ Examples and applications ========================= \[exdisks\] We will construct a hyperbolic Riemann surface lamination such that the generic leaf is the disk, while the rest, countably many, are annuli. Let $\Gamma$ be a Fuchsian group such that $\triangle/\Gamma$ is a compact Riemann surface of genus greater than one, and let $\iota:\Gamma\rightarrow\mathrm{Diff}(S^1)$ be a representation of $\Gamma$ given by the identity map, *i.e.*, we simply restrict $\Gamma$ to $S^1$. Let $X$ be the quotient of $\triangle\times S^1$ by the group consisting of elements $(\gamma,\iota(\gamma))$ with $\gamma\in\Gamma$. Fix $s\in S^1$ and consider the image of $\triangle_s=\triangle\times\{s\}$ in $X$. For two points $(\zeta_1,s)$ and $(\zeta_2,s)$ to be identified in $X$ we need $\gamma\in\Gamma$ such that $\gamma(\zeta_1)=\zeta_2$ and $\gamma(s)=s$. So $\triangle_s$ is mapped injectively into $X$ for all wandering points $s$. These are all but countably many points. For a periodic point $s$ we let $\Gamma_s$ be the isotropy subgroup of $\{s\}$, and we get that $\triangle/\Gamma_s$ injects into $X$. That the rest of the leafs are annuli follows from the following lemma. Let $\Gamma$ be a hyperbolic Fuchsian group, *i.e.*, all elements have two fixed points on $b\triangle$, and let $s\in b\triangle$. Then the isotropy group $\Gamma_s$ is either empty or generated by a single element. Assume that $\Gamma_s$ is not empty, and fix $\gamma_1\in\Gamma_s$. After conjugation, we may assume that $s=-1$, and that the other fixed point of $\gamma_1$ is $1$. We first claim that for any other $\gamma_2\in\Gamma_s$ the point $1$ is also a fixed point. If not, let $p\notin\{-1,1\}$ be a fixed point for $\gamma_2$. Assuming that $1$ is attracting for $\gamma_1$ we set $p_j:=\gamma_1^j(p)$ to obtain a sequence of points $p_j$ converging to $1$. Set $\sigma_j:=\gamma_1^j\circ\gamma_2\circ\gamma_1^{-j}$. Then $\sigma_j$ has fixed points $-1$ and $p_j$, and all maps have the same multipliers $\pm\lambda$, those of $\gamma_2$. So the sequence $\sigma_j$ converges to the map fixing $\pm 1$ and with multipliers $\pm\lambda$. But since $p_j$ is never equal to one this contradicts the discreteness of $\Gamma$. So all elements of $\Gamma_s$ is of the form $$\gamma_r(z)=\frac{z+r}{1+rz},$$ with $r$ real. Discreteness of $\Gamma$ implies that there is a smallest positive $r$ such that $\gamma_r$ is in the group, and we claim that this element generates $\Gamma_s$. Assume to get a contradiction that $\sigma\in\Gamma_s$ is not in $[\gamma_r]$, and that $\sigma(0)>0$. Since $\sigma(0)>\gamma_r(0)$ there exists a largest integer $k>0$ such that $\gamma_r^k(0)<\sigma(0)$. Then $$\sigma^{-1}(\gamma_r^{k+1}(0))=\mathrm{d_M}(\sigma(0),\gamma^{k+1}_r(0))<\mathrm{d_M}(\gamma_r^k(0),\gamma_r^{k+1}(0))=\gamma_r(0),$$ which contradicts the minimality of $\gamma_r$. Although, as we just have seen, there exist hyperbolic laminations for which there is a countable dense set of leaves which are annuli, the following shows that the set of annuli cannot be too large: Let $(X,\mathcal L)$ be a minimal compact foliated metric space, such that a residual set $S$ of leaves have finite genus and at most countably many ends. Then if a generic leaf is not a topological disk, there is no leafwise complex structure on $X$ such that all leaves are hyperbolic. Assume that there is a hyperbolic structure on $X$, and let $L$ be a leaf of finite genus and at most countably many ends. Then by [@HeSchramm] $L$ is isomorphic to to a domain $\Omega\subset Y$, where $Y$ is a compact Riemann surface, and all boundary components of $b\Omega$ are smooth or points. Since there are only countably many boundary components, there is an isolated component, and since $(X,\mathcal L)$ is non-singular, this component cannot be a point, since we would find arbitrarily Kobayashi short non-trivial loops in $L$. So the limit set $\Lambda(\Gamma)$ of the group $\Gamma$ associated to $L$ is different from $b\triangle$, and so by Theorem \[main2\], the remark following it, and the proof of Theorem \[dichotomy\], the generic leaf is a disk. A contradiction. By considering for instance suspensions over tori, there are Riemann surface laminations all of whose leaves are topological cylinders, in this case all of them conformally isomorphic to $\mathbb C^*$; the point is that you can never give such a foliation a hyperbolic structure. Let $\triangle/\Gamma$ be a compact Riemann surface of genus two. Then $\Gamma$ has four generators $a_1,b_1,a_2,b_2$, and we have the relation $$\label{rel} a_1b_1a_1^{-1}b_1^{-1}a_2b_2a_2^{-1}b_2^{-1}=\mathrm{id},$$ and is the only relation. This means that we may define a group homomorphism $\phi:\Gamma\rightarrow\Gamma$ by $$\phi(a_1)=a_1, \phi(b_1)=a_1, \phi(a_2)=a_2, \mbox{ and } \phi(b_2)=a_2.$$ Now let $\tilde\Gamma$ be the group consisting of elements $(\gamma,\phi(\gamma)), \gamma\in\Gamma,$ acting on $\triangle\times S^1$. Let $X=(\triangle\times S^1)/\tilde\Gamma$ denote the quotient with a natural projection $\pi:X\rightarrow Y =\triangle/\Gamma$. The leaves of the foliation on $X$ are the images of disks $\triangle_s=\triangle\times\{s\}$ via the quotient map defined by $\tilde\Gamma$. As in Example \[exdisks\] the image of $\triangle_s$ is biholomorphic to $\triangle/\Gamma_s$ where $\Gamma_s$ is the stabiliser $$\Gamma_s:=\{\gamma\in\Gamma:\phi(\gamma)(s)=s\}.$$ For all but countably many $s$ we have that $\Gamma_s$ is simply $\mathrm{Ker}(\phi)$, this is the set of wandering points $s$. So the generic leaf is not only of a fixed topological type; all but countably many leaves have the same conformal type $\triangle/\Gamma_s$. In fact, if $T$ is a transversal and $L_1$ and $L_2$ are two such leaves passing through points $t_1,t_2\in T$, there is a biholomorphism $g:L_1\rightarrow L_2$ with $g(t_1)=t_2$. This means that for a point $y\in Y$, the functions $\alpha,\beta$ and $\rho$ are constant on the intersection of $\pi^{-1}(y)$ with the set $\mathcal G$ of generic leaves, *i.e.*, on $\mathcal G$ the functions $\alpha,\beta$ and $\rho$ are functions on $Y$, and there they are continuous. So both their suprema and infima will actually be reached on $\mathcal G$, and it is clear that the supremum is not one. Moreover, since $\mathrm{Ker}(\phi)\subset\Gamma_s$ for all $s$, the supremum is in fact a strict supremum on $X$. \[hypsing\] Let $(X,\mathcal L,E)$ be a holomorphic foliation on $\mathbb P^2$ with hyperbolic singularities. Then for any point $e\in E$ there exists a sequence $z_j\rightarrow e$ such that $\rho(z_j)\rightarrow 1$. For simplicity we assume that near a point $e$ the foliation is defined by the vector field $X(z)=\frac{\partial}{\partial z_1} + \lambda\frac{\partial}{\partial z_2}$, so that integral curves are given by $\phi_{(x_0,y_0)}(z)=(x_0e^z,y_0e^{\lambda z})$. To construct large injective Kobayashi disks, we consider first annuli in the separatrix $\{z_2=0\}$. For any $M>0$ there exist real numbers $a<x_0<b$, with $b$ arbitrarily small, such that any curve $\gamma\subset A(a,b)$ connecting $x_0$ and $bA(a,b)$ has Kobayashi length at least $M$. Furthermore, we may choose an integer $N$, such that any curve starting at $x_0$ and circling the origin $N$ times has Kobayashi length at least $M$. Now consider $$S=\{z\in\mathbb C:\log a<\mathrm{Re}(z)<\log b, -2\pi N<\mathrm{Im}(z)<2\pi N\}.$$ If $y_0>0$ is chosen small enough, then $\phi_{(1,y_0)}:S\rightarrow X$ is an injective parametrisation of a piece of a leaf, and there is a $2N-1$ covering map $\pi_1:\phi_{1,y_0}(S)\rightarrow A(a,b)$. Now if $y_0$ is small enough, the Kobayashi length of a curve $\gamma$ in $\phi_{x_0,y_0}(S)$ is roughly the same as $\pi_1(\gamma)$, by the continuity of the Kobayashi metric. If a loop $\gamma$ based at $(x_0,y_0)$ is non-trivial, then either $\pi_1(\gamma)$ will have to leave $A(a,b)$, or it has to circle the origin at least $N$ times. So its Kobayashi length is at least $M$. \[fullmeasure\] We will construct a Fuchsian group $\Gamma$ such that the following holds: - $\Lambda(\Gamma)=b\triangle$, - $L=\triangle/\Gamma$ has a Cantor set of ends, - There is a set $F\subset b\triangle$ of full measure, such that for each $\zeta\in F$ there is a horocycle $D_{r}((1-r)\zeta)$ on which the universal covering map $f:\triangle\rightarrow L$ is injective. In particular, if $L$ were a leaf of a generic foliation on $\mathbb P^2$, the generic leaf of the unique minimal set would be biholomorphic to the unit disk. The construction will be along the lines described in the next section. Let $\gamma_1$ by an element of the form and set $\Gamma_1=[\gamma_1]$. The limit set $\Lambda(\Gamma_1)$ consists of two points, and we may choose a closed set $F_1\subset b\triangle\setminus\Lambda(\Gamma_1)$ of length $2\pi-1$ and an $r_1>0$ such that for each point $\zeta\in F^1_1$, the group $\Gamma_1$ does not identify points on the horocycle $D_r((1-r)\zeta)$. To construct $\gamma_2$ we write $b\Omega_1\cap b\triangle=c_1^1\cup c_1^2$, a union of two arcs. Choose $a_1\in c^1_1$ such that $a_1$ divides $c^1_1$ into two pieces of the same length. For any $\epsilon_2>0$ there exists $\gamma_2$ with a fundamental domain $\Omega_2$ such that $D_2=\triangle\setminus\overline{\Omega_2}\subset D_{\epsilon_2}(a_1)$. Set $\Gamma_2=[\Gamma_1,\gamma_2]$. Then $\Gamma_2$ makes no further identifications on $\triangle\setminus D_{\epsilon_2}(a_1)$, so for any $\delta_2>0$ we may choose $\epsilon_2$ small enough such that there exists $F_1^2\subset F_1^1$ of length $2\pi-1-\delta_2$, such that for each point $\zeta\in F^2_1$, the group $\Gamma_2$ does not identify points on the horocycle $D_{r_1}((1-r_1)\zeta)$. Now $\Lambda(\Gamma_2)$ has measure zero (see *e.g.* [@Beardon], Theorem 4), so there exists a closed set $F_2^1\subset b\triangle$ of length $2\pi-1/2$ not intersecting $\Lambda(\Gamma_2)$, and $r_2>0$ such that for each point $\zeta\in F^1_2$, the group $\Gamma_2$ does not identify points on the horocycle $D_{r_2}((1-r_2)\zeta)$. To construct $\gamma_3$ we consider a point $a_2\in c^1_2$, and repeat the argument to find $\gamma_3$, arbitrarily small $\delta_3>0$, and $F_1^3\subset F_1^2, F_2^2\subset F_2^1$, of length $2\pi-1-\delta_2-\delta_3$ and $2\pi-1/2-\delta_3$ respectively, such that on $F_1^3$ one finds injective horocycles of radius $r_1$, and on $F_2^2$ one finds injective horocycles of radius $r_2$. Set $\Gamma_3=[\Gamma_2,\gamma_3]$. Again $\Lambda(\Gamma_3)$ has measure zero, so there exists a closed set $F_3^1\subset\triangle\setminus\Lambda(\Gamma_3)$ of length $2\pi-1/3$ on which we can find injective horocycles of radius $r_3>0$ for some $r_3>0$. At this point it is clear how to continue constructing $\gamma_j,\delta_j,F_i^l,r_j$ such that for each group $\Gamma_m=\{\gamma_1,...\gamma_m\}$ with fundamental domain $\Omega_m$, we have that - $F_i^{m-i+1}$ has length $2\pi-1/i - \sum_{j=i-1}^m\delta_j$, - on each $F_i^{m-i+1}$ there are injective horocycles of radius $r_i$ for the group $\Gamma_m$. - $b(\cap_m\Omega_m)$ contains no open interval on the intersection with $b\triangle$. A construction of infinite Fuchsian Groups ========================================== In this section we will describe a general inductive construction of an infinite Fuchsian group. Recall that a fundamental domain for $\Gamma$ is an open set $\Omega\subset\triangle$ such that all points in $\triangle$ has its equivalent in $\overline{\Omega}$, and no two points in $\Omega$ are equivalent. A standard fundamental domain for $\Gamma$ is $$\Omega:=\{z\in\triangle:[z,0]<[z,\gamma(z)] \mbox{ for all } \gamma\neq\mathrm{id}\}.$$ Here $[\cdot,\cdot]$ denotes the Möbius distance. Recall that an element $\gamma\in\mathrm{Aut_{hol}}\triangle$ is hyperbolic if it has exactly two fixed points on $b\triangle$. Our basic example is a mapping $$\label{hyperbolic} \gamma(\zeta)=\frac{\zeta+r}{1+r\zeta},$$ and it is easy to see that any hyperbolic element is conjugate to a mapping on the form .   If we let $l_{+}$ denote the geodesic connecting $e^{i(\frac{\pi}{2}-\arcsin r)}$ and $e^{i(\arcsin r - \frac{\pi}{2})}$, and let $l_{-}$ denote the geodesic connecting$e^{i(\frac{\pi}{2}+\arcsin r)}$ and $e^{i(-\arcsin r - \frac{\pi}{2})}$, the fundamental domain $\Omega$ for $\Gamma=[\gamma]$ is the domain in $\triangle$ bounded by $l_+$ and $l_-$. The lines $l_{\pm}$ are geodesics passing through the points $\frac{1-\sqrt{1-r^2}}{r}$ and $\frac{\sqrt{1-r^2}-1}{r}$ on the real line, and the two lines are identified by $\gamma_1$, making $\triangle/\Gamma$ an annulus. Denote by $D_1$ and $D_2$ the two connected components of $\triangle\setminus\overline{\Omega}$. We may now describe an inductive procedure to construct an infinite Fuchsian group. Assume that we have constructed hyperbolic elements $\gamma_1,...,\gamma_n$, such that $\Gamma_m=[\gamma_1,...,\gamma_m]$ is a Fuchsian group with fundamental domain $\Omega_m$, such that $b\Omega_m\cap b\triangle$ consists of a union of arcs, *i.e.*, the underlying Riemann surface $L_m=\triangle/\Gamma_m$ is a bordered Riemann surface. Let $a$ be an interior point of one of the boundary arcs in $b\Omega_m\cap b\triangle$, and choose $0<\epsilon_m<<1$. It is clear that there exists a Möbius transformation $\phi(z)$ such that if we define $\gamma_{m+1}(z)=\phi^{-1}(\gamma(\phi(z)))$, then $[\gamma_{m+1}]$ has a fundamental domain $U_{m+1}$ with $\triangle\setminus\overline{U_{m+1}}\subset D_{\epsilon_m}(a)$. It follows from Lemma \[limitgroup\] that if $\epsilon_m$ is sufficiently small, then $[\Gamma_m,\gamma_{m+1}]$ is a Fuchsian group with fundamental domain $\Omega_m\cap U_{m+1}$. Set $\Gamma=[\gamma_1,\gamma_2,...]$ for a sequence defined inductively like this. If $\epsilon_m\searrow 0$ sufficiently fast, it is easy to see that $\Gamma(\overline\Omega)=\triangle$, where $\Omega=\cap_m\Omega_m$, since $\Gamma_m(\overline\Omega_m)=\triangle$, and so the argument in the proof of Lemma \[limitgroup\] gives the discreteness of $\Gamma$. \[limitgroup\] Let $\Gamma$ be a finitely generated hyperbolic Fuchsian group with fundamental domain $\Omega$. Furthermore, let $\gamma\in\mathrm{Aut_{hol}}\triangle$ by a hyperbolic element with a fundamental domain $U$ such that $D=\triangle\setminus U\subset\Omega$, the boundary $bU$ is contained in $\Omega$, and such that $bU\cap\triangle$ is bounded away from $b\Omega\cap\triangle$. Then $\tilde\Gamma=[\Gamma,\gamma]$ is a hyperbolic Fuchsian group, with a fundamental domain $\tilde\Omega=\Omega\cap U$. We will first show that if $p\in\overline{\tilde\Omega}$ and if $\gamma(p)=p$ for $\gamma\in\tilde\Gamma$, then $\gamma=\mathrm{id}$. Assume first that $p\in b\tilde\Omega$, and furthermore that $p\in b\Omega$. Choose $\delta>0$ such that $\Gamma(B_\delta(p))\cap\overline D=\emptyset$. Write $$\gamma_k\circ\gamma_{k-1}\circ\cdot\cdot\cdot\circ\gamma_1,$$ where the $\gamma_j$’s alter in belonging to either $\Gamma$ or $[\gamma]$, and non of them are the identity map. Assume first that $\gamma_1\in\Gamma$. Then by our assumption above, we have that $\gamma_2(\gamma_1(B_\delta(p)))\subset D$. We now prove by induction that for any $m\geq 1$ we have that $\gamma_{2m}\circ\cdot\cdot\cdot\circ\gamma_1(B_\delta(p))\subset D$, and $\gamma_{2m+1}\circ\cdot\cdot\cdot\circ\gamma_1(B_\delta(p))\subset\triangle\setminus\overline\Omega$. If the first claim holds for some $m$ then clearly the second claim holds, since a non-trivial element of $\Gamma$ will map $D$ outside $\Omega$, $\Omega$ being a fundamental domain for $\Gamma$ and $\overline D\subset\Omega$. By the same reasoning, if the second claim holds for some $m$, then the first claim holds for $m+1$. So $\gamma$ cannot have a fixed point. Similar arguments work if $p\in\tilde\Omega$ or $p\in bU$. Next we let $p\in\triangle$. We now show that if $\gamma_j\in\tilde\Omega$ for $j=1,2,...,$ and if $\gamma_j(p)\rightarrow q\in\overline{\tilde\Omega}$, then $\gamma_j=\mathrm{id}$ for $j\geq N$, for some $N>0$. Assume that $q\in b\Omega$, the case $q\in bU$ will be completely analagous. Choose $\delta>0$ such that $\Gamma(B_\delta(q))\cap\overline D=\emptyset$. We will consider the maps $\alpha_j=\gamma_{j+1}\gamma_j^{-1}$, such that, setting $q_j=\gamma_j(p)$, we have $q_j\rightarrow q$ and $\alpha_j(q_j)=q_{j+1}$. Now if $\alpha_j$ is eventually in $\Gamma$ for large $j$, then $\alpha_j=\mathrm{id}$ for large $j$. So $\gamma_j=\gamma_{j+1}$ for large $j$, hence $\gamma_j(q)=q$ for large $j$, and so $\gamma_j=\mathrm{id}$ for large $j$, since no other element of $[\Gamma,\gamma]$ can fix an element of $\overline{\tilde\Omega}$. We may finally show that $\tilde\Gamma$ is discrete. Assume that $\gamma_j\in\tilde\Gamma$ and $\gamma_j\rightarrow\gamma\in\mathrm{Aut_{hol}}\triangle$. Then $\gamma_j(0)\rightarrow\gamma(0)$. By the lemma below there exists $\phi\in\tilde\Gamma$ such that $\phi(\gamma(0))\in\overline{\tilde\Omega}$, and so $\phi\circ\gamma_j(0)\rightarrow q\in\overline{\tilde\Omega}$. So $\phi\circ\gamma_j=\mathrm{id}$ for large $j$. Let $\Gamma$ be a finitely generated hyperbolic Fuchsian group with fundamental domain $\Omega$. Furthermore, let $\gamma$ by a hyperbolic element with a fundamental domain $U$ such that the complement $\triangle\setminus\overline U$ is contained in $\Omega$. Then $[\Gamma,\gamma](\overline{\Omega'})=\triangle$, where $\Omega'=\Omega\cap U$. Let $A_1,A_2$ denote the two components of $\triangle\setminus\overline U$; these are the intersection of $\triangle$ and two disjoint euclidean disks $D_1$ and $D_2$ in $\mathbb C$. It suffices to show that $U\subset [\Gamma,\gamma](\overline\Omega)$. We consider what we can reach with compositions $$\gamma_k\circ\gamma_{k-1}\circ\cdot\cdot\cdot\circ\gamma_1,$$ where the $\gamma_j$’s alter in belonging to $\Gamma$ and $[\gamma]$, and $\gamma_1\in\Gamma$, and we let $\mathcal A_k$ denote the set of points we can reach with a composition of length $k$. Note first that $\mathcal A_1$ is all of $U$ except the full orbit $\mathcal F_1=\Gamma(A)$ where $A=A_1\cup A_2$. Next, by composing by elements of $[\gamma]$ we can reach all points in $A$ except the full orbit $\mathcal F_2=[\gamma](\mathcal F_1)$, but no additional points in $U$. But now $\mathcal F_2$ is a countable family of pieces of Euclidean disks contained in $A$, and $\mathcal A_3$ will consist of all points in $U$ except the full orbit $\mathcal F_3=\Gamma(\mathcal F_2)$. Continuing in this fashion we get a family $\mathcal F_{2i-1}$ of open sets, $\mathcal F_{2i+1}\subset\mathcal F_{2i-1}$, each $\mathcal F_{2i-1}$ is a countable family of pieces of Euclidean disk, each disk splitting into a countable family of smaller disks in the next $\mathcal F_{2i+1}$. Next, assume to get a contradiction that $\cap_i\mathcal F_{2i-1}$ is not empty. This means that there exists a sequence $$\alpha_j:=\gamma_{2j-1}\circ\gamma_{2j-2}\circ\cdot\cdot\cdot\circ\gamma_1,$$ such that the images $\alpha_j(A_1)$ (or $A_2$) decrease to a nontrivial intersection of $\triangle$ with a a Euclidean disk $D_3\subset\mathbb C$. We may now assume that $\alpha_j\rightarrow\alpha:D_1\rightarrow D_3$ uniformly on compacts. Now $\alpha$ cannot be constant, otherwise $\alpha_j(A_1)$ could not decrease to a nontrivial intersection with $\triangle$, hence $\alpha$ is a biholomorphism. Then for any composition $\beta:=\tilde\gamma_2\tilde\gamma_1$ with $\tilde\gamma_1\in\Gamma$ and $\tilde\gamma_2\in [\gamma]$ such that $\beta(D_1)\subset\subset D_1$ we get that $\alpha(\beta(D_1))\subset\subset D_3$, and so for a large enough $j$ we have that $\alpha_{j}(\beta(D_1))\subset\subset D_3$. This is a contradiction. [10]{} Beardon, A. F.; Inequalities for certain Fuchsian groups. *Acta Math.* [**127**]{} 1971 221–258. Brunella, M.; Inexistence of invariant measures for generic rational differential equations in the complex domain. *Bol. Soc. Mat. Mexicana* [**3**]{} 12 (2006), no. 1, 43–49. Candel, A. and Gómez-Mont, X; Uniformization of the leaves of a rational vector field. *Ann. Inst. Fourier* (Grenoble) [**45**]{} (1995), no. 4, 1123–1133. Candel, A. and Conlon, L.; Foliations I. Graduate Studies in Mathematics. Volume 23. Cantwell, J. and Conlon, L.; Generic leaves. *Comment. Math. Helv.* [**73**]{} (1998) 306–336. Cantwell, J. and Conlon, L.; Every surface is a leaf. *Topology* [**26**]{} (1987) 265–285. Dinh, T.C., Nguyen, N., and Sibony. N.; Heat equation and ergodic theorems for Riemann surface laminations. *Math. Ann.* [**354**]{} (2012) 331–376. Epstein, D. B. A., Millett, K. C., and Tischler, D.; Leaves without holonomy. *J. London Math. Soc.* (2) [**16**]{} (1977), no. 3, 548–552. Forstnerič, Franc; Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 56. Springer, Heidelberg, 2011. Fornæss, J. E. and Sibony, N.; Unique ergodicity of harmonic currents on singular foliations of $\mathbb P^2$. *GAFA* [**19**]{} (2010) 1334–1377 Fornæss, J. E. and Sibony, N.; Riemann surface laminations with singularities. J. Geom. Anal. [**18**]{} (2008), 400–442. Fornæss, J. E. and Sibony, N.; Harmonic Currents of Finite Energy and Laminations. GAFA [**15**]{} (2005), no.5, 962–1003. Fornæss, J. E., Sibony, N. and Wold, E. F.; Examples of minimal laminations and associated currents. *Math. Z.* [**269**]{} (2011), no. 1-2, 495–520. Garnett, L.; Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. [**51**]{} (1983), no. 3, 285–311. Ghys, E.; Laminations par surfaces de Riemann. Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthéses, 8, Soc. Math. France, Paris, 1999. Ghys, E.; Topologie des feuilles génériques. Ann. of Math. (2) [**141**]{} (1995), no. 2, 387–422. Goncharuk, N. and Kundryashov, Y.; Genera of non algebraic leaves of polynomial foliations in $\mathbb C^2$. arXiv 14017878. He, Z-X and Schramm, O; Fixed points, Koebe uniformization and circle packings. *Ann. of Math.* (2) [**137**]{} (1993), no. 2, 369–406. Hector, G.; Architecture des feuilletages de classe $C2$.Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), 243–258, Ilyashenko, Y.; Some open problems in real and complex dynamical systems. Nonlinearity [**21**]{} (2008), no. 7. Myrberg, P. J.; Über die Existenz Der Greenschen Funktionen auf Einer Gegebenen Riemannschen Fläche. *Acta Math.* [**61**]{} (1933), no. 1, 39–79. Păun, M. and Sibony, N.; Value distribution theory for parabolic Riemann surfaces. arXiv:1403.6596. Rajeswara Rao, K.V.; Fuchsian groups of convergence type and PoincarŽé series of dimension -2, *J. Math. Mech.* [**18**]{} (1968/1969), 629–644. Suita, N; Capacities and Kernels on Riemann Surfaces. *Arch. Rational Mech. Anal.* [**46**]{} (1972), 212–217. Royden, H. L.; The extension of regular holomorphic maps. *Proc. AMS* [**43**]{}, (1974), 306–310. Tsuji, M.; Potential theory in modern function theory. Maruzen Co., Ltd., Tokyo 1959 [^1]: The second author is supported by NRC grant number 240569 [^2]: Part of this work was done during the international research program “Several Complex Variables and Complex Dynamics” at the Centre for Advanced Study at the Academy of Science and Letters in Oslo during the academic year 2016/2017.

--- abstract: 'We present a wide dataset of $\gamma$-ray, X-ray, UVOIR, and radio observations of the [[*Swift*]{}]{} GRB100814A. At the end of the slow decline phase of the X-ray and optical afterglow, this burst shows a sudden and prominent rebrightening in the optical band only, followed by a fast decay in both bands. The optical rebrightening also shows chromatic evolution. Such a puzzling behaviour cannot be explained by a single component model. We discuss other possible interpretations, and we find that a model that incorporates a long-lived reverse shock and forward shock fits the temporal and spectral properties of GRB100814 the best.' author: - | Massimiliano De Pasquale$^{1,2,3}$, N. P. M. Kuin$^1$, S. Oates$^1$, S. Schulze$^{4,5,6}$, Z. Cano$^{4,7}$, C. Guidorzi$^8$, A. Beardmore$^9$, P. A. Evans$^9$, Z. L. Uhm$^2$, B. Zhang$^{2}$, M. Page$^{1}$, S. Kobayashi$^9$, A. Castro-Tirado$^{10}$, J. Gorosabel$^{10,11,12}$, T. Sakamoto$^{13}$, T. Fatkhullin$^{14}$, S. B. Pandey$^{15}$, M. Im$^{16}$, P. Chandra$^{17}$, D. Frail$^{18}$, H. Gao$^{2}$, D. Kopač$^{7}$, Y. Jeon$^{16}$, C. Akerlof$^{20}$, K. Y. Huang$^{21}$, S. Pak$^{22}$, W.-K. Park$^{16,23}$, A. Gomboc$^{19,24}$, A. Melandri$^{25}$, S. Zane$^1$, C. G. Mundell$^7$, C. J. Saxton$^{1,26}$, S. T. Holland$^{27}$, F. Virgili$^7$, Y. Urata$^{28}$, I. Steele$^7$, D. Bersier$^7$, N. Tanvir$^9$, V. V. Sokolov$^{14}$, A. S. Moskvitin$^{14}$\ $^1$Mullard Space Science Laboratory, University College London Dorking, Holmbury St. Mary, Dorking Surrey, RH5 6NT, United Kingdom\ $^2$Department of Physics, University of Nevada, Las Vegas, United States\ $^3$INAF/IASF, Via Ugo La Malfa 153, 90146, Palermo, Italy\ $^4$Centre for Astrophysics and Cosmology, Science Institute, University of Iceland, Dunhagi 5, 107 Reykjavík, Iceland\ $^5$Pontificia Universidad Católica de Chile, Departamento de Astronomía y Astrofísica, Casilla 306, Santiago 22, Chile\ $^6$Millennium Institute of Astrophysics, Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile\ $^7$Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool, L3 5RF United Kingdom\ $^8$Department of Physics and Earth Sciences, University of Ferrara, via Saragat 1, I-44122 Ferrara, Italy\ $^9$University of Leicester, University Rd, Leicester LE1 7RH, United Kingdom\ $^{10}$Instituto de Astrofísica de Andalucía (CSIC), P.O. Box, E-18080 Granada, Spain\ $^{11}$Ikerbasque, Basque Foundation for Science, Alameda de Urquijo 36-5, E-48008 Bilbao, Spain\ $^{12}$Unidad Asociada Grupo Ciencia Planetarias UPV/EHU-IAA/CSIC, Departamento de Física Aplicada I, E.T.S. Ingeniería,\ Universidad  del País-Vasco UPV/EHU, Alameda de Urquijo s/n, E-48013 Bilbao, Spain\ $^{13}$NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA\ $^{14}$Special Astrophysical Observatory, Russian Academy of Science, Russia\ $^{15}$ARIES, Manora Peak, Nainital, Uttarakhand, India, 263129\ $^{16}$CEOU-Astronomy Program, Dept. of Physics & Astronomy, FPRD, Seoul National University, Seoul, Republic of Korea\ $^{17}$National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, Pune University, Ganeshkhind, Pune 411 007, India\ $^{18}$NRAO, P.O. Box 0, Socorro, NM 87801, USA\ $^{19}$Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia\ $^{20}$500 East University, Ann Arbor, University of Michigan, MI 48109-1120 USA\ $^{21}$Department of Mathematics and Science, National Taiwan Normal University, Lin-kou District, New Taipei City 24449, Taiwan\ $^{22}$Kyung Hee University, 1 Seocheon-dong, Giheung-gu, Yongin-si Gyeonggi-do 446-701, Republic of Korea\ $^{23}$Korea Astronomy & Space Science Institute, 776 Daedukdae-ro, Yuseong-Gu, Daejeon 305-348, Republic of Korea\ $^{24}$Centre of Excellence Space-SI, Aškčereva cesta 12, SI-1000 Ljubljana, Slovenia\ $^{25}$INAF - Brera Astronomical Observatory, via E. Bianchi 46, I-23807, Merate (LC), Italy\ $^{26}$Physics Department, Technion - Israel Institute of Technology, Haifa 32000, Israel\ $^{27}$Space Telescope Science Institute, Baltimore, 3700 San Martin Dr. Baltimore, MD 21218, USA\ $^{28}$Institute of Astronomy, National Central University, Chung-Li 32054, Taiwan\ date: 'Accepted...Received...' title: 'Are plateaux and optical rebrightenings in GRB afterglows produced by an interplay of Forward and Reverse Shocks? the case of [*Swift*]{} GRB100814A.' --- \[firstpage\] Gamma-Ray Bursts. INTRODUCTION {#intro} ============ Research on gamma-ray bursts (GRBs) has greatly benefitted of the [[*Swift*]{}]{} mission (@ge04). This space observatory carries three scientific instruments: the Burst Alert Telescope (BAT; Barthelmy et al. 2005), the X-ray telescope (XRT; @ba05), and the Ultra-Violet Optical Telescope (UVOT; @rom05). When BAT detects a GRB, [*Swift*]{} slews towards the source position within 1-2 minutes, and follows up the GRB afterglow emission [@cos97; @gal98] until it becomes too weak to be detected, usually a few days after the trigger. [*Swift*]{} also delivers the position of a newly discovered source promptly to ground based observatories, which can observe the optical and radio afterglows in bands and sensitivities which cannot be achieved by the space facility. Therefore, GRB observations in the [*Swift*]{} age cover the temporal behaviour of GRBs in many different electromagnetic bands from $\sim100$ s after the trigger onwards. Moreover, [[*Swift*]{}]{} has dramatically increased the statistics of GRB afterglows observed (about 90 GRBs per year) from the past. Such comprehensive coverage and statistics have shown that the light curves of GRBs at different wavelengths can be surprisingly diverse. During the afterglow, changes of the flux decay-rate or even rebrightenings can occur in some electromagnetic bands but not in others. An obvious example is the X-ray flares, which do not usually show an optical counterpart (Falcone et al. 2006). Conversely, a few authors have examined GRBs with episodes of optical rebrightening which have no clear equivalent in the X-ray band, such as GRB081029 (Nardini et al. 2011, Holland et al. 2012), and GRB100621A (Greiner et al. 2013). Another less clear-cut case may be GRB050401 (De Pasquale et al. 2006). These events are particularly puzzling since, after the optical rebrightening, the X-ray and optical light curves resume similar behaviour, with simultaneous change of slope. This has called for a deep revision of the emission models of GRB afterglows, which in the past mostly involved a single emission component. Observations indicate that a single component cannot be responsible for the observed features, but all the components producing the afterglow may still be connected, and possibly have a common origin. According to the most accepted scenario, the initial phase of $\gamma$-ray emission arises when dissipation process(es) occur in ultra-relativistic shells emitted by a central engine (Rees & Mészáros 1994). The afterglow arises when the burst ejecta interact with the surrounding medium and produce two shocks; one moving forward in the medium (forward shock, or FS) and another one inward into the ejecta (reverse shock, or RS), causing their deceleration (Mészáros & Rees 1993; Sari & Piran 1999). Both shocks energize the electrons of the medium in which they propagate. The electrons in turn cool by synchrotron emission and produce the observed afterglow light. It is therefore possible that FS and RS can jointly contribute to the observed emission and, since their emissions peak at different wavelengths, produce the puzzling chromatic behaviour observed (e.g. Perley et al. 2014, Urata et al. 2014). Other scenarios put forward involve a residual ‘prompt’ emission producing the X-rays (Ghisellini et al. 2007), up-scattering of the photons produced by FS by fast ejecta (Panaitescu 2008), evolution of the physical parameters of the blast waves (Panaitescu et al. 2006), and two-component jet (De Pasquale et al. 2009; Liang et al. 2013). In this article, we present an ample dataset of the [*Swift*]{} GRB100814A and discuss the remarkable temporal properties of this event. GRB100814A shows a conspicuous rebrightening in the optical bands between $\sim15$ and $\sim200$ ks after the burst trigger. Such a rise of the optical flux has no clear counterpart in the X-ray light curve. However, the flux in both bands shows a similar quick decay after 200 ks. Radio observations show a broad peak about $10^6$ s after the trigger, followed by a slow decay which is different from the rapid fall of the flux visible in the X-ray and optical at the same epoch. Finally, we mention other [*Swift*]{} GRBs that show comparable features and how the modeling adopted in this paper might be applied to their cases. Throughout this paper, we use the convention $F_{\nu} \sim t^{-\alpha} \nu^{-\beta}$, where $F_{\nu}$ is the flux density, $t$ is the time since the BAT trigger, $\nu$ the frequency, $\alpha$ and $\beta$ are the temporal and spectral indices. The errors indicated are at $1~\sigma$ confidence level (68% C.L.), unless otherwise indicated. Reduction and analysis of data ============================== [*Swift*]{} $\gamma$-ray data ----------------------------- GRB100814A triggered the BAT instrument at T$_0$ = 03:50:11 UT on August 14, 2010 (@bea10). The refined BAT position is R.A. (J2000) =$01^{\rm h} 29^{\rm m} 55^{s}$, Dec. (J2000) =-$17^{\circ} 59' 25.7''$ with a position uncertainty of 1’ (90% C.L., Krimm et al. 2010). The GRB onset occurred 4 seconds before the BAT trigger time and it shows 3 main peaks (see Fig. \[100814a\_prompt\_lc\]). From the ground analysis of the BAT data ($15-350~\mathrm{keV}$ energy band) we found that the GRB duration is [*T*]{}$_{90}=174.5 \pm 9.5$ s by [battblocks]{} (v1.18). As for the spectral analysis, we will only consider results obtained in the $15-150~\mathrm{keV}$ band, because the mask weighted technique was used to subtract the background. In this case, it is not possible to use the data above $150$ keV where the mask starts to become transparent to the radiation. The BAT spectrum was extracted using [batbinevt]{} (v1.48). The time-averaged spectrum from T$_0$-3 to T$_0$+235 s is best fitted by a simple power law model. The photon index is $1.47 \pm 0.04$ (90% C.L.). This value is between the typical low energy photon index, $\simeq1$, and the high energy photon index, $\simeq2$ of GRB prompt emission described by the Band model [@ban93]. This suggests that that peak energy $E_{\mathrm{peak}}$ is likely to be inside the BAT energy range. The time-averaged $E_\mathrm{peak}$ is estimated to be $110_{-40}^{+335}~\mathrm{keV}$ using the BAT $E_\mathrm{peak}$ estimator (Sakamoto et al. 2009) The fluence in the $15-150~\mathrm{keV}$ band is $(9.0 \pm 1.2) \times 10^{-6}$ erg cm$^{-2}$. The BAT 1-s peak photon flux is $2.5 \pm 0.2$ ph cm $^{-2}$ s$^{-1}$ in the $15-150~\mathrm{keV}$ band. This corresponds to a peak energy flux of $(2.8 \pm 0.2) \times 10^{-7}$ erg cm$^{-2}$ s$^{-1}$ ($15-150~\mathrm{keV}$). This 1-s peak flux is measured from $T_0$(BAT) $-$0.06 s. [*Swift*]{} began to slew to repoint the sources with the XRT and UVOT 18 seconds after the trigger, when the prompt emission had not yet ended. The prompt emission of GRB100814A was also detected by [*Konus-Wind*]{} [@gol10], [*Fermi*]{} [@vok10], and [*Suzaku/WAM*]{} [@nis10]. As observed by [*Konus*]{}, the event had a duration of $\sim150$ seconds and fluence of $(1.2\pm0.2) \times 10^{-5}$ erg cm$^{-2}$ in the 0.02-2 MeV band (90% C.L.). The spectrum is best fitted by a power law plus exponential cut off model. The best fit parameters are a low-energy photon index $\Gamma_{1}=0.4 \pm 0.2$, and a cut off energy $E_{\mathrm{p}}=128\pm 12~\mathrm{keV}$. The value of this parameter is similar to $E_{\mathrm{peak}}$ drawn from BAT data. Assuming a redshift of $z=1.44$ [@ome10] and an isotropic emission, this corresponds to a $\gamma$-ray energy release of $\simeq 7\times10^{52}$ erg between 1 and 10000 $\mathrm{keV}$ in the cosmological rest frame of the burst. We derived this value using the $k$-correction of Bloom et al. (2001). X-ray data ---------- XRT initially found an uncatalogued bright X-ray source $48"$ from the BAT position. The ground-processed coordinates are R.A. (J2000) = $01^{\rm h} 29^{\rm m} 53.54^{\rm s}$, Dec.(J2000) = -17$^{\circ}$ 59’ 42.1” with an uncertainty of 15 (90 % C.L.). This source subsequently faded, indicating that it was the X-ray counterpart of GRB100814A. Windowed Timing (WT) mode data (with ms time resolution but only 1-D spatial information) were gathered up to $600$ s after the trigger, after which the data were gathered in Photon Counting (PC) mode (with 2.5-s time resolution and 2-D spatial information). For both the spectral and temporal analysis, we considered counts within the $0.3-10~\mathrm{keV}$ band. For the temporal analysis, we used the automated XRT GRB light curve analysis tools of Evans et al. (2009, 2007). At late times, we noticed the presence of a nearby source $11"$ away from the GRB position, contributing a count-rate of $\sim 8\times 10^{-4} \thinspace {\rm counts \thinspace s^{-1}}$ (corresponding to a $0.3-10\thinspace {\rm keV}$ flux of $\sim 4.5 \, \times 10^{-14} {\rm erg \thinspace cm^{-2} \thinspace s^{-1}}$), which caused the light curve to flatten to a roughly constant level beyond $\sim 9\times 10^5$ s after the trigger. To mimimise the effect of this nearby source on the GRB light curve at late times (after $2\times 10^5 \thinspace {\rm s}$) we used a fixed position extraction region (to prevent the automatic analysis software centroiding on the non-GRB source location), with a reduced extraction radius (of 23 arcsec) and ignored the data beyond $9\times10^5$ s after the trigger. The count-rate light curve was converted to a flux density light curve at 10 keV following Evans et al. (2010), which accounts for spectral evolution as the burst decays. Fig. 2 shows the X-ray light curve of GRB100814A, as well as the UV/optical/IR and radio ones. At the beginning of the XRT light curve we clearly distinguish a sequence of flares, the last one peaking at $\sim 220$ s, followed by a steep decay with slope $\alpha=4.65\pm0.08$, which we interpret as the end of the prompt emission phase. Unfortunately observations made during the first orbit end at $\sim 750$ s, which limits our ability to better define this phase of the emission, although the last data points seem to show a flattening of the light curve. During the second orbit observations, starting at $\sim 3000$ s, the flux decays at a much slower rate. This phase seems to last until $\sim 10^5$ s, when the decay of the X-ray flux becomes much steeper. This second phase of steep decay ends at $\sim 9\times10^5$ s after the trigger, followed by a phase of roughly constant flux. This flux, however, is not due to the GRB afterglow, but to the unrelated source 10 arcseconds from the burst position. The presence of a break at late time is obvious: if we try to fit the $0.3-10~\mathrm{keV}$ light curve from the beginning of the second orbit to $9\times10^5$ s with a single power law we get an unsatisfactory result ($\chi^2_{\nu}=494.5/306$ degrees of freedom, d.o.f.), while the use of a broken power law ($A t ^{-\alpha1}$ for $t\leq t_{\rm b}$; $A t_{\rm b}^{\alpha_{2} - \alpha_{1}} t^{-\alpha2}$ for $t\geq t_{\rm b}$) gives a very significant improvement ($\chi^2 =183.8/304$ d.o.f.). In this case, the best fit parameters are: decay indices $\alpha_{1}=0.52\pm0.03$ and $\alpha_{2}=2.11^{+0.15} _{-0.13}$, $t_{\mathrm{break}} = 133.1^{+7.9} _{-6.4}$ ks. In both cases, we added a constant to the broken power law model to take into account the presence of the serendipitous source. We also tried to fit the light curve with a smoothly joined broken power law model [@beu99], which enables us to examine different “sharpness” of the X-ray light curve break. We have found that the data do not discriminate between a smooth and a sharp transition. If all parameters are allowed to vary, a model with a sharp break ($n=10$ in the Beuermann et al. 1999 formula) produces a marginally better fit. We also note that the possible “dip" at $\sim 6\times10^4$ s is not statistically significant. We fitted the PC spectral data from the second orbit up to $9\times10^5$ s after the trigger with an absorbed power law model, by accounting separately for the Galactic and intrinsic absorption columns (the latter at $z=1.44$). The Galactic column density was fixed at $1.75\times 10^{20}$ cm$^{-2}$ [@kal05]. The fit is statistically satisfactory ($\chi^2 _{\nu} = 182.3/218$ d.o.f.). The best fit value for the column density of the extragalactic absorber is N$_{\mathrm{H}}= 1.18 ^{+0.29} _{-0.28} \times 10^{21}$ cm$^{-2}$, which is significantly different from zero, and the energy index of the power law is $\beta_\mathrm{X}=0.93\pm0.03$. We find no evidence for spectral evolution: parameters consistent with those given above are obtained when fitting the spectra taken before and after the $133$ ks break. When compared to other X-ray afterglows detected by [*Swift*]{}, the X-ray afterglow of GRB100814A has an average flux around $10^4$ s. However, the long X-ray plateau makes GRB100814A move to the bright end of the flux distribution at $\sim0.5$ day after the trigger in the cosmological rest frame (Fig. \[ag\_lc.eps\]). The $0.3-10~\mathrm{keV}$ X-ray flux normalized at 11 hr after the burst is $\simeq 10^{-11}$ erg cm$^{-2}$ s$^{-1}$ in the observer’s frame. UVOT and ground optical observatories data. {#opticaldata} ------------------------------------------- [*Swift*]{}/UVOT observations started 77 s after the trigger, with a 11 s exposure taken in the [*v*]{} band while the spacecraft was still slewing. A grism exposure followed, from which we derive a [*b*]{} magnitude [@kuin2015]. The first settled imaging exposure, in the [*u*]{} filter, started 153 s after the trigger and lasted 250 s; this exposure was obtained in event mode so that the position and arrival time of each photon was recorded. Immediately afterwards, UVOT took a sequence of 20 s exposures, cycling through its colour filters. After the first orbit of [*Swift*]{} observations finished at $\sim$ 700s UVOT switched to longer cadence observations, including the [*white*]{} filter in its sequence. GRB100814A was observed with 10 different ground-based optical telescopes (see Table 1) in a range of photometric bands. To minimise systematics between the different observatories and bands, where possible the same stars in the field surrounding GRB100814A were used as secondary standards for the different photometric bands and instruments. There are some practical limitations to this approach: the fields of view of some instruments are smaller than the basic set of secondary standards and the sensitivities of the instruments are limited to different brightness ranges. That means that usually a subset of the calibration stars was used for a particular instrument, and sometimes additional calibration stars were used to complement the common set. The secondary standards were calibrated in [*B*]{} and [*V*]{} using the UVOT [*b*]{} and [*v*]{} observations and the transformation equations provided by @Poole. The secondary standards were calibrated in $R$, $r'$ and $i'$ using the CQUEAN observations (see below), using the transformations from @jordi06 to obtain $R$ magnitudes. The $r'$ and $i'$ magnitudes of the secondary standards were verified using the 1-m Lulin Optical telescope observations (see below). The photometric errors which were assigned to the data include both the random error and the systematic error from the calibration of the secondary standards. The GRB was observed with the CQUEAN instrument [@park12; @kim11] mounted on the McDonald 2.1m Otto Struve telescope for five nights. During that time observations of two SDSS photometric standards, BD+17 4708 and SA113-260 [@JSmith] were obtained, and used to calibrate both the GRB photometry and the surrounding field stars down to the 22nd magnitude. The GRB was also followed with the 1-m Lulin Optical Telescope (LOT; Kinoshita et al. 2005, Huang et al. 2005). The secondary standards in the LOT field were calibrated independently of the CQUEAN observations, using LOT observations of four SDSS fields at a range of airmass on Sept 14, 2010. The magnitudes of the secondary standards were cross checked with the corresponding CQUEAN magnitudes and were found to be consistent within the errors. Observations with the Robotic Optical Transient Search [ROTSE; @Akerlof] IIIc site, located at the H.E.S.S. site at Mt Gamsberg, Namibia, were obtained starting 290 s after the trigger time. Unfortunately, the light of a nearby variable star (position R.A.(J2000)=$1^{\rm h} 29^{\rm m} 53.978^{\rm s}$, Dec.(J2000)=-17$^{\circ}$ 59’ 35.5", USNO R2=19.58 mag) contaminated the observations past 1000 s. We removed the star which caused problems by using image subtraction, to confirm the data prior to 1000 s are valid and uncontaminated. Although the observations were taken without an optical filter, the peak response is in the R band, and the ROTSE data were calibrated against the USNO-B1 [*R2*]{} magnitudes of 29 sources within $10'$ of the transient. We obtained late-time GRB observations with the Scorpio instrument [@scorpio] mounted on the Russian BTA 6-m telescope which were calibrated using the CQUEAN secondary standards. Observations from the Liverpool Telescope [@LT LT] and Faulkes Telescope North (FTN) were calibrated using a subset of the CQUEAN secondary standards which are within the LT and FTN fields of view. GRB100814A was also observed in the $R$ band with the 1.23-m Calar Alto Astronomical Observatory (CAHA), the IAC-80 Telescope of the Observatory del Teide, Tenerife, and the Gran Telescopio Canarias (GTC), La Palma. In all three cases the data were calibrated using the CQUEAN secondary standards. The GRB was also observed in $R$, $V$, and $B$ bands with the Northern Optical Telescope (NOT) in La Palma. The images were calibrated using the CQUEAN and UVOT secondary standards. The resulting optical light curves are shown in Fig \[all\_LCs\]. In order to improve our understanding of the behaviour of the optical light curve and check for the presence of chromatic evolution of the emission, we followed two approaches. In the first approach we normalised all of the light curves to a single filter, and in the second approach we analysed the light curves in different bands separately. The first approach, as described in @oat09, consists of renormalizing the light curves to a single filter. The very early optical light curve varies rapidly. An optical flare peaks at $\sim 180$ s and then rapidly decays, basically giving no contribution after $\simeq 375$ s. After this early flare, we have a phase in which the optical flux is roughly constant, followed by a decay starting at $\sim 1000$ s in all filters. To investigate the early plateau and the following decay more throughly, we have renormalized the early data to the [[*Swift*]{}]{} $u$ band filter. We then fitted the $375-11000$ s data points with a smooth broken power law, and the best fit parameters are $\alpha_{\mathrm{opt,2}}=0.03^{+0.16} _{-0.20}$, break time $t_{\mathrm{break}}=856^{+260} _{-190}$ s, $\alpha_{\mathrm{opt},3}=0.72\pm0.06$, with $\chi^2/\mathrm{d.o.f} = 52.5 / 33$. We show in Fig. \[fig:early\_LCs\] the renormalized early optical light curves. The initial optical flare may be produced by the same process responsible for the early flaring activity in the X-ray, since the temporal behaviour is roughly similar. Flares are likely produced by internal dissipation mechanisms, such as internal shocks, which occur when the ultrarelavitistic ejecta shells interact with each other. It is possible that the plateau we see between $375$ and $1000$ s is due to a decreasing emission from internal dissipation and rising emission from the external shock. Alternatively, the plateau might be due to a slow rise of the external shock emission only. We note that an initial plateau or shallow decay phase are associated to the external shock onset, as observed in several $\it{Swift}$ bursts [@oat09]. The origin of external shock emission is different from that due to internal dissipation mechanism. External shocks are produced by the interaction between ejecta and the circumburst medium and are likely to produce the long-lived and slowly varying afterglow emission. After $t_{\mathrm{break}}$, the optical flux follows the typical power law decay of GRB afterglows; the deceleration of the leading shell of the ejecta must have occurred at this time or earlier. In the following, we will assume that break time $t_{\mathrm{break}}$ marks the deceleration time, and will investigate the GRB afterglow from this time onwards. We will return to how the results of this article are affected if the actual deceleration is slightly earlier. Since the study of internal dissipation mechanisms is not the goal of this paper, we will not discuss the initial optical flare any further. Unfortunately, the sampling of our optical light curve between 10 and 20 ks is not good enough to ascertain precisely when the optical emission stops decaying and begins to rise. What we can say is that the rebrightening approximately started about 15 ks and culminated about 100 ks after the trigger, although there is no strong variation in flux between 50 and 200 ks, during which the light curves seem to form a plateau. Between 15 ks and 200 ks, during the optical rebrightening, the light curves do not seem to align. This might indicate that a process of chromatic evolution is taking place during the afterglow of GRB100814A. In more detail, the rebrightening appears to be bluer than the other portions of the light curves. These results indicate that the optical spectrum during the rebrightening is different from that found before and after the rebrightening. This feature reinforces the idea that the rebrightening is due to different emission components. It is also possible that there is a chromatic evolution [*during*]{} the rebrightening itself. In fact, if we renormalize the light curves during the rise between 15 and 60 ks, the light curves during the plateau do not match one another, with the data points of the redder filters being systematically above those of the bluer filter. Conversely, the light curves of the rise do not match one another if we renormalize them in the interval between 50 and 200 ks. However, the plateau phase, although being redder than the rise one, still shows a spectrum which is bluer than that of the following fast decay. In summary, it is possible that the rebrightening spectrum gets redder with time. This trend is found in other GRBs, such as GRB120404A (Guidorzi et al. 2014). We renormalized the late optical light curves to the $i'$ band, since we have a good coverage in this filter in late observations. This technique was applied to data points between $250$ ks, when the fast decay has clearly started, and $10^6$ s. A fit with a power law model yields an acceptable result: $\chi^2$/d.o.f. = 82.4/53. It provides a best fit decay slope of $\alpha = 2.00 \pm 0.07$. After $10^6$ s, the optical emission was very weak and difficult to constrain. At the time, contamination from the constant flux of the host galaxy may also be possible. This has been accounted for in the fit of the late decay by adding a constant in the model. We note that Nardini et al. (2014) found a late decay slope of $\alpha = 2.25 \pm 0.08$ between 200 ks and $10^6$ s, using GROND data (taken in $g'r'i'z'$ and $J$, $H$ and $K$ bands) and including the contribution of the host galaxy; such a value is consistent with our best fit value above. By means of observations of the Calar Alto 3.5-m telescope, 3 years after the event, we determined that the host galaxy of GRB100814 shows a magnitude $J=22.32 \pm0.32$ (Vega; error including calibration uncertainties). In order to obtain a clearer insight into the event, we studied the single light curves during the rebrightening (from 15 ks onwards) as well. Based on the densest sampled light curves, we find that the late-time evolution is characterised by two breaks. All UV and optical light curves are fit with a smoothly double broken power law [@Liang2008a; @Schulze2011a], using Simplex and Levenberg-Marquardt algorithms [@Press2002a]. The uncertainties in the data were used as weights. First, the parameters for each light curve were set to be identical, except for the normalisation constants. The quality of the fit is bad ($\chi^2 _{\rm red}$ = 5.66 for 148 d.o.f.); furthermore the residuals in the first two power law segments are not randomly distributed and show a trend with wavelength. The residuals around the second break and in the third power law segment are small and randomly distributed around the fit, implying that the evolution during the first power law segment is either chromatic or that the model used is just not good enough, and that the evolution after the second break is achromatic. Next, we allowed the decay slope $\alpha_1$ and the break time $t_\mathrm{b,\,1}$ to vary for each band independently. Not only did the fit statistics significantly improve ($\chi^2 _{\rm red}$/d.o.f. = 2.96/126), but also the amplitude of the residuals decreased substantially. We summarise the fit parameters in Table \[tab:ag\_fits\]. The behaviour in the first power law segment is strongly frequency dependent, since the peak time[^1] $t_\mathrm{peak}$, the peak flux density $F_{\nu,\,\rm p}$ are not the same for different frequencies $\nu$. To estimate the uncertainty, we only considered the error in the break time $t_{\rm{b},\,1}$. The uncertainties of the first power law segment are for most of our data sets too large to detect any trend. Estimating the correlation and linear regression coefficients is not trivial, because the uncertainties in all parameters ($\alpha_1$, $t_{{\rm peak}}$, $F_{\nu,\,p}$, $\nu$) are not small. Owing to this, we applied a Monte Carlo technique [@Varian2005a]. In this method, every data point is represented by a 2D Gaussian, where the centre of peaks in each dimension are the parameter estimates, and the corresponding $1\sigma$ errors are the width of the distributions. From these, we construct 10,000 resamples of the observed data sets, each of which is obtained by a random sampling with replacement from the original data set. For each of these data sets we compute the linear regression and correlation coefficients. The results are shown in Fig. \[fig:ag\_correlation\] and Table \[tab:ag\_correlations\]. From a statistical point of view, we do not find clear correlations. The most significant one is between $F_{\nu,\,\mathrm{p}}$ and $\nu$ with a correlation coefficient of -0.81 (Table \[tab:ag\_correlations\]), but even this correlation has significance of only $\simeq2\sigma$. The correlations between $F_{\nu,\,\mathrm{p}}$ and $t_{\rm peak}$, and $\nu$ and $t_{\rm peak}$ are not tight probably due to the large uncertainties in the break time. It is perhaps more correct to speak of ‘trends’ rather than correlations but, thanks to some small error bars of the parameters we have derived, we can still safely state that the light curves in redder filters have higher peak fluxes and later peak times than those in the bluer filters. Any theoretical interpretation should explain this chromatic behaviour of the optical rebrightening. We note that the second break time, when the optical flux starts to decay fast, is not consistent with the X-ray late break time, although the decay slopes are consistent.\ Radio data {#radio} ---------- GRB100814A was observed with the Expanded Very Large Array (EVLA) in wide C-band receiver with frequency at 4.5 and 7.9 GHz bands. The observations started on 2010 August 18 at 09:07 UT, $364.6$ ks after the burst. Ten epochs were taken in total, with the last being 744 days after the trigger. The first 4 epochs of observations were in EVLA C configuration, whereas the fifth epoch of observations was in hybrid DnC configuration. The sixth and seventh epochs of observations were made in EVLA lowest resolution D configuration mode. The flux density scale was tied to the extragalactic source 3C48 (J0137+331), whereas J0132-169 was used as flux calibrator. The observations were made for 1 hour at each epoch, including the calibrators. The data were analysed using standard AIPS routines. The GRB was detected at all the first 6 epochs. At the seventh epoch on 2010 Nov 21 (about 8700 ks after the trigger), the radio afterglow was detected at 7.9 GHz, but it was not detected at 4.5 GHz. The afterglow was not detected in either band in the remaining epochs. The peak flux was observed 11.32 days after the GRB. The peak flux densities were $582 \pm33$ $\mu$Jy and $534 \pm 27$ $\mu$Jy in the 4.5 and 7.9 GHz bands, respectively. The light curves in these two bands (visible in Fig. 2) show nearly simultaneous peaks, and their evolution afterwards looks similar, but the slopes before the peak different. We fitted both light curves with a smooth broken power law model, and we found the following best fit parameters: $\alpha_{4.5,1}= -1.27 ^{+0.20} _{-0.24}$, $t_{4.5,\mathrm{peak}} = 955.5^{+61.9} _{-56.0}$ ks, $\alpha_{4.5,2} = 0.89 ^{+0.11} _{-0.10}$; $\alpha_{7.9,1}= -0.19^{+0.12} _{-0.13}$, $t_{7.9, \mathrm{peak}} = 984.2^{+144.0} _{-116.2}$ ks, $\alpha_{7.9,2} = 0.77 ^{+0.09} _{-0.08}$. The two rise slopes are inconsistent at $\simeq5\sigma$, but the decay slopes are basically identical. Spectral Energy Distributions at several epochs {#SEDs} =============================================== To constrain the spectral indices of the optical and X-ray emission, we built and fitted the spectral energy distributions (hereafter SEDs) of the X-ray and optical emission. We chose the epochs of 500 s, 4.5 ks, 22 ks, 50 ks, and 400 ks. The methods used to construct the SEDs are described in Schady et al. (2007). For the optical parts of the SEDs, the UVOT photometry has been supplemented with ground based photometry when available. For data taken in the $g'$, $r'$, $i'$ and $z'$ bands, response functions have been taken from @fukugita96. The $R$ band data which have been used in the SEDs come from the IAC 80 telescope, and so for these data the response function was based on the IAC 80 $R$ filter and CCD response[^2]. We tried three fitting models, based on power law functions since the emission is synchrotron radiation. In the first one, the X-ray and optical were on the same power law segment. The second model is a broken power law. The third model is a broken power law with the difference between the spectral indices fixed to 0.5, as predicted in the case of a synchrotron emission cooling break. In all fitting models, we added two components of absorption. The first component is due to our Galaxy and fixed at the value given by the Leiden/Argentine/Bonn Survey, $N_H = 1.8\times10^{20}\mathrm{cm}^{-2}$. The second component represents the extragalactic absorption, with the redshift fixed at $z=1.44$. Similarly, we added three components for the extinction. The first component represents the Galactic extinction, fixed at the value given by Schlegel et al. (1998), $E(B-V) =0.02$ mag. The second component represents the extinction in the environment of the burst at redshift z=1.44; we chose the Small Magellanic Cloud extinction law, since it usually fits the extinction properties of the medium of GRB host galaxies (Schady et al. 2010). The third component is UV/optical attenuation by the intergalactic medium (Madau 1995). Since we do not detect any significant change in the X-ray spectrum from $\sim3000$ s to $\sim10^6$ s, we can assume that the X-ray spectral index is always the fairly constrained value determined using the whole dataset. Therefore, in all fits the spectral slope of the segment encompassing the X-ray band is forced between 0.84 and 1.02, i.e. within the best value of the fit of the X-ray data alone plus or minus 3$\sigma$. Given this constraint, no fits produced with a simple power law provide a statistically acceptable fit, with the exception of the 400 ks SED, and we do not consider them in the analysis below. The 500 s SED does not enable us to constrain fit results well, and we do not use it in our discussion. In the case of the 22 and 50 ks SEDs, we have also tried to fit the data with a model which is the sum of two broken power laws. This tested the possibility that two distinct components produce the optical and the X-ray flux and, given the chromatic behaviour of the optical afterglow, that the synchrotron peak frequency $\nu_{\mathrm{M}}$ is within or close to the optical band at these epochs. Thus, the low energy segment of the component producing the optical flux has been frozen to $\beta = -1/3$, while the component producing the X-ray flux has a break with differences in spectral slopes fixed to 0.5, as predicted by the external shock models (see Section 4). In the case of the $50$ ks SED, the sum of 2 broken power law models yields a slightly better fit than the model with a single broken power law and difference between the spectral indices fixed to 0.5: $\chi^2=111.6/112~\mathrm{d.o.f}$ versus $123.4 / 115~\mathrm{d.o.f}$. The best-fit break of the first component is $4.1^{+0.5} _{-0.6}$ eV. In the case of the $22$ ks SED, the fit becomes indistinguishable from a single broken power law model. We calculated the probability $P$ that the improvement in the fit of the 50 ks SED is given by chance by means of the F-test. We find that $P \simeq 1.1\times10^{-2}$. We tested these results by repeating the fit of the 50 ks SED with two broken power laws assuming Milky Way (MW) and Large Magellanic Cloud (LMC) extinction laws. In the case of the MW extinction law, the break of the first component is at $3.2^{+0.8} _{-0.4}$ eV, while the break is $4.8\pm1.0$ eV adopting a LMC extinction law. The two fits yield $\chi^2$ /d.o.f. = $106.9/112$ and $\chi^2$/d.o.f. = $107.9/112$. Fitting the 50 ks SED with a single broken power law model and difference between the spectral indices fixed to 0.5 with MW and LMC extinction laws yields $\chi^2$/d.o.f. = $113.1/115$ and $ \chi^2$/d.o.f. = 120.6/115, respectively.\ Thus the results do not depend sensitively on the choice of extinction law. In conclusion, broken power law and two-broken power law models are perfectly acceptable for the 50 ks SED, but the model with two broken power law components is preferred by the data; one of the two breaks is found in or near the optical band. Results are summarised in Tab. \[tab\_SEDs\] and shown in Figure \[fig:SEDs\]. The plot indicates changes in the spectral shape: while the 4.5 ks and the 400 ks SEDs show a normally steep optical spectrum, the 50 ks SED seems to have a flat optical emission. Furthermore, the 50 ks SED shows a steep optical-to-X index, which indicates that an additional optical component is needed with respect to other SEDs. Discussion ========== The most remarkable property of GRB100814A is the broad optical peak which started roughly 15 ks after the trigger and ended at about 200 ks, followed by a steep decay with a rate similar to that observed in the X-ray band at the same time. The rebrightening is chromatic, since throughout it the X-ray light curve keeps decaying at the same rate as it did before and shows no obvious counterpart of the rebrightening. When fitting the SED built at the peak of the rebrightening, we find a break frequency in the optical band. We also find that the peak time and maximum flux evolve with the frequency. Later on the optical flux starts decaying faster, and roughly at the same time the X-ray flux began to decay with approximately the same temporal slope. This however leads to critical questions regarding the sources of the emission in GRB100814A: if the X-ray and the optical fluxes are due to the same component, why do they behave so differently with the optical showing a rebrightening? And if the optical rebrightening is due to a different component, why does it end at about the time of the steep break in the X-ray?\ Single component FS model {#single} ------------------------- In GRB100814A both the X-ray flux and optical light curves initially show a shallow decay. Slow early decay has been seen commonly in GRB afterglows (Liang et al. 2007), both in the X-ray and in the optical. Its origin is still a matter of debate. One of the most popular explanations is a phase of energy injection into the ejecta, which may be due to Poynting flux emitted by the burst central engine or trailing shells of outflow that collide with the leading parts of it (Zhang et al. 2006). The steep, late decay observed in both the X-ray and in the optical bands at the late epoch could only be attributed to a jet phase in the context of the FS model. One can immediately check whether the standard FS model can explain the observed behaviour. The spectral and temporal indices of the flux of the observed bands are predicted by this model to be linked in relations which depend on the positions of the synchrotron self-absorption frequency $\nu_{\mathrm{SA}}$, the peak frequency $\nu_{\mathrm{M}}$ and cooling frequency $\nu_{\mathrm{C}}$ and the kind of expansion - collimated (jet) or spherical - and on the density profile of the surrounding medium, either constant (like in the interstellar medium, ISM) or decreasing with radius (like a stellar wind) (Sari, Piran & Narayan 1998; Sari Piran & Halpern 1999; Chevalier & Li 2000; Kobayashi & Zhang 2003a). The only ways to account for the rise of the optical light curves are to assume a transit of $\nu_{\mathrm{M}}$ throughout the optical band, or the onset of the FS emission. The former would also explain the chromatic nature of the event. We note that we can fit the X-ray light curve as the sum of two components: one rapidly decaying, likely connected with the prompt emission, and a rising component that peaks at $\simeq 900$ s, and successively produces the slow decay observed. If we assumed that this time were the peak time and the X-ray frequency $\nu_{\mathrm{X}} = 4.2\times10^{17}$ Hz ($1.73~\mathrm{keV}$) were the peak frequency, we would find that even the X-ray is consistent with the extrapolation of the relation between these two quantities from the optical band (bottom-left panel of Fig. 5). The X-ray peak would be shifted at much earlier time due to its higher frequency, but the X-ray and the optical would obey the same trends and be produced by the same component. However, if $\nu_{\mathrm{M}}$ were approaching the optical band, one should observe a flux rise from the beginning of observations in the ISM case or a decrease as $t^{-1/4}$ decay slope for stellar wind (with a density profile of $r^{-2}$, where $r$ is the distance from the progenitor, Kobayashi & Zhang 2003a). Neither of which are observed. Furthermore, to keep $\nu_\mathrm{M,\mathrm{FS}}$ in the optical band with a flat spectrum, one would require an extremely high value of kinetic energy of the ejecta (see Sect.s \[chromatic\] and \[RS-FS1\]). The optical bump cannot even be the onset of FS emission in the context of single component scenario, because one should not see the observed decrease of the X-ray and optical flux before it. The observed flux depends on parameters such as the fractions of blast wave energy given to radiating electrons and magnetic field $\epsilon_e$ and $\epsilon_B$, the circumburst medium density $n$, and the index of the power law energy distribution of radiating electrons $p$. A temporal evolution of such parameters might explain the observed behaviour. An example is a change of density of the environment $n$. For frequencies below the cooling break, the flux is proportional to $n$, while the flux in bands above the break does not depend on it. It is therefore possible that a rapid increase in $n$ causes an optical rebrightening and simultaneously leaves the X-ray flux decay unperturbed, as we observe. Does this explanation predict the spectral changes that we see in the GRB100814A rebrightening? Since $\nu_{\mathrm C} \sim n^{-1}$, one may think that $n$ could increase so much that $\nu_{\mathrm C}$ enters the optical band and changes the shape of the SED. However, several simulations have shown that the light curves do not show prominent rebrightening even if the blast-wave encounters an enhancement of density (Nakar & Granot 2007, Gat et al. 2013).\ We therefore conclude that a single component FS model cannot explain the GRB100814A observed behaviour. In the next section, we discuss a few multi-component models to interpret the behaviour of the afterglow of this burst. Two-component jet seen sideways ------------------------------- In this model, the prompt emission, the early optical and X-ray afterglow emission is produced by a wide outflow, while the late optical rebrightening is due to emission from a narrow jet seen off-axis. The emission from the latter is initially beamed away from the observer, however as the Lorentz factor decreases, more and more flux enters the line of sight. Such a scenario has been already invoked [@gra05] to explain late optical rebrightening features, so in principle it could explain the behaviour of GRB100814A. We note that Granot et al. (2005) interpret X-ray rich GRBs and X-ray flashes, which are events with peak energy of the prompt emission in the 10-100 and 1-10 keV ranges respectively, as GRBs seen off-axis. GRB100814A does not belong to such categories, having a peak energy above 100 keV. However, the shallow decay and the rebrightening feature of its afterglow may still be interpreted in the off-axis scenario. We shall now determine in more detail whether this scenario is plausible.\ ### Narrow jet {#Narrow} A relativistic jet initially observed off-axis will naturally produce a rising light curve; the exact slope depends on the ratio between the off-axis angle and the opening angle. Looking at the synthetic light curves created by the code in “afterglow library" of @van10 we notice that a jet seen at $\theta_\mathrm{obs} \sim 3\theta_j$ produces a rise with slope $\alpha \simeq -0.65$, and an initial decay with slope $\alpha \simeq 0.45$, which are similar to those we observe at the optical rebrightening (see also Granot et al. 2005). In this context, the peak luminsity observed at $\theta_\mathrm{obs}$ is related to that on axis by the formula $$L_{\theta_\mathrm{obs}, \mathrm{peak}} \simeq 2^{-\beta-3} (\theta_\mathrm{obs} / \theta_j - 1)^{-2\alpha} L_{0, t_{j}}$$ (Granot, Panaitescu, Kumar & Woosley 2002, hereafter GP2002), where $\theta_j$ is the opening angle and $t_j$ is the jet break time for an on-axis observer. For $\beta = 0.5$ and $\alpha = 2$, which are the typical values of these parameters, we have that $L_{\theta_\mathrm{obs},\mathrm{peak}} = 5.56\times10^{-2} L_{0, t_{j}}$. The peak time will be at $$T_\mathrm{peak} = [ 5 + 2\ln(\theta_{\mathrm{obs}}/\theta_j - 1)] (\theta_\mathrm{obs}/\theta_j - 1)^2 t_{j}~\mathrm{s}$$ for the values above, we have $T_\mathrm{peak} \simeq 25 \times t_{j}$. Since $T_\mathrm{peak} \simeq 90$ ks, $t_{j} \simeq 3.6$ ks. Now, defining $a \equiv (1+\Gamma^2 \theta^2)^{-1}$, we have (GP2002) $$\label{a} \nu(\theta_\mathrm{obs}) = a \nu (\theta=0) \\ ;\\ F_{\nu} (\nu,\theta_\mathrm{obs},t) = a^3 F_{\nu} (\nu/a,0,at)$$ where $\Gamma$ is the Lorentz factor. At the peak time we have $\Gamma^{-1} \sim \theta_\mathrm{obs} - \theta_{j} = 2 \theta_j$. By assuming $\theta$ is $\theta_\mathrm{obs} - \theta_{j}$, as GP2002 suggest, we have $a=0.5$ in the equations above. The peak frequency for $\theta_\mathrm{obs} = 0$ is given by\ $$\nu_\mathrm{M} = 3.3\times10^{14} (z+1)^{1/2} \epsilon_{B,-2}^{1/2} \left(\frac{p-2}{p-1}\right)^{2} \epsilon_e ^2 E_{K,52}^{1/2} t_\mathrm{d} ^{-3/2} \mathrm{Hz} \\ \label{nu_m}$$ where $t_\mathrm{d}$ indicates time in days. The maximum flux is\ $$F_{\nu} (\nu_\mathrm{M}) = 1600 (z+1) D_{28}^{-2} \epsilon_{B,-2} ^{1/2} E_{K,52} n^{1/2} (t/t_{j})^{-3/4}~\mathrm{\mu Jy} \\ \label{flux}$$ (Yost et al. 2003). $E_{K,52}$ is the kinetic energy of the ejecta, while $\epsilon_e$ and $\epsilon_{B,-2}$ are the fractions of shockwave energy given to radiating electrons and magnetic field respectively. $D_{28}$ is the luminosity distance of the burst, while $p$ is the index of the power law energy distribution of radiating electrons, $n$ the density in particles $\mathrm{cm}^{-3}$ of the circumburst medium. Subindices indicate normalized quantities, $Q_{x} = Q / 10^{x}$ in cgs units. Substituting the known parameters, taking $p=2.02$ to explain the flat X-ray spectrum, and remembering that for $\theta_\mathrm{obs} = 3\theta_j$ the observed $\nu_\mathrm{M}$ will be 1/2 of the $\nu_\mathrm{M}$ on-axis (see Eq. \[a\]), we have $$\label{power law} F_{\nu} (\nu_\mathrm{i},\theta_\mathrm{obs},t_\mathrm{peak}) = 0.17 E_{K,52} ^{1.27} \epsilon_{B,-2}^{0.77} \epsilon_{e} ^{1.02} n^{1/2}~\mathrm{\mu Jy}$$ where $\nu_\mathrm{i}$ is the flux in the $i'$ band ($3.9\times10^{14}$ Hz). At the peak of the rebrightening, we have $F_{\nu} \simeq200$ $\mu$Jy. Thus, we have the condition $$E_{K,52}^{1.27} \epsilon_{B,-2} ^{0.77} \epsilon_{e} ^{1.02} n^{1/2} \simeq 1200 \label{n}$$ ### Wide jet {#Wide} An off-axis model cannot explain the early shallow decay if the observer has $\theta_\mathrm{obs} < \theta_j$; the observer must be slightly outside the opening angle of the outflow (i.e., $\theta_\mathrm{obs}$ a bit larger than $\theta_j$). The time when the afterglow emission begins its typical power law decay, $t\simeq 860$ s, can be taken as the epoch when $\Gamma^{-1} \sim \theta_\mathrm{obs} - \theta_j$. The following decay, with $\alpha \simeq 0.6$, can be explained if $\theta_\mathrm{obs} \simeq 3/2 \theta_{j}$ (Van Eerten et al. 2010). Finally, a steeper decay will be visible when the observer will see the radiation from the far edge of the jet, when $\Gamma^{-1} \sim \theta_\mathrm{obs} - \theta_j + 2 \theta_j = 5/2~\theta_j$. Assuming that $\Gamma \propto t^{-3/8}$, this second break would be seen at $t_{2} \simeq 5^{8/3} \times 0.86 \simeq63$ ks. However, at this epoch the afterglow is dominated by the narrow jet emission. It is important though that $t_2$ occurs before the end of the rebrightening, otherwise this model would predict a return to shallow decay once the rebrightening were over. From Van Eerten et al. (2010), the brightness of an afterglow seen at 1.5 $\theta_j$ is $\sim 1/10$ of the brightness it would have if seen on-axis, in a given band. At 4500s, the $R$-band flux is $\simeq 100$ $\mu$Jy. If we assume $p=2.02$, we have $$E_{K,52} ^{1.27} \epsilon_e ^{1.02} \epsilon_{B,-2} ^{0.77} n^{1/2} \simeq 13.3\\ \label{w}$$ If we assume typical values $\epsilon_B=0.1$, $\epsilon_e=1/3$ and $n=10$ for both the narrow and wide jet, we obtain that the isotropic energetics of the narrow and the wide jet are $6.5\times10^{53}$ and $1.9\times10^{52}$ erg respectively. As for the half-opening angles of the outflow, a jet break at $\approx3.6$ks for the narrow jet would imply (Sari, Piran & Halpern 1999) $\theta\simeq0.027$ rad. The opening angle of the wide jet is 2/3 as much as the observing angle, while the opening angle of the narrow jet is 1/3 as much; thus the wide jet opening angle will be twice that of the narrow jet. The beaming-corrected energies are $2.3\times10^{50}$ and $2.7\times10^{49}$ erg respectively. These values of the parameters are not unusual for GRB modeling. In our model, the observed prompt $\gamma$-ray emission is dominated by the wide jet, since its edge is closer to the observer. To compute the prompt energy release in $\gamma$-rays that we would measure if we were within the opening angle of the wide jet, we can still use Eq. \[a\]. However, we must consider that, during the prompt emission, $\Gamma$ is much higher than during the afterglow emission; opacity arguments (Mészáros 2006) and measurements (Oates et al. 2009) indicate that initially $\Gamma \gtsim 100$. Assuming $\Gamma = 100$, one obtains $a\simeq0.12$. Granot et al. (2005), in their note 6, suggest that for $\Gamma^{-1} < (\theta_\mathrm{obs} - \theta_\mathrm{j} ) < \theta_\mathrm{j}$, the fluence roughly scales as $a^{2}$. Thus, an observer within the opening angle of the wide jet would detect a fluence $0.12^{-2} \times 1.2 \times10^{-5} = 8.25 \times 10^{-4}$ erg cm$^{-2}$. The corresponding energy emitted in $\gamma$-rays would be $E_\mathrm{iso} \simeq 5 \times 10^{54}$ erg. These values would already be very high. We know from our previous modeling, which takes into account the off-axis position of the observer, that the kinetic energy of the wide jet is $1.9\times10^{52}$  erg. Thus, the efficiency in converting the initial jet energy into $\gamma$-ray photons would be $\eta = E_\mathrm{iso} / (E_\mathrm{iso} + E_{K,52}) \simeq 99\%$. This inferred extreme efficiency is rather difficult to explain for all models of prompt emission, and it constitutes a problem for the off-axis model. We note, however, that the strong decrease of the observed fluence with off-axis angle may come from the assumption of a sharp-edge jet. For a structured jet with an energy and Lorentz factor profile, one may lessen the difficulty inferred above. Moreover, a lower efficiency would be derived if the kinetic energy of the outflow were higher than $1.9\times10^{52}$ erg; in turn a higher kinetic energy is possible assuming different values of the parameters $\epsilon_B$, $\epsilon_e$ and $n$. ### Chromatic behaviour {#chromatic} This modeling, however, does not yet take into account the presence of a spectral break during the rebrightening, which seems to cross the optical band from higher to lower frequencies. Such crossing may also explain the chromatic behaviour of the optical afterglow at the rebrightening. Taking into account equations (\[a\]) and (\[nu\_m\]), which give the value of $\nu_\mathrm{M}$ as observed on-axis and how its value is modified by observing the outflow off-axis, we find the condition $$E_{K,52} ^{1/2} \epsilon_{B,-2} ^{1/2} \epsilon_e ^{2} \simeq 4.2\times10^3$$ The high value for the right-hand is needed to have $\nu_\mathrm{M}$ in the optical range $\sim 10^5$ s after the trigger, even from a largely off-axis observer.\ Eq (\[n\]) has to be modified, because we are now assuming that at the rebrightening we are observing the peak flux $F_{\nu_\mathrm{M}}$. It becomes $$E_{K,52} \epsilon_{B,-2}^{1/2} n^{1/2} \simeq 28 \label{n2}$$ To satisfy these equations together, one would need the isotropic energy $E_{K,52} \sim 10^7$ and a value of density of $n \sim 10^{-14}$, both unphysical. As a further consequence of these extreme values for the energetics and densities, the Lorentz factor of the jets is also enormous. In fact, in order to be decelerated at $t_\mathrm{obs} \simeq 900$ s in such a thin medium, the initial Lorentz factor of the jet should be (Molinari et al. 2007) $\Gamma \sim 30000$. For these reasons, the model of the two-component jet seen sideways cannot be considered viable if, during the rebrightening, there is chromatic evolution due to the transit of $\nu_\mathrm{M}$. Reverse Shock and Forward Shock interplay ----------------------------------------- We now examine the possibility that some of emission of GRB100814A afterglow may be produced by the RS. We suppose that a process of energy injection, due to late shells piling up on the leading ones, lasts the whole duration of observations, producing a long-lived RS [@sm00; @zm01; @uhm07]. In such circumstances, the RS emission can be visible in the optical band and, under the right conditions, in the X-ray band as well. We explore two variants of this scenario. In the first, the early optical emission is RS, while the rebrightening and the X-ray emission is due to FS. In the second version, the RS generates the early optical and all the X-ray radiation we observe, while the the rebrightening is due to FS emission. ### Early optical from Reverse Shock, X-ray and optical rebrightening from Forward Shock {#RS-FS1} In this scenario, the break frequency determined by fitting the 50 ks SEDs is the synchrotron peak frequency $\nu_\mathrm{M,\mathrm{FS}}$ of the FS which is, initially, above the optical band. When $\nu_\mathrm{M,FS}$ approaches the optical band, the peak of the FS starts to dominate over the RS emission and produces the rebrightening and the chromatic behaviour we observe. After $\sim70$ ks, both X-ray and optical emissions are of the same origin, the FS.\ In the following, we shall be using the formulation of @sm00 (hereafter SM00) to predict the temporal evolution of the flux due to FS and RS. We assume that the circumburst medium density $n$ decreases with radius as $n \propto r^{{\it -g}}$,where $r$ is the radius reached by the shocks, while the mass $M$ of the late ejecta which pile up with the trailing shells obeys $M(>\Gamma) \propto \Gamma^{-s}$, where $\Gamma$ is the Lorentz factor of these late shells. This parameter, $s$, defines the energy injection into the ejecta (see also Zhang et al. 2006), which keeps the shocks (both reverse and forward) refreshed. The energy of the blast wave increases with time as $E \propto t^{1-q}$, where $q$ is linked to the parameter $s$ (Zhang et al. 2006). We note that SM00 take the approximation of a constant density throughout the shell crossed by the RS and do not take into account the $PdV$ (where $P$ stands for pressure and $dV$ the element of volume) work produced by the hot gas (Uhm 2011). Changes in the density and mechanical work should be taken into consideration in a more realistic scenario; we do that using numerical simulations (see below). However, this formulation enables us to use relatively easy closure relations that link the spectral and decay slopes to the parameter $s$ of energy injection and the density profile $g$ of the surrounding medium. At 4500 s, we assume $\nu_\mathrm{M,RS} < \nu_\mathrm{O} < \nu_\mathrm{C,RS}$, (where $\nu_\mathrm{O}$ is the frequency of optical bands) since $\nu_\mathrm{O} > \nu_\mathrm{C,RS} > \nu_\mathrm{M,RS}$ would imply an implausible index $p$ for the energy distribution of the electrons that produce the RS emission, $p \approx 1$. We also assume that the X-ray band is above the cooling frequency of the FS emission, i.e. $\nu_\mathrm{C,FS}$. To have spectral indices consistent with those observed, we assume $p_\mathrm{FS}=2.02$ and $p_\mathrm{RS}=2.20$ for the Forward and the Reverse Shock respectively. These values of $p$ would lead to spectral indexes $\beta_\mathrm{RS} = 0.60$ and $\beta_\mathrm{FS} = 1.01$, which are within $3\sigma$ of the spectral parameters obtained when fitting the various SEDs. We find that a uniform medium, $g=0$, cannot explain both the X-ray and early optical decay slopes. In fact, the amount of energy injection which would make the X-ray decay match the observed value produces too shallow an optical decay. Conversely, less energy injection, which would make the optical match the observation, would produce too steep an X-ray decay. Similarly, in the case of a wind-like circumburst medium with $g=2$, the amount of energy injection needed to model the observed optical decay would make the X-ray decay too slow. Instead, there exist solutions for “intermediate" profile density, $g=1.15$. Other similar cases, halfway between constant and stellar wind profiles, have been found in modeling of GRBs [@sth08]. For $g=1.15$, energy injection characterized by $s=2.75$ (or $q\simeq0.6$), requires the decay indices of the RS and the FS emissions to be $\alpha_{RS} = 0.58$, and $\alpha_{\mathrm{FS}} = 0.58$. We can also test whether this model predicts the correct rise and the decay slopes at the rebrightening (see Fig. \[all\_LCs\] and Table 2). For $g=1.15$ and $s=2.75$, $\nu_\mathrm{M,FS} \propto t^{-1.28}$ and $F_{\nu}(\nu_\mathrm{M,FS}) \propto t^{0.15}$ (see SM00). This implies that $F_{\nu}(\nu < \nu_\mathrm{M,FS})$ will rise as $\propto t^{+0.57}$ and decay as $t^{-0.51}$, in agreement with what is observed, except for a slightly shallower rise than observed. As for the steep decay at $t>2\times10^5$ s, assuming a sideways spreading jet and the same energy injection, the decay slope would be $\alpha\approx1.3$. This is not consistent with the observed X-ray and optical and may be an issue of the scenario at hand. We note that numerical simulations (e.g. Zhang & MacFadyen 2009; Wygoda et al. 2011; van Eerten & MacFadyen 2012) of jet breaks indicate that the ejecta undergo little sideways spreading, but the decay slope can be very steep because of jet edge effects. A degree of energy injection can moderate this fast decay and perhaps reproduce the observed behaviour, although this may be difficult to prove quantitatively. To summarize, this model naturally explains the presence of a break frequency at the optical rebrightening, and the chromatic behaviour as a consequence of the interplay of RS and FS. A similar two-component scenario has already been used to model a few [[*Swift*]{}]{} GRBs (e.g. Jélinek et al. 2006) and pre-[[*Swift*]{}]{} GRBs (see Kobayashi & Zhang. 2003b). However, in previous cases the RS was supposed to vanish within a few hundreds seconds; in the case of GRB100814A the RS emission can be long-lived due to the continuous process of energy injection.\ The model explains also why the rise and decay slopes in different filters are consistent. It explains also why the optical rebrightening has no X-ray counterpart and why the decay steepens first in X-rays and then in the optical band: the jet break takes longer to appear in the optical than in the X-ray band, because at 200 ks $\nu_\mathrm{M,\mathrm{FS}}$ is still close to the optical range, while $\nu_\mathrm{X} \gg \nu_\mathrm{M,FS}$. The decay slopes before the rebrightening and during the rebrightening itself are also roughly accounted for. In this scenario the X-ray and optical rebrightening are due to the same component. Thus, they should exhibit the same global temporal behaviour. If we extrapolate the peak time - peak frequency trend to X-ray frequencies, the peak time of the X-ray emission should have been observed several hundreds of seconds after the trigger (see Section 4.1). This agrees with observations, since the X-ray plateau appears to have started at that epoch. Finally, such a long lived RS scenario would produce a bright radio emission; radio observations started a few days after the trigger and managed to detect a measurable radio flux (see Section \[radio\]). However, a more serious issue we have yet to consider is whether $\nu_\mathrm{M,FS}$ can be in the optical band as late as $\sim90$ ks. We compute the value of $\nu_\mathrm{M}$ from 860 s, the earliest epoch when the emission of the FS shock is recorded. Since GRB100814A may be an intermediate case between constant density and stellar wind environment, we carry out our test using both equations (1) and (2) of Yost et al. (2003). We take $p_{\mathrm{FS}} \simeq 2.02$. Having derived the value of $\nu_\mathrm{M,FS}$ at 860 s, we follow its temporal evolution according to SM00 for $g=1.15$ and $s=2.75$. We find that $\nu_\mathrm{M,FS} \propto t^{-1.28}$. Thus, at 90 ks, we would have $$\epsilon_{B,-2} ^{1/2} \epsilon_e^2 E_{K,52} ^{1/2} \simeq 760\;. \label{model2a}$$ in the case of constant density and $$\epsilon_{B,-2} ^{1/2} \epsilon_e^2 E_{K,52} ^{1/2} \simeq 470\;. \label{model2b}$$ for stellar wind. Even assuming very large values for $\epsilon_{B,-2}$ and $\epsilon_e$, $33$ and $1/3$ respectively at equipartition, we would still need $E_K \sim 10^{58}$ erg for the case of a stellar wind. Such large energy is not predicted by any models of the GRB central engine. ### Early optical and X-ray emission from RS, rebrightening from FS {#RS-FS2} A more plausible variant of the previous model, which also keeps all the advantages described above, predicts that all the emission in the X-ray band is also produced by the RS, with $\nu_\mathrm{X} > \nu_\mathrm{C,RS}$, while the FS produces the rebrightening. In this case, we can choose a large value for the parameter $p$ of the Forward Shock, and this greatly eases the energy requirements. We find that for $g=1.25$, $s=2.65$, $p_\mathrm{RS} = 2.02$, $p_{\mathrm{FS}} = 2.85$, the predicted temporal slopes are $\alpha_\mathrm{O}=0.57$ before the rebrightening, $\alpha_\mathrm{X} = 0.60$; the slope of the optical rise is $-0.52$, while the successive decay between $\sim50$ and $\sim200$ ks would be $\alpha_\mathrm{O}=1.11$. All these values are within $2.8\sigma$ of the observed values, except the rise, which is slightly steeper than predicted, and decay slope after the rebrightening, which is slower than predicted. However, the decay slope may be shallower because $\nu_\mathrm{M,FS}$ is still close to the optical band and the model is approximated, thus we can consider this solution satisfactory. The spectral slopes are accounted for, too. Equation (\[model2b\]) becomes $$\epsilon_{B,-2} ^{1/2} \epsilon_e ^2 E_{K,52} ^{1/2} \simeq 0.58 \label{model3a}$$ We can derive, as we did for $\nu_\mathrm{M,FS}$, another condition. The maximum flux $F_{\nu} (\nu_{\mathrm{M},\mathrm{FS}})$ has to be equal to the peak flux reached at $\simeq 90$ ks, which is $\simeq 200$ $\mu$Jy. We find $$\epsilon_{B,-2} ^{1/2}E_{K,52} ^{1/2} A_{*} \simeq 1.6 \times 10^{-3}\;. \label{model3b}$$ Where $A_{*}$ defines the normalization of the density profile, i.e. $n = A_{*} r^{-g}$ (see Chevalier & Li 2000). These equations have to be solved jointly. Assuming the typical $\epsilon_e=1/3$, $\epsilon_{B,-2} ^{1/2} E_{K,52} ^{1/2} \simeq 5.3$. If we take $\epsilon_{B,-2}=33$ as well (these values of the $\epsilon$ parameters are reached at equipartition) then $E_{K,52}\simeq0.86$ at the onset of the external shock and energy injection. The medium is thin, with $A_{\ast} \simeq 3\times10^{-4}$.\ Using the values of $E_K$ and circumburst density we can also estimate the RS microphysical parameters. At $50~ks$ the X-ray emission is still dominated by the RS and, from the best fit model, there is a break at $92.4^{+42.6} _{-39.9} \mathrm{eV} \simeq 2.2\times10^{16} \mathrm{Hz}$, which must be the synchrotron cooling frequency $\nu_{C,RS}$. For the chosen values of $s$ and $g$, it decays as $t^{-0.06}$. Thus, we can compute it at $t_\mathrm{break}=860$ s as a function of the relevant parameters, multiply it by $(50/0.86)^{-0.06} \simeq 0.76$ and force the result to be equal to the break energy we find at 50 ks. For the value of $\nu_\mathrm{C,RS}$ at $t_\mathrm{break}$, which we have taken as the deceleration time $t_\mathrm{dec}$ (see Sect. \[opticaldata\]), we adopt the formulation of Kobayashi & Zhang (2003a), their Eq. 9, $$\nu_\mathrm{C,RS} = 2.12\times10^{11} \left( \frac{1+z}{2} \right)^{-3/2} \epsilon_{B,\mathrm{RS,-2}} ^{-3/2} E_{K,52} ^{1/2} A_{*} ^{-2} t_{\mathrm{dec}}^{1/2} \, \mathrm{Hz} \label{nu_C,RS}$$ For the above values of density and energy it is $\nu_\mathrm{C,RS} = 4.7\times10^{19} \epsilon_{B,\mathrm{RS,-2}}^{-3/2}$ Hz. Thus Eq.\[nu\_C,RS\] implies a very high value for $\epsilon_{B,\mathrm{RS,-2}}$; taking $\nu_\mathrm{C,RS} \simeq 2.2\times10^{16} \mathrm{Hz}$ would imply $\epsilon_{B,\mathrm{RS,-2}} \approx 100$. Such value is very large and would imply a very strong magnetization of the outflow, for which the RS emission may be suppressed. However, the error on the break energy is quite large, with a $3\sigma$ upper limit of $0.45~\mathrm{keV}$. We can thus assume that $\epsilon_{B,\mathrm{RS,-2}} \gtsim 47$. Such limit indicates that the ejecta carry a considerable magnetic field; we caution that, in such condition, our analytical formulation may not be the most correct way to predict the dynamics and the flux produced by the RS (Mimica, Giannios & Aloy 2009). However, for the sake of simplicity, we will assume that the theoretical derivation we have used so far still applies. In the following, we will assume $\epsilon_{B,\mathrm{RS,-2}} = 60$. This value of $\epsilon_{B,\mathrm{RS}}$ derived above enables us to explain the spectral break at 4.5 ks as $\nu_{C,RS}$ too. This model predicts the correct values for the late, post-jet break decay slopes, if one assumes that the jet is spreading sideways: from Table 1 of Racusin et al. (2009), for $p_{\mathrm{FS}}=2.85$, $q=0.6$, $\nu_\mathrm{O} < \nu_\mathrm{C,FS}$, the flux decays as $\alpha=2.07$, consistent with observations. Numerical simulations indicate that jets have little side ways spreading (see above) and the steep decay can be explained in terms of edge effect. However, by coincidence this effect seems to predict slopes consistent with those of the spreading jet model. As for the X-ray light curve, it is reasonable to assume that the RS emission post-jet slopes are similar to that of the FS after a jet break. Pressure and speeds of the RS and FS shocks should not change across the contact discontinuity that divides the two at the jet break time, so the sideways expansions due to overpressure in both regions should be similar and lead to comparable behaviours in terms of dynamics and related emission. Thus, the late X-ray decay slope can be explained by the model we are discussing. We can now determine $\epsilon_{e,\mathrm{RS}}$. The optical flux at 860 s is RS emission, and the flux density is $F_{\nu} \simeq 300 \mu Jy$. The optical emission is $$F_{\nu} (\nu_\mathrm{O}) = F \left( \nu_\mathrm{peak,RS} \right) (\frac {\nu_\mathrm{O}}{\nu_\mathrm{peak,RS}})^{-\beta}\;, \label{optical1}$$ where $\nu_\mathrm{peak} = \mathrm{max}(\nu_\mathrm{M,RS},\nu_\mathrm{SA,RS})$ [^3]. Now, we know that $$\nu_\mathrm{M,RS} = \Gamma^{-2} \nu_\mathrm{M,FS} \left( \frac{\epsilon_\mathrm{e,RS}}{\epsilon_\mathrm{e,FS}} \right)^{2} \left( \frac{\epsilon_\mathrm{B,RS}}{\epsilon_{B,\mathrm{FS}}} \right)^{1/2} R_\mathrm{p} ^2\;, \label{optical}$$ where $R_p$ = $g_\mathrm{RS}/ g_\mathrm{FS}$ with $g=(p-2)/(p-1)$. We first find $\Gamma$ at the deceleration time, $\Gamma_{\mathrm{dec}}$, using Eq. 2 of Molinari et al. (2007), $A_{\ast} = 3\times 10^{-4}$ and $E_K = 0.86\times10^{52}$ erg. In this calculation and in the following ones, we assume, as stated previously, that $t_{\mathrm{break}}$ is the deceleration time of the leading shell. We find that $\Gamma_\mathrm{{dec}} \simeq 125$, weakly depending on density and $E$. For the values of the RS parameters already defined, and even assuming a very high value for $\epsilon_{e,\mathrm{RS}}=0.4$, we have $\nu_\mathrm{M,RS} < \nu_\mathrm{SA,RS}$ at deceleration time. Thus, the peak flux of the RS will be reached at $\nu_\mathrm{SA,RS}$ and in Eq. \[optical1\] $\nu_\mathrm{peak}$ is the self-absorption frequency. We know that $$F_{\nu} (\nu_\mathrm{peak, RS}) = \Gamma F_{\nu} (\nu_\mathrm{M, FS} ) \left( \frac{\epsilon_\mathrm{B,RS}} {\epsilon_{B,\mathrm{FS}}} \right)^{1/2}\;, \label{RSpeak}$$ where $\Gamma$ is the Lorentz factor at any given time[^4]. For the values already found, we have $F_\nu (\nu_{\mathrm{peak,RS}}) = 2.2\times10^4~\mu$Jy at the onset of the deceleration. From the observed optical flux using Eq. \[optical1\] we find $\nu_\mathrm{SA,RS} \simeq 9.8 \times10^{10}$ Hz. Together with other parameters, from Eq. 9 of SM00 we also find $\epsilon_\mathrm{e,RS}$, which is the only remaining unknown. We find that $\epsilon_\mathrm{e,RS} \simeq 0.19$. We note that the observed spectral index in the optical $\beta_\mathrm{O}$ is not constrained toward low values at a few ks. Using multi filter GROND data, Nardini et al. (2014) find a value of $\beta_\mathrm{O} \sim 0.2-0.3$, which seems to decrease with time between $\sim 1$ and $\sim 10$ ks. Such value and behaviour cannot be explained in the standard external shock model, unless one assumes that the RS emission is in the fast cooling regime, $\nu_\mathrm{C} < \nu_\mathrm{O} < \nu_\mathrm{M}$, in a wind environment, so that $\nu_\mathrm{C}$ is rising. Since the synchrotron spectrum, around $\nu_\mathrm{C}$, is thought to be very smooth, one expects to see $\beta_\mathrm{O}$ to change from $\approx0.5$ to $\approx0$ when $\nu_\mathrm{C}$ approaches the optical band from redder frequencies. This configuration is not attainable in our scenario, in which the early emission is from RS. To estimate $\nu_\mathrm{M,RS}$, we start from $\nu_\mathrm{M,FS}$, and then use Eq. \[optical\]. We know already that $\Gamma_\mathrm{{dec}} \simeq 125$. Thus, we have $ \nu_\mathrm{M,RS} \sim 8.7 \times 10^{9} $ Hz at 860 s with the values of $\epsilon_\mathrm{B,-2,RS}=60$ and $\epsilon_\mathrm{e,RS} = 0.19$. According to SM00, with $g=1.25$ and $s=2.65$ it is $\nu_\mathrm{M,RS} \propto t^{-0.81}$. Thus, at 4500 s it is $ \nu_\mathrm{M,RS}\simeq 2.3\times10^9$ Hz. Even for higher values of $\epsilon_e$ of the RS, typical of a magnetized outflow, implausibly high values of $E$ or a much higher value of $p_\mathrm{RS}$ (which is however constrained to be $p_\mathrm{RS}<2.04$ by the X-ray spectral index) would be required to move $\nu_\mathrm{M,RS}$ above the optical band at 4500 s. In our scenario, a more reasonable hypothesis to explain the spectral evolution between 1 and 10 ks is that, as time goes by, the second component producing the rebrightening becomes more and more important. This component has a blue spectrum ($\beta < 0$) in this phase, thus the observed SED, which is a sum of the two components, gets gradually shallower with time and mimics the observed $\beta_O$. Modeling of the Radio Emission. ------------------------------- We shall now investigate the behaviour of the radio light curves in the context of this scenario. The radio flux is still rising after the putative jet break, peaking at $10^6$ s and decaying afterwards. The rise of the radio flux can be ascribed to a few possibilities: i) the same component responsible for the optical peak moves into the radio band. However, if the optical peak at $10^5$ s is caused by the transit of $\nu_\mathrm{M,FS}$, for the same peak frequency to cross the radio band a few $10^9$ Hz at $10^6$ s, would require that $\nu_\mathrm{M,FS}$ should evolve as $t^{-5}$. This is not possible even in the context of a jet break. ii) the radio peak marks the transit of $\nu_\mathrm{M,RS}$. At deceleration it is $\nu_\mathrm{M,RS} \simeq 8.7\times10^{9}$ Hz and decays as $t^{-0.8}$ for the chosen values of $s$ and $g$; at the jet break time $\nu_\mathrm{M,RS} \simeq 1.5\times10^{8}$ Hz, and it is likely to decay faster from this point. Thus, $\nu_\mathrm{M,RS}$ is not expected to transit in the $4.7$ and $7.9$ GHz bands as late as $10^6$ s. We are therefore left only with the possibility that the radio peak is due to the self-absorption frequency $\nu_\mathrm{SA}$, either of the RS or the FS, crossing the radio band from bluer frequencies. According to the analytical solution of a sideways spreading jet, the flux below $\nu_\mathrm{SA}$ is expected to become constant after the jet break; however numerical simulations [@van11a] have shown that the flux can still increase if the observing frequency is $\nu<\nu_\mathrm{SA}$. We will attempt to find an order of magnitude value of this parameter, since it is not easy to find its analytical expression for $0<g<2$. By adopting the $g=2$ case of Yost et al. (2003) and considering a very tenuous medium (see above), the self-absorption frequency of the FS is expected to be at $\sim2\times10^5$ Hz at $1.3\times10^5$ s. After the jet break, it is not expected to rise within this time up to $\sim10^9$ Hz, even in the case of energy injection. A similar result is derived if we use SM00, their Eq. 9, to obtain the value of $\nu_\mathrm{SA,FS}$ at the deceleration time of 860 s, and then we constrain its temporal evolution plugging a density profile[^5] of $n\propto t^{-0.64}$ and $E_{K,52}\propto t^{0.4}$. If $\nu_\mathrm{SA,FS}$ basically did not depend on $E_{K,52}$ for $g=1.25$ and we thus neglected this dependence, the self-absorption frequency would be even lower and make its transit in the radio band even more difficult to attain. Instead, the self-absorption frequency of the RS could be in the right range. We know already that $\nu_\mathrm{SA,RS} \simeq 9.8 \times 10^{10}$ Hz at deceleration time. From this epoch, we compute its evolution assuming, as above, that $n\propto t^{-0.64}$ and $E_{K,52}\propto t^{0.4}$. Thus, $\nu_\mathrm{SA,RS} \simeq 3\times10^{9}$ Hz at jet break time. To estimate $\nu_\mathrm{SA,RS}$ from this epoch onwards, we assume that $\nu_\mathrm{SA,RS} \sim \Gamma^{8/5} \nu_\mathrm{SA,FS}$ (SM00). In the jet break regime without energy injection, $\Gamma \propto t^{-1/2}$, while $\nu_\mathrm{SA,FS}\propto t^{-1/5}$, thus $\nu_\mathrm{SA,RS} \propto t^{-1}$. Thus, at $10^6$ s, $\nu_\mathrm{SA,RS}$ should be $\simeq 0.4$ GHz. However, because of the ongoing energy injection, $\Gamma$ will decrease more slowly, and it is not unreasonable to assume that $\nu_\mathrm{SA,RS}$ is still in the GHz range. A similar result can be obtained from Eq. 2 of Yost et al. (2003), if we determine $\nu_\mathrm{SA,FS}$ at deceleration, follow its temporal evolution as above, and derive $\nu_\mathrm{SA,RS}$ by multiplying by $\Gamma^{8/5}$. The peak flux, too, should be in the right range. For the values of $s$ and $g$ chosen, RS peak flux evolves as $t^{-0.16}$ until the jet break. After that, we use the relation $F_{\nu} (\nu_\mathrm{peak, RS}) \propto \Gamma F_{\nu} (\nu_\mathrm{peak, FS})$. In jet break regime, $\Gamma \propto t^{-1/2}$, while $F_{\nu} (\nu_\mathrm{peak, FS}) \propto t^{-1}$. The latter is proportional to $E_{K,52}^{1/2}$; since in our case $E_{K,52} \propto t^{0.4}$, it is reasonable to assume $F_{\nu} (\nu_\mathrm{peak, FS}) \propto t^{-0.8}$. Combining the two, we get $F_{\nu} (\nu_\mathrm{peak, RS}) \propto t^{-1.3}$. At the radio peak time $10^6$ s, the RS peak flux is thus expected to be $\sim 700$ $\mu$Jy, similar to what derived from observations. We therefore conclude, from this qualitative discussion, that the radio peak may be produced by the transit of the RS self-absorption frequency in this band. The fact that the $7.9$ GHz light curve is initially much flatter than the $4.7$ GHz one (see Sect. \[radio\]) might also be explained, as $\nu_\mathrm{SA,RS}$ is moving from bluer to redder frequencies.\ Comments on the physical parameters. ------------------------------------ There exists some degeneracy in the derived values of the physical parameters. Different pairings of $s$ and $g$ can account for similar decays in the X-ray, optical and radio afterglow bands. However, under the assumption that $\epsilon_{e,\mathrm{FS}} < 1/3$, we find $A_{\ast} < 3\times10^{-4}$ from Eqn.s \[model3a\] and \[model3b\]. Values of $E_{K,52}$ much higher than $\simeq1$ would imply higher $\Gamma$ and $F_{\nu} (\nu_\mathrm{peak})$; $\nu_\mathrm{SA,RS}$ should have to be lower to explain the flux at deceleration. This could be obtained by increasing the value of $\epsilon_{e,\mathrm{RS}}$.\ A value of $E_{K,52} \simeq 0.86$ may imply a rather high efficiency of the mechanism converting kinetic energy into the initial burst of $\gamma$-rays, $\eta = E_\mathrm{iso} / (E_\mathrm{iso} +E_{K,52}) \simeq 0.9$. Such value can hardly be obtained in most of the prompt emission models. However, it is worth noting that the value of $E_{K,52}$ is calculated at deceleration, when the energy injection begins. It is possible that the energy injection is due to trailing ejecta shells which have also produced the $\gamma-$ray emission. If this is the case, the efficiency should be calculated when the energy injection ends. In our model, this process goes on for at least until the last radio detection, $\simeq9\times10^6$ s; at this epoch, the kinetic energy associated to the blast wave will be $\simeq 3.4\times10^{53}$ erg. Thus the efficiency would be $\simeq 0.17$. To compute the beaming angle $\theta_\mathrm{j}$ of the outflow, we use the condition $\Gamma^{-1} \simeq \theta_\mathrm{j}$ which holds at jet break time, $\simeq 1.33\times10^5$ s. At this epoch, $\Gamma \simeq 36$; thus $\theta_\mathrm{j} \simeq 0.028$. At the end of observations, the beaming-corrected value for the kinetic energy is $\simeq 1.4\times10^{50}$ erg, typical of other GRBs (Frail et al. 2001; Ghirlanda et al. 2007). Another important feature of the scenario we are devising is the very low density of the environment, $A_{*} \simeq 3\times10^{-4}$, which corresponds to a mass loss rate of a few $\times10^{-9}$ solar masses year$^{-1}$ from the progenitor of GRB100814A. Comparably low values of $A_{*}$, however, are not unprecedented in GRB afterglow modeling (e.g. Cenko et al. 2011), and have been predicted for very low metallicity stars [@vin01]. For the value of $A_{*}$ at hand, the blast wave would reach densities comparable to the average density of the Universe at $z=1.44$ at $\sim 10^7$ s if it kept expanding radially. It is therefore possible that the density profile turns into a constant one before this happens, although the quality of late time data is not good enough to see the effects of this transition. We now briefly discuss how our modeling changes if the actual deceleration time is earlier than 856 s (see Sect. 2.3). We tested the hypothesis that the actual deceleration time is half this value, i.e. $428$ s. We find that equations \[model3a\] and \[model3b\] would change slightly, and we would find slightly different values of $A$ and $E$ to satisfy both equations; other microphysical parameters relative to the FS would stay the same. However, an earlier deceleration time would imply an higher Lorentz Factor, $\Gamma_\mathrm{dec} \simeq 150$ rather than $\simeq125$ as in the previous case with $t_\mathrm{dec}\simeq856$ s; Eq. \[RSpeak\] would thus imply a higher RS peak flux. The peak frequency for the RS would still be $\nu_\mathrm{SA,RS}$, but since it is inversely proportional to the deceleration time, it would be roughly twice the previous value. Taken together, these two differences would make an initial optical flux, at the deceleration time, too high and incompatible with observations. The only way to decrease $\nu_\mathrm{SA,RS}$ and thus the flux in the optical band would be to increase $\epsilon_{e,\mathrm{RS}}$, but it would have to be as high as $\epsilon_{e,\mathrm{RS}}\simeq0.5$, which is impossible because $\epsilon_{B,\mathrm{RS}}\simeq0.6$ already to make $\nu_\mathrm{C,RS}$ in the right range (see Eq. \[nu\_C,RS\]) and the sum $\epsilon_{B,\mathrm{RS}}+\epsilon_{e,\mathrm{RS}}$ cannot be more than 1. Acceptable solutions would be possible only if $t_\mathrm{dec} \gtsim 600$ s, and radio observations might be explained as well. We therefore conclude that, in our model, the deceleration time of the leading shell can occur before 856 s but not much before.\ We summarize a description of different models proposed so far, with their advantages and problems, as well as values of the physical parameters, in Table \[tab\_models\]. Numerical simulations {#simulations} --------------------- We try now to approach the properties of GRB100814A using the numerical modeling of Uhm (2011) and Uhm et al. (2012). This is not based on full-blown hydrodynamical simulations, but a semi-analytical formulation of a relativistic blast wave. It applies the conservation laws of energy-momentum and mass in the region between the FS and the RS. Such work also considers a variable adiabatic index for the shocked gas in the regions intersected by the FS and the RS; this is quite important in the case of RS, which evolves from a non-relativistic regime to a mildly or relativistic regime as the blast wave propagates. We note that our simulations also make use of radial stratifications of the ejecta which can be quite different from a constant or a simple power law. Under such conditions, the FS dynamics may deviate from the self-similar solution of Blandford & McKee (1976), but using the accurate numerical solutions of Uhm (2011), we can effectively predict the dynamics of the shocks.\ For the blast wave itself, we adopt a Lagrangian description (Uhm 2011, Uhm et al. 2012), which considers the blast wave as composed of many different Lagrangian shells all the way from the FS to the RS fronts. Each shell has its own physical parameters, such as energy density, radius, pressure, adiabatic index and, if necessary, magnetic field and electron energy distribution. This is rather different from the classical, simple analytical scenario of Sari et al. (1998), where the entire shocked region has the same radius, energy, pressure, magnetic field and power law distribution of electrons. The simulation shows the evolution of each shell, tracking the parameters such as energy, adiabatic index and magnetic field; it derives the minimum Lorentz factor and cooling Lorentz factor of the electrons by solving the full differential equations (Uhm et al. 2012) numerically. Curvature effects of each shell, which has its own radius, are taken into account as well. Finally, the afterglow light curves are calculated by integrating the photons emitted from all shells that arrive at the same observer’s time $t_\mathrm{obs}$.\ In the scenario we tested, the energy injection is due to late shells that collide with the trailing ones, and a long-lived RS develops. The flux is due to both FS and RS, whose relative contribution evolves with time and depends on the observing frequency. For simplicity, we consider the flux in the $R$ and X-ray bands only and ignore the light curves in other optical and radio bands. The physical parameters involved were changed manually a few times, keeping some parameters fixed and altering others, until we found a visual good agreement between the derived light curve and the observations. We did not derive error margins. The results of the numerical modeling are shown in Fig. \[cRX\], and the distribution of the Lorentz factor of the ejecta versus time $\tau$ of the ejection is shown in Fig. \[gej\_tau\]. The FS has $\epsilon_{e,\mathrm{\mathrm{FS}}} = 0.1$, $\epsilon_{B,\mathrm{\mathrm{FS}}} = 0.01$; the RS has $\epsilon_\mathrm{e,RS} = 0.1$, while $\epsilon_\mathrm{B,RS} = 0.05$. Both shocks create a population of radiating electrons whose energy distribution is a power law with index $p=2.1$. The isotropic kinetic energy involved is $10^{54}$ erg, and the ambient medium density is 1 cm$^{-3}$. The $\sim2\times10^5$ s jet break is caused by a jet opening angle of 0.07 rad. To provide a satisfactory picture, the RS needs to energize 100% of electrons of the ejecta while the FS is much less effective, providing energy only to $1.2\%$ of electrons of the medium it is moving into. The agreement between the predicted flux in the R and X-ray bands is subjectively good except for a slight ($\sim30\%$) overestimate of the optical flux at $\sim 25$ ks. The optical flux is due to declining RS emission up to $\sim10$ ks, when the FS emission takes over and dominates afterwards. The X-ray emission is always dominated by the FS, although a small increase in flux ($10\%$) in this band is visible around $80$ ks. This model also predicts a hardening of the spectrum around the peak time, as observed.[^6]\ We emphasize how it is possible, on the basis of agreement between the synthetic results and observations, to constrain the temporal evolution of the Lorentz factor of the material emitted by the central engine. Such a method opens interesting opportunities to explain diverse behaviours in GRBs and understand better the physics of the central engine. The rebrightening of GRB100814A occurs at $\sim 1$ day; it shows a slow rise slope and it looks smooth. A few GRBs show a much faster rise. A possibility, envisaged in Uhm et al. (2012) and Uhm & Belobedorov (2007), is that the central engine produces shells with a variety of Lorentz factors, evolving with time and more complicated than a simple power law. In these circumstances, it is possible to reproduce faster rises and decays which are otherwise difficult to explain with the external shock model.\ We point out that our numerical simulations have confirmed the basic scenario drawn from the analytical model. In order to have the X-ray emission and the optical rebrightening produced by FS, with $\nu_\mathrm{M,FS}$ crossing the optical band as late as $\sim 1$ day and $p\simeq2.1$, one needs either an extreme value of kinetic energy imparted to the whole bulk of the emitting medium, or a more realistic value of kinetic energy somehow imparted only to a tiny fraction of the medium. It is not clear how one could attain either. Other possibilities ------------------- We shall now briefly discuss other possible scenarios to explain the behaviour of GRB100814A, in connection with other GRBs showing the same phenomenology. ### Changes of the other microphysical parameters The fact that the rebrightening is not visible in the X-ray requires strong [*ad hoc*]{} assumptions regarding the evolution of these parameters, which makes the whole scenario contrived and implausible (Panaitescu et al. 2006; see, however, Filgas et al. 2011). ### End of energy injection. The rebrightening is produced when the energy injection, in form of late shells which pile up on the leading ones, ends, and bright FS and RS reverberate throughout the ejecta themselves (Zhang & Mészáros 2002, Vlasis et al. 2011). Before and after the rebrightening, the emission comes only from FS of the leading shell.\ It has been found that the rebrightening is prominent, as in the case of 100814A, only if the ejecta are collimated. This would explain why we see, shortly after the rebrightening, a jet break and why the break times are not simultaneous. The spectral evolution observed during the rebrightening can be explained if we assume that a RS spectrum, with its peak frequency crossing the optical band, is outshining the FS emission. This model predicts a late radio peak, more or less simultaneous with the optical peak. However, we have no radio observations at the epoch of the optical peak, so this prediction could not be tested. The late radio peak, which occurred $\simeq 13$ times later than the optical peak, was likely due to the behaviour of critical frequencies and dynamics. ### Internal dissipation emission We shall now discuss the possibility that the optical emission of GRB100814A is not being produced by external shocks, but it is an outcome of dissipation processes occurring inside the ejecta themselves.\ First, optical flares may have already been found in GRB afterglows (Roming et al. 2006, Swenson et al. 2013, Kopač et al. 2013) and at least some of them are likely to be produced by internal dissipation processes, like their X-ray counterparts. Therefore, internal dissipation processes could generate late optical emission in GRBs. Second, in addition to GRB100814A, other events like GRB081029 (Nardini et al. 2011; Holland et al. 2012) and GRB100621A (Greiner et al. 2013) show sudden optical rebrightening towards the end of the X-ray slow decay phase. Another similarity to the case of GRB100814 is that the X-ray light curves of these GRBs do not seem to be altered much during the optical rebrightening: the flux in this higher energy band does not exhibit any clear analog rise. A difference is that, in these events, the rise of the optical flux is much steeper than in 100814A, approaching $\alpha_\mathrm{O} \simeq - 10$. Furthermore, there is spectral variability and, sometimes, rapid temporal variability during the rebrightening itself.\ While a complicated distribution of Lorentz factor of the shells can reproduce slopes steeper than those detected for GRB100814A, it may be nevertheless difficult to explain such extreme slopes and variability in the context of external shock mechanism. Now, if what we see in GRB100814A is only a “mild" version of the same phenomenon registered in other GRBs, one may thus need to abandon the external shock scenarios and study the behaviour in the context of internal dissipation models, in which fast variability is allowed by high bulk Lorentz Factors. The X-ray afterglow of GRB100814A is among the brightest of any observed by [*Swift*]{} during the end of the plateau phase (see Fig.\[ag\_lc.eps\]). According to Panaitescu & Verstrand (2011), the X-ray afterglow of bursts with chromatic behaviour is on average brighter than that of bursts that do not show it. This might indicate that in these events the origin of at least the X-ray emission is not from the FS, but some other mechanism, such as internal dissipation.\ A drawback of this scenario is that we do not yet understand well the behaviour of the internal dissipation emission. Thus, such identification is rather [*ad hoc*]{}, and not much susceptible to testing. The chromatic behaviour at the optical rebrightening of GRB100814A is not clearly accounted for, nor is the late steep decay similar to that observed in the X-rays. Conclusions =========== We have reduced and examined an ample set of data on GRB100814A, observed by $\it{Swift}$, $\it{Fermi}$, and several ground optical and radio facilities. A prominent feature of this burst is an optical rebrightening, starting around $15 - 20$ ks after the burst trigger, which follows a typical early phase of slow decay of the flux. Such a rebrightening is not present in the X-ray light curve. However, when the optical rebrightening gives way to a steep decay, the X-ray light curve shows a break and a steepening as well. The radio emission, instead, peaks around $10^6$ s.\ The optical rebrightening has a chromatic behaviour. This is already evident in the analysis of light curves; furthermore, a study of the spectral energy distributions shows a possible spectral break in the optical band, which is consistent with the transit of the synchrotron peak frequency $\nu_\mathrm{M}$ through it.\ We have discussed a few models to interpret the behaviour of GRB100814A. The first model theorizes a double component jet; initially, both X-ray and optical emission are produced by a wide outflow component, seen just off-axis. A narrow component produces the optical rebrightening when its emission enters the line of sight of the observer. While this model can reproduce the temporal behaviour observed, the occurrence of a spectral break in the optical band at $\sim1$ day after the trigger would require an unphysical value of kinetic energy.\ A second model assumes that the observed emission is a combination of a long-lived RS, caused by continuous energy injection in the form of late shells, and FS. For a configuration of the circumburst medium density profile and strength of energy injection, simple analytical calculations show that the X-ray emission and the optical rebrightening can be attributed to FS, while the RS produces the early optical shallow decay. The late steepening is due to a jet break. This model explains why the X-ray light curve shows no sign of the flux rebrightening seen in the optical, while it breaks to a steeper decay at an epoch similar to that of the optical. However, this model has again difficulty in explaining the presence of $\nu_\mathrm{M}$ crossing the optical band during the optical peak since it requires a very high value of energy $E$ of the ejecta.\ More detailed, numerical calculations based on the the modeling of Uhm et al. (2012) indicate that the general behaviour can be described with the interplay of FS and RS, and more reasonable values of energy. Furthermore, this numerical modeling enables us to constrain how the Lorentz factor of the shells emitted by the GRB central engine evolves in time, thus shedding light on the still poorly known physics of this object. On the other hand, in the case at hand, one would require that the FS accelerates only $\simeq1\%$ of the electrons of the surrounding medium, which may be difficult to explain.\ A variant of this model which keeps its advantages and sidesteps its problem is one in which all the emission, both in the X-ray and optical, is actually due to RS, while the optical bump is due to the emergence of a FS component with steep spectrum. In this case, a very high value of energy is not needed: $E\sim10^{50}$ erg after correction for beaming. Furthermore, this model predicts the correct optical post-jet break slopes if one assumes that the jet edge effect produces decay slopes similar to those expected for jets with sideways expansion.\ The interplay between FS and RS emission may explain other GRBs that have an optical bump and chromatic behaviour. For different strengths of energy injection and density profile of the medium, a variety of behaviours, either chromatic or achromatic, can be reproduced. However, it is difficult to explain events which have a steep optical rebrightening with external shock scenarios. This is especially true when rapid flux fluctuations are present at the top of the rebrightening, for example in GRB081029 or GRB10621A. Therefore, a possibility we cannot exclude is that either or both the X-ray and optical emission are due to some internal dissipation mechanism.\ GRB100814A belongs to the growing family of events whose afterglow cannot be explained by a simple component FS emission, but requires a superposition of more components, either produced by different regions of the ejecta or due to different blast waves. This category of events includes bursts with chromatic behaviour and rebrightenings at the end of the slow decline phase such as GRB100814A. Detailed temporal and spectral analyses of multi-wavelength data is needed in order to test the different scenarios, identify and characterize the different components present in afterglows. Thankfully, the combination of [*Swift*]{} and ground based facilities allows observers to produce an ample and extended coverage of GRBs and shed light on their complex and intriguing behaviour. Acknowledgements ================ MDP, MJP, NPK and SRO acknowledge United Kingdom Space Agency (UKSA) funding. MDP thanks M. A. Aloy, F. Daigne and A. Mizuta for insightful discussions at “Supernovae and Gamma-Ray Burst 2013” conference, Kyoto. CGM thanks the Royal Society, the Wolfson Foundation and the Science and Technology Facilities Council (STFC) for support. FG acknowledges support from STFC. APB and PAE acknowledge UKSA support. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. AG acknowledges funding from the Slovenian Research Agency and from the Centre of Excellence for Space Sciences and Technologies SPACE-SI, an operation partly financed by the European Union, European Regional Development Fund and Republic of Slovenia“. MI, YJ, and S. Pak acknowledge the support from the Creative Initiative program, grant No. 2008-0060544 of the National Research Foundation of Korea (NRF), funded by the Korea government (MSIP). SS acknowledges financial support from support by a Grant of Excellence from the Icelandic and the Iniciativa Cientifica Milenio grant P10-064-F (Millennium Center for Supernova Science), with input from ”Fondo de Innovación para la Competitividad, del Ministerio de Economía, Fomento y Turismo de Chile", and Basal-CATA (PFB-06/2007). The Liverpool Telescope is operated by Liverpool John Moores University at the Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. The Faulkes Telescopes, now owned by the Las Cumbres Observatory Global Telescope network, are operated with support from the Dill Faulkes Educational Trust. {width="77.50000%"} {width="75.00000%"} [**Telescope or Observatory**]{} Telescope Aperture Filter or freq./energy band Notes ---------------------------------- --------------------------------------------------- ------------------------------------------ ------- -- EVLA 4.7, 7.9 GHz BTA/Scorpio 6.0 m $R,I$ CQUEAN 2.1 m $r$,$i$ IAC80 0.82 m $R$ CAHA 1.23 m $R$ NOT 2.56 m $R,V,B$ GTC 10.4 m $r$, R500B, R500R 1 LOT 1 m $g'$,$r'$,$i'$,$z'$ LT 2.0 m $R$,$i$ FTN 2.0 m $R$,$i$ ROTSE 0.45 m unfilt 2 UVOT 0.30 m $wh$,$v$,$b$,$u$,$uvw1$,$uvm2$,$uvw2$,gu 3 XRT 0.3-10 $\mathrm{keV}$ BAT 15-150 $\mathrm{keV}$ notes 1 - R500B, and R500R are spectroscopic observations 2 - calibrated to $R$c 3 - gu is uv grism [llll]{}\ Band & $\alpha_1$ & $t_{\rm{b},\,1}$ (ks)\ $uvw2$ &$ \approx-0.60 $&$ \approx37.50 $\ $uvm2$ &$ -0.66 \pm 0.29 $&$ 51.39 \pm 7.92 $\ $uvw1$ &$ -0.56 \pm 0.25 $&$ 55.49 \pm 6.65 $\ $u$ &$ -0.59 \pm 0.22 $&$ 56.11 \pm 4.69 $\ $b$ &$ -0.65 \pm 0.57 $&$ 55.17 \pm 11.64 $\ $v$ &$ -0.60 \pm 0.14 $&$ 83.71 \pm 7.33 $\ $g'$ &$ \approx-0.68 $&$ 65.83 \pm 35.98 $\ $white$ &$ \approx-0.69 $&$ \approx 48.66 $\ $r'$ &$ -0.66 \pm 0.02 $&$ 73.53 \pm 1.6 $\ $R_c$ &$ \approx-0.68 $&$ 74.92 \pm 1 $\ $i'$ &$ -0.59 \pm 0.02 $&$ 92.50 \pm 2.74 $\ $z'$ &$ \approx- 0.68 $&$ \approx 79.39 $\ \ Parameter& Value & Parameter & Value\ $\alpha_2 $&$ 0.48\pm0.02 $&$ \alpha_3 $&$ 1.97\pm0.02 $\ $t_{\rm{b},\, 2}\,(\rm{ks}) $&$ 217.7\pm2.4 $&$n_1=n_2 $&$ 10 $\ Filter $t_\mathrm{peak}$ (ks) --------- ------------------------ $w2$ $36.54 \pm 6.48$ $m2$ $51.78 \pm 5.48$ $w1$ $ 53.87 \pm 3.49$ $u$ $54.87 \pm 1.88$ $b$ $55.06 \pm 4.09$ $v$ $82.20 \pm 5.77$ $white$ $48.96 \pm 6.33$ $g'$ $ 65.98 \pm 1.60$ $r'$ $ 75.33 \pm 1.54$ $i'$ $ 87.84 \pm 1.76$ $z'$ $ 80.22 \pm 3.79$ ------------------------------------- ---------------- ------------------ ------------------ ---------------- ------------------ ---------------- ------------------ ---------------- -- -- Significance Significance Significance $F_{\nu,\,\rm{p}} = N t^{-\alpha}$ $0.87\pm0.40$ $-5.09 \pm 1.98$ $0.55\pm 0.25$ $1.12\,\sigma$ $0.55 \pm 0.25$ $1.14\,\sigma$ $0.44 \pm 0.23$ $1.17\,\sigma$ $F_{\nu,\,\rm{p}} = N \nu^{-\beta}$ $-0.72\pm0.16$ $9.93 \pm 2.39$ $-0.81 \pm 0.11$ $2.03\,\sigma$ $-0.82 \pm 0.09$ $2.15\,\sigma$ $-0.70 \pm 0.14$ $2.03\,\sigma$ $\nu = N t^{-\gamma}$ $-1.45\pm0.39$ $21.85 \pm 1.93$ $-0.80 \pm 0.12$ $1.97\,\sigma$ $-0.80 \pm 0.14$ $1.93\,\sigma$ $-0.65 \pm 0.16$ $1.82\,\sigma$ ------------------------------------- ---------------- ------------------ ------------------ ---------------- ------------------ ---------------- ------------------ ---------------- -- -- [cccccc]{} & 500 s & 4.5 ks & 22 ks & 50 ks & 400 ks\ \ Simple power law & & & & &\ $E(B-V)$ (mag) & & & & & $(2.6\pm0.8)\times10^{-2}$\ $\beta $ & & & & & $0.95\pm0.01$\ $\chi^2/$ d.o.f. & & & & & $58.7/45$\ \ Broken power law & & & & &\ $E(B-V)$ (mag) & [**$(6_{-2.6} ^{+2.5})\times10^{-2}$**]{} & [**$(4.6^{+3.4} _{-2.8})\times10^{-2}$**]{} & [**$(0 ^{+0.2}) \times 10^{-2}$**]{} & [**$(4.8^{+0.3} _{-1.8})\times10^{-2}$**]{} &\ $\beta_1$ & $0.07^{+0.33} _{-0.30} $ & $ 0.52 ^{+0.07} _{-2.30} $ & $0.59^{+0.01} _{-0.19}$ & $0.10^{+0.22} _{-0.04}$ &\ $E_{break} \mathrm{(eV)}$ & $90.4 ^{+910} _{-86.8}$ & $641 ^{+313} _{-640}$ & $1240 ^{+230} _{-1170} $ & $10.0^{+9.6} _{-3.8}$ &\ $\beta_2$ & $0.89 ^{+0.13} _{-0.05}$ & $1.02 _{-0.08}$ & $0.93\pm0.09$ & $1.02 _{-0.02}$ &\ $\chi^2/$ d.o.f. & $5.8/4$ & $11.8/14$ & $55.2/34$ & $119.5/114$ &\ \ Broken power law & & & & &\ with $\Delta\beta =1/2$ & & & & &\ $E(B-V)$ (mag) & [**$(4.4 _{-1.4} ^{+2.5})\times10^{-2}$**]{} & **[$(4.7^{+3.3} _{-2.8})\times10^{-2}$]{} & [**$(0^{+1.5})\times10^{-2}$**]{} & [**$(1.4\pm0.5)\times10^{-2}$**]{} &\ $\beta_1$ & $0.34^{+0.06} $ & $0.52 _{-0.06}$ & $0.39 ^{+0.13} _{-0.04}$ & $0.50^{+0.02} _{-0.04}$ &\ $E_{break} \mathrm{(eV)}$ & $583 ^{+515} _{-259}$ & $655 ^{+305} _{-390}$ & $86 ^{+193} _{-66}$ & $47.1\pm22.0$ &\ $\beta_2$ & $0.84^{+0.06}$ & $1.02 _{-0.06}$ & $0.89 ^{+0.13} _{-0.04}$ & $1.00^{+0.02} _{-0.04}$ &\ $\chi^2/$ d.o.f. & $5.95/5$ & $11.8/15$ & $56.1/35$ & $123.4/115$ &\ \ Sum of two broken power laws & & & & &\ $E(B-V)$ (mag) & & & & [**$(1.1^{+2.1} _{-0.4})\times10^{-2}$**]{} &\ $\beta_{1,I}$ & & & & $-0.33$ &\ $E_{break,I} \mathrm{(eV)}$ & & & & $4.1^{+0.5} _{-0.6}$ &\ $\beta_{2,I}$ & & & & $8.5 ^{+unconstrained}_{-6.3}$ &\ $\beta_{1,II}$ & & & & $0.52 _{-0.04}$ &\ $E_{break,II} \mathrm{(eV)}$ & & & & $92.4^{+42.6} _{-39.9}$ &\ $\beta_{2,II}$ & & & & $1.02 _{-0.04}$ &\ $\chi^2/$ d.o.f. & & & & $111.6/112$ &\ ** [ccc]{}\ Model & Advantages & Problems\ Single component jet & Shallow decay and optical rise in principle explained & Behaviour for energy injection and FS onset\ & by several possibilities: energy injection, FS onset, & cannot be chromatic in X and optical bands.\ & transit of $\nu_\mathrm{M}$ through the optical band. & Transit of $\nu_\mathrm{M}$ is not possible because the\ & & optical flux should evolve with a slope\ & & between $t^{+0.5}$ and $t^{-0.25}$. The power law\ & & index of radiating electrons is $p\sim2$, and one\ & & would require an unfeasibly high ejecta\ & & kinetic energy to have $\nu_\mathrm{M}$ in the optical\ & & band as late as $\sim 1$ day.\ Single component jet with & Higher density may enhance the flux for $\nu<\nu_\mathrm{C}$, & Simulations show that flux rebrightening\ density rise in the circumburst medium & i.e. the optical band, and leave the flux for $\nu>\nu_\mathrm{C}$ & is not prominent if the blast-wave encounters\ & unchanged; $\nu_\mathrm{C}$ may move into the optical band & a density enhancement.\ & and cause the chromatic behaviour. &\ Two-component jet: wide outflow producing & It has already been invoked and reasonable & Chromatic behaviour during the\ the early optical and X-ray, optical & physical parameters are needed to explain & rebrightening is not explained.\ rebrightening and late X-ray from narrow & the observed light curves. & Extremely high efficiency required.\ jet observed off-axis & &\ & &\ $p$ & 2.02 &\ $\theta_\mathrm{j,wide}$, $\theta_\mathrm{j,narrow}$, $\theta_\mathrm{obs}$ & 0.054, 0.027, 0.081\ $\epsilon_{e}$, $\epsilon_{B}$, $n$ & 1/3, 0.1, 10\ $E_\mathrm{K,52,narrow}$, $E_\mathrm{K,52,wide}$ & $65$, $1.9$\ $E_\mathrm{K,52,narrow,corr}$, $E_\mathrm{K,52,wide,corr}$ & $0.023$, $0.0027$\ As above, two-component jet with $\nu_\mathrm{M}$ & Reasonable physical parameters are required & Unreasonable kinetic energy of the narrow\ transiting the optical band at 90 ks. & for the wide jet. & jet and circumburst medium density are\ & &required.\ $p$ & 2.02\ $\epsilon_{e}$, $\epsilon_B$, $n$ & 1/3, 0.1, $\sim10^{-14}$\ $E_\mathrm{K,52,narrow}$ & $\sim10^7$\ Interplay between RS and FS. Early optical & High value of $\nu_\mathrm{M,FS}$ explains why the & With $p_\mathrm{FS} \leq 2.04$, an inconceivably high value\ light curve from RS, all X-ray and optical & rebrightening is present in the optical band but & of kinetic energy of the ejecta is required to\ rebrightening from FS. & not in the X-ray band. Same rise and decay & keep $\nu_\mathrm{M,FS}$ in the optical band $\sim1$ day after\ & slopes in different filters during the rebrightening & the trigger. Decay slope after the jet break is\ & are accounted for. This model explains why the & not correctly predicted.\ & late X-ray and optical light curve show the similar &\ & decay slopes and why the X-ray break is earlier. &\ & Radio emission expected. &\ & &\ $s$ & 2.75 &\ $g$ & 1.15 &\ $p_{RS}$ & 2.20 &\ $p_{FS}$ & 2.02 &\ $E_\mathrm{K,52}$ & $\sim10^6$ &\ Interplay between RS and FS. Early optical & All the advantages above; the radio light curves & Optical rise slope during the\ light curve and all X-ray from RS, optical & and the late jet break slope are & rebrightening slightly under predicted.\ rebrightening from FS. & predicted too (with some &\ & assumptions). &\ $s$ & 2.65&\ $g$ & 1.25&\ $p_\mathrm{RS}$ & 2.02&\ $\epsilon_{e,\mathrm{RS}}$, $\epsilon_{B,\mathrm{RS}}$ & 0.60, 0.19&\ $p_\mathrm{FS}$ & 2.85 &\ $\epsilon_{e,\mathrm{FS}}$, $\epsilon_{B,\mathrm{FS}}$ & 1/3, 1/3 &\ $E_\mathrm{K,52}$ & 0.86 &\ Afanasiev V.L., & Moiseev A.V., 2005, PaZh, 31, 214; and Astronomy Letters, 31, 194. Akerlof C.W. et al. , 2003, PASP, 115, 132. Band D., Matteson J., Ford L. et al., 1993, , 413, 281 Barthelmy S. D., Barbier L.M., Cumming J. R. et al., 2004, Space Science Reviews 120, 143 Blandford. R. D., McKee C. F., 1976, PhFl, 19, 1130 Beardmore A. et al., 2010, GCN 11087 Bertin, E. SExtractor v2.5 Users Manual (draft), 2005, http://terapix.iap.fr/rubrique.php?id\_rubrique=91 Beuermann K. et al. 1999, A&A 352, 26L Bloom J. S., Frail D. A. & Sari R., 2001, AJ, 121, 2879 Burrows D.N. et al., 2005, Space Science Review, 120, 165. Cenko S. B., Frail D. A., Harrison F. A. et al., 2011, , 732, 29 Chevalier R. A. & Li, Z.-Y., 2000, , 536, 195 Costa E., Frontera F., Heise J. et al., 1997, Nature, 387, 783 De Pasquale M., Beardmore A., Barthelmy S. et al, 2006, MNRAS, 365, 1013 De Pasquale M., Evans P., Oates S. et al, 2009, MNRAS, 392, 153 Evans P. A., Willingale, R., Osborne J. et al., 2010, MNRAS, 519, 102 Evans P. A., Beardmore A., Page K. et al., 2009, MNRAS, 397, 1177 Evans P. A., Beardmore A., Page K. et al., 2007, A&A, 469, 379 Falcone A. D, Burrows D. N., Romano P., et al., 2006, AIP Conference Proceedings, 836, 386. Filgas R., Greiner, J., Schady, P., et al., 2011, A&A, 535 57 Frail D. A., Kulkarni S. R., Sari R. et al., 2001, , 611, 1005 Fukugita M., Ichikawa, T., Gunn J.E., et al., 1996, . 111, 1748 Galama T. J., Wijers R. A. M., Bremer M. et al., 1998, , 500, 97 Gat I., van Eerten H., & MacFadyen A. 2013, , 773, 2 Gehrels N. et al., 2004, , 611, 1005 Ghirlanda G., Nava L., Ghisellini G. et al., 2007, A&A, 466, 127 Ghisellini G., Ghirlanda G., Nava L. et al., 2007, , 658L, 75 Golenetskii S. et al., 2010, GCN 11119 Granot J., Ramirez-Ruiz E., Perna R., 2005, , 630, 1003 Granot J., Panaitescu A., Kumar P, Woosley S. E., 2002, 570L, 61 Greiner J., Kruehler T., Nardini M. et al., 2013, A&A, 560, 70 Guidorzi C., Mundell C.G., Harrison R. et al. 2014, MNRAS, 438, 752 Holland S. T., De Pasquale M., Mao J. et al., 2012, , 745, 41 Huang, K.Y., Urata, Y., Filippenko, A.V. et al., 2005, , 628, L93 Jélinek M., Prouza M., Kubánek, et al., 2006, , 454L, 119 Jordi K., Grebel E. K., Ammon K., 2006, A&A, 460,339. Kalberla P. M. W., Burton W. B., Hartmann D. et al. 2005, A&A, 440, 775 Kim E., Park W.-K., Jeong H. et al. 2011, JKAS, 44, 115 Kinoshita D. et al., 2005, ChJAA, 5, 315. Kobayashi S. & Zhang B., 2003a, , 597, 455 Kobayashi S. & Zhang B., 2003b, , 582, 75 Kopač D., Kobayashi S., Gomboc A. et al. 2013, , 772, 73 Krimm H.A. et al. 2010, GCN 11094 Kuin, N.P.M., et al., 2015, ArXiv 1501.02433. Liang E.-W., Zhang B.-B., Zhang B. et al., 2007, , 670, 565 Liang E.-W., Racusin J., Zhang B. et al., 2008, , 675, 528 Liang E.-W., Li L., Gao H. et al., 2013, , 774, 13 Mészáros P. & Rees M., 1993 , 418, 59. Mészáros P., 2006 Rep. Prog. Phys., 69, 2259 Madau P., 1995, , 441, 18 Mimica P., Giannios D., Aloy M. A., 2009, A&A, 494, 879 Mimica P., Giannios D., Metzger B., 2012, EPJ Web of Conferences, 39, id.04003 Molinari E. et al., 2007 A&A 469, L13 Nakar E. & Granot J. 2007, MNRAS, 380, 1744 Nardini M., Greiner J., Krueler, T. et al., 2011, A&A , 531, 39 Nardini M., Elliott, J., Filgas, R. et al., 2014, A&A , 562, 29 Nishioka Y. et al., 2010, GCN 11134 O’Meara J. et al. 2010, GCN 11089 Oates S., Page M.J., Schady P. et al., 2009, MNRAS, 395, 490. Perley, D.A., Cenko, S.B., Corsi, A. et al., 2014, , 781, 37 Park W.-K. et al., PASP, 2012, 124, 839 Poole T. et al., 2008, MNRAS, 383, 627. Panaitescu A. & Verstrand T. 2011, MNRAS, 414, 3537 Panaitescu A., 2008, MNRAS, 383, 1143 Panaitescu A., Mészáros P., Burrows, D. et al., 2006, MNRAS, 369, 2059 Numerical recipes in C++ : the art of scientific computing, 2002, Press Racusin J., Liang E.-W, Burrows D. et al., 2009, , 698, 43 Rees M & Mészáros 1994 , 693, 922. Roming P. et al., 2005, Space Science Reviews, 120, 95. Roming P., Van Den Berk D., Pal’shin, V. et al. 2006, , 651, 985 Sakamoto T. et al., 2009 , 693, 922. Sari R. & Mészáros P., 2000, , 535, 33L Sari R., Piran T. & Narayan R., 1998, , 497, L17 Sari R., Piran T. & Halpern J. 1999, , 519L, L17 Sari R. & Piran T. 1999, , 138 537 Schady P., Mason K.O., Page M. et al., 2007, MNRAS, 377, 273 Schady P., Page, M. J, Oates, S.R. et al., 2010, MNRAS, 401, 2773 Schlegel D. J., Finkbeiner D. P. & Davis M. 1998, , 500, 525 Schulze S., Klose S., Bjornsson G. et al., 2011, A&A, 526, 23 Smith J. A. et al., 2002, AJ, 123, 2121. Starling R.L.C., van der Horst A., Rol E. et al., 2008, , 672, 433 Steele I.A., Smith R.J., Rees P.C. et al., 2004, Ground-based Telescopes. Edited by Oschmann, J. M., Jr., Proceedings of the SPIE, 5489, 679. Swenson C., Roming P., De Pasquale M. et al., 2013, , 774, 2 Uhm Z.-L.& Belobedorov 2007, ApJ, 665, L93 Uhm Z.-L. 2011, ApJ, 733, 86 Uhm Z.-L., Zhang B. et al. 2012, , 761, 147 Urata, Y., Huang, K., Takahashi, S., et al. 2014, , 789, 146 van Eerten H., Zhang W., & MacFadyen A., 2010, ApJ 722, 235 van Eerten H., Meliani Z.; Wijers R.A.M. J. et al. 2011, MNRAS, 410, 2016 van Eerten H., & MacFadyen et al. 2012, , 751, 155 Varian H. 2005, Mathematica Journal, 9, 768 Vlasis A., van Eerten H., Meliani Z. et al. 2011, MNRAS, 415, 279 Vink J.S, De Koter A., & Lamers H.J. 2001, A&A, 369, 574 von Kienlin A., on behalf of the [*Fermi*]{}-GBM team, 2010, GCN 11099 Yost S., Harrison F. A., Sari R. et al., 2003, , 597, 459 Wygoda, N., Waxman, E. & Frail, D.A., 2011, , 738, 23 Zhang B. & Mészáros P., 2001, , 552, 35L Zhang B. & Mészáros P., 2002, , 566, 712 Zhang B., Fan Y.-Z.; Dyks J. et al., MNRAS, 2006, , 642, 354 Zhang W.-Q. & MacFadyen, A. 2009, , 698, 1261 [^1]: The peak time was computed from $dF_{\nu}/dt = 0$ (see @Molinari2007a for an explicit expression) [^2]: http://www.iac.es/telescopes/pages/en/home/telescopes/iac80.php [^3]: We do not know, at this stage, whether the peak flux of the RS will be reached at the synchrotron peak frequency or at the synchrotron self-absorption frequency [^4]: This condition is valid at any given time, not only at deceleration as usually assumed. The component moving at $\Gamma$ is responsible for the energy injection and just decelerates at the moment. [^5]: Derived from Eq. 2 of Sari & Mészáros (2000) [^6]: Relativistic hydrodynamic 1D and 2D simulations (Mimica et al. 2012) have shown under certain conditions the relativistic ejecta may undergo a total or partial [*lateral collapse*]{} and be (totally or partially) disrupted by the circumburst matter. If a fraction of the jet would be choked due to this effect, less energy may reach the working surface of the jet, leading to light curves different from those predicted without such jet disruption. Certainly, the possibility of the jet collapsing laterally depends on a delicate balance between the external medium ram pressure and the jet total pressure. In order to elucidate whether this effect is truly relevant or simply produces a small readjustment of the ejecta in the transversal direction one may require detailed 2D and 3D simulations, which are however out of the scope of this paper.

--- abstract: 'Spectra taken with the Space Telescope Imaging Spectrograph (STIS) allow accurate location and extraction of the nuclear spectrum of NGC 4151, with minimal contamination by extended line emission and circumnuclear starlight. Spectra since 1997 show that the P Cygni Balmer and He I absorption seen previously in low nuclear states, is present in higher states, with outflow velocity that changes with the nuclear flux. The phenomenon is discussed in terms of some of the absorbers seen in the UV resonance lines, and outflows from the central source and surrounding torus.' author: - 'J. B. Hutchings[^1]' - 'D.M. Crenshaw' - 'S.B. Kraemer, J.R. Gabel' - 'M.E. Kaiser' - 'D. Weistrop' - 'T.R. Gull' title: 'Balmer and He I absorption in the nuclear spectrum of NGC 4151 [^2]' --- Introduction ============ NGC 4151 is the brightest Seyfert 1 type galaxy, and has been studied in considerable detail at all wavelengths. In reference to the nuclear region, the Hubble Space Telescope has been instrumental in resolving the innermost few arcseconds, and revealing the spatial and velocity structure of the narrow emission line gas (see e.g. Hutchings et al 1998, Kaiser et al 2000, Nelson et al 2000, Crenshaw et al 2000). The detailed picture that emerged, for NGC 4151 and other Seyferts, is of a hollow biconical outflow of narrow-line clouds. In the case of NGC 4151, the high velocity radio jets lie along the cone axis, and our line of sight lies close to the edge of the approaching cone. This scenario was put forward earlier by, for example, Pedlar et al (1993) and Boksenberg et al (1995). NGC 4151 is also known to show flux variations over a factor of ten or more, and has been the subject of echo-mapping observational campaigns. These have shown the inner broad emission line region to have an extent of several light days (see overview by Peterson et al 1998). The brightness and spatial extent of the narrow emission lines has made it difficult to isolate the nuclear spectrum and emission lines. It was noted originally by Anderson and Kraft (1969) that there are shortward shifted absorptions in H$\gamma$ and the metastable He I $\lambda$3888 line. Anderson (1974) followed up with further data and a discussion that suggested a connection between the continuum flux and the absorption strength. This has been poorly documented since, but Sergeev, Pronik, and Sergeeva (2001) give a summary of observations over 11 years that show the absorptions are present in a nuclear low state in 1999. No systematic study has been made of the absorptions, perhaps because ground-based observing conditions cause a large range of contamination by the circumnuclear flux, both line and continuum. The long slit (or slitless) spectroscopic capability of STIS, along with the spatial resolution of the HST, has made it possible to obtain and study the nuclear spectrum consistently and cleanly. There is extended narrow line emission with many velocity components, even within the central arcsec, which can affect the overall line profiles, if included. In this paper we discuss the series of visible range nuclear spectra from STIS, that fortuitously cover a wide range of nuclear flux variations. We are particularly interested in the outflow absorption that is seen in the strong Balmer and the metastable He I line. Outflow is also seen in higher velocity emission line clouds near the nucleus (Hutchings et al 1999), multiple shifted absorption lines in C IV and other UV resonance lines (Weymann et al 1997, Crenshaw et al 2000, Kriss et al 2002), and warm absorbers seen in X-ray data (see e.g. Schurch and Warwick 2002). A full picture of the different outflows has yet to emerge, and this paper adds further information to the inventory. Observations and data ===================== Table 1 shows the observations used in this paper, along with some principal measures. With one exception (June 2000), the observations were executed as programs by the authors, so the data are known to be suitable for this investigation. In addition to the spectra listed in table 1, we inspected and measured associated STIS spectra covering the far-UV to 1 micron, to obtain a complete picture of the nuclear spectrum at the same times. The spectral resolutions at H$\beta$ are $\sim$800 for the G430L and $\sim$8000 for G430M spectra. The nuclear flux varied considerably over the timespan covered, including some unusually low states. The nuclear spectrum was extracted from the long slit (or slitless) data by using the detector rows that covered the continuum, clearly detected. These were collapsed to a single spectrum, and used for further measurements. The data were retrieved from the CADC with on-the-fly calibration, and also extracted using CALSTIS from the STIS team database. Two observations were made nominally offset by 0.09" from the nucleus (June 1998 and June 1999). The nuclear continuum is clearly present in the spectra, but the slit should have lost some of the flux. From the nuclear cross-sections in the other spectra, we estimate that the continuum fluxes for these spectra may be underestimated by a factor 2.5, and this factor is included in the Table 1 values. We note that the overall correlation with nuclear flux is not altered by this correction, or by an uncertainty of a further factor two in either direction. Figure 1 shows the average of the five G430L spectra as extracted this way, after each had been normalised to the continuum, and Figure 2 shows the comparison of the two G430M spectra (not normalised, as the coverage was not sufficient to establish the continuum level). The asymmetry of the H$\beta$ profiles (and other Balmer lines) is apparent in Figures 1 and 2. The profiles consist of a broad emission, a narrow emission, and a shortward absorption trough. The H$\gamma$ line is blended with \[O III\] 4363Å  emission, and H$\delta$ is blended with \[S II\] emission. We also note the absorption feature near 3900Å, which is unique in the nuclear spectrum. In Figure 3, the Balmer lines H$\beta$, H$\gamma$, and H$\delta$ are superposed in velocity space, after scaling to the same broad emission line profile peaks. The Figure also sketches in a symmetrical broad emission profile for the profiles from each observation. This was derived by folding the profile about velocities near to zero and matching the unblended parts of them on each side. The agreement among the three Balmer lines is notable and lends confidence in the result. Note that in all cases there is an apparent shortward absorption, and that it is much more obvious in the later spectra, when the continuum level was very low. In the June 1999 spectrum, the Balmer absorption extends to higher velocities, beyond the deep minimum that corresponds with the main feature seen in other low-state spectra. We measured this absorption as two separate features in this spectrum. We also note that the centres of the symmetrical broad Balmer emission profiles are consistently longward of the 1000 km s$^{-1}$ generally quoted for NGC 4151, by an average of 350 km s$^{-1}$. (This standard value is presumably derived from the mean of many blended narrow emission peaks of different velocity over the nuclear region and hence somewhat arbitrary.) It may thus be that the velocities recorded in Table 1 should be more negative by this amount, to represent velocities with respect to the central BLR. However, we note that in earlier bright epochs, the broad emission line profiles are stronger on the shortward side of the nominal redshift (see Sergeev et al 2001), so that there may be changes in the broad profiles (which may be caused by changing obscuration of the redshifted outflowing matter on the other side of the nucleus), and our symmetrical assumptions in Figure 3 may not apply to other epochs. Table 1 shows the measures of the Balmer absorption, as well as the equivalent width of the He I absorption, and measures of the He II 4686Å  peak. This latter has a broad emission component too, but the peak (as all other narrow emissions) largely varies in EW as a result of the continuum changes, and the values serve as a consistency check on the continuum flux numbers. We define the V$_{min}$ value as the turning point of a parabola fit to the absorption profile, and V$_{edge}$ as the shortward limit of absorption as illustrated in Figure 3. Finally, we measured the velocity of the minimum of the $\sim$3900Å  absorption, assuming the identification is He I 3888Å. The decrement of the Balmer absorption and emission suggest this identification, plus the facts that this metastable state line is known to arise in high density outflows, and that the velocities do not agree with the other Balmer absorptions. Anderson and Kraft (1969) made the same argument. The absorption lies between two emission lines (see Fig 1) so a concern is that changes may be distorted by blending. However, we find no changes in the absorption or emission FWHM in the sense that the velocity changes would require if the lines are blended, so conclude that the absorption feature is resolved with the G430L spectra. We measured the absorption EW with respect to the continuum beyond the neighbouring emissions, which may underestimate the true value. However, the measurements are well defined and consistent, and also show a similar variation to the H$\beta$ absorption. If we use the \[Ne III\] or \[O III\] emission as a wavelength fiducial instead of the data calibration, we find the absorption velocities may change by up to 50 km s$^{-1}$. This will not reduce the significance of the H$\beta$ changes, but are comparable with the He I range in Table 1. Variable inclusion of different narrow emission components in the different spectra are a more likely explanation of their velocity scatter, however. In Figure 4 we show the line velocities and absorption strengths as a function of the continuum flux at 4800Å. We discuss the correlations further below. In addition to the spectra listed in Table 1, we extracted spectra from the G750L, G230M, G140L, and G140M gratings, taken at the same times. These spectra show that the derived fluxes show no discontinuities from 1200Å  to 1 micron. We also inspected and measured major line features in these wavelengths, including H$\alpha$, Mg II 2800Å, and C IV, Si IV, N V, and Ly$\alpha$ in the far UV. The H$\alpha$ line is strong and blended with \[N II\], so is not useful for studying the absorption (although it clearly is present in the form of asymmetry of the shortward side of the peak). The far UV lines have been discussed in detail by several other authors, and we discuss below possible correlation with the varying Balmer and He I absorption in the G430 spectra. The C IV profile shows many absorption components shortward of line centre, and these appear as a single smooth profile in the low dispersion spectra. However, our G430M spectra show clearly that the H$\beta$ absorption is a single broad feature and not resolvable into sharp components as seen in the UV resonance lines. This too relates to our discussion below. We measured the absorption FWHM values where possible - i.e. the He I absorption, and the deep absorption profiles in the 1999 and 2000 spectra. The He I line is consistent with a value of 460 km s$^{-1}$ for all cases, while the Balmer absorption is 340 km s$^{-1}$ in the G430L spectra and 420 km s$^{-1}$ in the July 2000 G430M spectrum. Smoothing the G430M spectrum to the resolution of the G430L does not alter the FWHM value, so the profiles are resolved in all spectra. Discussion ========== The presence of Balmer and He I $\lambda$3888 absorption indicates the presence of relatively high density and low ionisation outflowing material. Furthermore, the outflow is apparently connected to the variations in the continuum flux. While the H$\beta$ absorption has been noted before in low continuum states, it has not been isolated well from extended line emission, or correlated with the nuclear variations. We find that there is an asymmetry in the Balmer profiles at all nuclear flux states, that may be measured as an outflow (P Cygni) absorption, and that in fact the absorbed flux is largest when nucleus is in a high state. We also find that the velocity of the outflow is highest in the high nuclear state. The He I absorption shows a similar correlation but at lower outflow velocities. While the Balmer line measures in the high nuclear states depend on assuming the broad profile is symmetrical (Figure 3), and as the broad profiles do on other occasions have considerable blue-ward asymmetry, we may be wary of these measured values. However, the correlation with nuclear flux, the close agreement among 3 Balmer lines, and the changes in the (broad-component-free) He I line, suggest the effects are real. There are outflow absorptions seen in the far-UV resonance lines, that have been discussed in detail (e.g. Weymann et al 1997, Crenshaw et al 2000, Kraemer et al 2001). Most of these absorbers are narrow and do not change by much, if at all, and may arise in the clouds similar to those responsible for the narrow emission lines, from an extended region outside the BLR. However, there are some broader absorbers in the UV lines - in particular component D+E in Kraemer et al (2001), which has velocity -490 km s$^{-1}$ and FWHM 435 km s$^{-1}$. This component has a high density of absorbing material and may give rise to Balmer absorption too. The H$\beta$ EW of 3.2 (July 2000) implies a column of 1.5 x 10$^{14}$cm$^{-2}$, while the EW for He I of 1.4 (May 2000) implies 1.8 x 10$^{14}$cm$^{-2}$ if they are associated with the UV component D+E responding to changes in the ionising continuum. It seems likely that this UV absorber is the same as that causing the low velocity strong Balmer absorption in the nuclear low state. However, the connection with the higher velocity Balmer absorption, and the lower velocity He I absorption is not clear. In the higher nuclear states, the Balmer and He I velocities increase, as seen in Figure 4. These Balmer profiles do not appear to be composed of two or more components, and the complex absorption spectra in the UV resonance lines do not appear to include such changes. On the other hand the He I absorber shows much less change in velocity. Thus, association of the Balmer and He I absorbers with components of the highly ionised species of C IV, Si IV, N V is unclear. The FUSE spectrum of NGC 4151 in a low flux state shows smooth broad absorption profiles in O VI. These will be discussed in detail by Kriss et al (2002), but for this discussion, we assume they arise in an accelerating flow from the central disk, as discussed for NGC 3516 by Hutchings et al (2001). NGC 4151 is often noted as being a marginal Sy 1 type and the outflow models for the NLR gas suggest that the line of sight lies close to the edge of the opening cone of the ionising radiation (see e.g. Crenshaw et al 2001). This means that whatever is defining the cone lies close to the line of sight. This is generically referred to as the obscuring torus. It is very reasonable to propose that the nuclear activity is causing some erosion of the edge of the torus, and this may be where the outflows we see in H and He I arise. The radiation effects that drive the flow will vary with the nuclear flux, and lead to the velocity/flux correlation we see in Figure 4, with some time delay. A sudden drop in nuclear flux may leave a weakening broad absorption profile that lasts until the high velocity flow has dispersed: this is possibly what we see in the June 1999 profiles, since UV spectra from a few months earlier show the nuclear flux to be much higher. By contrast, the low state of June 2000, was preceded by low flux in April 2000. The lower outflow velocity seen in the He I absorption must be significant. It suggests an acceleration in a cooling medium, as in stellar winds. It is interesting that Anderson and Kraft (1969) saw the same velocity difference, reinforcing our conclusion that the He I absorption is not significantly blended with the neighbouring \[Ne III\] emission line. Anderson (1974) reports structure within the He I absorption, which is comparable to the noise in his spectra (and may also involve variable off-nuclear contamination). This is not resolvable in our G430L spectra. However, we note that the ten times higher resolution G430M spectra of H$\beta$ show very smooth absorption profiles (see Fig 2). The flow velocities are similar to those seen in the NLR and the associated sharp absorptions in the UV resonance lines. They are smaller than the higher velocity flows seen in the inner NLR (Hutchings et al 1999), even without the projection effects that must apply to emission line clouds. They are also somewhat smaller than the flow velocities seen in massive star winds (e.g. Fullerton et al 2000). The flux from the nucleus of NGC 4151 is in the range of 100 to 10$^5$ times that of an OB star, so the radiation pressure would be similar at a distance of 3.5x10$^{15}$ cm, or 0.001 pc, if we want the same process to apply. The time lag over this distance is of order one day, which places it within the BLR by the echo mapping results. However, we see no unambiguous evidence that the higher velocity flow shows up in the UV resonance lines, which are the principal drivers of stellar winds. Thus, it is possible that the flow reported here may arise in a more distant location, and that some other force than central radiation may drive it. It thus seems possible that the outflow arises in a high density region (such as the torus edge) which is heated by both the nuclear radiation and outflowing material. The mild heating and acceleration we see may arise by entrainment in the biconical outflow. In very high nuclear states (not sampled in the data in this paper), the broad emission line profile changes asymmetrically, and perhaps the low ionisation outflow velocity is higher, so that it may be more difficult to identify it in the line profiles. It will be instructive to continue to monitor the nuclear visible spectrum through higher nuclear flux states. The special NGC 4151 line of sight geometry may be a valuable clue on the origins of the nuclear outflows. Figure captions 1\. Average low dispersion (G430L) nuclear spectrum of NGC 4151 from data January 1998 to June 2000, normalized to the continuum defined by line-free regions. The principal emission lines, and He I 3888Å, are identified, redshifted by 1000 km s$^{-1}$. The spectra exclude all light beyond $\sim$0.15 arcsec from the nucleus. 2\. Comparison of the nuclear spectra from the G430M grating, from high and low nuclear flux states. Note that the H$\beta$ absorption is a single broad feature and not composed of several narrow absorptions, as seen in the UV resonance lines. 3\. Normalised Balmer profiles in velocity space, flux-scaled to match their broad line components. The narrow line components are relatively stronger in the later spectra, when the nuclear flux was low. In these, the \[O III\] $\lambda$4959 line is truncated for easier viewing. Similarly, the \[O III\] $\lambda$4363 and \[Si II\] emissions, near H$\gamma$ and H$\delta$ respectively, are truncated. A symmetrical broad emission profile has been sketched in to outline the Balmer absorption. This absorption is seen to agree well between the three Balmer lines. The average centre of the symmetrical broad profiles is marked, which is somewhat longward of the nominal NGC 4151 redshift of 1000 km s$^{-1}$. The numbers in parentheses next to the dates give the nuclear flux at 4800Å  in units of 10$^{-14}$ erg Å$^{-1}$ cm$^{-2}$ sec$^{-1}$. 4\. Absorption line measures from Table 1 plotted against continuum flux at 4800Å. Note the trend to higher outflow velocity with increasing nuclear flux. While the absorption equivalent widths drop, the absorbed flux rises with increasing nuclear flux. The circled points refer to the high velocity wing of the absorption in the June 1999 spectrum. The lines are fits to the data as labelled, except for H$\beta$ absorption, where the points are joined. References Anderson K.S., and Kraft R.P., 1969, ApJ, 158, 859 Anderson K.S., 1974, ApJ, 189, 195 Boksenberg A., et al, 1995, ApJ, 440, 151 Crenshaw D.M. et al, 2000, AJ, 120, 1731 Fullerton A.W. et al, 2000, ApJ, 538, L43 Hutchings J.B. et al 1998, ApJ, 492, L115 Hutchings J.B. et al 1999, AJ, 118, 2101 Hutchings J.B. et al 2001, ApJ, 559, 173 Kaiser M.E. et al, 2000, ApJ, 528, 260 Kraemer S.B. et al, 2001, ApJ, 551, 671 Kriss G.A. et al 2002 (in preparation) Nelson C.H. et al, 2000, ApJ, 531, 257 Peterson B.M. et al, 1998, PASP, 110, 660 Pedlar A., et al, 1993, MNRAS, 263, 471 Sergev S.G., Pronik V.I., and Sergeeva E.A., 2001, ApJ, 554, 245 Schurch N.J. and Warwick R.S., 2002, astro-ph/0204057 (MNRAS in press) Weymann R.J., Morris S.L., Gray M.E., Hutchings J.B., 1997, ApJ, 483, 717 [rcccccccccc]{} 15 Jul 1997 &0645&M, 50 &6.e-14 &0.6 &1.3e-13 &-1850 &-1200 &- &- &1.18 Jan 1998 &0822&L, 0.1 &7.e-14 &0.4 &1.4e-13 &-2000 &-1200 &-530 &0.6 &1.110 Feb 1998 &0855&L, 0.1 &6.e-14 &0.6 &1.4e-13 &-2200 &-1100 &-580 &0.8 &1.31 Jun 1998 &0966&L, 0.1 &2.5e-14 &0.8 &1.1e-14 &-1600 &-750 &-365 &1.5 &1.44 Jun 1999 &1334&L, 0.1 &7.e-15 &2.6 &2.0e-14 &-920 &-460 &-370 &1.8 &4.3 &&&&6.2 &4.3e-14 &-2250 24 May 2000 &1689&L, 0.1 &1.e-14 &1.8 &4.0e-14 &-1135 &-605 &-495 &1.4 &5.02 Jul 2000 &1728&M, 0.2 &6.e-15 &3.2 &7.0e-14 &-990 &-500 &- &- &- [^1]: Guest user, Canadian Astronomy Data Centre, which is operated by the HIA, NRC of Canada [^2]: Based on observations made with the NASA/ESA Hubble Space Telescope. STScI is operated by AURA Inc, under NASA contract NAS5-26555.

--- abstract: 'The frustrated pyrochlore magnet Yb$_2$Ti$_2$O$_7$ has the remarkable property that it orders magnetically, but has no propagating magnons over wide regions of the Brillouin zone. Here we use inelastic neutron scattering to follow how the spectrum evolves in cubic-axis magnetic fields. At high fields we observe in addition to dispersive magnons also a two-magnon continuum, which grows in intensity upon reducing the field and overlaps with the one-magnon states at intermediate fields leading to strong renormalization of the dispersion relations, and magnon decays. Using heat capacity measurements we find that the low and high field regions are smoothly connected with no sharp phase transition, with the spin gap increasing monotonically in field. Through fits to an extensive data set we re-evaluate the spin Hamiltonian finding dominant quantum exchange terms, which we propose are responsible for the anomalously strong fluctuations and quasiparticle breakdown effects observed at low fields.' author: - 'J. D. Thompson,$^{1}$ P. A. McClarty,$^{2,3}$ D.Prabhakaran,$^{1}$ I. Cabrera,$^{1}$ T. Guidi,$^{2}$ and R.Coldea$^{1}$' bibliography: - 'library.bib' title: | Quasiparticle Breakdown and Spin Hamiltonian of the\ Frustrated Quantum Pyrochlore Yb$_2$Ti$_2$O$_7$ in Magnetic Field --- The lattice of corner-shared tetrahedra realized in cubic A$_2$B$_2$O$_7$ pyrochlores and AB$_2$O$_4$ spinels, is a canonical lattice to explore correlated magnetism in the presence of strong geometric frustration effects. In the strongly spin-orbit coupled rare earth pyrochlores, experiment has uncovered materials offering a tremendously rich spectrum of magnetic behavior. Notable examples include classical spin ice physics as in the rare-earth titanates (Ho/Dy)$_2$Ti$_2$O$_7$ where Ising antiferromagnetism leads to an emergent classical electrostatics at low temperatures [@Castelnovo2012] and “order-by-disorder” in XY antiferromagnets where thermal and quantum fluctuations lift a large frustration-induced degeneracy resulting in unconventional magnetic order as in Er$_2$Ti$_2$O$_7$ [@Savary2012; @Zhitomirsky2012; @Savary2012b; @Ross2014]. Currently, much of the interest in this field concentrates on a handful of materials that seem to fall outside a semiclassical understanding of these systems. The pyrochlore Yb$_2$Ti$_2$O$_7$ [@Blote; @Yasui2003; @Bonville2004; @Gardner2004; @Ross2009; @Thompson2011a; @Ross2011a; @Chang2012; @Ross2012a; @Applegate2012a; @PhysRevB.93.064406; @hayre2013thermodynamic; @Pan2014; @Lhotel2014; @Chang2014; @Gaudet; @Pan2015; @Robert2015; @Tokiwa2016; @yaouanc2016novel], where Kramers Yb$^{3+}$ ions behave as effective spin 1/2 moments, is quite unique in its behavior: in high applied magnetic fields dispersive magnons were observed [@Ross2011a], which are apparently replaced by a broad continuum of scattering at zero field [@PhysRevB.93.064406] despite the presence of ferromagnetic order. This exotic behavior is not yet understood. To make progress one would like to know i) how the broad scattering continuum in zero field originates from quantum fluctuations, whether those fluctuations are also present at high field and, if so, how they manifest themselves, ii) how the sharp magnons “disappear” over a wide range of the Brillouin zone as the field is lowered. Here we experimentally answer those questions by studying the behavior in a magnetic field applied along the cubic \[001\] direction, which has not been explored in detail before and which, we will show, allows for a transparent interpretation of the phase diagram and evolution of the spectrum in a magnetic field. The experiment also allows us to re-visit the parametrization of the magnetic exchange, which is a critical ingredient for any future theoretical understanding. The magnetic ground state of Yb$_2$Ti$_2$O$_7$ has ferromagnetic polarization along one of the cubic axes and moments canted towards the local $\langle 111\rangle$ axes [@Yasui2003; @Chang2012; @PhysRevB.93.064406; @yaouanc2016novel]. The transition to this magnetic order is observed in the range 0.2-0.26 K depending on precise synthesis conditions, and is absent altogether in some samples [@Blote; @Ross2012a]. Here we report inelastic neutron scattering (INS) and heat capacity studies on large single crystals with a sharp magnetic transition near $T_{\rm N}=0.214(2)$ K, similar in behavior to that observed in the “best” crystals [@Chang2012; @Arpino2017], so we believe the magnetic properties are representative of the high-purity limit. We find clear evidence that quantum fluctuations are present even at the highest fields probed (9 T) and become progressively stronger upon lowering field. Using an extensive data set on the high-field magnon dispersions for two distinct field orientations combined with magnetization data we re-evaluate the spin Hamiltonian with freely-refined nearest-neighbor exchange parameters and two $g$-tensor terms. We find a spin Hamiltonian that is more “quantum” than suggested by earlier studies [@Ross2011a], and we propose that such dominant quantum exchange terms may provide the mechanism to explain the unexpectedly strong dispersion renormalization and magnon decay effects at low fields, unique among all quantum pyrochlore magnets studied so far. {width="\textwidth"} The spin dynamics in a 6.3 g single crystal of Yb$_2$Ti$_2$O$_7$ were probed using the neutron spectrometer LET [@Bewley2011] at the ISIS neutron source, at a temperature of $0.15$ K and in magnetic fields up to 9 T along the \[001\] axis (for more details see [@sm]). An estimated non-magnetic background was subtracted from the raw neutron data and intensities were corrected for absorption effects. Unless explicitly stated otherwise, field values quoted throughout refer to the externally applied fields $\mu_0 H_{\rm app}$, whereas the spin-wave calculations are performed for the corresponding demagnetization-corrected (internal) fields $\mu_0 H_{\rm int}$ [@sm]. We first discuss the results in high field where the spectrum is dominated by strongly dispersive modes. Fig. \[fig:1\]d) shows representative data at 5 T collected with neutrons of incident energy $E_i=2.5$ meV for a fixed sample orientation that probed the scattering at low energies for wavevectors near ($\bar{2}00$). Sharp modes are observed, and a representative lineshape profile is shown in Fig. \[fig:1\]f), which reveals four well-defined peaks, whose positions were extracted using Gaussian fits (solid line). By fitting similar energy scans extracted from a volume of data collected for multiple sample orientations an extensive data set on the dispersion relations was obtained for many reciprocal space directions (shown in Figs. \[fig:HoraceSlices\] and \[fig:HoraceSlices2\] panels a-d) and m-p) in the Supplemental Material [@sm]). The observed sharp modes are physically attributed to magnons originating on the four sublattices of the pyrochlore structure, which can be regarded as an FCC Bravais lattice of corner-shared tetrahedra. Following [@Ross2011a] we compare the observed dispersion relations to spin-wave modes of a nearest-neighbor Hamiltonian with four symmetry-allowed, anisotropic exchange terms and two $g$-tensor terms ($g_{\parallel}$ and $g_{\perp}$, along and transverse to the local three-fold $\langle 111 \rangle$ axes) to describe the Zeeman interaction. The exchange Hamiltonian reads [@Ross2011a] $$\begin{aligned} {\cal H}_{\rm Exchange} & = & \sum_{\langle ij \rangle} \big\{ J_{zz} \mathsf{S}^z_i \mathsf{S}^z_j - J_{\pm} \left( \mathsf{S}^+_i \mathsf{S}^-_j + \mathsf{S}^-_i \mathsf{S}^+_j \right) \nonumber \\ &+& J_{\pm\pm} \left( \gamma_{ij} \mathsf{S}^+_i \mathsf{S}^+_j + \gamma_{ij}^{*} \mathsf{S}^-_i \mathsf{S}^-_j \right) \nonumber \\ &+& J_{z\pm} \left[ \mathsf{S}^z_i \left( \zeta_{ij} \mathsf{S}^+_j + \zeta_{ij}^{*} \mathsf{S}^-_i \right) + \left( i \leftrightarrow j \right) \right] \big\}, \label{eq:Ham}\end{aligned}$$ where the sum extends over all nearest-neighbor $ij$ pairs counted once ($\gamma$ and $\zeta$ are geometric terms given in [@sm]). $\mathsf{S}$ denotes a spin-1/2 operator with components expressed in a local frame where the $z$-axes point along the local \[111\] direction. Following [@Ross2011a] we calculate the spin-wave spectrum at high fields after finding the mean-field ground state numerically. Varying any of the six parameters separately ($J_{zz},J_{\pm},J_{\pm\pm},J_{z\pm},g_{\parallel},g_{\perp}$) modifies the equilibrium spin orientation, this affects the spin-wave expansion of the Hamiltonian and in turn the dispersion relations, with the consequence that the parameters are all strongly coupled making it challenging to obtain the values independently in an unbiased way. Strongly coupled parameters generically occur when the spin Hamiltonian does not have rotational symmetry around the field axis, as is the case here [@note]. To provide enough constraints we have extracted over 550 dispersion points combined from our data in field along $[001]$ and previous data in field along $[\bar{1}10]$ from [@Ross2011a] (the two data sets provide complementary projections of the $g$-tensor and exchange matrices) and also magnetization data near saturation (this further constrains the absolute $g$-values). Using this extended data set a free fit to the six parameters converged to a unique solution (for details see [@sm]) $$\begin{aligned} J_{zz} = & ~0.026(3)~{\rm meV}, &J_{\pm\pm} = 0.048(2)~{\rm meV}, \nonumber \\ J_{\pm} = & ~ 0.074(2)~{\rm meV}, &J_{z\pm} = -0.159(2)~{\rm meV}, \nonumber \\ g_\parallel = & \!\!\!\!\!\! 2.14(3), &~~g_\perp = 4.17(2).\label{eq:pars}\end{aligned}$$ We find one dominant coupling, the “transverse” exchange $J_{z\pm}$, all other exchanges are much smaller. As a consistency check of the applicability of the spin-wave approximation we have verified that the dispersions at 5 and 9 T are described by the same parameters. This confirms that 5 T is a sufficiently high field that dispersion renormalization effects beyond linear spin wave order are negligible. The above Hamiltonian provides an excellent description of all the data available (compare Figs. \[fig:1\]d and g), Figs. \[fig:HoraceSlices\] and \[fig:HoraceSlices2\] panels a-d) with e-h) [@sm]. The earlier exchange parameters proposed in [@Ross2011a], deduced assuming a larger $g$-tensor anisotropy, did not fit the \[001\] field data well \[compare Fig. \[fig:HoraceSlices\]a-d) with i-l)\]. The revised parameters fit well the dispersions data for [*both*]{} field directions, and furthermore reproduce THz data on the zone-centre magnon energies at high field \[see Fig. \[fig:THz\]\] and the most recent estimate of the $g$-tensor anisotropy $g_{\perp}/g_{\parallel}$ deduced from crystal-field studies [@Gaudet]; the exchange parameters are also consistent with a recent parameterization of the zero-field quasielastic diffuse scattering pattern at higher temperature (0.4 K)[@Robert2015]. A semi-classical analysis of this Hamiltonian puts the system in a canted ferromagnetic phase, as seen experimentally. However, the system is located very close in parameter space to a phase boundary with an ordered XY phase [@sm]. This fact may prove to be of significance [@jaubert2015multiphase] in understanding the anomalously large fluctuation effects at low field (discussed below). The INS data in high field contains, in addition to sharp one-magnon modes, also a weak scattering continuum in an energy range corresponding to twice the one-magnon energies. This is illustrated in Fig. \[fig:1\]h) at 7 T. The strong signal in the range 1.3-2.2 meV is due to one-magnon excitations. Note the broad signal in the range 2.75-4 meV, not present at low field (filled circles, 0.21 T). The magnetic character of this continuum scattering is confirmed by a strong field dependence, its energy boundaries move in magnetic field at twice the rate compared to the extreme energies of one-magnon states \[see Fig. \[fig:1\]i)\] and its integrated intensity increases strongly upon decreasing field \[see Fig. \[fig:ContinuumField\]\]. Those features are characteristic for two-magnon excitations. If the Hamiltonian does not have rotational invariance around the field as is the case here, then zero-point quantum fluctuations are present at all fields and reduce the ordered moment from its maximum value and magnetization saturation is reached only asymptotically (as shown in Fig. \[fig:Magnetization\]). In the presence of such zero-point fluctuations neutrons can also scatter by simultaneously creating two magnons that share the energy and momentum transfer, thus appearing as a continuum contribution in the INS. Upon lowering the field zero-point quantum fluctuations are expected to grow, the magnetization to decrease and the two-magnon scattering intensity to increase, as indeed observed. Having established the physical origin of the continuum scattering we note that its intensity is underestimated compared to one-magnon states and there is more intensity at the lower boundary than predicted by a non-interacting spin-wave model (solid line in Fig. \[fig:1\]h), implying that magnon-magnon interactions are important for describing the lineshapes quantitatively. Upon decreasing field the one-magnon energies decrease linearly and the two-magnon continuum boundaries decrease at twice that rate, see Fig. \[fig:1\]i). Qualitative changes in the spectrum occur when the highest-energy magnon branch overlaps with the continuum, expected to occur near 2.25 T. Fig. \[fig:1\]c) shows data well below this field at 1.5 T, the highest-energy magnon can no longer be distinguished from the continuum (suggesting strong one$\rightarrow$two-magnon decay processes) and the dispersion bandwidth of the lower three modes is strongly renormalized (suppressed), suggesting strong interactions between one and two-magnon states even when overlap does not occur. Upon further lowering the field to 0.5 T, Fig. \[fig:1\]b), the three sharp modes appear squeezed into a single, gapped, almost dispersionless branch followed by strong continuum scattering at higher energies At 0.21 T a clear gapped sharp mode can still be observed near 0.22 meV \[see Fig. \[fig:1\]e) black circles\], and at zero field (red symbols) there is a relatively broad maximum near 0.15 meV and a continuum lineshape extending up to $1.5$ meV. Such dramatic quasi-particle breakdown effects over a large part of the Brillouin zone are very unusual in three-dimensional ordered magnets and demonstrate anomalously strong quantum fluctuations. To obtain complementary information on the lowest-energy magnetic excitations and evolution of the spin gap in magnetic field we have performed AC heat capacity measurements down to 0.1 K and fields up to 1.5 T $\parallel$ \[001\] on a 9.7 mg rectangular single crystal (for details see [@sm]). In zero field, a sharp anomaly is observed near 0.214(2) K \[see Fig. \[fig:2\] top trace, blue symbols\]. This anomaly is attributed [@Chang2012] to the transition to canted ferromagnetic order with spontaneous ferromagnetic polarization along a cubic axis. We observe that in a very small applied field of 0.05 T (green trace) the anomaly is significantly reduced and a broad hump appears at higher temperatures near 0.25 K. Increasing the field to 0.1 T (cyan trace) the anomaly is almost completely suppressed and has disappeared at 0.125 T, where only a broad Schottky feature is observed, which moves to higher temperature upon increasing the field. Plotting the location of the sharp anomaly on a field-temperature phase diagram in Fig. \[fig:1\]j) shows that the phase transition from paramagnet to the low-temperature canted ferromagnet exists only over a very small field range, above which no phase transition boundary exists. This can be naturally understood as follows. The transition to the canted ferromagnet spontaneously picks the direction of the ferromagnetic polarization along one of the cubic axes (6 domains) and the canting of moments is uniquely determined for each site in relation to the orientation of the local 3-fold axes. In an external magnetic field along \[001\], the direction of the ferromagnetic polarization is picked by the field $--$ the paramagnet and the ordered phase have the same symmetry $--$ so there is no longer a need for a phase transition. If the transition in zero field were continuous, one would expect it to occur only at strictly zero field. If it were first order, as is believed to be the case here, one would expect it to survive for a small, but finite field range, which is fully consistent with our data. The phase diagram in Fig. \[fig:1\]j) shows that in \[001\] field the canted ferromagnet and the high-field-polarized state are continuously connected, without encountering a phase transition. This is further supported by the field dependence of the heat capacity data. The strong suppression of heat capacity at low temperatures and presence of a Schottky anomaly that moves to higher temperatures upon increasing field are characteristic signatures of a spin gap that increases monotonically upon increasing field. To parameterize this behavior we have compared the rising part of the $C(T)$ data (up to the broad peak) to the form expected for a two-level system (for details of fits see [@sm]). The extracted gap is plotted in Fig. \[fig:2\](inset) and shows a rapid increase in field. The monotonic gap increase is consistent with no phase boundary between the canted ferromagnet and the high-field-polarized state in Fig. \[fig:1\]j). ![Heat capacity as a function of temperature, $C(T)$, at various fields $\parallel$ \[001\], vertical dashed line at $T_{\rm N}$ is a guide to the eye. (inset) Gap extracted from the heat capacity data as described in the text, dashed line is guide to the eye.[]{data-label="fig:2"}](fig2.pdf){width="\columnwidth"} To summarize, we have reported high-resolution INS measurements of the spin dynamics as a function of applied magnetic field in the frustrated pyrochlore Yb$_2$Ti$_2$O$_7$. We have observed direct evidence for coherent quantum fluctuations manifested in a field-dependent two-magnon scattering continuum at high fields, and strong magnon decay and dispersion renormalization effects at low fields. Through fits to an extensive data set of the high-field magnon dispersions and magnetization data we have re-evaluated the spin Hamiltonian finding dominant quantum exchange terms, which we propose are responsible for the anomalously strong quantum fluctuation effects observed at low field. It may be the case that those effects may be understood as being related to the close proximity of the material to the semi-classical phase boundary between canted ferromagnet and an order-by-disorder antiferromagnet, or potentially a quantum spin liquid phase. We propose that the experimental strategy employed here of probing quasi-particle breakdown in fields comparable to the exchange strength will be a useful experimental tool in future studies of quantum spin liquid candidates. Our results emphasize the need for theoretical efforts to understand the quantum phase diagram of effective spin-1/2 pyrochlore Hamiltonians away from the well-understood semiclassical limit. JDT was supported by the University of Oxford Clarendon Fund Scholarship and NSERC of Canada. PAM acknowledges support from an STFC Keeley-Rutherford fellowship. Work in Oxford was partially supported by EPSRC (UK) through grants No. EP/H014934/1 and EP/M020517/1. Supplemental Material ===================== Here we provide additional technical details on 1) the anisotropic spin exchange Hamiltonian on the pyrochlore lattice, 2) the linear spin-wave formalism to derive the magnon dispersion relations at high fields, 3) time-of-flight inelastic neutron scattering (INS) experiments to probe the spin dynamics, 4) the fitting procedure to extract the Hamiltonian parameters from high-field one-magnon dispersion relations and magnetization, and comparison with THz data, 5) semiclassical calculations for the fitted Hamiltonian: mean-field ordering temperature, Curie-Weiss temperature, mean-field phase diagram, proximity of Yb$_2$Ti$_2$O$_7$ to the phase boundary between canted ferromagnetic and antiferromagnetic orders, 6) observation of a two-magnon scattering continuum at high field and comparison with spin-wave predictions, 7) magnon dispersion renormalization and decay effects when one- and two-magnon phase spaces overlap, 8) ac heat capacity measurements and spin gap dependence in field, 9) magnetization measurements. S1. Spin Hamiltonian ==================== Pyrochlore Lattice and Local Frame {#sec:lattice-frame} ---------------------------------- The pyrochlore lattice may be viewed as an FCC Bravais lattice with a tetrahedral basis. The basis is taken to be $\bm{r}_1 = (a/8)(1,1,1)$, $\bm{r}_2 = (a/8)(1,-1,-1)$, $\bm{r}_3 = (a/8)(-1,1,-1)$, $\bm{r}_4 = (a/8)(-1,-1,1)$ with coordinates given in the global (cubic axes) frame, where $a$ is the cubic lattice parameter. A general site on the pyrochlore lattice labelled $i$ is located at position $\bm{R}_i + \bm{r}_n$, where $\bm{R}_i$ is the FCC lattice position and $n=1$ to $4$ is the sublattice index. A natural choice of coordinate frame on the pyrochlore lattice has the local $z$ axis at each site along the local $[111]$ direction, distinguished by the local oxygen environment. In particular, we take the local right-handed frames for the four sublattices with unit vectors as $$\begin{aligned} \bm{\hat{z}}_1 & = \frac{1}{\sqrt{3}} (1,1,1), ~~~ \bm{\hat{z}}_2 & = \frac{1}{\sqrt{3}} (1,-1,-1) \nonumber\\ \bm{\hat{z}}_3 & = \frac{1}{\sqrt{3}} (-1,1,-1),~~~ \bm{\hat{z}}_4 & = \frac{1}{\sqrt{3}} (-1,-1,1) \label{eq:z}\end{aligned}$$ and $$\begin{aligned} \bm{\hat{x}}_1 & = \frac{1}{\sqrt{6}} (-2,1,1), ~~~ \bm{\hat{x}}_2 & = \frac{1}{\sqrt{6}} (-2,-1,-1) \nonumber\\ \bm{\hat{x}}_3 & = \frac{1}{\sqrt{6}} (2,1,-1), ~~~ \bm{\hat{x}}_4 & = \frac{1}{\sqrt{6}} (2,-1,1). \label{eq:x}\end{aligned}$$ We define a set of rotation matrices to transform vector components from the global frame to those in the local frame of each sublattice as $R_{n}^{x\beta}=\bm{\hat{x}}_n\cdot\bm{\hat{e}}_\beta$, $R_{n}^{y\beta}=\bm{\hat{y}}_n\cdot\bm{\hat{e}}_\beta$, $R_{n}^{z\beta}=\bm{\hat{z}}_n\cdot\bm{\hat{e}}_\beta$. Here $\bm{\hat{e}}_\beta$ stands for a unit vector axis of the global frame, $\beta=x,y$ or $z$. Anisotropic Exchange Couplings ------------------------------ The most general, symmetry-allowed exchange matrix between nearest-neighbor spins on the pyrochlore lattice is uniquely defined by four independent terms [@McClarty2009; @Thompson2011a; @Ross2011a; @Onoda2011a]. Following Ref. [@Ross2011a] the exchange Hamiltonian expressed in the local frames has the form in (\[eq:Ham\]) where $$\zeta = \left( \begin{array}{cccc} 0 & -1 & e^{i\pi/3} & e^{-i\pi/3} \\ -1 & 0 & e^{-i\pi/3} & e^{i\pi/3} \\ e^{i\pi/3} & e^{-i\pi/3} & 0 & -1 \\ e^{-i\pi/3} & e^{i\pi/3} & -1 & 0 \end{array} \right) \nonumber$$ and $\gamma=-\zeta^{*}$. In a more abbreviated notation, ${\cal H}_{\rm Exchange} = \sum_{\langle ij \rangle} \mathcal{J}_{ij}^{\alpha\beta} \mathsf{S}^\alpha_i \mathsf{S}^\beta_j$ where throughout we use the notation convention that repeated axes indices are summed over. The spin operators are understood to operate on the effective spin one-half doublet at each magnetic site that characterizes the ground state of the crystal field split $J=7/2$ multiplet of the Kramers Yb$^{3+}$ ions. We may neglect the influence of excited crystal field levels because the crystal field gap to the first excited level far exceeds the exchange coupling strengths [@Gaudet]. We neglect the long-range dipolar interaction because it is small compared to the exchange. The quality of our fits (presented below) is such that we do not require the inclusion of exchange couplings beyond nearest neighbor. The total Hamiltonian including the Zeeman coupling to an external magnetic field is $${\cal H} = {\cal H}_{\rm Exchange} + {\cal H}_{\rm Zeeman} \label{eq:Hamiltonian}$$ with $${\cal H}_{\rm Zeeman} = - \mu_{\rm B} \mu_0 H_{\rm int}^{\mu} \sum_{i} g^{\mu\nu}_{n} \mathsf{S}_{i}^{\nu}, \label{eq:zeeman}$$ where the magnetic field components $\mu_{0}H_{\rm int}^{\mu}$ are given in the global frame and $g_n^{\mu\nu}$ is the $g$-tensor for sublattice $n$. S2. Spin Wave Theory {#sec:SWT} ==================== Starting from the spin Hamiltonian with effective spin one-half moments in (\[eq:Hamiltonian\]), we first compute the ground state in a $[001]$ applied magnetic field within mean field theory, assuming that the magnetic structure can always be described using the primitive tetrahedral structural cell (magnetic propagation vector $\bm{q}=\bm{0}$), as is the case semiclassically at both zero and very large fields. The orientation of the effective spin one-half moments within the ground state ${\bm{\hat{u}}}_n$ allows us to specify a local quantization frame on each sublattice, where $\bm{\hat{\tilde{z}}}_n={\bm{\hat{u}}}_n$, $\bm{\hat{\tilde{x}}}_n={\bm{\hat{u}}}_n \times [1,1,1]/\parallel {\bm{\hat{u}}}_n \times [1,1,1] \parallel$ and $\bm{\hat{\tilde{y}}}_n=\bm{\hat{\tilde{z}}}_n \times \bm{\hat{\tilde{x}}}_n$. In addition to the rotation matrices $R_n^{\alpha\beta}$ defined above, which rotate from the global to the local frame, rotations from the local frame to the quantization frame are given by $\bar{R}_n^{\mu\alpha}$ where $\mu$ is the quantization frame index, $\alpha$ is the local frame index and $n$ is the sublattice label. In the local quantization frame, we write the spin operators in terms of Holstein-Primakoff (HP) bosons and expand around the classical ground state in powers of $1/S$. $$\begin{aligned} \tilde{\mathsf{S}}^{z}_i & = S - a^\dagger_i a_i \\ \tilde{\mathsf{S}}^+_i & = \tilde{\mathsf{S}}^x_i+ i \tilde{\mathsf{S}}^y_i = \sqrt{ 2S - a^\dagger_i a_i } a_i \approx \sqrt{2S} a_i \\ \tilde{\mathsf{S}}^-_i & = \tilde{\mathsf{S}}^x_i- i \tilde{\mathsf{S}}^y_i = a_i^\dagger \sqrt{ 2S - a^\dagger_i a_i }\approx \sqrt{2S} a_i^\dagger.\end{aligned}$$ The commutation relations satisfied by the boson operators are $$\left[ a_i , a_j^\dagger \right] = \delta_{ij}.$$ Other commutators vanish. The leading term in this expansion is the classical Hamiltonian $$\mathcal{H}_0 = S(S+1) \sum_{\langle ij \rangle} \mathcal{J}_{ij}^{\alpha\beta} u_n^\alpha u_{n'}^\beta,$$ where the subscripts $n$ and $n'$ are the sublattice indices of the $i$ and $j$ sites, respectively. Here the term of order $S^2$ is the mean-field exchange energy and the term of order $S$ comes from symmetrizing the exchange Hamiltonian quadratic in the boson operators. We write the boson operators in Fourier space. The interaction matrix is then $$\mathcal{J}_{nn'\bm{k}}^{\alpha\beta} = \frac{1}{N} \sum_{ij} \mathcal{J}_{ij}^{\alpha\beta} \exp\left[ i \bm{k}\cdot (\bm{R}_i - \bm{R}_j +\bm{r}_n -\bm{r}_{n'}) \right],$$ where $N$ is the number of FCC lattice sites. In the quantization frame, the interaction matrix becomes $\tilde{\mathcal{J}}_{nn'\bm{k}}^{\mu\nu}=\bar{R}_n^{\mu\alpha}\bar{R}_{n'}^{\nu\beta}\mathcal{J}_{nn'\bm{k}}^{\alpha\beta}$. The terms linear in the bosons vanish in the ground state computed from the minimum semiclassical energy and fluctuations around the ground state may be computed from the quadratic Hamiltonian $\mathcal{H}_2$. We introduce operators $$\begin{aligned} \bm{\sigma}^x_{\bm{k}} \equiv \sqrt{ \frac{S}{2} } \left( \bm{a}_{\bm{k}} + \bm{a}^\dagger_{-\bm{k}} \right) \nonumber \\ \bm{\sigma}^y_{\bm{k}} \equiv i \sqrt{\frac{S}{2} } \left( \bm{a}_{\bm{k}} - \bm{a}^\dagger_{-\bm{k}} \right) \nonumber \end{aligned}$$ with $\bm{a}_{\bm{k}}=(a_{1\bm{k}},a_{2\bm{k}},a_{3\bm{k}},a_{4\bm{k}})$ so that $$\begin{aligned} \mathcal{H}_2 & = \sum_{\bm{k}} \left( \begin{array}{cc} (\bm{\sigma}^x_{\bm{k}})^\dagger & (\bm{\sigma}^y_{\bm{k}})^\dagger \end{array} \right) \left( \begin{array}{cc} \bm{P}^x_{\bm{k}} & \bm{T}_{\bm{k}} \\ \bm{T}^\star_{\bm{k}} & \bm{P}^y_{\bm{k}} \end{array} \right) \left( \begin{array}{c} \bm{\sigma}_{\bm{k}}^x \\ \bm{\sigma}_{\bm{k}}^y \end{array} \right), \nonumber \\ & \equiv \sum_{\bm{k}} \left( \begin{array}{cc} (\bm{\sigma}^x_{\bm{k}})^\dagger & (\bm{\sigma}^y_{\bm{k}})^\dagger \end{array} \right) \boldsymbol{H} \left( \begin{array}{c} \bm{\sigma}_{\bm{k}}^x \\ \bm{\sigma}_{\bm{k}}^y \end{array} \right), \label{eq:H2}\end{aligned}$$ where the sum runs over all wavevectors $\bm{k}$ in the first Brillouin zone of the FCC lattice and where $$\begin{aligned} P^x_{nn'\bm{k}} & = \mathcal{R}_{n} \delta_{nn'} + \tilde{\mathcal{J}}_{nn'\bm{k}}^{xx} + \frac{\mu_{\rm B}}{2} \mu_0 H_{\rm int}^{\alpha}R_n^{\beta\alpha} g^{\beta} \bar{R}_n^{z\beta} \delta_{nn'} \\ P^y_{nn'\bm{k}} & = \mathcal{R}_{n} \delta_{nn'} + \tilde{\mathcal{J}}_{nn'\bm{k}}^{yy} + \frac{\mu_{\rm B}}{2} \mu_0 H_{\rm int}^{\alpha}R_n^{\beta\alpha} g^{\beta} \bar{R}_n^{z\beta} \delta_{nn'} \\ T_{nn'\bm{k}} & = \tilde{\mathcal{J}}_{nn'\bm{k}}^{xy}\end{aligned}$$ and $\mathcal{R}_n = -\sum_{n'} \tilde{\mathcal{J}}_{nn',\bm{k}=\bm{0}}^{zz} $. Since the $g$-tensor is diagonal in the local frame, with the form $g={\rm diag}(g_{\perp},g_{\perp},g_{\parallel})$, we write $g$ with a single index denoting the diagonal element. The diagonalization of the quadratic Hamiltonian $\mathcal{H}_2$ in (\[eq:H2\]) to find the magnon wavefunctions and energies must preserve the boson commutation relations, which take the form $$\left[ \bm{\sigma}_{\bm{k}}^\alpha,\left( \bm{\sigma}_{\bm{k}}^\beta \right)^\dagger \right]= S \bm{\eta}^{\alpha\beta}$$ with $$\bm{\eta} \equiv \left( \begin{array}{cc} 0 & -i \bm{I} \\ i\bm{I} & 0 \end{array} \right),$$ where $\bm{I}$ is the $4\times 4$ identity matrix. Then the spin wave energies $\omega_{m\bm{k}}$ for modes $m=1$ to 4 are the positive semidefinite set of eigenvalues of the matrix $2\bm{\eta H}$ and the right eigenvectors $\bm{v}_{m\bm{k}}$ of $\bm{\eta H}$ preserve the commutation relation among the $\bm{\sigma}$ operators provided that $\bm{v}^\dagger \bm{\eta} \bm{v}=\bm{g}$, where $\bm{g} = {\rm diag}(1,1,1,1,-1,-1,-1,-1)$. The neutron scattering intensity is proportional to $$\left\vert F(\bm{Q}) \right\vert^{2} \sum_{\alpha,\beta} \left( \delta_{\alpha\beta}- \hat{Q}_{\alpha} \hat{Q}_{\beta} \right) S^{\alpha\beta}\left( \bm{Q},\omega \right),$$ where $\bm{Q}=\bm{k}_i-\bm{k}_f$ is the wavevector transfer, $\bm{k}_i$ ($\bm{k}_f$) is the incident (final) neutron wavevector, $\omega$ is the neutron energy transfer and $F(\bm{Q})$ is the magnetic form factor of Yb$^{3+}$ ions, assumed spherically symmetric. The indices $\alpha$ and $\beta$ are components of the moments in the global frame. We evaluate the inelastic part of the scattering function at zero temperature $$\begin{aligned} & S^{\alpha\beta}_{\rm inelas}\left( \bm{Q},\omega \right)= \nonumber \\ & ~~~~\sum_{n,n'} \sum_E \langle 0 \vert \mathsf{J}^{\alpha}_n(-\bm{Q}) \vert E \rangle \langle E \vert \mathsf{J}^{\beta}_{n'}(\bm{Q}) \vert 0 \rangle \delta\left( \omega - \omega_E\right), \label{eq:Sinelas}\end{aligned}$$ where the second sum is over all excited states $|E\rangle$ of energy $\omega_E$ above the ground state $|0\rangle$. The physical moment $\mathsf{J}^{\alpha}_n$ is related to the effective spin one-half moment components in the quantization frame $\tilde{\mathsf{S}}_n^\gamma$ through $\mathsf{J}^{\alpha}_n = \Gamma_n^{\alpha\gamma} \tilde{\mathsf{S}}^{\gamma}_n$ where we have defined $\Gamma_n^{\alpha\gamma}\equiv R_n^{\beta\alpha} g^{\beta} \bar{R}_n^{\gamma\beta}$. One-magnon excited states are accessed via spin fluctuations with transverse polarization ($\mu,\nu=x$ or $y$) in the quantization frame, and the corresponding inelastic scattering function is $$\begin{aligned} S^{\alpha\beta}_{\rm inelas, 1M}&\left( \bm{Q},\omega \right) = \frac{S}{2} \sum_{n,n'} \sum_{\mu\nu =x,y} \Gamma_n^{\alpha\mu} \Gamma_{n'}^{\beta\nu} \times \nonumber \\ & \sum_{m} v^{\mu \dagger}_{mn,-\bm{k} } v^{\nu}_{m n' \bm{k}} e^{-i\bm{\tau}\cdot \bm{r}_{nn'} } \delta(\omega-\omega_{m\bm{k}}),\end{aligned}$$ where $\bm{r}_{nn'}=\bm{r}_n-\bm{r}_{n'}$ and the eigenvectors are written as $v^{\mu}_{mn\bm{k}}$, indexed by the transverse spin deviation component $\mu=x,y$. Here $\bm{k}$ is the wavevector transfer reduced to the first Brillouin zone, i.e. $\bm{Q}=\bm{\tau}+\bm{k}$ where $\bm{\tau}$ is closest reciprocal lattice vector to $\bm{Q}$. Two-magnon excited states are accessed via fluctuations polarized longitudinally in the quantization frame. To see this we consider the longitudinal scattering function for the effective spin in the quantization frame, which has an analogous form to (\[eq:Sinelas\]) $$S_{{\rm eff},nn'}^{zz}(\bm{Q},\omega) \equiv \sum_E \langle 0 \vert \tilde{ \mathsf{S}}^{z}_n (-\bm{Q}) \vert E \rangle \langle E \vert \tilde{ \mathsf{S}}^{z}_{n'} (\bm{Q}) \vert 0 \rangle \delta(\omega - \omega_E), \label{eq:longitudinal}$$ where $$\begin{aligned} \tilde{ \mathsf{S}}^{z}_n (\bm{Q}) & = \sqrt{N}S \delta(\bm{k}) e^{-i\bm{\tau}\cdot \bm{r}_n} \nonumber \\ & - \frac{1}{\sqrt{N}}\sum_{\bm{q},\bm{q'}} \delta(\bm{k}-\bm{q}+\bm{q'}) a_{n\bm{q}}^\dagger a_{n\bm{q'}} e^{-i\bm{\tau}\cdot \bm{r}_n} \label{eq:szz}\end{aligned}$$ is the component of the effective spin along the quantization direction. The first term leads to a contribution of to order $S^2$ in the structure factor (\[eq:longitudinal\]), which is the elastic, Bragg scattering in the ordered phase. The second term in (\[eq:szz\]) leads to an inelastic contribution of order $S^0$ in the structure factor (\[eq:longitudinal\]), due to magnon pair creation/annihilation. We introduce $$\begin{aligned} \Lambda_{mn\boldsymbol{q}} & \equiv (v^{x }_{mn\boldsymbol{q}} -i v^{y }_{mn\boldsymbol{q}})/2 \nonumber \\ \bar{\Lambda}_{mn\boldsymbol{q}} & \equiv (v^{x }_{mn\boldsymbol{q}} + i v^{y }_{mn\boldsymbol{q}})/2. \label{eq:lambda}\end{aligned}$$ It is also convenient to introduce $\Omega_{\bm{q}\bm{q'}nmm'} \equiv \Lambda_{mn\bm{q}} \bar{\Lambda}_{m'n\bm{q'}} + \bar{\Lambda}_{mn\bm{q}} \Lambda_{m'n\bm{q'}} (1-(2-\sqrt{2})\delta_{mm'}\delta(\bm{q'}-\bm{q}))$ such that the inelastic part of (\[eq:longitudinal\]) becomes $$\begin{aligned} S&_{{\rm eff, inelas}, nn'}^{zz}(\bm{Q},\omega) = \frac{1}{N} \sum_{\bm{q},\bm{q'}} \sum_{m\geq m'} \Omega^{\star}_{\bm{q}\bm{q'}nmm'} \Omega_{\bm{q}\bm{q'}n'mm'} \times \\ & ~~~~~e^{-i\bm{\tau}\cdot \bm{r}_{nn'}} \delta(\bm{k}-\bm{q}-\bm{q'}) \delta\left(\omega-\omega_{m\bm{q}}-\omega_{m'\bm{q'}}\right).\end{aligned}$$ The two-magnon scattering from the physical moment is given by $$S^{\alpha\beta}_{\rm inelas,2M}\left( \bm{Q},\omega \right) = \sum_{n,n'} \Gamma_n^{\alpha z} \Gamma_{n'}^{\beta z} S_{{\rm eff, inelas},nn'}^{zz}(\bm{Q},\omega),$$ which we have evaluated numerically using a Monte Carlo integration over the Brillouin zone in order to compare with the experimental data. S3. Inelastic Neutron Scattering Experiments {#InelasticNeutrons} ============================================ Experimental Details {#neutronexperimentdets} -------------------- {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} A single crystal of Yb$_2$Ti$_2$O$_7$ was grown as described in Ref. [@Prabhakaran2011] using a four-mirror optical floating-zone furnace (Crystal System Inc.) in an argon rich atmosphere with a growth rate of 1-2 mm/h, similar to the conditions reported in Ref. [@Chang2012]. Due to the argon atmosphere, the as-grown crystal was oxygen deficient and dark in color. In order to improve the oxygen stoichiometry, the as-grown crystal was annealed at 1200$^{\circ}$C for 5 days under oxygen flow atmosphere and the crystal become transparent and almost colorless. Several pieces were cut from this larger crystal and used for all the different measurements reported here. ![Schematic of the ($hk0$) horizontal scattering plane indicating where dispersion maps were extracted from Horace scans (red lines) and plotted in Figs. \[fig:HoraceSlices\] and \[fig:HoraceSlices2\] panels a-d) and m-p). Solid black lines show intersections with the FCC Brillouin zone boundaries.[]{data-label="fig:BZmap"}](S4.pdf){width="0.8\columnwidth"} The spin dynamics was probed using the direct-geometry time-of-flight chopper spectrometer LET at the ISIS neutron source [@Bewley2011] using a cylindrical-shaped $6.3$ g single crystal of Yb$_2$Ti$_2$O$_7$ aligned in the $(hk0)$ horizontal scattering plane. The sample was cooled using a dilution fridge insert with a base temperature of 0.15 K where all measurements were performed, this temperature is well below the zero-field magnetic ordering transition temperature of 0.214(2) K \[see phase diagram in Fig. \[fig:1\]j)\]. Magnetic fields up to 9 T were applied along the crystal \[001\] axis using a vertical cryomagnet. Applied fields were corrected for demagnetization effects as discussed in Sec. S9. To avoid complexities associated with multiple (ferro)magnetic domains the sample was cooled to base temperature in a finite magnetic field (0.21 T) to ensure a single magnetic domain with ferromagnetic polarization along the field. The zero-field data was collected last, after reducing the field to zero from finite values. The inelastic scattering was probed for neutrons with incident energy $E_i=1.34$, $2.5$, 4 and $6.3$ meV, with measured energy resolutions of $0.023(1)$, $0.055(2)$, $0.094(1)$ and $0.220(5)$ meV (Full Width Half Maximum), respectively, on the elastic line. For the majority of the measurements LET was operated in multi-repetition mode, allowing the mapping of the inelastic scattering with $E_i=1.34$, 2.5 and 6.3 meV simultaneously. For an overview of how the spin dynamics evolves as a function of field, measurements were performed up to 9 T for a fixed sample orientation that probed the scattering at low energies for wavevectors near ($\bar{2}00$), with a typical counting time of $2.5$ h per setting; those results are summarized in Fig. \[fig:FieldSlices2.5\] (top and every other subsequent row). To obtain an extended data set on the wavevector-dependence of the spin dynamics in the Brillouin zone the inelastic scattering was measured at a selection of fixed applied fields (0, 0.21, 1.5 and 5 T) for a range of sample rotation angles around the vertical \[001\] direction spanning $90^\circ$ in steps of $1^\circ$ (Horace scan), each position counted for approximately $7$ minutes. This gave access to a wide range of wavevectors in the ($hk0$) plane and typical intensity maps extracted from this data volume are shown in Fig. \[fig:HoraceSlices\]a-d). We estimated the non-magnetic energy-dependent background using Fig. \[fig:1\]i) as a guide to indicate where no magnetic scattering is expected at different fields, i.e. at high fields no magnetic signal is expected below the one-magnon gap and in the interval between the one- and two-magnon energy ranges (shaded regions in Fig. \[fig:1\]i), whereas at low field no magnetic signal is expected at very high energies. The estimated non-magnetic background was subtracted from the raw intensities to obtain the purely magnetic signal, which was afterwards corrected for neutron absorption effects using a numerical Monte Carlo routine for a tilted cylindrical sample (assuming an inverse velocity dependence of the neutron absorption cross-section). Throughout this paper, wavevectors are given as $(h,k,l)$ in reciprocal lattice units of $2\pi/a$ of the structural cubic unit cell with lattice parameter $a=10.026$ Å [@Gardner2004]. ![Dispersion maps along various directions in the ($hhl$) plane for a field of 5 T along \[$\bar{1}10$\] (first column is INS data from [@Ross2011a]), compared with spin-wave calculations for the refined Hamiltonian in (\[eq:pars\])(middle column) and the model in Ref [@Ross2011a]. Both calculations include the magnetic form factor and convolution with a finite energy resolution of 0.09 meV (FWHM). Spin-wave calculations using the applied field value of 5 T or an estimated demagnetization-corrected field of $\mu_0H_{\rm int}=4.93$ T gave essentially indistinguishable results (the latter is plotted in the figure).[]{data-label="fig:RossComparison"}](S5.pdf){width="\columnwidth"} {width="\textwidth"} High-field Spin-wave Dispersions -------------------------------- Representative intensity maps for some high-symmetry directions in the ($hk0$) plane are shown in Figs. \[fig:HoraceSlices\] and \[fig:HoraceSlices2\] panels a-d) and m-p). In those figures the row of panels immediately below the raw data, e-h) and q-t), are the spin-wave calculation for the fitted Hamiltonian (\[eq:pars\]), and demonstrate an excellent agreement with the data along all directions probed. For comparison, the subsequent row of panels, i-l) and u-x), show the calculation for the parameters in Ref. [@Ross2011a], in this case systematic differences are seen in terms of shifts of the dispersion modes between the data and the model predictions, compare for example Fig. \[fig:HoraceSlices\]a-d) with i-l), from which we conclude that the earlier proposed model cannot account for the observed dispersions in field along \[$001$\], whereas the current refined model can account for all the dispersion data, even for field along \[$\bar{1}10$\], see Fig. \[fig:RossComparison\]. S4. Fitting Procedure to Determine the Spin Hamiltonian ------------------------------------------------------- In this section we give details of the fitting procedure used to obtain the Hamiltonian parameters from experimentally measured magnon dispersion relations at high fields. From the INS data in applied fields of $5$ and $9$ T $\parallel[001]$ we extracted energy scans \[as in Fig. \[fig:1\]e)\] and determined mode energies $\omega_m$, where the subscript $m=1$ to 4 labels the modes at a given ($h,k,l$) in order of increasing energy. For some scans it was not possible to detect four distinct modes, but by continuity with neighboring regions in reciprocal space, we were able to determine the appropriate labels of all the visible modes. This way we obtained a list of dispersion points $(h,k,l,\omega,m)$. In order to provide multiple and independent constraints on the fits, we also extracted dispersion points from the INS data reported in [@Ross2011a] in a field of $5$ T$\parallel[\bar{1}10]$. We also included in the fitting procedure the requirement for the model to reproduce the measured magnetization value at $\mu_0H_{\rm int}=6.86$ T $\parallel$ \[001\], at which field the magnetization is almost saturated (see Sec. S9). By computing the magnon dispersion relations within linear spin wave theory, we carried out a least squares minimization allowing all six parameters of the Hamiltonian to vary independently. The minimization was based on a simulated annealing algorithm that was run several thousand times, initialized every time with different random starting parameters, for an extensive sampling of the parameter space to detect the global minimum. As a first test, we fixed the $g$ tensor ratio $g_{\perp}/g_{\parallel}=2.4$ as in [@Ross2011a] and the minimization procedure using only the $[\bar{1}10]$ dispersion data set converged, within error, to the parameter values given in that paper. We also found that the $5$ T$\parallel[001]$ dispersions data alone were insufficiently constraining to determine both $g$ factor elements. In other words, the minimization procedure on this data set alone, without a constraint on the $g$-factor components leads to almost degenerate solutions with different sets of parameters. However, by taking the $5$ T$\parallel [\bar{1}10]$ dispersion data, and the $5$ and $9$ T $\parallel[001]$ dispersions data sets, together with the $7$ T$\parallel [001]$ magnetization, we obtained a unique, optimum solution, with a clear global minimum in the goodness of fit, that can account for all the data. We emphasize that when not all the neutron data and magnetization constraints are included in the fitting procedure, there is a family of almost degenerate solutions to the optimization problem. The existence of a set of parameters that nearly satisfy all the fitting constraints is a consequence of the strong correlation between certain parameters. In Fig. \[fig:chisquared\]d), we show the magnitude of the correlation coefficient between the six parameters. All $6$ parameters are significantly correlated with particularly large correlations between the two $g$ factor parameters, which can be traced back to the constraint on the magnetization. In the absence of the magnetization constraint, the strongest correlation is between $J_{zz}$ and $g_{\parallel}$. The refined Hamiltonian parameters obtained using all the above mentioned data points are given in (\[eq:pars\]), where the quoted uncertainties were estimated as follows. Each dispersion point ($h,k,l,w,m$) had an energy uncertainty $\sigma$, which ranged between $0.01$ and $0.02$ meV. The larger error bars are for the magnon energies determined visually from the digitized INS intensity maps reported in Ref. [@Ross2011a]. We created a list of $10^4$ sets of mode energies, where each set is shifted from the original by a gaussian random number times the standard deviation of the energy of that mode. We carried out the least square minimization to determine the optimum model parameters for each such set of slightly shifted mode energies. The errors on the model parameters were then extracted from the resulting distributions of those fitted parameters. [*Field Misalignment*]{} $-$ In the INS measurements the magnetic field was slightly misaligned relative to the $[001]$ crystallographic axis by $1.0(1)^{\circ}$. We have verified that assuming perfect alignment $\parallel[001]$ or including this small misalignment resulted in essentially identical values for the optimized parameters, within the uncertainties listed in (\[eq:pars\]). ![THz spectrum as a function of magnetic field along \[001\] from Ref. [@Pan2014], symbols correspond to excitation energies observed using different experimental geometries and polarizations as listed in the legend. Solid lines are the calculated spin-wave energies at the zone center, $\omega_{1-4,\bm{k}=\bm{0}}$, for the model Hamiltonian in (\[eq:pars\]), assuming quoted field values have negligible demagnetization corrections.[]{data-label="fig:THz"}](S7.pdf){width="0.95\columnwidth"} Comparison with THz data ------------------------ As a further test of the determined Hamiltonian we have verified consistency also with the recently reported THz spectrum for field $\parallel[001]$. THz spectroscopy measures the long wavelength ($\bm{k}=\bm{0}$) excitation spectrum and in total four main families of peaks are observed using different experimental setups, see Fig. \[fig:THz\]. The energies of the observed modes are well reproduced (solid lines) for fields above 3 T by the spin wave energies calculated for the Hamiltonian in (\[eq:pars\]) with no adjustable parameters. At lower fields linear spin wave theory fails owing to the proximity of single and two-magnon states (as discussed later in Sec. S7). We have also verified the applicability of the THz selection rules to the observation of spin wave modes. THz spectroscopy measures the complex transmittance $t(\omega)$ of initially polarized THz radiation through a single crystal sample in the presence of a static magnetic field. The transmittance is proportional to the imaginary part of the susceptibility $\chi''_{\alpha\beta}(\omega)$, which we calculate using a random phase approximation. The Faraday geometry with the incident THz pulse in the $z$ direction and linearly polarized in the $x$ direction and the Voigt geometry with the field in the $y$ direction, where all axes labels refer to the global (cubic axes) frame. In the case of the Faraday geometry, Ref. [@Pan2014] concentrated on the transmittance of circularly polarized radiation. We find that the spectrum in the Voigt geometry, obtained from $\chi''_{xx}(\omega)$ and $\chi''_{yy}(\omega)$, is sensitive to three magnon modes - the lowest and two highest modes, as indeed observed (blue symbols in Fig. \[fig:THz\]). In the Faraday geometry, the lowest mode should appear in the right circularly polarized channel, from $\chi''_{xx}(\omega)-\chi'_{xy}(\omega)$, and the highest mode in the left circularly polarized channel, $\chi''_{xx}(\omega)+\chi'_{xy}(\omega)$. However, the second-to-lowest mode should not be visible using the Faraday geometry although, apparently, this mode is visible in the data. However, if we allow for a small field misalignment away from the $[001]$ direction then the second-to-lowest mode exhibits a peak in the THz spectrum, making the Faraday spectrum also consistent with the model predictions. S5. Semiclassical Properties of the Spin Hamiltonian ==================================================== Quantum Fluctuations {#sec:DeltaS} -------------------- The role of quantum fluctuations on the size of the ordered moment may be computed within linear spin wave theory. The departure of the effective spin one-half moment from its fully available value $S=1/2$ is given by $$\Delta S \equiv \frac{1}{4N} \sum_{i} \langle a_{i}^{\dagger}a_{i} \rangle = \frac{1}{4N} \sum_{\bm{k}} \sum_{n,m} {\Lambda}^{*}_{mn\bm{k}} {\Lambda}_{mn\bm{k}} \label{eq:DeltaS}$$ in terms of the rotated eigenvectors $\Lambda$, defined in (\[eq:lambda\]). At 5 T we obtain a relatively small reduction of 2%, providing at least a partial consistency check to justify the applicability of the linear spin-wave approach to parameterize the dispersion relations and extract the Hamiltonian. We note however that even though quantum fluctuations are small at those fields, they are still present and are ultimately responsible for the observation of a weak, but finite intensity two-magnon continuum in addition to the dominant one-magnon excitations in INS (to be discussed in detail in Sec. S6). ![Spin reduction $\Delta S$ as a function of applied field calculated using linear spin-wave theory, eq. (\[eq:DeltaS\]). \[fig:DeltaS\]](S8.pdf){width="0.95\columnwidth"} Total Moment Sum Rule --------------------- The total moment sum rule is a single constraint on the structure factor computed for a model with spin $S$ $$\frac{1}{N} \sum_{\bm{k},\alpha} \int S_{\rm eff}^{\alpha\alpha}(\bm{k},\omega)~d\omega= S(S+1) \label{eq:sum_rule}$$ where the structure factor $S_{\rm eff}$ is computed for the effective spin one-half moment. This is to be distinguished from the experimental structure factor for Yb$_2$Ti$_2$O$_7$, which is computed for the true $\mathsf{J}$ moment in the ground state crystal field doublet. In this section, we compile various contributions to the total scattering sum rule for the exchange parameters determined from experiment. We concentrate on the 9 T magnon spectrum because, of all the measured fields, this one should be the closest match to linear spin wave theory. In the previous section we discussed the leading order quantum correction to the ordered spin, which is $\Delta S =0.0030$ at 9 T. The total elastic, one- and two-magnon contributions are listed in Table \[table:TMSR\]. The combined total differs from the sum rule in (\[eq:sum\_rule\]) by less than 0.25%, such small violations of the sum rule are generally expected in linear spin wave theory, higher order contributions in $1/S$ are generally needed to renormalize the intensities to agree with the sum rules. --------------------- --------- --------- -------- --------- Elastic 1M 2M Total \[0.5ex\] Intensity 0.241 0.50244 0.0051 0.74854 --------------------- --------- --------- -------- --------- : Calculated contributions to total (effective spin) sum rule at $9$ T. The elastic (Bragg) contribution is $(S-\Delta S)^2$ where $\Delta S$ is the zero-point spin reduction in (\[eq:DeltaS\]). The single magnon (1M) and two magnon (2M) contributions are obtained from numerical integration of the transverse and longitudinal (effective spin) dynamical correlations, respectively. \[table:TMSR\] ![Semiclassical phase diagram in the space of anisotropic exchange parameters given in Eq. (\[eq:Ham\]). The origin of the phase diagram is a Coulomb phase at the classical level. Three long-ranged ordered phases appear in its vicinity: a splayed (canted) ferromagnet (FM) with spontaneous polarization along one of the cubic directions, the $\psi_4$(Palmer-Chalker) [@McClarty2009] antiferromagnetic state and a ground state with an accidental degeneracy in the $\Gamma_5$ manifold, resolved by quantum fluctuations into long-range ordered antiferromagnetic states $\psi_2$ (as in the ordered phase of Er$_2$Ti$_2$O$_7$) or $\psi_3$, both with only a discrete degeneracy [@McClarty2009]. All these phases have ordering wavevector $\bm{q}=\bm{0}$. Two points are marked on the phase diagram corresponding to the exchange values proposed by Ross [*et al.*]{} [@Ross2011a] (blue circle) and those obtained here (red square). The $X_1$ and $X_2$ coordinates are linear combinations of the primary couplings chosen such that the phase diagram includes both sets of Hamiltonian parameters and also the Coulomb phase at the origin. In particular, $X_1=\left\{-0.9198, -0.3834, -0.084, 0 \right\}$ and $X_2=\left\{0.349, -0.7233, -0.5204, 0.2904\right\}$ in the space of couplings $\left\{J_{zz}, J_{\pm\pm}, J_{\pm},J_{z\pm} \right\}$.[]{data-label="fig:PhaseDiagram"}](S9.pdf){width="0.95\columnwidth"} Mean-Field Phase Diagram ------------------------ The fit to the exchange parameters fixes the semiclassical ground state of Yb$_2$Ti$_2$O$_7$ to be a canted ferromagnet with an ordering wavevector $\bm{q}=\bm{0}$ and spontaneously chosen net polarization along one of the cubic axes. The moments are non-collinear, tilted towards the the local \[111\] axes. The mean field $T_c \approx 2.95$ K is far in excess of the observed transition temperature in the material. We have computed the ground state phase diagram for the exchange model Eq. (\[eq:Ham\]) in the vicinity of these exchange parameters in order to point out proximate phases. To this purpose, we choose a highly symmetric point in the space of anisotropic couplings that harbors a Coulomb phase at the classical level corresponding to couplings $J_{zz}=1$, $J_{\pm\pm}=0.5$, $J_{\pm}=0.25$ and $J_{z\pm}=-1/\sqrt{8}$ [@Benton2016]. We rescale these couplings so that $J_{z\pm}$ matches the value extracted from the experimental data namely $-0.162$ meV. Then we choose to plot the phase diagram in Fig. \[fig:PhaseDiagram\] in a plane through the space of couplings containing both the exchange values proposed by Ross [*et al.*]{} [@Ross2011a] (blue circle) and those obtained here (red square). Evidently, both sets of parameters place Yb$_2$Ti$_2$O$_7$ in the same phase, which is the same as the one determined experimentally. The principal difference between the two sets of exchange parameters is that the set determined in this work lies very close to the phase boundary with the $\psi_3$ state with antiferromagnetic order. While apparent from the phase diagram, one may confirm that the difference between the mean field energy of the ground state and the energy of the $\psi_3$ state is smaller for the exchange parameters determined in this work. Curie-Weiss Temperature ----------------------- The Curie-Weiss temperature obtained from a high temperature expansion for the Hamiltonian in (\[eq:Hamiltonian\]) is [@Ross2011a] $$\begin{aligned} & 2k_{\rm B} (2g_{\perp}^2 + g^2_{\parallel}) \Theta_{\rm CW} \\ & = \left( g_{\parallel}^2 J_{\rm{zz}} - 4g_{\perp}^2 \left( J_{\pm} + 2J_{\pm\pm} \right) - 8\sqrt{2} g_{\perp} g_{\parallel} J_{z\pm} \right).\end{aligned}$$ For the parameters in Ref. [@Ross2011a], $\Theta_{\rm CW}=312$ mK whereas we find $641$ mK for the parameters extracted in this study. This calculated value is to be compared with experimental values of $400$ mK [@Blote] and $750$ mK [@Hodges2001]. Magnon Decay Matrix Element --------------------------- The spin structure of Yb$_2$Ti$_2$O$_7$ is noncollinear for the entire range of fields explored in the experiment reported here. Then, in the local quantization frame there are terms for an isotropic exchange Hamiltonian that couple $x$ and $z$ components of the spins. As a consequence, the $1/S$ expansion in terms of Holstein-Primakoff (HP) bosons contains cubic interaction terms which take one magnon into two or [*vice versa*]{}. In our case, such terms arise as a consequence of noncollinearity [*and*]{} anisotropic exchange in the global frame. {width="\textwidth"} The spin wave dispersions, computed to quadratic order in HP bosons, are renormalized by interaction terms. In the case of cubic terms, there is a self-energy contribution $\boldsymbol{\Sigma}\left( \bm{k}, \omega_{\bm{k}} \right)$ to the spectrum coming from bubble diagrams. Schematically $$\boldsymbol{\Sigma}\left( \bm{k}, \omega_{\bm{k}} \right) \sim \sum_{\bm{q}} \frac{ \vert \Gamma\left( \bm{k}, \bm{q} \right)\vert^2 } { \omega_{\bm{k}} - \omega_{\bm{q}} - \omega_{\bm{k}-\bm{q}} + i0^+ }$$ where $\Gamma$ is the amplitude of the cubic vertex and for clarity we have omitted the individual mode labels of the three magnon energies, which in principle can each belong to a different dispersion mode $m$. There are singularities in the integrand whenever the single magnon energy $\omega_{\bm{k}}$ overlaps with the two-magnon continuum $\omega_{\bm{q}} + \omega_{\bm{k}-\bm{q}}$. When this is the case, magnon decay processes become kinematically allowed and one expects a renormalization of the spectrum and also magnons acquire a finite lifetime resulting in a broadening of the magnon peaks at the nominal energy $\omega_{\bm{k}}$. The extent of these effects depends strongly on the density of states of two magnon decays. In Yb$_2$Ti$_2$O$_7$ the estimated threshold field below which the two-magnon continuum overlaps with the highest-energy one-magnon dispersion mode is $2.3$ T and below this field experiments observe a substantial broadening of the highest energy magnon lineshape in the overlap region (see Sec. S7). S6. Two-Magnon Scattering Continuum at High Fields ================================================== The INS intensity maps at high field showed a distinct continuum intensity signal at energies corresponding to twice the one-magnon energies, identified with neutrons scattering by creating a pair of magnons. The intensity map in Fig. \[fig:Intensity6.3meV\]e) shows this scattering contribution at 6 T, note the weak band of scattering intensity centered around 3 meV, which shifts in energy upon varying field (compare with data in panels at other fields). Energy scans observing directly the shift in energy and intensity increase upon lowering field are shown in Fig. \[fig:ContinuumField\]a) (shaded areas). A quantitative comparison with spin-wave theory for non-interacting magnons in Fig. \[fig:1\]h)(solid line) shows that the intensity of the continuum scattering relative to the one-magnon intensity is underestimated, and also that the continuum lineshape profile is different (more intensity at the lower boundary), suggesting that inclusion of magnon-magnon interactions may be required to account for those features. Apart from an overall renormalization of the relative continuum intensity its strong increase upon lowering field is well captured by spin-wave theory (solid line) in Fig. \[fig:ContinuumField\]b). The wavevector-dependence of the continuum intensity at 5 T is shown in Fig. \[fig:ContinuumSlice\]a), the intensity has a local minimum near ($\bar{2}00$), and this feature is well captured by the spin-wave prediction for the two-magnon intensity (panel b). ![(a) Energy scans through the intensity maps in Fig. \[fig:Intensity6.3meV\] ($h=[-4,0]$) showing the two-magnon scattering continuum increasing in intensity and being displaced to lower energies upon decreasing field from top to bottom traces (shading emphasizes the magnetic intensity). For clarity traces are offset vertically (by +150) as a function of increasing field. Dashed lines indicate the linear field dependence of the extracted continuum boundaries. Intensities in the lower trace are multiplied by 0.5 to fit on the same scale. b) Experimentally-extracted energy integrated area of the continuum scattering (from scans such as in Fig. \[fig:1\]h) as a function of internal field and the corresponding spin-wave prediction for two-magnon scattering (solid line) times an overall scaling factor given in the legend.[]{data-label="fig:ContinuumField"}](S11.pdf){width="\columnwidth"} S7. Magnon Decay and Dispersion Renormalization at Intermediate Fields ====================================================================== An overview of the field-dependence of the spin dynamics is plotted in Fig. \[fig:FieldSlices2.5\] as a function of increasing field in the top and every other subsequent row, whereas the rows of panels immediately below show the spin-wave calculation. Good agreement is found above $\sim$$3$ T, below this field one- and two-magnon phase spaces overlap \[see Fig. \[fig:1\]i)\] and more complex behavior occurs. The energy scan in Fig. \[fig:EnergyScan\_MagnonDecay\]a) at 3 T shows four well-resolved sharp peaks followed by a weak scattering continuum (shaded area) centered near 1.5 meV. Upon lowering field to 2 T (panel b) the lower three peaks have shifted to lower energies, whereas the fourth peak has merged with the continuum with a lower boundary near 1 meV. We interpret this “disappearance” of the highest-energy magnon mode as being due to its spontaneous decay into two-magnon states. Upon lowering the field further to 1.5 T (panel c) the whole pattern shifts to lower energies and furthermore the energy spacing between the lower three peaks is clearly smaller than at 3 T (panel a). At 1 T (d) the peaks 1 and 2 have almost merged and the spacing 1-3 is reduced further. This magnon bandwidth narrowing is not captured by the (linear) spin-wave approach, which predicts almost field-independent bandwidths, compare Fig. \[fig:FieldSlices2.5\]c-d) with g-h). ![(a) Wavevector-dependence of the continuum scattering intensity in the $(hk0)$ plane at 5 T. Data comes from a Horace scan averaged over $E=[2.03,3.06]$ meV and $l=[-0.2,0.2]$. b) Corresponding intensity map of the two-magnon continuum scattering in spin-wave theory, with the same overall scale factor as in Fig. \[fig:ContinuumField\]. Solid lines show intersections with the FCC Brillouin zone boundaries.[]{data-label="fig:ContinuumSlice"}](S12.pdf){width="\columnwidth"} ![Energy scan illustrating magnon decay and dispersion renormalization effects below the field where one- and two-magnon phase spaces overlap. The spacing between the lowest three sharp modes (labelled 1-3) is significantly reduced as the whole group shifts to lower energies upon reducing field (top to bottom). At the same time the relative separation to the high-energy continuum scattering (shaded area) reduces, at 2 T (panel b) the fourth magnon mode has already “dissolved” in the continuum, which rapidly grows in intensity upon lowering field. Data in all scans comes from the intensity maps at the corresponding fields in Fig. \[fig:FieldSlices2.5\] for $h=[-2.2,-1.5]$ and $l=[-0.4,0.4]$. Filled triangles in bottom panel show 9 T data to illustrate the quality of the non-magnetic background subtraction.[]{data-label="fig:EnergyScan_MagnonDecay"}](S13.pdf){width="\columnwidth"} We attribute the dispersion renormalization to the increase of quantum fluctuations upon lowering field, also manifested in the continuum intensity now becoming comparable to that of the sharp modes. Physically, a reduction in the magnon bandwidth could be interpreted as a reduction in the kinetic energy that one-spin-flip states gain from coherently hopping across the lattice sites, or an effective inhibition of such coherent propagation due to the increased quantum fluctuations at low field. Given the close proximity between the sharp modes and the continuum boundary at those low fields it may be possible that the dispersion renormalization effects observed might be captured, at least partly, by including interactions between the one-magnon states and the higher-energy continuum scattering. Scattering Continuum in Zero Field ---------------------------------- The discrepancy between the linear spin-wave prediction and the data becomes even more dramatic in the region of very low fields. In zero field the spin-wave model predicts sharp modes in the range 0.21-0.61 meV, whereas the data shows a broad scattering continuum throughout this range with considerable scattering weight at lower energies and also extending up to 1.5 meV, compare Fig. \[fig:FieldSlices2.5\]a and e). The continuum scattering lineshapes are clearly apparent in Fig. \[fig:1\]e) (red symbols) with no clear sharp modes seen in this energy range, in clear contrast with the spin-wave prediction. An overview of how the spectrum evolves as a function of field for several wavevector directions in the ($hk0$) plane extracted from a Horace data volume is shown in Fig. \[fig:horace\_overview\]. Notice the contrast between the 5 T data (top row) dominated by well-defined sharp modes with a large gap, and zero field (bottom row) where an extended scattering continuum dominates. At a relatively small applied field of 0.21 T a clear sharp mode is stabilized near 0.22 meV for small $|\bm{Q}|$ see Fig. \[fig:horace\_overview\]h) (bottom left), but the continuum scattering up to 1.5 meV is essentially unchanged compared to zero field (compare with panel k). Continuum scattering is observed at all wavevectors probed, with some clear intensity modulations illustrated in Fig. \[fig:continuum\_horace\_slices\], the intensity is strongest near the ($hh0$) line for energies below $\sim$0.4 meV (panel a), at higher energies the intensity appears more uniformly distributed (panels b-c). The zero and low field spin dynamics is in sharp contrast with linear spin-wave theory, which would predict a spectrum dominated by dispersive sharp magnon modes, compare Figs. \[fig:horace\_overview\] and  \[fig:horace\_overview\_LSWT\] (bottom rows). We interpret this dramatic quasiparticle breakdown over a large part of the Brillouin zone as an indication that fluctuations become very strong in the low field regime, presumably due to the dominant “quantum” exchange term $J_{z\pm}$, with the consequence that a semiclassical linear spin-wave description becomes inadequate to capture the spin dynamics. S8. Heat Capacity Measurements {#heatcapacity} ============================== Heat capacity measurements were collected using a home-made setup based on the AC technique [@Rost; @Sullivan1968]. The sample was a flat-plate 9.7 mg single crystal of Yb$_2$Ti$_2$O$_7$ with approximate dimensions $1.85\times1.61\times0.45$ mm$^3$ cut from the same piece as the crystal used for the INS measurements. Cooling was provided by an Oxford Instruments Kelvinox25 dilution refrigerator, of base {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} temperature $\sim$$30$ mK, equipped with a $16$ T superconducting magnet, with the sample aligned such that the magnetic field was applied perpendicular to the sample plate, along the \[001\] crystal direction. The heat capacity setup contained a Lakeshore RX-$102$A-BR Ruthenium Oxide thermometer mounted on the sample using Apiezon N grease, which was then connected to a $5\times5\times0.150$ mm$^3$ $99.95\%$ pure Ag platform. A $120~\Omega$ strain gauge was connected to the bottom of the platform using GE $7031$ varnish and used as a heater to apply an oscillating temperature at a frequency of $5\times10^{-3}$ Hz. A $1$ cm $99.99\%$ pure Pt wire was used to establish a heat link between the platform and an OFHC Copper heat sink connected to the mixing chamber of the dilution fridge. The thermometer was calibrated down to $30$ mK in zero field against a calibrated Lakeshore RX-$102$B-CB thermometer. Field calibrations of the thermometer were performed using constant temperature magnetic field sweeps. The measured heat capacity was normalized into absolute units by calibration against the known specific heat of a standard sample measured in the same setup. The applied magnetic field values $\mu_0H_{\rm app}$ were corrected for demagnetization effects to obtain the net internal fields $\mu_0H_{\rm int}$ as discussed in Sec. S9. The heat capacity was measured at fixed applied field as a function of increasing temperature. Fig. \[fig:HC1\] displays the obtained heat capacity in zero field (blue symbols). The behavior is comparable to that reported in Ref. [@Chang2012] (red symbols) on a crystal where the sharp low-temperature anomaly has been identified with the onset of spontaneous canted ferromagnetic order. Systematic studies of samples of various purities have shown that more stoichiometric samples[@Ross2014; @Arpino2017] display a single sharp peak in the heat capacity (at temperatures up to 0.26 K), whereas samples believed to be affected by substantial structural disorder/stuffing/oxygen non-stoichiometry show rather different behavior, with only a broad peak or multiple peaks[@Ross2012a]. The presence of a single sharp peak in the heat capacity of our sample is indicative of a sharp transition to a well-developed magnetic order, suggesting that structural disorder effects are rather small and the magnetic behavior is representative of the high-purity limit. Fig. \[fig:2\] shows heat capacity measurements in applied magnetic field. Above $0.1$ T the sharp peak observed in zero field is completely suppressed, replaced by a broad Schottky feature [@Gopal1966]. Upon increasing field the low-temperature tail of the specific heat is progressively suppressed and the Schottky anomaly moves to higher temperatures, both are indications of a gap in the excitation spectrum, which increases upon increasing field. To capture this trend we compare the data to the behavior expected for a system with an excited level at energy $\Delta$ above the ground state, $$C(T)=R \left( \frac{\Delta}{k_{\rm B}T}\right)^2 \frac{e^{-\Delta/k_{\rm B}T}}{\left( 1+e^{-\Delta/k_BT}\right)^2}. \label{eq:two-level}$$ ![Specific heat as a function of temperature in zero magnetic field (blue symbols) compared to earlier reports [@Chang2012] (red symbols).[]{data-label="fig:HC1"}](S17.pdf){width="\columnwidth"} ![a-e) Specific heat as a function of temperature and magnetic field $\parallel [001]$. Solid lines show fits of the rising part of $C(T)$ to a two-level system, eq. (\[eq:two-level\]), solid green lines show the spin-wave prediction. Data points are the raw specific heat minus an estimate of the non-magnetic contribution obtained from measurements at high field (lower trace in Fig. \[fig:2\]). f) Same data as in d) plotted now as $\ln\left(CT^2\right)$ vs. $1/T$ to expose the near-linear dependence (dashed line) predicted by (\[eq:two-level\]) for $T\ll \Delta/k_{\rm B}$.[]{data-label="fig:fitsHC"}](S18.pdf){width="\columnwidth"} This form was fitted to the measured $C(T,B)$ data in the temperature region up to the broad peak maximum and avoiding the very low-temperature part, where quadrupolar contributions lead to a $1/T^2$ behavior, seen already in earlier measurements [@Blote]. The non-magnetic contribution was estimated from the measured $C(T)$ at high field (lowest trace in Fig. \[fig:2\]) when the spin gap is $\sim$0.4 meV, such that the population of thermally excited magnon states over the whole temperature range of the heat capacity measurements (up to $0.7$ K) is negligible. The fits are illustrated in Fig. \[fig:fitsHC\] ![Magnetization as a function of magnetic field along \[001\] (blue points, 1.8 K). Green points are lower temperature (0.09 K) data from Ref. [@Lhotel2014]. The red solid line is the zero-temperature mean-field calculation, which reproduces well the observed magnetization value near saturation. \[fig:Magnetization\]](S19.pdf){width="\columnwidth"} and give a good parameterization of the rising part of $C(T)$. The fitted pre-factor $R$ is systematically lower than the expected molar gas constant, a reduction would be expected as INS measurements \[see Fig. \[fig:1\]b)\] show a density of states that is not concentrated solely in a single level at the gap energy $\Delta$ as assumed by (\[eq:two-level\]), but is in fact extended over a wide energy range above the gap. The gap extracted from the heat capacity data is plotted in Fig. \[fig:2\](inset) and shows a monotonic increase in field. $\mu_0H_{\rm app}$(T) $0.05$ $0.075$ $0.1$ $0.125$ $0.15$ $0.21$ $0.30$ $0.5$ $1.5$ ------------------------ ---------- ------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ -- -- -- -- $\mu_0H_{\rm int}$ (T) $<0.001$ $0.0013(2)$ $0.007(1)$ $0.019(2)$ $0.035(3)$ $0.084(3)$ $0.165(4)$ $0.354(4)$ $1.330(5)$ $\mu_0H_{\rm app}$ (T) $0.21$ $0.5$ $0.75$ $1$ $1.5$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ ------------------------ --------- -------- --------- --------- -------- -------- -------- -------- -------- -------- -------- -------- -------- $\mu_0H_{\rm int}$ (T) $0.166$ $0.45$ $0.698$ $0.946$ $1.44$ $1.94$ $2.94$ $3.94$ $4.93$ $5.93$ $6.93$ $7.93$ $8.93$ As already shown by the INS data the low-field behavior of the excitations cannot be captured by a spin-wave approach, a scattering continuum dominates instead of sharp modes and moreover there is a large density of states at energies much below the predicted spin-wave gap in zero field of 0.2 meV, compare Fig. \[fig:FieldSlices2.5\]a) with f). The failure of linear spin-wave theory to capture the low-field behavior is also dramatically illustrated by the specific heat data. Fig. \[fig:fitsHC\]a-e) shows that the calculated spin-wave heat capacity $C_{\rm SW}$ (solid green curves) significantly underestimates the low-temperature heat capacity at low fields, as expected if excited states existed below the predicted spin-wave gap. The comparison also shows that upon increasing field the spin-wave calculation becomes progressively closer to the data, this is expected as in the limit of high enough fields where all the low-energy excitations are well-defined, sharp magnons, then we know from comparison to the INS data (at 5 T) that spin-wave theory provides a very good description of the low-energy states, compare Fig. \[fig:1\]d) and g). The comparison in Fig. \[fig:fitsHC\]a-e) is consistent with the expectation that $C_{\rm SW}$ approaches the measured $C(T)$ at high enough fields. S9. Magnetization Measurements {#magnetization} ============================== Magnetization data was collected in order to provide further constraints on the overall scale of the $g$-tensor. Measurements were performed at 1.8 K using a SQUID magnetometer (Quantum Design MPMS) on a near-rectangular 27.50(2) mg single crystal of Yb$_2$Ti$_2$O$_7$ with approximate dimensions $2.09\times1.81\times1.01$ mm$^3$ cut from the same crystal piece used for the heat capacity and INS experiments. The sample was aligned such that the magnetic field was applied normal to the largest sample face, along the $[001]$ crystallographic axis. Fig. \[fig:Magnetization\] shows the obtained magnetization curve, which gives a magnetic moment at the highest field probed of $\mu_0H_{\rm int}=6.86$ T of $1.745(4)\mu_{B}$ per Yb$^{3+}$ ion. Field values were corrected for demagnetization effects as discussed below. Demagnetization corrections --------------------------- The applied magnetic fields in the magnetization, specific heat and neutron scattering measurements were corrected for demagnetization effects to obtain the internal fields $H_{\rm int}=H_{\rm app}-NM$ where $N$ is the demagnetization factor and $M$ is the magnetization volume density. For the heat capacity and magnetization samples $N$ was calculated as $0.64$ and 0.49, respectively, using analytical results for a rectangular prism [@demag_prism]. The sample used for the INS measurements was a cylinder with the magnetic field applied at an angle $\theta\simeq32^\circ$ to the cylinder axis. Here the internal field was approximated considering only the projection of the demagnetization field along the applied field axis, giving an effective $N \simeq N_{\parallel}\cos^2\theta+N_{\perp}\sin^2\theta$, where the demagnetization factors for the directions along and transverse to the cylinder axis were calculated as $N_{\parallel}=0.13$ and $N_{\perp}=0.44$, respectively, using analytical results for a cylinder [@demag_cylinder]. The estimated internal fields for the heat capacity and neutron data are listed in Tables \[tab:demag\_hc\] and  \[tab:demag\_ins\], where we used as the reference magnetization curve $M$ vs. $\mu_0H_{\rm int}$ the reported data at 90 mK up to 4 T $\parallel[001]$ from [@Lhotel2014] supplemented with our own magnetization data extended up to 7 T at 1.8 K \[see Fig. \[fig:Magnetization\]\]. The quoted errors in the tables include an uncertainty in matching the absolute scales of the above two magnetization curves such that the estimated extrapolation of the 90 mK data to high fields overlaps with the 1.8 K data above 5 T. Since the neutron scattering data was collected at a slightly higher temperature of 150 mK where the magnetization is expected to be somewhat reduced compared to 90 mK in the limit of low fields, in the region of low fields the demagnetization corrections are therefore overestimated, so the quoted internal fields in Table \[tab:demag\_ins\] are to be interpreted as a lower bound and to become more accurate at high fields where the magnetization is less sensitive to temperature. Similarly, for the heat capacity temperature scans at constant applied field in Fig. \[fig:2\], the internal fields quoted in Table \[tab:demag\_hc\] are to be interpreted as the values at the lowest temperatures at the start of the scans, the actual internal fields are expected to increase towards the applied field value upon increasing temperature (as the magnetization decreases, so demagnetization corrections reduce).

--- author: - 'O. Cucciati' - 'B. C. Lemaux' - 'G. Zamorani' - 'O. Le Fèvre' - 'L. A. M. Tasca' - 'N. P. Hathi' - 'K-G. Lee' - 'S. Bardelli' - 'P. Cassata' - 'B. Garilli' - 'V. Le Brun' - 'D. Maccagni' - 'L. Pentericci' - 'R. Thomas' - 'E. Vanzella' - 'E. Zucca' - 'L. M. Lubin' - 'R. Amorin' - 'L. P. Cassarà' - 'A. Cimatti' - 'M. Talia' - 'D. Vergani' - 'A. Koekemoer' - 'J. Pforr' - 'M. Salvato' bibliography: - 'biblio.bib' date: 'Received - ; accepted -' title: 'The progeny of a Cosmic Titan: a massive multi-component proto-supercluster in formation at z=2.45 in VUDS[^1]' --- Introduction {#intro} ============ Proto-clusters are crucial sites for studying how environment affects galaxy evolution in the early universe, both in observations (see e.g. [@steidel05; @peter07; @miley08; @tanaka10; @strazzullo13]) and simulations (e.g. [@chiang17; @muldrew18]). Moreover, since proto-clusters mark the early stages of structure formation, they have the potential to provide additional constraints on the already well established probes on standard and non-standard cosmology based on galaxy clusters at low and intermediate redshift (see e.g. [@allen11; @heneka18; @schmidt09; @roncarelli15], and references therein). Although the sample of confirmed or candidate proto-clusters is increasing in both number (see e.g. the systematic searches in [@diener2013_list; @chiang2014_list; @franck16_CCPC; @lee2016_colossus; @toshikawa18_goldrush]) and maximum redshift (e.g. [@higuchi18_silverrush]), our knowledge of high-redshift ($z>2$) structures is still limited, as it is broadly based on heterogeneous data sets. These structures span from relaxed to unrelaxed systems, and are detected by using different, and sometimes apparently contradicting, selection criteria. As a non-exhaustive list of examples, high-redshift clusters and proto-clusters have been identified as excesses of either star-forming galaxies (e.g. [@steidel00; @ouchi05; @lemaux09; @capak11]) or red galaxies (e.g. [@kodama07; @spitler12]), as excesses of infrared(IR)-luminous galaxies [@gobat11], or via SZ signatures [@foley11_SZ] or diffuse X-ray emission [@fassbender11]. Other detection methods include the search for photometric redshift overdensities in deep multi-band surveys [@salimbeni09; @scoville13_env] or around active galactic nuclei (AGNs) and radio galaxies [@pentericci00; @galametz12], the identification of large intergalactic medium reservoirs via Ly$\alpha$ forest absorption [@cai16_method; @lee2016_colossus; @cai17_z23], and the exploration of narrow redshift slices via narrow band imaging [@venemans02; @lee14_NB]. The identification and study of proto-structures can be boosted by two factors: 1) the use of spectroscopic redshifts, and 2) the use of unbiased tracers with respect to the underlying galaxy population. On the one hand, the use of spectroscopic redshifts is crucial for a robust identification of the overdensities themselves, for the study of the velocity field, especially in terms of the galaxy velocity dispersion which can be used as a proxy for the total mass, and finally for the identification of possible sub-structures. On the other hand, if such proto-structures are found and mapped by tracers that are representative of the dominant galaxy population at the epoch of interest, we can recover an unbiased view of such environments. In this context, we used the VUDS (VIMOS Ultra Deep Survey) spectroscopic survey [@lefevre2015_vuds] to systematically search for proto-structures. VUDS targeted approximately $10000$ objects presumed to be at high redshift for spectroscopic observations, confirming over $5000$ galaxies at $z>2$. These galaxies generally have stellar masses $\gtrsim 10^{9} {\rm M}_{\odot}$, and are broadly representative in stellar mass, absolute magnitude, and rest-frame colour of all star-forming galaxies (and thus, the vast majority of galaxies) at $2 \lesssim z \lesssim 4.5$ for $i\leq25$. We identified a preliminary sample of $\sim50$ candidate proto-structures (Lemaux et al, in prep.) over $2<z<4.6$ in the COSMOS, CFHTLS-D1 and ECDFS fields (1 deg$^2$ in total). With this ‘blind’ search in the COSMOS field we identified the complex and rich proto-structure at $z\sim2.5$ presented in this paper. This proto-structure, extended over a volume of $\sim60\times60\times150$ comoving Mpc$^3$, has a very complex shape, and includes several density peaks within $2.42<z<2.51$, possibly connected by filaments, that are more dense than the average volume density. Smaller components of this proto-structure have already been identified in the literature from heterogeneous galaxy samples, like for example Ly$\alpha$ emitters (LAEs), three-dimensional (3D) Ly$\alpha$-forest tomography, sub-millimetre starbursting galaxies, and CO-emitting galaxies (see [@diener2015_z245; @chiang2015_z244; @casey2015_z247; @lee2016_colossus; @wang2016_z250]). Despite the sparseness of previous identifications of sub-clumps, a part of this structure was already dubbed “Colossus” for its extension [@lee2016_colossus]. With VUDS, we obtain a more complete and unbiased panoramic view of this large structure, placing the previous sub-structure detections reported in the literature in the broader context of this extended large-scale structure. The characteristics of this proto-structure, its redshift, its richness over a large volume, the clear detection of its sub-components, the extensive imaging and spectroscopy coverage granted by the COSMOS field, provide us the unique possibility to study a rich supercluster in its formation. From now on we refer to this huge structure as a ‘proto-supercluster’. On the one hand, throughout the paper we show that it is as extended and as massive as known superclusters at lower redshift. Moreover, it presents a very complex shape, which includes several density peaks embedded in the same large-scale structure, similarly to other lower-redshift structures defined superclusters. In particular, one of the peaks has already been identified in the literature [@wang2016_z250] as a possibly virialised structure. On the other hand, we also show that the evolutionary status of some of these peaks is compatible with that of overdensity fluctuations which are collapsing and are foreseen to virialise in a few gigayears. For all these reasons, we consider this structure a proto-supercluster. In this work, we aim to characterise the 3D shape of the proto-supercluster, and in particular to study the properties of its sub-components, for example their average density, volume, total mass, velocity dispersion, and shape. We also perform a thorough comparison of our findings with the previous density peaks detected in the literature on this volume, so as to put them in the broader context of a large-scale structure. The paper is organised as follows. In Sect. \[data\] we present our data set and how we reconstruct the overdensity field. The discovery of the proto-supercluster, and its total volume and mass, are discussed in Sect. \[supercluster\]. In Sect. \[3D\_peaks\] we describe the properties of the highest density peaks embedded in the proto-supercluster (their individual mass, velocity dispersion, etc.) and we compare our findings with the literature. In Sect. \[discussion\] we discuss how the peaks would evolve according to the spherical collapse model, and how we can compare the proto-supercluster to similar structures at lower redshifts. Finally, in Sect. \[summary\] we summarise our results. Except where explicitly stated otherwise, we assume a flat $\Lambda$CDM cosmology with $\Omega_m=0.25$, $\Omega_{\Lambda}=0.75$, $H_0=70\kms {\rm Mpc}^{-1}$ and $h=H_0/100$. Magnitudes are expressed in the AB system [@oke74; @fukugita96]. Comoving and physical Mpc(/kpc) are expressed as cMpc(/ckpc) and pMpc(/pkpc), respectively. The data sample and the density field {#data} ===================================== VUDS is a spectroscopic survey performed with VIMOS on the ESO-VLT [@lefevre2003], targeting approximately $10000$ objects in the three fields COSMOS, ECDFS, and VVDS-2h to study galaxy evolution at $2 \lesssim z \lesssim 6$. Full details are given in [@lefevre2015_vuds]; here we give only a brief review. VUDS spectroscopic targets have been pre-selected using four different criteria. The main criterion is a photometric redshift ($z_p$) cut ($z_p+1\sigma \geq 2.4$, with $z_p$ being either the $1^{st}$ or $2^{nd}$ peak of the $z_p$ probability distribution function) coupled with the flux limit $i\leq 25$. This main criterion provided 87.7% of the primary sample. Photometric redshifts were derived as described in [@ilbert2013] with the code [*Le Phare[^2]*]{} [@arnouts99; @ilbert2006_pz]. The remaining targets include galaxies with colours compatible with Lyman-break galaxies, if not already selected by the $z_p$ criterion, as well as drop-out galaxies for which a strong break compatible with $z>2$ was identified in the $ugrizYJHK$ photometry. In addition to this primary sample, a purely flux-limited sample with $23 \leq i \leq 25$ has been targeted to fill-up the masks of the multi-slit observations. VUDS spectra have an extended wavelength coverage from 3600 to $9350\AA$, because targets have been observed with both the LRBLUE and LRRED grisms (both with R$\sim230$), with 14h integration each. With this integration time it is possible to reach S/N $\sim 5$ on the continuum at $\lambda\sim8500$Å  (for $i = 25$), and for an emission line with flux $F = 1.5\times10^{-18}$erg s$^{-1}$ cm$^{2}$. The redshift accuracy is $\sigma_{zs}= 0.0005 (1 + z)$, corresponding to $\sim150\kms$ (see also [@lefevre2013a]). We refer the reader to [@lefevre2015_vuds] for a detailed description of data reduction and redshift measurement. Concerning the reliability of the measured redshifts, here it is important to stress that each measured redshift is given a reliability flag equal to X1, X2, X3, X4, or X9[^3], which correspond to a probability of being correct of 50-75%, 75-85%, 95-100%, 100%, and $\sim80$% respectively. In the COSMOS field, the VUDS sample comprises 4303 spectra of unique objects, out of which 2045 have secure spectroscopic redshift (flags X2, X3, X4, or X9) and $z\ge 2.$ Together with the VUDS data, we used the zCOSMOS-Bright [@lilly2007; @lilly2009] and zCOSMOS-Deep (Lilly et al, in prep., [@diener2013_list]) spectroscopic samples. The flag system for the robustness of the redshift measurement is basically the same as in the VUDS sample, with very similar flag probabilities (although they have never been fully assessed for zCOSMOS-Deep). In the zCOSMOS samples, the spectroscopic flags have also been given a decimal digit to represent the level of agreement of the spectroscopic redshift ($z_s$) with the photometric redshift ($z_p$). A given $z_p$ is defined to be in agreement with its corresponding $z_s$ when $|z_s-z_p|<0.08(1+z_s)$, and in these cases the decimal digit of the spectroscopic flag is ‘5’. For the zCOSMOS samples, we define secure $z_s$ those with a quality flag X2.5, X3, X4, or X9, which means that for flag X2 we used only the $z_s$ in agreement with their respective $z_p$, while for higher flags we trust the $z_s$ irrespectively of the agreement with their $z_p$. With these flag limits, we are left with more than 19000 secure $z_s$, of which 1848 are at $z\geq 2$. We merged the VUDS and zCOSMOS samples, removing the duplicates between the two surveys as follows. For each duplicate, that is, objects observed in both VUDS and zCOSMOS, we retained the redshift with the most secure quality flag, which in the vast majority of cases was the one from VUDS. In case of equal flags, we retained the VUDS spectroscopic redshift. Our final VUDS$+$zCOSMOS spectroscopic catalogue consists of 3822 unique secure $z_s$ at $z\geq 2$. We note that we did not use spectroscopic redshifts from any other survey, although other spectroscopic samples in this area are already publicly available in the literature (see e.g. [@casey2015_z247; @chiang2015_z244; @diener2015_z245; @wang2016_z250]). These samples are often follow-up of small regions around dense regions, and we did not want to be biased in the identification of already known density peaks. Unless specified otherwise, our spectroscopic sample always refers only to the good quality flags in VUDS and zCOSMOS discussed above. We also did not include public $z_s$ from more extensive campaigns, like for example the COSMOS AGN spectroscopic survey [@trump09], the MOSDEF survey [@kriek15], or the DEIMOS 10K spectroscopic survey [@hasinger18]. We matched our spectroscopic catalogue with the photometric COSMOS2015 catalogue [@laigle2016]. The matching was done by selecting the closest source within a matching radius of $0.55^{\prime \prime}$. Objects in the COSMOS2015 have been detected via an ultra-deep $\chi^2$ sum of the $YJHK_s$ and $z^{++}$ images. $YJHK_s$ photometry was obtained by the VIRCAM instrument on the VISTA telescope (UltraVISTA-DR2 survey[^4], [@mcCracken12]), and the $z^{++}$ data, taken using the Subaru Suprime-Cam, are a (deeper) replacement of the previous $z-$band COSMOS data [@taniguchi2007; @taniguchi2015]. With this match with the COSMOS2015 catalogue we obtained a uniform target coverage of the COSMOS field down to a given flux limit (see Sect.\[method\]), using spectroscopic redshifts for the objects in our original spectroscopic sample or photometric redshifts for the remaining sources. The photometric redshifts in COSMOS2015 are derived using $3^{\prime \prime}$ aperture fluxes in the 30 photometric bands of COSMOS2015. According to Table 5 of [@laigle2016], a direct comparison of their photometric redshifts with the spectroscopic redshifts of the entire VUDS survey in the COSMOS field (median redshift $z_{\rm med}=2.70$ and median $i^+-$band $i^+_{\rm med}=24.6$) gives a photometric redshift accuracy of $\Delta z = 0.028(1+z)$. The same comparison with the zCOSMOS-Deep sample (median redshift $z_{\rm med}=2.11$ and median $i^+-$band $i^+_{\rm med}=23.8$) gives $\Delta z = 0.032(1+z)$. {width="6.cm"} {width="6.cm"} {width="6.cm"} The method to compute the density field and identify the density peaks is the same as described in [@lemaux2018_z45]; we describe it here briefly. The method is based on the Voronoi Tessellation, which has already been successfully used at different redshifts to characterise the local environment around galaxies and identify the highest density peaks, including the search for groups and clusters (see e.g. [@marinoni02; @coooper05; @cucciati10; @gerke12; @scoville13_env; @darvish15; @smolcic17]). Its main advantage is that the local density is measured both on an adaptive scale and with an adaptive filter shape, allowing us to follow the natural distribution of tracers. In our case, we worked in two dimensions in overlapping redshift slices. We used as tracers the spectroscopic sample complemented by a photometric sample which provides us with the photometric redshifts of all the galaxies for which we did not have any $z_s$ information. For each redshift slice, we generated a set of Monte Carlo (MC) realisations. Galaxies (with $z_s$ or $z_p$) to be used in each realisation were selected observing the following steps, in this order: - irrespectively of their redshift, galaxies with a $z_s$ were retained in a percentage of realisations equal to the probability associated to the reliability flag; namely, in each realisation, before the selection in redshift, for each galaxy we drew a number from a uniform distribution from 0 to 100 and retained that galaxy only if the drawn number was equal to or less than the galaxy redshift reliability; - galaxies with only $z_p$ were first selected to complement the retained spectroscopic sample (i.e. the photometric sample comprises all the galaxies without a $z_s$ or for which we threw away their $z_s$ for a given iteration), then they were assigned a new photometric redshift $z_{\rm p,new}$ randomly drawn from an asymmetrical Gaussian distribution centred on their nominal $z_p$ value and with negative and positive sigmas equal to the lower and upper uncertainties in the $z_p$ measurement, respectively; with this approach we do not try to correct for catastrophic redshift errors, but only for the shape of the PDF of each $z_p$; - among the samples selected at steps 1 and 2, we retained all the galaxies with $z_s$ (from step 1) or $z_{\rm p,new}$ (from step 2) falling in the considered redshift slice. We performed a 2D Voronoi tessellation for each $i^{th}$ MC realisation, and assigned to each Voronoi polygon a surface density $\Sigma_{VMC,i}$ equal to the inverse of the area (expressed in Mpc$^2$) of the given polygon. Finally, we created a regular grid of $75\times75$ pkpc cells, and assigned to each grid point the $\Sigma_{VMC,i}$ of the polygon enclosing the central point of the cell. For each redshift slice, the final density field $\Sigma_{VMC}$ is computed on the same grid, as the median of the density fields among the realisations, cell by cell. As a final step, from the median density map we computed the local over-density at each grid point as $\delta_{\rm gal} = \Sigma_{VMC}/ \tilde{\Sigma}_{VMC} -1 $, where $\tilde{\Sigma}_{VMC}$ is the mean $\Sigma_{VMC}$ for all grid points. In our analysis we are more interested in $\delta_{\rm gal}$ than in $\Sigma_{VMC}$ because we want to identify the regions that are overdense with respect to the mean density at each redshift, a density which can change not only for astrophysical reasons but also due to characteristics of the imaging/spectroscopic survey. Moreover, as we see in the following sections, the computation of $\delta_{\rm gal}$ is useful to estimate the total mass of our proto-cluster candidates and their possible evolution. Proto-cluster candidates were identified by searching for extended regions of contiguous grid cells with a $\delta_{\rm gal}$ value above a given threshold. The initial systematic search for proto-clusters in the COSMOS field (which will be presented in Lemaux et al., in prep.) was run with the following set of parameters: redshift slices of 7.5 pMpc shifting in steps of 3.75 pMpc (so as to have redshift slices overlapping by half of their depth); 25 Monte Carlo realisations per slice; and spectroscopic and photometric catalogues with $[3.6]\leq 25.3$ (IRAC Channel 1). With this ‘blind’ search we re-identified two proto-clusters at $z\sim3$ serendipitously discovered at the beginning of VUDS observations [@lemaux2014_z33; @cucciati2014_z29], together with other outstanding proto-structures presented separately in companion papers ([@lemaux2018_z45], Lemaux et al. in prep.). Discovery of a rich extended proto-supercluster {#supercluster} =============================================== The preliminary overdensity maps showed two extended overdensities at $z\sim2.46$, in a region of $0.4\times0.25$ deg$^2$. Intriguingly, there were several other smaller overdensities very close in right ascension (RA), declination (Dec), and redshift. We therefore explored in more detail the COSMOS field by focusing our attention on the volume around these overdensities. This focused analysis revealed the presence of a rich extended structure, consisting of density peaks linked by slightly less dense regions. The method {#method} ---------- We re-ran the computation of the density field and the search for overdense regions with a fine-tuned parameter set (see below), in the range $2.35 \lesssim z \lesssim 2.55$, which we studied by considering several overlapping redshift slices. Concerning the angular extension of our search, we computed the density field in the central $\sim1\times1$ deg$^2$ of the COSMOS field, but then used only the slightly smaller 0.91 deg$^2$ region at $149.6\leq RA \leq 150.52$ and $1.74 \leq Dec \leq2.73$ to perform any further analysis (computation of the mean density etc.). This choice was made to avoid the regions close to the field boundaries, where the Voronoi tessellation is affected by border effects. In this smaller area, considering a flux limit at $i=25$, about 24% of the objects with a redshift ($z_s$ or $z_p$) falling in the above-mentioned redshift range have a spectroscopic redshift. If we reduce the area to the region covered by VUDS observations, which is slightly smaller, this percentage increases to about 28%. We also verified the robustness of our choices for what concerns the following issues: [**Number of Monte Carlo realisations**]{}. With respect to Lemaux et al. (in prep.), we increased the number of Monte Carlo realisations from the initial 25 to 100 to obtain a more reliable median value (similarly to, e.g. [@lemaux2018_z45]). We verified that our results did not significantly depend on the number of realisations $n_{\rm MC}$ as long as $n_{\rm MC}\geq 100$, and, therefore, all analyses presented in this paper are done on maps which used $n_{\rm MC}=100$. This high number of realisations allowed us to produce not only the median density field for each redshift slice, but also its associated error maps, as follows. For each grid cell, we considered the distribution of the 100 $\Sigma_{VMC}$ values, and took the $16^{th}$ and $84^{th}$ percentiles of this distribution as lower and upper limits for $\Sigma_{VMC}$. We produced density maps with these lower and upper limits, in the same way as for the median $\Sigma_{VMC}$, and then computed the corresponding overdensities that we call $\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$. [**Spectroscopic sample.**]{} As in [@lemaux2018_z45], we assigned a probability to each spectroscopic galaxy to be used in a given realisation equal to the reliability of its $z_s$ measurement, as given by its quality flag. Namely, we used the quality flags X2 (X2.5 for zCOSMOS), X3, X4, and X9 with a reliability of 80%, 97.5%, 100% and 80% respectively (see Sect. \[data\]; here we adopt the mean probability for the flags X2 and X3, for which [@lefevre2015_vuds] give a range of probabilities). These values were computed for the VUDS survey, but we applied them also to the zCOSMOS spectroscopic galaxies in our sample, as discussed in Sect. \[data\]. We verified that our results do not qualitatively change if we choose slightly different reliability percentages or if we used the entire spectroscopic sample (flag=X2/X2.5, X3, X4, X9) in all realisations instead of assigning a probability to each spectroscopic galaxy. The agreement between these results is due to the very high flag reliabilities, and to the dominance of objects with only $z_p$. With the cut in redshift at $2.35 \leq z \leq 2.55$, the above-mentioned quality flag selection, and the magnitude limit at $i\leq 25$ (see below), we are left with 271 spectroscopic redshifts from VUDS and 309 from zCOSMOS, for a total of 580 spectroscopic redshifts used in our analysis. This provides us with a spectroscopic sampling rate of $\sim24\%$, considering the above mentioned redshift range and magnitude cut. We remind the reader that we use only VUDS and zCOSMOS spectroscopic redshift, and do not include in our sample any other $z_s$ found in the literature. [**Mean density.**]{} To compute the mean density $\tilde{\Sigma}_{VMC}$ we proceeded as follows. Given that $\Sigma_{VMC}$ has a log-normal distribution [@coles91], in each redshift slice we fitted the distribution of ${\rm log}(\Sigma_{VMC})$ of all pixels with a $3\sigma$-clipped Gaussian. The mean $\mu$ and standard deviation $\sigma$ of this Gaussian are related to the average density $\langle \Sigma_{VMC} \rangle$ by the equation $\langle \Sigma_{VMC}\rangle = 10^{\mu}e^{2.652\sigma^{2}}$. We used this $\langle \Sigma_{VMC} \rangle$ as the average density $\tilde{\Sigma}_{VMC}$ to compute the density contrast $\delta_{gal}$. $\tilde{\Sigma}_{VMC}$ was computed in this way in each redshift slice. [**Overdensity threshold.**]{} In each redshift slice, we fitted the distribution of ${\rm log}(1+\delta_{\rm gal})$ with a Gaussian, obtaining its $\mu$ and $\sigma$. We call these parameters $\mu_{\rm \delta}$ and $\sigma_{\rm \delta}$, for simplicity, although they refer to the Gaussian fit of the ${\rm log}(1+\delta_{\rm gal})$ distribution and not of the $\delta_{\rm gal}$ distribution. We then fitted $\mu_{\rm \delta}$ and $\sigma_{\rm \delta}$ as a function of redshift with a second-order polynomial, obtaining $\mu_{\rm \delta,fit}$ and $\sigma_{\rm \delta,fit}$ at each redshift. Our detection thresholds were then set as a certain number of $\sigma_{\rm \delta,fit}$ above the mean overdensity $\mu_{\rm \delta,fit}$, that is, as $ {\rm log}(1+\delta_{\rm gal}) \geq \mu_{\rm \delta,fit}(z_{\rm slice}) + n_{\sigma}\sigma_{\rm \delta,fit}(z_{\rm slice})$, where $z_{\rm slice}$ is the central redshift of each slice, and $n_{\sigma}$ is chosen as described in Sects. \[3D\] and \[3D\_peaks\]. From now, when referring to setting a ‘$n_{\sigma}\sigma_{\rm \delta}$ threshold’ we mean that we consider the volume of space with $ {\rm log}(1+\delta_{\rm gal}) \geq \mu_{\rm \delta,fit}(z_{\rm slice}) + n_{\sigma}\sigma_{\rm \delta,fit}(z_{\rm slice})$. [**Slice depth and overlap.**]{} We used overlapping redshift slices with a full depth of 7.5 pMpc, which corresponds to $\delta z \sim 0.02$ at $z\sim2.45$, running in steps of $\delta z \sim 0.002$. We also tried with thinner slices (5 pMpc), but we adopted a depth of 7.5 pMpc as a compromise between i) reducing the line of sight (l.o.s.) elongation of the density peaks (see Sect. \[3D\]) and ii) keeping a low noise in the density reconstruction. We define ‘noise’ as the difference between $\delta_{\rm gal}$ and its lower and upper uncertainties $\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$[^5]. The choice of small steps of $\delta z \sim 0.002$ is due to the fact that we do not want to miss the redshift where each structure is more prominent. [**Tracers selection**]{} We fine-tuned our search method (including the $\delta_{\rm gal}$ thresholds etc...) for a sample of galaxies limited at $i=25$. We verified the robustness of our findings by using also a sample selected with $K_S\leq 24$ and one selected with ${\rm[3.6]}\leq24$ (IRAC Channel 1). With these two latter cuts, in the redshift range $2.3\leq z \leq 2.6$ we have a number of galaxies with spectroscopic redshift corresponding to $\sim87\%$ and $\sim94\%$ of the number of spectroscopic galaxies with $i\leq25$, respectively, but not necessarily the same galaxies, while roughly 65% and 85% more objects, respectively, with photometric redshifts entered in our maps than did with $i\leq25$. Although the $K_S\leq 24$ and ${\rm[3.6]}\leq24$ samples might be distributed in a different way in the considered volume because of the different clustering properties of different galaxy populations, with these samples we recovered the overdensity peaks in the same locations as with $i\leq25$. Clearly, the $\delta_{\rm gal}$ distribution is slightly different, so the overdensity threshold that we used to define the overdensity peaks (see Sect. \[3D\_peaks\]) encloses regions with slightly different shape with respect to those recovered with a sample flux-limited at $i\leq25$. We defer a more precise analysis of the kind of galaxy populations which inhabit the different density peaks to future work. Figure \[2D\_maps\] shows three 2D overdensity ($\delta_{\rm gal}$) maps obtained as described above, in the redshift slices $2.422<z<2.444$, $2.438<z<2.460$, and $2.454<z<2.476$. We can distinguish two extended and very dense components at two different redshifts and different RA-Dec positions: one at $z\sim 2.43$, in the left-most panel, that we call the “South-West” (SW) component, and the other at $z\sim 2.46$, at higher RA and Dec, that we call here the “North-East” (NE) component (right-most panel). The NE and SW components seem to be connected by a region of relatively high density, shown in the middle panel of the figure. This sort of filament is particularly evident when we fix a threshold around $2\sigma_{\rm \delta}$, as shown in the figure. For this reason, we retained the $2\sigma_{\rm \delta}$ threshold as the threshold used to identify the volume of space occupied by this huge overdensity. As a reference, a $2\sigma_{\rm \delta}$ threshold corresponds to $\delta\sim0.65$, while 3, 4, and 5$\sigma_{\rm \delta}$ thresholds correspond to $\delta\sim1.1$, $\sim1.7$, and $\sim2.55$, respectively To better understand the complex shape of the structure, we performed an analysis in three dimensions, as described in the following sub-section. The 3D matter distribution {#3D} -------------------------- We built a 3D overdensity cube in the following way. First, we considered each redshift slice to be placed at $z_{\rm slice}$ along the line of sight, where $z_{\rm slice}$ is the central redshift of the slice. All the 2D maps were interpolated at the positions of the nodes in the 2D grid of the lowest redshift ($z=2.35$). This way we have a 3D data cube with RA-Dec pixel size corresponding to $\sim75\times75$ pkpc at $z=2.38$, and a l.o.s. pixel size equal to $\delta z \sim 0.002$ (see Sect. \[method\]). From now on we use ‘pixels’ and ‘grid cells’ with the same meaning, referring to the smallest components of our data cube. We smoothed our data cube in RA and Dec with a Gaussian filter with sigma equal to 5 pixels. Along the l.o.s., we used instead a boxcar filter with a depth of 3 pixels. The shape and dimension of the smoothing in RA-Dec was chosen as a compromise between the two aims of i) smoothing the shapes of the Voronoi polygons and ii) not washing away the highest density peaks. The smoothing along the l.o.s. was done to link each redshift slice with the previous and following slice. Different choices on the smoothing filters do not significantly affect the 2D maps in terms of the shapes of the over-dense regions, and have only a minor effect on the values of $\delta_{\rm gal}$, even if the highest-density peaks risk to be washed away in case of excessive smoothing. We produced data cubes for the lower and upper limits of $\delta_{\rm gal}$ ($\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$) in the same way. These two latter cubes are used for the treatment of uncertainties in our following analysis. Figure \[2D\_maps\] shows that around the main components of the proto-supercluster there are less extended density peaks. Since we wanted to focus our attention on the proto-supercluster, we excluded from our analysis all the density peaks not directly connected to the main structure. To do this, we proceeded as follows: we started from the pixels of the 3D grid which are enclosed in the $2\sigma_{\rm \delta}$ contour of the “NE” region in the redshift slice $2.454<z<2.476$ (right panel of Fig. \[2D\_maps\]). Starting from this pixel set, we iteratively searched in the 3D cube for all the pixels, contiguous to the previous pixels set, with a ${\rm log}(1+\delta_{\rm gal})$ higher than $2\sigma_{\rm \delta}$ above the mean, and we added those pixels to our pixel set. We stopped the search when there were no more contiguous pixels satisfying the threshold on ${\rm log}(1+\delta_{\rm gal})$. In this way we define a single volume of space enclosed in a $2\sigma_{\rm \delta}$ surface, and we define our proto-supercluster as the volume of space comprised within this surface. The final 3D overdensity map of the proto-supercluster is shown in Fig. \[3D\_cube\], with the three axes in comoving megaparsecs. The 3D shape of the proto-supercluster is very irregular. The NE and SW components are clearly at different average redshifts, and have very different 3D shapes. Figure \[3D\_cube\] also shows that both components contain some density peaks (visible as the reddest regions within the $2\sigma_{\rm \delta}$ surface) with a very high average $\delta_{\rm gal}$. We discuss the properties of these peaks in detail in Sect. \[3D\_peaks\]. The volume occupied by the proto-supercluster shown in Fig. \[3D\_cube\] is about $9.5\times 10^4$ cMpc$^3$ (obtained by adding up the volume of all the contiguous pixels bounded by the 2$\sigma_{\rm \delta}$ surface), and the average overdensity is $\langle \delta_{\rm gal} \rangle \sim 1.24$. We can give a rough estimate of the total mass $M_{\rm tot}$ of the proto-supercluster by using the formula (see [@steidel98]): $$\displaystyle M_{\rm tot}=\rho_{\rm m} V (1+\delta_{\rm m}), \label{eq_mass}$$ where $\rho_m$ is the comoving matter density, $V$ the volume[^6] that encloses the proto-cluster and $\delta_{\rm m}$ the matter overdensity in our proto-cluster. We computed $\delta_{\rm m}$ by using the relation $\delta_{\rm m}=\langle \delta_{\rm gal} \rangle/b$, where $b$ is the bias factor. Assuming $b=2.55$, as derived in [@durkalec15b] at $z\sim2.5$ with roughly the same VUDS galaxy sample we use here, we obtain $M_{\rm tot} \sim 4.8\times10^{15}{\rm M}_{\odot}$. There are at least two possible sources of uncertainty in this computation[^7]. The first is the chosen $\sigma_{\rm \delta}$ threshold. If we changed our threshold by $\pm0.2\sigma_{\rm \delta}$ around our adopted value of $2\sigma_{\rm \delta}$, $\langle \delta_{\rm gal} \rangle$ would vary by $\sim \pm 10\%$ and the volume would vary by $\sim \pm 17\%$, for a variation of the estimated mass of $\sim \pm 15\%$ (a higher threshold means a higher $\langle \delta_{\rm gal} \rangle$ and a smaller volume, with a net effect of a smaller mass; the opposite holds when we use a lower threshold). Another source of uncertainty is related to the uncertainty in the measurement of $\delta_{\rm gal}$ in the 2D maps. If we had used the 3D cube based on $\delta_{\rm gal,16}$(/$\delta_{\rm gal,84}$), we would have obtained $\langle \delta_{\rm gal} \rangle \sim 1.23(/1.26)$ and a volume of 1.06(/0.75)$\times 10^5$ cMpc$^3$, for an overall total mass $\sim10$% larger (/ $\sim20$% smaller). If we sum quadratically the two uncertainties, the very liberal global statistical error on the mass measurement is of about $+18\%/-25\%$. Irrespectively of the errors, it is clear that this structure has assembled an immense mass ($> 2\times10^{15}{\rm M}_{\odot}$) at very early times. This structure is referred to hereafter as the “Hyperion proto-supercluster”[^8] or simply “Hyperion” (officially PSC J1001$+$0218) due to its immense size and mass and because one of its subcomponents (peak \[3\], see Sect. \[peak3\]) is broadly coincident with the Colossus proto-cluster discovered by [@lee2016_colossus]. We remark that the volume computed in our data cube is most probably an overestimate, at the very least because it is artificially elongated along the l.o.s. This elongation is mainly due to 1) the photometric redshift error ($\Delta z\sim0.1$ for $\sigma_{zp}=0.03(1+z)$ at $z=2.45$), 2) the depth of the redshift slices ($\Delta z\sim0.02$) used to produce the density field, and 3) the velocity dispersion of the member galaxies, which might create the feature known as the Fingers of God ($\Delta z\sim0.006$ for a velocity dispersion of $500\kms$). Although the velocity dispersion should be important only for virialised sub-structures, these three factors should all work to surreptitiously increase the dimension of the structure along the l.o.s. and at the same time decrease the local overdensity $\delta_{\rm gal}$. In this transformation there is no mass loss (or, equivalently, the total galaxy counts remain the same, with galaxies simply spread on a larger volume). Therefore, the total mass of our structure, computed with Eq. \[eq\_mass\], would not change if we used the real (smaller) volume and the real (higher) density instead of the elongated volume and its associated lower overdensity. We also ran a simple simulation to verify the effects of the depth of the redshift slices on the elongation. We built a simple mock galaxy catalogue at $z=2.5$ following a method similar to that described in [@tomczak17], a method which is based on injecting a mock galaxy cluster and galaxy groups onto a sample of mock galaxies that are intended to mimic the coeval field. As in [@tomczak17], the three dimensional positions of mock field galaxies are randomly distributed over the simulated transverse spatial and redshift ranges, with the number of mock field galaxies set to the number of photometric objects within an identical volume in COSMOS at $z\sim2.5$ that is devoid of known proto-structures. Galaxy brightnesses were assigned by sampling the $K-$band luminosity function of [@cirasuolo10], with cluster and group galaxies perturbed to slightly brighter luminosities (0.5 and 0.25 mag, respectively). Member galaxies of the mock cluster and groups were assigned spatial locations based on Gaussian sampling with $\sigma$ equal to 0.5 and 0.33 h$_{70}^{-1}$ pMpc, respectively, and were scattered along the l.o.s. by imposing Gaussian velocity dispersions of 1000 and 500 km s$^{-1}$, respectively. We then applied a magnitude cut to the mock catalogue similar to that used in our actual reconstructions, applied a spectroscopic sampling rate of 20%, and, for the remainder of the mock galaxies, assigned photometric redshifts with precision and accuracy identical to those in our photometric catalogue at the redshift of interest. We then ran the exact same density field reconstruction and method to identify peaks as was run on our real data, each time varying the depth of the redshift slices used. Following this exercise, we observed a smaller elongation for decreasing slice depth, with a $\sim40\%$ smaller elongation observed when dropping the slice size from 7.5 to 2.5 pMpc. This result confirmed that we need to correct for the elongation if we want to give a better estimate of the volume and/or the density of the structures in our 3D cube. We will apply a correction for the elongation to the highest density peaks found in the Hyperion proto-supercluster, as discussed in Sect. \[3D\_peaks\]. ![3D overdensity map of the Hyperion proto-supercluster, in comoving megaparsecs. Colours scale with $\rm log(\sigma_{\rm \delta})$, exactly as in Fig. \[2D\_maps\], from blue ($2\sigma_{\rm \delta}$) to the darkest red ($\sim8.3\sigma_{\rm \delta}$, the highest measured value in our 3D cube). The $x-$, $y-$ and $z-$axes span the ranges $149.6 \leq RA \leq 150.52$, $1.74 \leq Dec \leq2.73$ and $2.35 \leq z \leq 2.55$. The NE and SW components are indicated. We highlight the fact that this figure shows only the proto-supercluster, and omits other less extended and less dense density peaks which fall in the plotted volume (see discussion in Sect. \[3D\].)[]{data-label="3D_cube"}](./fig2.ps){width="9.0cm"} ![Zoom-in of Fig. \[3D\_cube\]. The angle of view is slightly rotated with respect to Fig. \[3D\_cube\] so as to distinguish all the peaks. The colour scale is the same as in Fig. \[3D\_cube\], but here only the highest density peaks are shown, that is, the 3D volumes where ${\rm log}(1+\delta_{\rm gal})$ is above the $5\sigma_{\rm \delta}$ threshold discussed in Sect. \[3D\_peaks\]. Peaks are numbered as in Fig. \[3D\_sph\_peaks\] and Table \[peaks\_tab\]. []{data-label="3D_cube_peaks"}](./fig3.ps){width="8.0cm"} The highest density peaks {#3D_peaks} ========================= We identified the highest density peaks in the 3D cube by considering only the regions of space with ${\rm log}(1+\delta_{\rm gal})$ above $5\sigma_{\rm \delta}$ from the mean density. In our work, this threshold corresponds to $\delta_{\rm gal} \sim 2.6$, which corresponds to $\delta_m\sim 1 $ when using the bias factor $b=2.55$ found by [@durkalec15b]. We also verified, [*a posteriori*]{}, that with this choice we select density peaks which are about to begin or have just begun to collapse, after the initial phase of expansion (see Sect. \[discussion\]). This is very important if we want to consider these peaks as proto-clusters. With the overdensity threshold defined above, we identified seven separated high-density sub-structures. We show their 3D position and shape in Fig. \[3D\_cube\_peaks\]. We computed the barycenter of each peak by weighting the $(x,y,z)$ position of each pixel belonging to the peak by its $\delta_{\rm gal}$. For each peak, we computed its volume, its $\langle \delta_{\rm gal} \rangle$, and derived its $M_{\rm tot}$ using Eq. \[eq\_mass\] (the bias factor is always $b=2.55$, found by [@durkalec15b] and discussed in Sect. \[3D\]). Table \[peaks\_tab\] lists barycenter, $\langle \delta_{\rm gal} \rangle$, volume, and $M_{\rm tot}$ of the seven peaks, numbered in order of decreasing $M_{\rm tot}$. We applied the same peak-finding procedure on the data cubes with $\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$, and computed the total masses of their peaks in the same way. We used these values as lower and upper uncertainties for the $M_{\rm tot}$ values quoted in the table. From Table \[peaks\_tab\] we see that the overall range of masses spans a factor of $\sim30$, from $\sim0.09$ to $\sim2.6$ times $10^{14}$M$_\odot$. The total mass enclosed within the peaks ($\sim5.0 \times 10^{14}$M$_\odot$) is about 10% of the total mass in the Hyperion proto-supercluster, while the volume enclosing all the peaks is a lower fraction of the volume of the entire proto-supercluster ($\sim6.5\%$), as expected given the higher average overdensity within the peaks. The most massive peak (peak \[1\]) is included in the NE structure, together with peak \[4\] which has one fifth the total mass of peak \[1\]. Peak \[2\], which corresponds to the SW structure, has a $M_{\rm tot}$ comparable to peak \[4\], and it is located at lower redshift. Peak \[3\], with a $M_{\rm tot}$ similar to peaks \[2\] and \[4\], is placed in the sort of filament shown in the middle panel of Fig. \[2D\_maps\]. At smaller $M_{\rm tot}$ there is peak \[5\], with the highest redshift ($z=2.507$), and peak \[6\], at slightly lower redshift. They both have $M_{\rm tot} \sim0.2 \times 10^{14}$M$_\odot$. Finally, peak \[7\] is the least massive, and is very close in RA-Dec to peak \[2\], and at approximately the same redshift. In Appendix \[app\_Mtot\_sigma\] we show that the computation of $M_{\rm tot}$ is relatively stable if we slightly change the overdensity threshold used to define the peaks, with the exception of the least massive peak (peak \[7\]). Figure \[3D\_cube\_peaks\] shows that the peaks have very different shapes, from irregular to more compact. We verified that their shape and position are not possibly driven by spectral sampling issues, by checking that the peaks persist through the $2^{\prime}$ gaps between the VIMOS quadrants from VUDS. This also implies that we are not missing high-density peaks that might fall in the gaps. We remind the reader that the zCOSMOS-Deep spectroscopic sample, which we use together with the VUDS sample, has a more uniform distribution in RA-Dec, and does not present gaps. Concerning the shape of the peaks, we tried to take into account the artificial elongation along the l.o.s.. As mentioned at the end of Sect. \[3D\], this elongation is probably due to the combined effect of the velocity dispersion of the member galaxies, the depth of the redshift slices, and the photometric redshift error (although we refer the reader to e.g. [@lovell18] for an analysis of the shapes of proto-clusters in simulations). We used a simple approach to give an approximate statistical estimate of this elongation, starting from the assumption that on average our peaks should have roughly the same dimension in the $x$, $y,$ and $z$ dimensions[^9], and any measured systematic deviation from this assumption is artificial. In each of the three dimensions we measured a sort of effective radius $R_e$ defined as $R_{e,x}=\sqrt{ \sum _{i}w_i(x_{i}-x_{peak})^2 / \sum_i(w_i) }$ (and similarly for $R_{e,y}$ and $R_{e,z}$), where the sum is over all the pixels belonging to the given peak, the weight $w_i$ is the value of $\delta_{\rm gal}$, $x_{i}$ the position in cMpc along the $x-$axis and $x_{peak}$ is the barycenter of the peak along the $x-$axis, as listed in Table \[peaks\_tab\]. We defined the elongation $E_{\rm z/xy}$ for each peak as the ratio between $R_{e,z}$ and $R_{e,xy}$, where $R_{e,xy}$ is the mean between $R_{e,x}$ and $R_{e,y}$. The effective radii and the elongations are reported in Table \[peaks\_elongation\_tab\]. If the measured volume $V_{\rm meas}$ of our peaks is affected by this artificial elongation, the real corrected volume is $V_{\rm corr} = V_{\rm meas} / E_{\rm z/xy}$. Moreover, given that the elongation has the opposite and compensating effects of increasing the volume and decreasing $\delta_{\rm gal}$, as discussed at the end of Sect. \[3D\], $M_{\rm tot}$ remains the same. For this reason, inverting Eq. \[eq\_mass\] it is possible to derive the corrected (higher) average overdensity $\langle \delta_{\rm gal, corr} \rangle$ for each peak, by using $V_{\rm corr}$ and the mass in Table \[peaks\_tab\]. $V_{\rm corr}$ and $\langle \delta_{\rm gal, corr} \rangle$ are listed in Table \[peaks\_elongation\_tab\]. We note that by definition $R_e$ is smaller than the total radial extent of an overdensity peak, because it is computed by weighting for the local $\delta_{\rm gal}$, which is higher for regions closer to the centre of the peak. For this reason, the $V_{\rm corr}$ values are much larger than the volumes that one would naively obtain by using $R_{e,xy}$ as intrinsic total radius of our peaks. We use $\langle \delta_{\rm gal, corr} \rangle$ in Sect. \[discussion\] to discuss the evolution of the peaks. We refer the reader to \[app\_elongation\] for a discussion on the robustness of the computation of $E_{\rm z/xy}$ and its empirical dependence on $R_{e,xy}$. ![Same volume of space as Fig. \[3D\_cube\_peaks\], but in RA-Dec-$z$ coordinates. Each sphere represents one of the overdensity peaks, and is placed at its barycenter (see Table \[peaks\_tab\]). The colour of the spheres scales with redshift (blue = low $z$, dark red = high $z$), and the dimension scales with the logarithm of $M_{\rm tot}$ quoted in Table \[peaks\_tab\]. Small blue dots are the spectroscopic galaxies which are members of each overdensity peak, as described in Sect.\[3D\_peaks\].[]{data-label="3D_sph_peaks"}](./fig4.ps){width="8.0cm"} We also assigned member galaxies to each peak. We defined a spectroscopic galaxy to be a member of a given density peak if the given galaxy falls in one of the $\geq 5 \sigma_{\delta}$ pixels that comprise the peak. The 3D distribution of the spectroscopic members is shown in Fig. \[3D\_sph\_peaks\], where each peak is schematically represented by a sphere placed in a $(x,y,z)$ position corresponding to its barycenter. It is evident that the 3D distribution of the member galaxies mirrors the shape of the peaks (see Fig. \[3D\_cube\_peaks\]). The number of spectroscopic members $n_{\rm zs}$ is quoted in Table \[peaks\_tab\]. The most extended and massive peak, peak \[1\], has 24 spectroscopic members. All the other peaks have a much smaller number of members (from 7 down to even only one member). We remind the reader that these numbers depend on the chosen overdensity threshold used to define the peaks, because the threshold defines the volume occupied by the peaks. Moreover, here we are counting only spectroscopic galaxies with good quality flags (see Sect. \[data\]) from VUDS and zCOSMOS, excluding other spectroscopic galaxies identified in the literature (but see Sect. \[vel\_disp\] for the inclusion of other samples to compute the velocity dispersion). Velocity dispersion and virial mass {#vel_disp} ----------------------------------- We computed the l.o.s. velocity dispersion $\sigma_{\rm v}$ of the galaxies belonging to each peak. For this computation we used a more relaxed definition of membership with respect to the one described above, so as to include also the galaxies residing in the tails of the velocity distribution of each peak. Basically, we used all the available good-quality spectroscopic galaxies within $\pm2500\kms$ from $z_{\rm peak}$ comprised in the RA-Dec region corresponding to the largest extension of the given peak on the plane of the sky. Moreover, we did not impose any cut in $i-$band magnitude, because, in principle, all galaxies can serve as reliable tracers of the underlying velocity field. We also included in this computation the spectroscopic galaxies with lower quality flag (flag = X1 for VUDS, all flags with X1.5$\leq$flag$<$2.5 for zCOSMOS), but only if they could be defined members of the given peak, with membership defined as at the end of the previous section. This less restrictive choice allows us to use more galaxies per peak than the pure spectroscopic members, although we still have only $\leq 4$ galaxies for three of the peaks. We quote these larger numbers of members in Table \[peaks\_tab\_veldisp\]. With these galaxies, we computed $\sigma_{\rm v}$ for each peak by applying the biweight method (for peak \[1\]) or the gapper method (for all the other peaks), and report the results of these computations in Table \[peaks\_tab\_veldisp\]. The choice of these methods followed the discussion in [@beers90], where they show that for the computation of the scale of a distribution the gapper method is more robust for a sample of $\lesssim 20$ objects (all our peaks but peak \[1\]), while it is better to use the biweight method for $\gtrsim 20$ objects (our peak \[1\]). We computed the error on $\sigma_{\rm v}$ with the bootstrap method, which was taken as the reference method in [@beers90]. In the case of peak \[7\], with only three spectroscopic galaxies available to compute $\sigma_{\rm v}$, we had to use the jack-knife method to evaluate the uncertainty on $\sigma_{\rm v}$; see also Sect. \[app\_vdisp\_sigma\] for more details on $\sigma_{\rm v}$ of peak \[7\]. We found a range of $\sigma_{\rm v}$ between $320\kms$ and $731\kms$. The most massive peak, peak \[1\], has the largest velocity dispersion, but for the other peaks the ranking in $M_{\rm tot}$ is not the same as in $\sigma_{\rm v}$. The uncertainty on $\sigma_{\rm v}$ is mainly driven by the number of galaxies used to compute $\sigma_{\rm v}$ itself, and it ranges from $\sim12$% for peak \[1\] to $\sim65$% for peak \[7\], for which we used only three galaxies to compute $\sigma_{\rm v}$. As we see below, other identifications in the literature of high-density peaks at the same redshift cover broadly the same $\sigma_{\rm v}$ range. As we already mentioned, there are some works in the literature that identified/followed up some overdensity peaks in the COSMOS field at $z\sim2.45$, such as for example [@casey2015_z247], [@diener2015_z245], [@chiang2015_z244], and [@wang2016_z250]. Moreover, the COSMOS field has also been surveyed with spectroscopy by other campaigns, such as for example the COSMOS AGN spectroscopic survey [@trump09], the MOSDEF survey [@kriek15], and the DEIMOS 10K spectroscopic survey [@hasinger18]. We collected the spectroscopic redshifts of these other samples (including in this search also much smaller samples, like e.g. the one by [@perna15]), removed the possible duplicates with our sample and between samples, and assigned these new objects to our peaks, by applying the same membership criterion as applied to our VUDS$+$zCOSMOS sample. We re-computed the velocity dispersion using our previous sample plus the new members found in the literature. We note that many objects in the COSMOS field have been observed spectroscopically multiple times, and in most of the cases the new redshifts were concordant with previous observations. This is a further proof of the robustness of the $z_s$ we use here. In the literature we only find new members for the peaks \[1\], \[3\], \[4\], and \[5\]. For each of these peaks, Table \[peaks\_tab\_veldisp\] reports the number $n_{\rm lit}$ of spectroscopic redshifts added to our original sample, together with the new estimates of $\sigma_{\rm v}$ and $M_{\rm vir}$. The new $\sigma_{\rm v}$ is always in very good agreement (below $1\sigma$) with our previous computation, but it has a smaller uncertainty. We will see that this translates into new $M_{\rm vir}$ values which are in very good agreement with those based on the original $\sigma_{\rm v}$. As a by-product of the use of the spectroscopic member galaxies, we also computed a second estimate of the redshift of each peak (after the barycenter, see above). [@beers90] show that the biweight method is the most robust to compute the central location of a distribution of objects (in our case, the average redshift) also in the case of relatively few objects ($5-50$). This central redshift, $z_{\rm BI}$, is reported in Table \[peaks\_tab\_veldisp\], and is in excellent agreement with $z_{\rm peak}$, that is, the barycenter along the l.o.s. quoted in Table \[peaks\_tab\]. The use of the gapper and/or biweight methods is to be favoured when estimating the scale of a distribution also because they apply when the distribution is not necessarily a Gaussian, and certainly the shape of the galaxy velocity distribution in a proto-cluster may not follow a Gaussian distribution. In addition, it is questionable to assume that proto-clusters are virialised systems. Nevertheless, a crude way to estimate the mass of the peaks is to assume the validity of the virial theorem. In this way we can estimate the virial mass $M_{\rm vir}$ by using the measured velocity dispersion and some known scaling relations. We follow the same procedure as [@lemaux12], where $M_{\rm vir}$ is defined as: $$\displaystyle M_{\rm vir}=\frac{3 \sqrt{3} \sigma_{\rm v}^{3}}{\alpha~ 10~ G~ H(z)}. \label{mvir}$$ In Eq. \[mvir\], $\sigma_{\rm v}$ is the line of sight velocity dispersion, $G$ is the gravitational constant, and $H(z)$ is the Hubble parameter at a given redshift. Equation \[mvir\] is derived from i) the definition of the virial mass, $$\displaystyle M_{\rm vir}=\frac{3}{G}\sigma_{\rm v}^{2}~R_{\rm v} , \label{vir_theo}$$ where $R_{\rm v}$ is the virial radius; ii) the relation between $R_{\rm 200}$ and $R_{\rm v}$, $$\displaystyle R_{\rm 200}=\alpha ~ R_{\rm v}, \label{rvir_r200}$$ where $R_{\rm 200}$ is the radius within which the density is 200 times the critical density, and iii) the relation between $R_{\rm 200}$ and $\sigma_{\rm v}$, $$\displaystyle R_{\rm 200}=\frac{\sqrt{3}~\sigma_{\rm v}}{10~H(z)}. \label{r200_sigma}$$ Equations \[vir\_theo\] and \[r200\_sigma\] are from [@carlberg97]. Differently from [@lemaux12], we use $\alpha\simeq0.93$, which is derived comparing the radii where a NFW profile with concentration parameter $c=3$ encloses a density 200 times ($R_{\rm 200}$) and 173 times ($R_{\rm v}$) the critical density at $z\simeq2.45$. Here we consider a structure to be virialised when its average overdensity is $\Delta_{\rm v} \simeq 173$, which corresponds, in a $\Lambda$CDM Universe at $z\simeq2.45$, to the more commonly used value $\Delta_{\rm v} \simeq 178$, constant at all redshifts in an Einstein-de Sitter Universe (see the discussion in Sect. \[collapse\]). The virial masses of our density peaks, computed with Eq. \[mvir\], are listed in Table \[peaks\_tab\_veldisp\], together with the virial masses obtained from the $\sigma_{\rm v}$ computed by using also other spectroscopic galaxies in the literature. Figure \[virial\_mass\] shows how our $M_{\rm vir}$ compared with the total masses $M_{\rm tot}$ obtained with Eq. \[eq\_mass\]. For four of the seven peaks, the two mass estimates basically lie on the 1:1 relation. In the three other cases, the virial mass is higher than the mass estimated with the overdensity value: namely, for peaks \[4\] and \[5\] the agreement is at $<2\sigma$, while for peak \[7\] the agreement is at less than $1\sigma$ given the very large uncertainty on $M_{\rm vir}$. The overall agreement between the two sets of masses is surprisingly good, considering that $M_{\rm vir}$ is computed under the strong (and probably incorrect) assumption that the peaks are virialised, and that $M_{\rm tot}$ is computed above a reasonable but still arbitrary density threshold. Indeed, although the adopted density threshold corresponds to selecting peaks which are about to begin or have just begun to collapse (see Sect. \[3D\_peaks\]), the evolution of a density fluctuation from the beginning of collapse to virialisation can take a few gigayears (see Sect. \[discussion\]). Moreover, the galaxies used to compute $\sigma_v$ and hence $M_{\rm vir}$ are drawn from slightly larger volumes than the volumes used to compute $M_{\rm tot}$, because we included galaxies in the tails of the velocity distribution along the l.o.s., outside the peaks’ volumes. We also find that $M_{\rm tot}$ continuously varies by changing the overdensity threshold to define the peaks (see Appendix \[app\_Mtot\_sigma\]), while the computation of the velocity dispersion in our peaks is very stable if we change this same threshold (see Appendix \[app\_vdisp\_sigma\]). As a consequence, we do not expect the estimated $M_{\rm vir}$ to change either. In addition to these caveats, peaks \[1\], \[2\] and \[3\] show an irregular 3D shape (see Appendix \[app\_peaks\]), and they might be multi-component structures. In these cases, the limited physical meaning of $M_{\rm vir}$ is evident. We also note that peak \[5\] has already been identified in the literature as a virialised structure (see [@wang2016_z250] and our discussion in Sect. \[peak5\]), meaning that its $M_{\rm vir}$ is possibly the most robust among the peaks, but in our reconstruction it is the most distant from the 1:1 relation between $M_{\rm vir}$ and $M_{\rm tot}$. This might suggest that our $M_{\rm tot}$ is underestimated, at least for this peak. We also remark that there is not a unique scaling relation between $\sigma_{\rm v}$ and $M_{\rm vir}$. For instance, [@munari13] study the relation between the masses of groups and clusters and their 1D velocity dispersion $\sigma_{\rm 1D}$. Clusters are extracted from $\Lambda$CDM cosmological N-body and hydrodynamic simulations, and the authors recover the velocity dispersion by using three different tracers, that is, dark-matter particles, sub-halos, and member galaxies. They find a relation in the form: $$\displaystyle \sigma_{\rm 1D}=A_{1D}\left[ \frac{h(z)~M_{\rm 200}}{10^{15}M_{\odot}} \right]^{\alpha} , \label{m200_munari}$$ where $A_{1D} \simeq 1180 \kms$ and $\alpha \simeq 0.38$, as from their Fig. 3 for $z=2$ (the highest redshift they consider) and by using galaxies as tracers for $\sigma_{\rm 1D}$. [@evrard08] find a relation based on the same principle as Eq. \[m200\_munari\], but they use DM particles to trace $\sigma_{\rm 1D}$. On the observational side, [@sereno15] find a relation in perfect agreement with [@munari13] by using observed data, with cluster masses derived via weak lensing. We also used Eq. \[m200\_munari\] to compute $M_{\rm vir}$[^10]. We found that the $M_{\rm vir}$ computed via Eq. \[m200\_munari\] are systematically smaller (by 20-40%) than the previous ones computed with Eq. \[mvir\]. This change would not appreciably affect the high degree of concordance between $M_{\rm vir}$ and $M_{\rm tot}$ for our peaks. In summary, the comparison between $M_{\rm vir}$ and $M_{\rm tot}$ is meaningful only if we fully understand the evolutionary status of our overdensities and know their intrinsic shapes (and we remind the reader that in this work the shape of the peaks depends at the very least on the chosen threshold, and it is not supposed to be their intrinsic shape). On the other hand, it would be very interesting to understand whether it is possible to use this comparison to infer the level of virialisation of a density peak, provided that its shape is known. This might be studied with simulations, and we defer this analysis to a future work. ![Virial mass $M_{\rm vir}$ of the seven identified peaks, as in Table \[peaks\_tab\_veldisp\], vs. the total mass $M_{\rm tot}$ as in Table \[peaks\_tab\]. We show both the virial mass computed only with our spectroscopic sample (red dots, column 6 of Table \[peaks\_tab\_veldisp\]) and how it would change if we add to our sample other spectroscopic sources found in the literature (black crosses, column 9 of Table \[peaks\_tab\_veldisp\]). Only peaks \[1\], \[3\], \[4\], and \[5\] have this second estimate of $M_{\rm vir}$. The dotted line is the bisector, as a reference.[]{data-label="virial_mass"}](./fig5.ps){width="9.0cm"} The many components of the proto-supercluster {#peak_list} --------------------------------------------- The COSMOS field is one of the richest fields in terms of data availability and quality. It was noticed early on that it contains extended structures at several redshifts (see e.g. [@scoville07_env; @guzzo07_z07; @cassata07_z07; @kovac10_density; @delatorre10_clustering; @scoville13_env; @iovino16_wall]). Besides using galaxies as direct tracers, as in the above-mentioned works, the large-scale structure of the COSMOS field has been revealed with other methods like weak lensing analysis (e.g. [@massey07]) and Ly$\alpha$-forest tomography [@lee2016_colossus; @lee2017_clamato]. Systematic searches for galaxy groups and clusters have also been performed up to $z\sim1$ (for instance [@knobel09_groups] and [@knobel12_groups20k]), and in other works we find compilations of candidate proto-groups [@diener2013_list] and candidate proto-clusters [@chiang2014_list; @franck16_CCPC; @lee2016_colossus] at $z\gtrsim1.6$. In some cases, the search for (proto-)clusters was focused around a given class of objects, like radio galaxies (see e.g. [@castignani14]). In particular, it has been found that the volume of space in the redshift range $2.4 \lesssim z \lesssim 2.5 $ hosts a variety of high-density peaks, which have been identified by means of different techniques/galaxy samples, and in some cases as part of dedicated follow-ups of interesting density peaks found in the previous compilations. Some examples are the studies by [@diener2015_z245], [@chiang2015_z244], [@casey2015_z247], [@lee2016_colossus], and [@wang2016_z250]. In this paper, we generally refer to the findings in the literature as density peaks when referring to the ensemble of the previous works; we use the definition adopted in each single paper (e.g. ‘proto-groups’, ‘proto-cluster candidates’, etc.) when we mention a specific study. We note that in the vast majority of these previous works there was no attempt to put the analysed density peaks in the broader context of a large-scale structure. The only exceptions are the works by [@lee2016_colossus] and [@lee2017_clamato], based on the Ly$\alpha$-forest tomography. [@lee2016_colossus] explore an area of $\sim14\times16$ h$^{-1}$ cMpc, which is roughly one ninth of the area covered by Hyperion, while [@lee2017_clamato] extended the tomographic map up to an area roughly corresponding to one third of the area spanned by Hyperion. Both these works do mention the complexity and the extension of the overdense region at $z\sim 2.45$, and the fact that it embeds three previously identified overdensity peaks [@diener2015_z245; @casey2015_z247; @wang2016_z250]. Nevertheless, they did not expand on the characteristics of this extended region, and were unable to identify the much larger extension of Hyperion, because of the smaller explored area. In this section we describe the characteristics of our seven peaks, and compare our findings with the literature. The aim of this comparison is to show that some of the pieces of the Hyperion proto-supercluster have already been sparsely observed in the literature, and with our analysis we are able to add new pieces and put them all together into a comprehensive scenario of a very large structure in formation. We also try to give a detailed description of the characteristics (such as volume, mass, etc.) of the structures already found in the literature, with the aim to show that different selection methods are able to find the same very dense structures, but these methods in some cases are different enough to give disparate estimates of the peaks’ properties. For this comparison, we refer to Fig. \[3D\_map\_lit\] and Table \[literature\_tab\], as detailed below. Moreover, in Appendix \[app\_peaks\] we show more details on our four most massive peaks, which we dub “Theia”, “Eos”, “Helios”, and “Selene”[^11]. Among the previous findings, we discuss only those falling in the volume where our peaks are contained. We remind that we did not make use of the samples used in these previous works. The only exception is that the zCOSMOS-Deep sample, included in our data set, was also used by [@diener2013_list]. ### Peak \[1\] - “Theia” {#peak1} Peak \[1\] is by far the most massive of the peaks we detected. Figure \[3D\_cube\_peaks\] shows that its shape is quite complex. The peak is composed of two substructures that indeed become two separated peaks if we increase the threshold for the peak detection from $5\sigma_{\delta}$ to $6.6\sigma_{\delta}$. In Fig. \[peak1\_fig\] of Appendix \[app\_peaks\] we show two 2D projections of peak \[1\], which indicate the complexity of the 3D structure of this peak. Figure \[3D\_map\_lit\] is the same as Fig. \[3D\_cube\_peaks\], but we also added the position of the overdensity peaks found in the literature. We verified that our peak \[1\] includes three of the proto-groups in the compilation by [@diener2013_list], called D13a, D13b, and D13d in our figure. Proto-goups D13a and D13b are very close to each other ($\sim 3$ arcmin on the RA-Dec plane) and together they are part of the main component of our peak \[1\]. D13d corresponds to the secondary component of peak \[1\], which detaches from the main component when we increase the overdensity threshold to $6.6\sigma_{\delta}$. Another proto-group (D13e) found by [@diener2013_list] falls just outside the westernmost and northernmost border of peak\[1\]. It is not unexpected that our peaks (see also peaks \[3\] and \[4\]) have a good match with the proto-groups found by [@diener2013_list], given that their density peaks have been detected using the zCOSMOS-Deep sample, which is also included in our total sample[^12]. In our peak \[1\] we find 24 spectroscopic members (see Table \[peaks\_tab\]), 14 of which come from the VUDS survey and 10 from the zCOSMOS-Deep sample. The shape of peak\[1\] (a sort of ‘L’, or triangle) is mirrored by the shape of the proto-cluster found by [@casey2015_z247], as shown in their Fig. 2. In our Fig. \[3D\_map\_lit\] their proto-cluster is marked as Ca15, and we placed it roughly at the coordinates of the crossing of the two arms of the ‘L’ in their figure, where they found an X-ray detected source. In their figure, the S-N arm extends to the north and has a length of $\sim14$ arcmin, and the E-W arm extends towards east and its length is about 10 arcmin. They also show that their proto-cluster encloses the three proto-groups D13a, D13b, and D13d. Although we found a correspondence between the position/extension of our peak \[1\] and the position/extension of some overdensities in the literature, it is harder to compare the properties of peak \[1\] and such overdensities. This difficulty is given mainly by the different detection techniques. We attempted this comparison and show the results in Table \[literature\_tab\]. In this table, for each overdensity in the literature we show its redshift, $\delta_{\rm gal}$, velocity dispersion, and total mass, when available in the respective papers. We also computed its total volume, based on the information in its respective paper, and computed its $\delta_{\rm gal}$ and total mass (using Eq. \[eq\_mass\]) in that same volume in our 3D cube. In the case of a 1:1 match with our peak (like in the case of Ca15 and our peak \[1\]), we also reported the properties of our matched peak. In the case of the proto-groups D13a, D13b, D13d and D13e, we found in the literature only their $\sigma_{\rm v}$, which we cannot compare directly with our peak \[1\] given that there is not a 1:1 match. The $\delta_{\rm gal}$ recovered in our 3D cube in the volumes corresponding to the four proto-groups are broadly consistent with the typical $\delta_{\rm gal}$ of our peaks, with the exception of D13e which in fact falls outside our peak \[1\]. These proto-groups have all relatively small volumes and masses compared to our peaks. At most, the largest one (D13a) is comparable in volume and mass with our smallest peaks (\[5\],\[6\], and \[7\]). The average difference in volume between our peaks and the proto-groups found in [@diener2013_list] might be due to the fact that they identified groups with a Friend-of-Friend algorithm with a linking length of 500 pkpc, i.e. $\sim1.7$ cMpc at $z=2.45$, which is smaller than the effective radius of our largest peaks (although their linking lengths and our effective radii do not have the same physical meaning). The properties of Ca15 were computed in a volume almost three times as large as our peak \[1\]. Nevertheless, its $\delta_{\rm gal}$ is much higher, probably because of the different tracers (they use dusty star forming galaxies, ‘DSFGs’). Despite our lower density in the Ca15 volume, we find a higher total mass ($M_{\rm tot} = 4.82 \times 10^{14} M_{\odot}$ instead of their total mass of $>0.8 \times 10^{14} M_{\odot}$). This is probably due to the different methods used to compute $M_{\rm tot}$: we use Eq. \[eq\_mass\], while [@casey2015_z247] use abundance matching techniques to assign a halo mass to each galaxy, and then sum the estimated halo masses for each galaxy in the structure. Moreover, they state that their mass estimate is a lower limit. ### Peak \[2\] - “Eos” {#peak2} As peak \[1\], this peak seems to be composed by two sub-structures, as shown in details in Fig. \[peak2\_fig\]. The two substructures detach from each other when we increase the overdensity threshold to $5.3\sigma_{\delta}$. On the contrary, by decreasing the overdensity threshold to $4.5\sigma{\delta}$ we notice that this peak merges with the current peak \[7\]. We did not find any direct match of peak \[2\] with previous detections of proto-structures in the literature. We note that this part of the COSMOS field is only partially covered by the tomographic search performed by [@lee2016_colossus] and [@lee2017_clamato]. This could be the reason why they do not find any prominent density peak there. ### Peak \[3\] - “Helios” {#peak3} The detailed shape of peak \[3\] is shown in Fig. \[peak3\_fig\]. From our density field, it is hard to say whether its shape is due to the presence of two sub-structures. Even by increasing the overdensity threshold, the peak does not split into two sub-components. Peak \[3\] is basically coincident with the group D13f from [@diener2013_list], and its follow-up by [@diener2015_z245], which we call D15 in our Fig. \[3D\_map\_lit\]. The barycenter of our peak \[3\] is closer to the position of D13f than to the position of D15, on both the RA-Dec plane ($<8^{\prime\prime}$ to D13f, $\sim50^{\prime\prime}$ on the Dec axis to D15) and the redshift direction ($\Delta z \sim 0.004$ with D13f, and $\Delta z \sim 0.05$ with D15). This very good match is possibly due also to the fact that our sample includes the zCOSMOS-Deep data (see comment in Sect. \[peak1\]). Indeed, out of the seven spectroscopic members that we identified in peak \[3\], five come from the zCOSMOS-Deep sample and two from VUDS. We note that the list of candidate proto-clusters by [@franck16_CCPC] includes a candidate that corresponds, as stated by the authors, to D13f. Interestingly, [@diener2015_z245] mention that D15 might be linked to the radio galaxy COSMOS-FRI 03 [@chiaberge09], around which [@castignani14] found an overdensity of photometric redshifts. Although the overdensity of photometric redshifts surrounding the radio galaxy is formally at slightly lower redshift than D15 (see also [@chiaberge10]), it is possibly identifiable with D15, given the photometric redshift uncertainty. Table \[literature\_tab\] shows that the velocity dispersion found by [@diener2015_z245] for D15 is very similar to the one we find for our peak \[3\], although the density that they recover is much larger ($\delta_{\rm gal}= 10$ vs $\delta_{\rm gal} \sim 3.$). We note that D15 is defined over a volume which is almost twice as large as peak \[3\]. The velocity dispersion of F16 is instead almost double the one we recover for peak \[3\]. Their search volume is huge ($\sim10000$ cMpc$^3$) compared to the volume of peak \[3\]. Considering that they also find quite high $\delta_{\rm gal}$, they compute a total mass of $\sim15\times10^{14}$ M$_\odot$, which is approximately three times larger than the one we find in our data in their same volume ($4.89\times10^{14}$ M$_\odot$), but about a factor of 30 larger than the mass of our peak \[3\]. Very close to peak \[3\] there are the three components of the extended proto-cluster dubbed ‘Colossus’ in [@lee2016_colossus][^13]. Here we call the three sub-structures L16a, L16b and L16c, in order of decreasing redshift. This proto-cluster was detected by IGM tomography (see also [@lee2017_clamato]) performed by analysing the spectra of galaxies in the background of the proto-cluster. The three peaks form a sort of chain from $z\sim2.435$ to $z\sim2.45$, which extends over $\sim2^{\prime}$ in RA and $\sim6^{\prime}$ in Dec. We derived the positions of the first and third peaks from Fig. 12 of [@lee2016_colossus], and assumed that the intermediate peak was roughly in between (see their Figs. 4 and 13). Neither L16a, L16b, or L16c coincide precisely with one of our peaks, but they fall roughly 3 arcmin eastwards of the barycenter of our peak \[3\]. The declination and redshift of the intermediate component correspond to those of our peak \[3\]. Given the extension of the three peaks in RA-Dec (they have a radius from $\sim2$ to $\sim4$ arcmin) and the extension of our peak \[3\] ($\sim2$ arcmin radius), the ‘Colossus’ overlaps with, and it might be identified with, our peak \[3\]. [@lee2016_colossus] compute the total mass of their overdensity, and find that it is $1.6\pm0.9\times10^{14}$ M$_\odot$. Computing the overall mass in the volumes of the three components L16a, L16b, and L16c in our data cube, we find a smaller mass ($0.83\times10^{14}$ M$_\odot$), which is still consistent with the value found by [@lee2016_colossus]. We additionally compared our results with those by [@lee2016_colossus] by directly using the smoothed IGM overdensity, $\delta_F^{\rm sm} $, estimated from the latest tomographic map [@lee2017_clamato]. We measured their average $\delta_F^{\rm sm}$ in the volume enclosing our peak \[3\] and found that this volume of space corresponds to an overdense region with respect to the mean intergalactic medium (IGM) density at these redshifts. Specifically, using the definition in [@lee2016_colossus], for which negative values of $\delta_F^{\rm sm} $ signify overdense regions, we found that our peak has $\langle \delta_F^{\rm sm} \rangle \sim -2.4\sigma_{\rm sm}$, with $\sigma_{\rm sm}$ denoting the effective sigma of the $\delta_F^{\rm sm} $ distribution. We repeated the same analysis in the volumes enclosed by our other peaks (with the exception of peak \[2\], which lies almost entirely outside the tomographic map), and we found that their $\langle \delta_F^{\rm sm} \rangle$ fall in the range from $-1.9\sigma_{\rm sm}$ to $-1\sigma_{\rm sm}$ meaning that all of our peaks appear overdense with respect to the mean IGM density at these redshifts. This persistent overdensity measured across the six peaks that we are able to measure in the tomographic map strongly hint at a coherent overdensity also present in the IGM maps. Further, all peaks have measured $\langle \delta_F^{\rm sm} \rangle$ values consistent with the expected IGM absorption signal due to the presence of at least some fraction of simulated massive ($M_{\rm tot, z=0}>10^{14} M_{\odot}$) proto-clusters (see section 4 of [@lee2016_colossus]). We note, however, that none of our peaks have $\langle \delta_F^{\rm sm} \rangle < -3\sigma_{\rm sm}$, which is the threshold suggested by [@lee2016_colossus] to safely identify proto-clusters (see their Fig. 6) with IGM tomography. Additionally, the level of the galaxy overdensity or $M_{\rm tot}$ from our galaxy density reconstruction does not necessarily correlate well with the $\langle \delta_F^{\rm sm} \rangle$ measured for the ensemble of proto-supercluster peaks likely due to a variety of astrophysical reasons as well as reasons drawing from the slight differences in the samples employed and reconstruction method. Regardless, this comparison demonstrates the complementarity of our method and IGM tomography to identify proto-clusters. This comparison will be expanded in future work to investigate differences in the signals in the two types of maps according to physical properties (like gas temperature, etc.) of individual proto-clusters. [@lee2016_colossus] identify their proto-cluster with one of the candidate proto-clusters found by [@chiang2014_list] (proto-cluster referred to here as Ch14). These latter authors systematically searched for proto-clusters using photometric redshifts and [@chiang2015_z244] presented a follow-up of Ch14, presenting a proto-cluster that we refer to here as Ch15. From [@chiang2015_z244], it is not easy to derive an official RA-Dec position of Ch15, so we assume it is at the same RA-Dec coordinates as Ch14. The redshifts of Ch14 and Ch15 are slightly different ($z=2.45$ and $z=2.445$, respectively). Our peak \[3\] is $\lesssim5$ arcmin away on the plane of the sky from Ch14 and Ch15, and this is in agreement with the distance that [@chiang2015_z244] mention from their proto-cluster to the proto-group D15, which matches with our peak \[3\]. Moreover, [@chiang2015_z244] associate a size of $\sim10\times7$ arcmin$^2$ to Ch15, which makes Ch15 overlap with peak \[3\]. According to [@chiang2015_z244], Ch15 has an overdensity of LAEs of $\sim4$ , computed over a volume of $\sim12000$ cMpc$^3$. Over this volume, the overdensity in our data cube is very low ($\delta_{\rm gal} = 0.53$), because it encompasses also regions well outside the highest peaks and even outside the proto-supercluster. Despite the low density, the volume is so huge that the mass of Ch15 that we compute in our data cube exceeds $5\times10^{14}$ M$_\odot$. [@chiang2015_z244] do not mention any mass estimate for Ch15. ### Peak \[4\] - “Selene” {#peak4} Peak \[4\] seems to be composed of a main component, which includes most of the mass/volume, and a tail on the RA-Dec plane, which is as long as about twice the length of the main component. This is shown in Fig. \[peak4\_fig\]. We did not find spectroscopic members in the tail. The barycenter of peak \[4\], centred on its main component, is coincident with the position of the proto-group D13c from [@diener2013_list]. Their distance on the plane of the sky is $\lesssim 30^{\prime\prime}$ arcsec, and they have the same redshift. Also in this case, this perfect agreement might be due to our use of the zCOSMOS-Deep sample (see Sect. \[peak1\]), although only half (2 out of 4) of the spectroscopic members of peak \[4\] come from the zCOSMOS-Deep survey. [@diener2013_list] compute a velocity dispersion of 239 km s$^{-1}$ for D13c, while we measured $\sigma_{\rm v}=672$ km s$^{-1}$ for peak \[4\]. This discrepancy, which holds even if we consider our uncertainty of $\sim 150$ km s$^{-1}$, might be due to the larger number of galaxies that we use to compute $\sigma_{\rm v}$ (9 vs. their 3 members). Moreover, the volume over which their proto-group is defined is much smaller (one seventh) than the volume covered by peak \[4\]. ### Peak \[5\] {#peak5} Peak \[5\] has a regular roundish shape on the RA-Dec plane, so we do not show any detailed plot in Appendix \[app\_peaks\]; it corresponds to the cluster found by [@wang2016_z250], which we call W16 in this work. We remark that [@wang2016_z250] find an extended X-ray emission associated to this cluster, and indeed they define W16 as a ‘cluster’ and not a ‘proto-cluster’ because they claim that there is evidence that it is already virialised. We refer to their paper for a more detailed discussion. The RA-Dec coordinates of W16 are offset by $\sim30^{\prime\prime}$ on the RA axis and $\sim5^{\prime\prime}$ on the Dec axis from peak \[5\]. The redshift of our peak \[5\] is $\Delta z=0.001$ higher than the redshift of W16. The velocity dispersion of our peak \[5\] is in remarkably good agreement with the one computed by [@wang2016_z250] (533 and 530 km s$^{-1}$, respectively), and, as a consequence, there is a very good agreement between the two virial masses. We note that peak \[5\] is one of the cases in our work where the total mass computed from $\delta_{\rm gal}$ is much smaller than the virial mass computed from the $\sigma_{\rm v}$. What is interesting in W16 is that it is extremely compact: the extended X-ray detection has a radius of about $24^{\prime \prime}$, and the majority of its member galaxies are also concentrated on the same area. Should we consider this small radius, its volume would be five times smaller than the one of our peak \[5\]. Instead, in Table \[literature\_tab\] we used a larger volume for the comparison (429 cMpc$^3$), derived from the maximum RA-Dec extension of the member galaxies quoted in [@wang2016_z250]. ### Peak \[6\] {#peak6} Peak \[6\] has a regular shape on the plane of the sky. We did not find any other overdensity peak or proto-cluster detected in the literature matching its position. ### Peak \[7\] {#peak7} Peak \[7\] has also a roughly round shape on the RA-Dec plane. It merges with peak \[2\] if we decrease the overdensity threshold to $4.5\sigma{\delta}$. We could not match it with any previous detection of proto-structures in the literature. ---------------------------- ----------------- ------------------ ---------------- -------------- ------------------------------------- -------------- ------------------------- ID RA$_{\rm peak}$ Dec$_{\rm peak}$ $z_{\rm peak}$ n$_{\rm zs}$ $\langle \delta_{\rm gal} \rangle $ Volume $M_{\rm tot}$ (Fig. \[3D\_cube\_peaks\]) \[deg\] \[deg\] \[cMpc$^3$\] \[$10^{14}$ M$_\odot$\] (1) (2) (3) (4) (5) (6) (7) (8) 1 150.0937 2.4049 2.468 24 3.79 3134 2.648$_{-1.39}^{+0.56}$ 2 149.9765 2.1124 2.426 7 2.89 951 0.690$_{-0.51}^{+0.84}$ 3 149.9996 2.2537 2.444 7 3.03 805 0.598$_{-0.37}^{+0.24}$ 4 150.2556 2.3423 2.469 4 3.20 720 0.552$_{-0.30}^{+0.40}$ 5 150.2293 2.3381 2.507 1 3.11 252 0.190$_{-0.16}^{+0.09}$ 6 150.3316 2.2427 2.492 4 3.12 251 0.190$_{-0.13}^{+0.06}$ 7 149.9581 2.2187 2.423 1 2.58 134 0.092$_{-0.09}^{+0.11}$ ---------------------------- ----------------- ------------------ ---------------- -------------- ------------------------------------- -------------- ------------------------- ---------------------------- ---------------- ----------- ----------- ----------- ---------------- ------------------------------------------- ---------------- ID $z_{\rm peak}$ R$_{e,x}$ R$_{e,y}$ R$_{e,z}$ $E_{\rm z/xy}$ $\langle \delta_{\rm gal, corr} \rangle $ V$_{\rm corr}$ (Fig. \[3D\_cube\_peaks\]) cMpc cMpc cMpc \[cMpc$^3$\] (1) (2) (3) (4) (5) (6) (7) (8) 1 2.468 3.37 4.07 7.76 2.09 10.84 1500 2 2.426 2.31 3.25 5.18 1.87 7.74 509 3 2.444 1.94 1.82 6.15 3.26 15.92 247 4 2.469 2.77 2.12 6.00 2.45 11.73 294 5 2.507 1.05 1.27 4.07 3.52 17.70 72 6 2.492 0.88 1.05 5.83 6.03 32.29 42 7 2.423 1.22 0.90 2.71 2.55 10.73 53 ---------------------------- ---------------- ----------- ----------- ----------- ---------------- ------------------------------------------- ---------------- ---------------------------- ---------------- ----------------- -------------- --------------------- ------------------------- --------------- --------------------- ------------------------- ------- ID $z_{\rm peak}$ n$_{zs,\sigma}$ $z_{\rm BI}$ $\sigma_{\rm v}$ $M_{\rm vir}$ n$_{\rm lit}$ $\sigma_{\rm v}$ $M_{\rm vir}$ Ref. (Fig. \[3D\_cube\_peaks\]) \[km s$^{-1}$\] \[$10^{14}$ M$_\odot$\] \[km s$^{-1}$\] \[$10^{14}$ M$_\odot$\] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 2.468 29 2.467 731$_{-92}^{+88}$ 2.16$_{-0.71}^{+0.88}$ 11 737$_{-86}^{+85}$ 2.21$_{-0.69}^{+0.85}$ 1,2,3 2 2.426 8 2.426 474$_{-144}^{+129}$ 0.60$_{-0.40}^{+0.63}$ - - - - 3 2.444 7 2.445 417$_{-121}^{+91}$ 0.41$_{-0.26}^{+0.33}$ 7 500$_{-87}^{+79}$ 0.70$_{-0.30}^{+0.39}$ 4,5,6 4 2.469 9 2.467 672$_{-162}^{+145}$ 1.68$_{-0.94}^{+1.33}$ 1 644$_{-158}^{+142}$ 1.47$_{-0.84}^{+1.21}$ 1 5 2.507 4 2.508 533$_{-163}^{+87}$ 0.82$_{-0.55}^{+0.49}$ 13 472$_{-80}^{+86}$ 0.57$_{-0.24}^{+0.37}$ 7 6 2.492 4 2.490 320$_{-151}^{+56}$ 0.18$_{-0.15}^{+0.11}$ - - - - 7$^*$ 2.423 3 2.428 461$_{-304}^{+304}$ 0.55$_{-0.53}^{+1.97}$ - - - - ---------------------------- ---------------- ----------------- -------------- --------------------- ------------------------- --------------- --------------------- ------------------------- ------- ![Same as Fig. \[3D\_cube\_peaks\], but in RA-Dec-z coordinates. Moreover, we overplot the location of the overdensity peaks/proto-clusters/proto-groups detected in other works in the literature (blue and green cubes, and blue and cyan crosses). Different colours and shapes are used for the symbols for clarity purposes only. Labels correspond to the IDs in Table \[literature\_tab\]. The dimensions of the symbols are arbitrary and do not refer to the extension of the overdensity peaks found in the literature. []{data-label="3D_map_lit"}](./fig6.ps){width="9.0cm"} ------------------------- ------ ------- ------------------------ ------------------ ------------------------- ------------- ------------------------------------ ------------------------- ------------ ------------------------------------ ---------- ------------------------- ------------------------ ------------------------- ID Ref. z $\delta_{\rm gal}$ $\sigma_{\rm v}$ M$_{\rm tot}$ Volume $\langle \delta_{\rm gal} \rangle$ M$_{\rm tot}$ Match with $\langle \delta_{\rm gal} \rangle$ Volume M$_{\rm tot}$$^{e}$ $\sigma_{\rm v}$$^{e}$ M$_{\rm vir}$$^{e}$ (Fig. \[3D\_map\_lit\]) \[km s$^{-1}$\] \[$10^{14}$ M$_\odot$\] cMpc$^3$ \[$10^{14}$ M$_\odot$\] this work cMpc$^3$ \[$10^{14}$ M$_\odot$\] \[km s$^{-1}$\] \[$10^{14}$ M$_\odot$\] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) L16a 4 2.450 - - 1.6$\pm$0.9$^{b}$ 1568$^{b}$ 1.50$^{b}$ 0.83$^{b}$ \[3\]\* - - - - - L16b 4 2.443 - - 1.6$\pm$0.9$^{b}$ 1568$^{b}$ 1.50$^{b}$ 0.83$^{b}$ \[3\]\* - - - - - L16c 4 2.435 - - 1.6$\pm$0.9$^{b}$ 1568$^{b}$ 1.50$^{b}$ 0.83$^{b}$ \[3\]\* - - - - - W16 5 2.506 - 530$\pm120$ 0.79$^{+0.46}_{-0.29}$ 429 2.46 0.29 \[5\] 3.11 252 0.190 533 0.82 F16 8 2.442 9.27$\pm$4.93 770 15.5/14.1$^{d}$ $\sim10000$ 1.04 4.89 \[3\] 3.03 805 0.598 417 0.41 D15 1 2.450 10 426 - 1513 1.99 0.92 \[3\] 3.03 805 0.598 417 0.41 Ca15 2 2.472 11$^{c}$ - $>0.8\pm0.3$ 8839 1.55 4.82 \[1\] 3.79 3134 2.648 731 2.16 Ch15 3 2.440 4$^{a}$ - - $\sim12000$ 0.53 $\sim5.6$ \[3\]\* - - - - - Ch14 7 2.450 1.34$^{+0.49}_{-0.40}$ - - $\sim23000$ 0.37 $\sim9.1$ \[3\]\* - - - - - D13a 6 2.476 - 264 - 87 3.12 0.07 \[1\]\* - - - - - D13b 6 2.469 - 488 - 253 3.73 0.21 \[1\]\* - - - - - D13c 6 2.469 - 239 - 108 4.26 0.10 \[4\] 3.20 720 0.552 672 1.68 D13d 6 2.463 - 30 - 26 4.08 0.02 \[1\]\* - - - - - D13e 6 2.452 - 476 - 38 0.89 0.02 \[1\]\* - - - - - D13f 6 2.440 - 526 - 425 2.87 0.31 \[3\] 3.03 805 0.598 417 0.41 ------------------------- ------ ------- ------------------------ ------------------ ------------------------- ------------- ------------------------------------ ------------------------- ------------ ------------------------------------ ---------- ------------------------- ------------------------ ------------------------- Discussion ========== The detection of such a huge, massive structure, caught during its formation, poses challenging questions. On the one hand, one would like to know whether we can predict the evolution of its components. On the other, it would be interesting to understand whether at least some of these components are going to interact with one another, or at the very least, how much they are going to interact with the surrounding large-scale structure as a whole. Moreover, the existence of superclusters at lower redshifts begs the question of whether this proto-structure will evolve to become similar to one of these closer superclusters. We address these issues below in a qualitative way, and defer any further analysis to a future work. The evolution of the individual density peaks. {#collapse} ---------------------------------------------- Assuming the framework of the spherical collapse model, we computed the evolution of our overdensity peaks as if they were isolated spherical overdensities. This is clearly a significant assumption (see e.g. [@despali13] for the evolution of ellipsoidal halos), but it can help us in roughly understanding the evolutionary status of these peaks, and how peaks with similar overdensities would evolve with time. According to the spherical collapse model, any spherical overdensity will evolve like a sub-universe, with a matter-energy density higher than the critical overdensity at any given epoch. In our case, we reasonably assume that the average matter overdensity $\langle \delta_{\rm m} \rangle $ in our peaks corresponds to a non-linear regime, because it is already well above 1. We report $\langle \delta_{\rm m} \rangle $ in Table \[peaks\_evol\_tab\] as $\langle \delta_{\rm m,corr} \rangle $, given that we define it as $\langle \delta_{\rm m,corr} \rangle = \langle \delta_{\rm gal,corr} \rangle / b$, with $\langle \delta_{\rm gal,corr} \rangle $ as reported in Table \[peaks\_elongation\_tab\] and $b$ the bias measured by [@durkalec15b] as in Sect. \[3D\]. Given that it is much easier to compute the evolution of an overdensity in linear regime than in non-linear regime, we transform [@padmanabhan] our $\langle \delta_{\rm NL} \rangle $ into their corresponding values in linear regime, $\langle \delta_{\rm L} \rangle $, and make them evolve according to the spherical linear collapse model. In particular, the overdense sphere passes through three specific evolutionary steps. The first one is the point of turn-around, when the overdense sphere stops expanding and starts collapsing, becoming a gravitationally bound structure. This happens when the overdensity in linear regime is $\delta_{\rm L,ta}\simeq 1.062$ (in non-linear regime it would be $\delta_{\rm NL,ta}\simeq 4.55$). After the turn-around, when the radius of the sphere becomes half of the radius at turn-around, the overdense sphere reaches the virialisation. In this moment, we have $\delta_{\rm L,vir}\simeq 1.58$ and $\delta_{\rm NL,vir}\simeq 146$. The sphere then continues the collapse process, till the moment of maximum collapse which theoretically happens when its radius becomes zero with an infinite density. In the real universe the collapse stops before the density becomes infinite, and at that time the system, which still satisfies the virial theorem, reaches $\delta_{\rm L,c}\simeq 1.686$ ($\delta_{\rm NL,c}\simeq 178$). In our work we are interested in the moments of turn-around and collapse. Here we will follow the formalism as in [@pace10], and we will use the symbol $\delta_{\rm c}$ for $\delta_{\rm L,c}\simeq 1.686$ and the symbol $\Delta_{\rm V}$ for $\delta_{\rm NL,c}\simeq 178$. When we refer to the time(/redshift) of turn-around and collapse, we use $t_{\rm ta}$(/$z_{\rm ta}$) and $t_{\rm c}$(/$z_{\rm c}$). We reiterate that $\delta_{\rm c}$ and $\Delta_{\rm V}$ are constant with redshift in an Einstein - de Sitter (EdS) Universe, while they evolve with time in a $\Lambda$CDM cosmology, and their evolution depends on the relative contribution of $\Omega_{\rm \Lambda}(z)$ and $\Omega_{\rm m}(z)$ to $\Omega_{\rm tot}(z)$. At high redshift (e.g. $z=5$) when $\Omega_{\rm \Lambda}(z)$ is small, $\delta_{\rm c}$ and $\Delta_{\rm V}$ are close to their EdS counterparts. As time goes by, $\Omega_{\rm \Lambda}(z)$ increases and both $\delta_{\rm c}$ and $\Delta_{\rm V}$ decrease. This is shown, for instance, in [@pace10], where they show that $\delta_{\rm c}$ decreases by less than 1% from $z=5$ to $z=0$, while in the same timescale $\Delta_{\rm V}$ decreases from $\sim178$ to $\sim 100$ (see also [@bryan_norman98], where they use the symbol $\Delta_{\rm c}$ instead of $\Delta_{\rm V}$). In our work we allow our overdensities to evolve in the linear regime, so we are interested at the time when they reach $\delta_{\rm c}$. Given its small evolution with redshift, we consider it a constant, set as in the EdS universe. The evolution of a fluctuation is given by its growing mode $D_{\rm +}(z)$. At a given redshift $z_{\rm 2}$, the overdensity $\delta_{\rm L}(z_{\rm 2})$ can be computed knowing the overdensity at another redshift $z_{\rm 1}$ and the value of the growing mode at the two redshifts, as follows: $$\displaystyle \delta_{\rm L}(z_{\rm 2}) = \delta_{\rm L}(z_{\rm 1}) \frac{D_{\rm +}(z_{\rm 2})}{D_{\rm +}(z_{\rm 1})} \label{delta_evol} .$$ In a $\Lambda$CDM universe, we define the linear growth factor $g$ as $g \equiv D_{\rm +}(z)/a$, where $a=(1+z)^{-1}$ is the cosmic scale factor. By using an approximate expression for $g$ (see e.g. [@carroll92] and [@hamilton01]), which depends on $\Omega_{\rm \Lambda}(z)$ and $\Omega_{\rm m}(z)$, we can recover $D_{\rm +}(z)$ and with equation \[delta\_evol\] derive the time when our peaks reach $\delta_{\rm L,ta}$ and $\delta_{\rm c}$, starting from the measured values of $\delta_{\rm L}(z_{\rm obs})$, with $z_{\rm obs}$ being the redshifts given in Table \[peaks\_tab\]. Figure \[delta\_evol\_ps\] shows the evolution of the density contrast of our peaks. In Table \[peaks\_evol\_tab\] we list the values of $z_{\rm ta}$ and $z_{\rm c}$, together with the time elapsed from $z_{\rm obs}$ to these two redshifts. As a very rough comparison, if we considered the entire Hyperion proto-supercluster with its $\langle \delta_{\rm gal} \rangle \sim 1.24$ (Sect. \[3D\]), and assumed an elongation equal to the average elongation of the peaks to derive its $\langle \delta_{\rm gal,corr} \rangle$ and then its $\langle \delta_{\rm m,corr} \rangle$, the proto-supercluster would have $\delta_{\rm L} \lesssim 0.8$ at $z=2.46$ (to be compared with the y-axis of Fig. \[delta\_evol\_ps\]). We note that the evolutionary status of the peaks depends by definition on their average density, that is, the higher the density, the more evolved the overdensity perturbation. The most evolved is peak \[6\], which has $\langle \delta_{\rm m,corr} \rangle = 12.66$, almost twice as large as the second densest peak (peak \[5\]). According to the spherical collapse model, peak \[6\] will be a virialised system by $z\sim1.7$, that is, in $1.3$ Gyr from the epoch of observation. The least evolved is peak \[2\], that will take 0.6 Gyr to reach the turn-around and then another $\sim3.8$ Gyr to virialise. This simple exercise, which is based on a strong assumption, shows that the peaks are possibly at different stages of their evolution, and will become virialised structures at very different times. In reality, the peaks’ evolution will be more complex, given that they will possibly accrete mass/subcomponents/galaxies during their lifetimes, and these results make it desirable to study how we can combine the density-driven evolution of the individual peaks with the overall evolution of the Hyperion proto-supercluster as a whole. Moreover, by comparing the evolutionary status of each peak with the average properties of its member galaxies, it will be possible to study the co-evolution of galaxies and the environment in which they reside. We defer these analyses to future works. ![Evolution of $\delta_m$ for the seven peaks listed in Table \[peaks\_evol\_tab\], with different line styles as in the legend. The evolution is computed in a linear regime for a $\Lambda$CDM Universe. For each peak, we start tracking the evolution from the redshift of observation (column 2 in Table \[peaks\_evol\_tab\]), and we consider as starting $\delta_{\rm m}$ the one computed from the corrected $\langle \delta_{\rm gal,corr} \rangle$ (column 7 in Table \[peaks\_elongation\_tab\]) and transformed into linear regime. The horizontal lines represent $\delta_{\rm L,ta}\simeq 1.062$, $\delta_{\rm L,vir}\simeq 1.58$ and $\delta_{\rm L,c}\simeq 1.686$. See Sect. \[collapse\] for more details.[]{data-label="delta_evol_ps"}](./fig7.ps){width="9.0cm"} ----- ------- ---------------------------------------- ---------------- ------------- --------------------- -------------------- ID $z$ $\langle \delta_{\rm m,corr} \rangle $ $z_{\rm ta}$ $z_{\rm c}$ $\Delta t_{\rm ta}$ $\Delta t_{\rm c}$ \[Gyr\] \[Gyr\] (1) (2) (3) (4) (5) (6) (7) 1 2.468 4.25 2.402 1.054 0.08 3.16 2 2.426 2.04 2.001 0.781 0.60 4.37 3 2.444 6.24 $>z_{\rm obs}$ 1.282 - 2.32 4 2.469 4.60 $>z_{\rm obs}$ 1.108 - 2.95 5 2.507 6.94 $>z_{\rm obs}$ 1.388 - 2.07 6 2.492 12.66 $>z_{\rm obs}$ 1.675 - 1.33 7 2.423 4.21 2.347 1.017 0.10 3.26 ----- ------- ---------------------------------------- ---------------- ------------- --------------------- -------------------- : Evolution of the density peaks according to the spherical collapse model in linear regime. Columns (1) and (2) are the ID and the redshift of the peak, as in Table \[peaks\_elongation\_tab\]. Column (3) is the average matter overdensity derived from the average galaxy overdensity of column (7) of Table \[peaks\_elongation\_tab\]. Columns (4) and (5) are the redshifts when the overdensity reaches the overdensity of turn-around and collapse, respectively. Columns (6) and (7) are the corresponding time intervals $\Delta t$ since the redshift of observation $z_{\rm obs}$ (column 2) to the redshifts of turn-around and collapse. When $z_{\rm ta} < z_{\rm obs}$ the turn-around has already been reached before the redshift of observation, and in these cases the corresponding $\Delta t $ have not been computed. See Sect. \[collapse\] for more details.[]{data-label="peaks_evol_tab"} The proto-supercluster as a whole. {#whole_psc} ---------------------------------- In the previous section we pretended that the peaks were isolated density fluctuations and traced their evolution in the absence of interactions with other components of the proto-supercluster. This is an oversimplification, because several kinds of interactions are likely to happen in such a large structure, such as for example accretion of smaller groups along filaments onto the most dense peaks, as expected in a $\Lambda$CDM universe. For instance, for what concerns merger events between proto-clusters, [@lee2016_colossus] examined the merger trees of some of the density peaks that they identified in realistic mock data sets by applying the same 3D Ly$\alpha$ forest tomographic mapping that they applied to the COSMOS field. They found that in the examined mocks, very few of the proto-structures identified by the tomography at $z\sim2.4$ and with an elongated shape (such as the ‘chain’ of their peaks L16a, L16b, and L16c discussed in Sect. \[peak3\]) are going to collapse to one single cluster at z=0. Similarly, [@topping18] analysed the Small MultiDark Planck Simulation in search for $z\sim3$ massive proto-clusters with a double peak in the galaxy velocity distribution and with the two peaks separated by about $2000\kms$, like the one they identified in previous observations [@topping16]. They found that such double-peaked overdensities are not going to merge into a single cluster at $z=0$. The structures found by [@lee2016_colossus] and [@topping16] are much smaller and with simpler shapes compared to the Hyperion proto-supercluster, and yet they are unlikely to form a single cluster at z=0, according to simulations. Interestingly, [@topping18] also found that in their simulation the presence of two massive peaks separated by $2000\kms$ is a very rare event (one in $7.4h^3$Gpc$^{-3}$) at $z\sim3$. These findings indicate that the evolution of the Hyperion proto-supercluster cannot be simplified as series of merging events, and that the identification of massive/complex proto-clusters at high redshift could be useful to give constraints on dark matter simulations. Indeed, it would be interesting to know whether or not Hyperion could be the progenitor of known lower-redshift superclusters. One difficulty is that there is no unique definition of a supercluster (but see e.g. [@chon15] for an attempt), and the taxonomy of known superclusters up to $z\sim1.3$ spans wide ranges of mass (from a few $10^{14}M_{\odot}$ as in [@swinbank07] to $>10^{16}M_{\odot}$ as in [@bagchi17]), dimension (a few cMpc as in [@rosati99_lynx] or $\sim 100$ cMpc as in [@kim16_sc]), morphology (compact as in [@gilbank08], or with multiple overdensities as in [@lubin00; @lemaux12]), and evolutionary status (embedding collapsing cores as in [@einasto16_SGW] or already virialised clusters as in [@rumbaugh18]). This holds also for the well-known superclusters in the local universe (see e.g. [@shapley30; @shapley34_hercules; @delapparent86_greatwall; @haynes86_pp]), not to mention the category of the so-called Great Walls, which are sometimes defined as systems of superclusters (like e.g. the Sloan Great Wall, [@vogeley04; @gott05], and the Boss Great Wall, [@lietzen16_sc]). Clearly, Hyperion shares many characteristics with the above-mentioned superclusters, making it likely that its eventual fate will be to become a supercluster. A further step would be identifying which known supercluster is most likely to be similar to the potential descendant(s) of Hyperion. This would be surely an important step in understanding how the large-scale structure of the universe evolves and how it affects galaxy evolution. On the other hand, it is also interesting to study the likelihood of such (proto-) superclusters existing in a given cosmological volume, given their volumes and masses (see e.g. [@sheth11]). For instance, [@lim14] show that the relative abundance of rich superclusters at a given epoch could be used as a powerful cosmological probe. From [@lim14] we can qualitatively assess how many superclusters of the kind that we detect are expected in the volume probed by VUDS. [@lim14] derive the mass function of superclusters, defined as clusters of clusters according to a Friend of Friend algorithm. Since the supercluster mass function at $z\sim2.5$ was not explicitly studied, we adopt here expectations from their study of the $z=1$ supercluster mass function keeping in mind that this expectation will be a severe upper limit given that the halo mass function at the high-mass end decreases by a factor of $\ga100$ from $z=1$ to $z=2.5$ (see, e.g. [@percival05]). With this in mind, we estimate, using those results of [@lim14] that employ a similar cosmology to the one used in this study, the extreme upper limit to the number of superclusters with a total mass $>5\times10^{14}$ M$_{\odot}$ expected within the RA-Dec area studied in this paper and in the redshift range $2<z<4$ to be $\sim$4. We consider this mass limit, $>5\times10^{14}$ M$_{\odot}$, because it is the sum of the masses of our peaks, similarly to how they compute the masses of their superclusters. The extremeness of this upper limit is such that much more precise comparisons need to be made. We defer the detailed analysis of number counts and evolution of proto-superclusters at $z\sim2.5$ in simulated cosmological volumes to a future work. Summary and conclusions {#summary} ======================= Thanks to the spectroscopic redshifts of VUDS, together with the zCOSMOS-Deep spectroscopic sample, we unveiled the complex shape of a proto-supercluster at $z\sim2.45$ in the COSMOS field. We computed the 3D overdensity field over a volume of $\sim100\times100\times250$ comoving Mpc$^3$ by applying a Voronoi tessellation technique in overlapping redshift slices. The tracers catalogue comprises our spectroscopic sample complemented by photometric redshifts for the galaxies without spectroscopic redshift. Both spectroscopic and photometric redshifts were treated statistically, according to their quality flag or their measurement error, respectively. The main advantage of the Voronoi Tessellation is that the local density is measured both on an adaptive scale and with an adaptive filter shape, allowing us to follow the natural distribution of tracers. In the explored volume, we identified a proto-supercluster, dubbed “Hyperion" for its immense size and mass, extended over a volume of $\sim60\times60\times150$ comoving Mpc$^3$. We estimated its total mass to be $\sim 4.8\times 10^{15}{\rm M}_{\odot}$. Within this immensely complex structure, we identified seven density peaks in the range $2.40<z<2.5$, connected by filaments that exceed the average density of the volume. We analysed the properties of the peaks, as follows: - We estimated the total mass of the individual peaks, $M_{\rm tot}$, based on their average galaxy density, and found a range of masses from $\sim 0.1\times 10^{14}{\rm M}_{\odot}$ to $\sim 2.7\times 10^{14}{\rm M}_{\odot}$. - By assigning spectroscopic members to each peak, we estimated the velocity dispersion of the galaxies in the peaks, and then their virial mass $M_{\rm vir}$ (under the admittedly strong assumption that they are virialised). The agreement between $M_{\rm vir}$ and $M_{\rm tot}$ is surprisingly good, considering that (almost all) the peaks are most probably not yet virialised. - If we assume that the peaks are going to evolve separately, without accretion/merger events, the spherical collapse model predicts that these peaks have already started or are about to start their collapse phase (‘turn-around’), and they will all be virialised by redshift $z\sim0.8$. - We finally performed a careful comparison with the literature, given that some smaller components of this proto-supercluster had previously been identified in other works using heterogeneous galaxy samples (LAEs, 3D Ly$\alpha$ forest tomography, sub-mm starbursting galaxies, CO emitting galaxies). In some cases we found a one-to-one match between previous findings and our peaks, in other cases the match is disputable. We note that a direct comparison is often difficult because of the different methods/filters used to identify proto-clusters. In summary, with VUDS we obtained, for the first time across the central $\sim1$ deg$^2$ of the COSMOS field, a panoramic view of this large structure that encompasses, connects, and considerably expands on all previous detections of the various sub-components. The characteristics of the Hyperion proto-supercluster (its redshift, its richness over a large volume, the clear detection of its sub-components), together with the extensive band coverage granted by the COSMOS field, provide us the unique possibility to study a rich supercluster in formation 11 billion years ago. This impressive structure deserves a more detailed analysis. On the one hand, it would be interesting to compare its mass and volume with similar findings in simulations, because the relative abundance of superclusters could be used to probe deviations from the predictions of the standard $\Lambda$CDM model. On the other hand, it is crucial to obtain a more complete census of the galaxies residing in the proto-supercluster and its surroundings. With this new data, it would be possible to study the co-evolution of galaxies and the environment in which they reside, at an epoch ($z\sim2-2.5$) when galaxies are peaking in their star-formation activity. We thank the referee for his/her comments, which allowed us to clarify some parts of the paper. This work was supported by funding from the European Research Council Advanced Grant ERC-2010-AdG-268107-EARLY and by INAF Grants PRIN 2010, PRIN 2012 and PICS 2013. This work was additionally supported by the National Science Foundation under Grant No. 1411943 and NASA Grant Number NNX15AK92G. OC acknowledges support from PRIN-INAF 2014 program and the Cassini Fellowship program at INAF-OAS. This work is based on data products made available at the CESAM data center, Laboratoire d’Astrophysique de Marseille. This work partly uses 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. This paper is also based in part on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data products produced by TERAPIX and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium. OC thanks M. Roncarelli, L. Moscardini, C. Fedeli, F. Marulli, C. Giocoli, and M. Baldi for useful discussions, and J.R. Franck and S.S. McGaugh for their kind help in unveiling the details of their work. Stability of the peaks properties {#app_stability} ================================= We investigated the extent to which the choice of a 5$\sigma_{\delta}$ threshold affects some of the properties of the identified peaks. Namely, we varied the overdensity threshold from 4.5$\sigma_{\delta}$ to 5.5$\sigma_{\delta}$, and verified the variation of $M_{\rm tot}$ (Table \[peaks\_tab\]), velocity dispersion (Table \[peaks\_tab\_veldisp\]) and elongation (Table \[peaks\_elongation\_tab\]) as a function of the used threshold. Total mass {#app_Mtot_sigma} ---------- Figure \[Mtot\_vs\_sigma\] shows the fractional variation of $M_{\rm tot}$ (Table \[peaks\_tab\]) as a function of the adopted threshold, which is expressed in terms of the corresponding multiple of $\sigma_{\delta}$. Five peaks out of seven show roughly the same variation, while peak \[1\] has a much smaller variation and peak \[7\] a much steeper one. This might imply that the (baryonic) matter distribution within peak \[7\] is less peaked toward the centre with respect to the other peaks, while the matter distribution within peak \[1\] is more peaked. Given that we are probing very dense peaks (they are about to collapse, see Sect. \[discussion\]), we expect the total mass enclosed above a given overdensity threshold to have large variations if we vary the overdensity threshold by much. If instead we focus on a small $n_{\sigma}$ range around our nominal value of $n_{\sigma}=5$, for instance the interval $5\pm0.2$, we see that the variation of the total mass is much smaller than the uncertainty on the total mass quoted in Table \[peaks\_tab\], which was computed by using the density maps obtained with $\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$ (see Sect. \[method\]). This means that, although the total mass of our peaks depends on the chosen overdensity threshold, because of the very nature of the mass distribution in these peaks, at the chosen threshold the uncertainty is dominated by the uncertainty on the reconstruction of the density field and not by our precise definition of ‘overdensity peak’. ![ Fractional variation of the total mass $M_{\rm tot}$ (Table \[peaks\_tab\]) for the seven peaks as a function of the overdensity threshold adopted to identify them, expressed in terms of the corresponding multiples $n_{\sigma}$ of $\sigma_{\delta}$. The reference total mass value is the one at the 5$\sigma_{\delta}$ threshold. The different lines correspond to the different peaks as in the legend. The filled symbols on the right, with their error bars, correspond to the fractional variation of $M_{\rm tot}$ calculate at $5\sigma_{\delta}$ resulting from the uncertainties on the density reconstruction quoted in Table \[peaks\_tab\]. The position of the error bars on the x-axis is arbitrary. In all cases, these errors are much larger than the uncertainty resulting from slightly modulating the overdensity threshold employed.[]{data-label="Mtot_vs_sigma"}](./figA1.ps){width="9.0cm"} Velocity dispersion {#app_vdisp_sigma} ------------------- Similarly to the variation of the total mass, we verified how the velocity dispersion $\sigma_{\rm v}$ varies as a function of the adopted overdensity threshold, for the seven identified peaks. For each threshold, the velocity dispersion and its error are computed as described in Sect. \[vel\_disp\], and only when we could use at least three spectroscopic galaxies. For all the peaks, $\sigma_{\rm v}$ is relatively stable in the entire range of the explored overdensity thresholds, and its small variations (due to the increasing or decreasing number of spectroscopic members) are always much smaller than the uncertainties computed on the velocity dispersion itself, at fixed $n_{\sigma}$. For this reason we consider the virial masses quoted in Table \[peaks\_tab\_veldisp\] to be independent from small variations of the overdensity threshold. We remind the reader that for the computation of the velocity dispersion we used a more relaxed definition of galaxy membership within each peak so as to increase the number of the available galaxies (see Sect. \[vel\_disp\]). Even with this broader definition, for peak \[7\] we had only two galaxies available if we used $n_{\sigma}=5$ to define the peak, while their number increased to four by using $n_{\sigma}=4.9$. For this reason, we decided that the most reliable value of $\sigma_{\rm v}$ for peak \[7\] is the one computed using $n_{\sigma}=4.9$, and we quote this $\sigma_{\rm v}$ in Table \[peaks\_tab\_veldisp\]. Elongation {#app_elongation} ---------- Here we approximately estimate how the elongation depends on the typical dimension of our density peaks. Our estimation is based on the following simplistic assumptions: 1) the intrinsic shape of a proto-cluster is a sphere with radius $r_{\rm int}$, and its measured dimensions on the $x-$ and $y-$axis ($r_x$ and $r_y$) correspond to the intrinsic dimension $r_{int}$, i.e. $r_x=r_y=r_{\rm int}$, and 2) the measured dimension on the $z-$axis ($r_z$) corresponds to $r_{int}$ plus a constant factor $\Delta r$, which is the result of the complex interaction among the several factors that might cause the elongation (the depth of the redshift slices, the photometric redshift error etc), i.e. $r_z=r_{int}+\Delta r$. From these assumptions it follows: $$\displaystyle \frac{r_z}{r_{xy}} = 1 + \frac{\Delta r}{r_{int}}, \label{elongation_eq}$$ where $r_{xy}$ is the average between $r_x$ and $r_y$, and in our example we have $r_{xy}=r_x=r_y=r_{\rm int}$. If we substitute $r_x$, $r_y$ and $r_z$ with $R_{e,x}$, $R_{e,y}$ and $R_{e,z}$ as defined in Sect. \[3D\_peaks\], from Eq. \[elongation\_eq\] follows: $$\displaystyle E_{\rm z/xy} = 1 + \frac{\Delta r}{R_{e,xy}}, \label{elongation_eq2}$$ with $E_{\rm z/xy}$ and $R_{e,xy}$ as defined in Sect. \[3D\_peaks\]. This means that the measured elongation depends on the circularised 2D effective radius as $y=1+A/x$. To verify this dependence, we measured $E_{\rm z/xy}$ and $R_{e,xy}$ for our seven peaks for different thresholds, expressed in terms of the multiples $n_{\sigma}$ of $\sigma_{\rm \delta}$. In this case, we made the threshold vary from 4.1 to 7 $\sigma_{\rm \delta}$, because the two peaks \[1\] and \[4\] merge in one huge structure if we use a threshold $<4.1\sigma_{\delta}$. We notice that peak \[5\] disappears for $\sigma_{\rm \delta}>5.8$ above the mean density, and peak \[7\] for $\sigma_{\rm \delta}>5.4$. The peaks \[1\], \[2\] and \[4\] are split into two smaller peaks when $\delta_{\rm gal}$ is above 6.5$\sigma_{\rm \delta}$, 5.2$\sigma_{\rm \delta}$ and 5.1$\sigma_{\rm \delta}$ above the mean density, respectively. Figure \[elongation\_fig\] shows how $E_{\rm z/xy}$ varies as a function of $R_{e,xy}$. The three curves with equation $y=1+A/x$ are shown to guide the eye, with $A$ tuned by eye to match the normalisation of some of the observed trends. It is evident that the foreseen dependence of $E_{\rm z/xy}$ on $R_{e,xy}$ is confirmed. In the Figure, $A$ increases by a factor of $\sim3$ from the lowest curve (corresponding e.g. to peak \[7\]) to the highest one (matching e.g. peak \[6\]). The specific value of $A$ is likely due to a complex combination of peculiar velocities, spectral sampling, reconstruction methods (e.g. slice size relative to the true l.o.s. extent), and photometric redshift errors. It is beyond the scope of this paper to precisely quantify the contribution of each for each individual peak. Nevertheless, although in some cases $E_{\rm z/xy}$ quickly vary for small changes of $R_{e,xy}$ (i.e. small changes in the threshold), this plot confirms that its measured values are reasonably consistent with our expectations. ![Elongation $E_{\rm z/xy}$ as a function of $R_{e,xy}$. The different colours refer to the different peaks as in the legend. $E_{\rm z/xy}$ and $R_{e,xy}$ are measured by fixing different thresholds (number of $\sigma_{\rm \delta}$ above the mean density) to define the peaks themselves, ranging from 4.1 to 7 $\sigma_{\rm \delta}$. $E_{\rm z/xy}$ and $R_{e,xy}$ measured at the $5\sigma_{\rm \delta}$ threshold are highlighted with a filled circle, and are the same quoted in Table \[peaks\_elongation\_tab\]. The peaks \[1\] and \[2\] are split into two smaller peaks when $\delta_{\rm gal}$ is above 5.5$\sigma_{\rm \delta}$ and 5.7$\sigma_{\rm \delta}$ above the mean density, respectively: this is shown in the plot by splitting the curve of the two peaks into two series of circles (filled and empty). The three dotted lines corresponds to the curves $y=1+A/x$, with $A=4.3,2.9,1.5$ from top to bottom. The values of A are chosen to make the curves overlap with some of the data, to guide the eye.[]{data-label="elongation_fig"}](./figA2.ps){width="9.0cm"} Details on individual peaks {#app_peaks} =========================== We show here the projections on the RA-Dec and $z$-Dec planes of the four most massive peaks (“Theia”, “Eos”, “Helios”, and “Selene”), to highlight their complex shape. The remaining peaks have very regular shapes on the RA-Dec and $z$-Dec planes, so we do not show them here. The projections that we show include the peak isodensity contours in the 3D cube and the position of the spectroscopic member galaxies. The $z$-Dec projection is associated to the velocity distribution of the spectroscopic members. ![For peak \[1\], “Theia”, the [*top-left*]{} panel show the projection on the RA-Dec plane of the $5\sigma_{\rm \delta}$ contours which identify the peak in the 3D overdensity cube; the different colours indicate the different redshift slices (from blue to red, they go from the lowest to the highest redshift). Filled circles are the spectroscopic galaxies which are members of the peak (flag=X2/X2.5, X3, X4, X9), with the same colour code as the the contours. The black cross is the RA-Dec barycenter of the peak. In the top-right and bottom-left corners we show the scale in pMpc and cMpc, respectively, for both RA and Dec. [*Top-right.*]{} Projections on the $z$-Dec plane of the same contours shown in the top-left panel, with the same colour code. The filled circles and the black cross are as in the top-left panel. On the top and on the bottom of the panel we show the scale in pMpc and cMpc, respectively. [ *Bottom-right*]{}. The black histogram represents the velocity distribution of the spectroscopic galaxies which fall in the same RA-Dec region as the proto-cluster. The red histogram includes only VUDS and zCOSMOS galaxies with reliable quality flag, and flags X1/X1.5 for galaxies within the peak volume (see Sect. \[vel\_disp\] for details). The vertical solid green line indicates the barycenter along the l.o.s (the top x-axis is the same as the one in the top-right panel), and the two dashed vertical lines the maximum extent of the peak. The dotted-dashed blue vertical line is the $z_{\rm BI}$ of Table \[peaks\_tab\_veldisp\], around which we center the Gaussian (blue solid curve) with the same $\sigma_v$ as in Table \[peaks\_tab\_veldisp\]. The two dotted blue curves are the uncertainties on the Gaussian due to the uncertainties on $\sigma_v$. In the [*bottom-left*]{} corner of the figure we summarise some of the peak properties, which are all already mentioned in the Tables or in the text. []{data-label="peak1_fig"}](./figB1.ps){width="9.0cm"} ![As in Fig.\[peak1\_fig\], but for Peak \[2\], “Eos”.[]{data-label="peak2_fig"}](./figB2.ps){width="9.0cm"} ![As in Fig.\[peak1\_fig\], but for Peak \[3\], “Helios”. []{data-label="peak3_fig"}](./figB3.ps){width="9.0cm"} ![As in Fig.\[peak1\_fig\], but for Peak \[4\], “Selene”.[]{data-label="peak4_fig"}](./figB4.ps){width="9.0cm"} [^1]: Based on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, under Large Program 185.A-0791. [^2]: http://www.cfht.hawaii.edu/$\sim$arnouts/LEPHARE/lephare.html [^3]: $X=0$ is for galaxies, $X=1$ for broad line AGNs, and $X=2$ for secondary objects falling serendipitously in the slits and spatially separable from the main target. The case $X=3$ is as $X=2$ but for objects not separable spatially from the main target. [^4]: https://www.eso.org/sci/observing/phase3/data\_releases/uvista\_dr2.pdf [^5]: In this work we neglect the correlations in the noise between the cells in the same slice and those in different slices. [^6]: In [@cucciati2014_z29] we corrected the volume of the proto-cluster under analysis by a factor which took into account the Kaiser effect, which causes the observed volume to be smaller than the real one, due to the coherent motions of galaxies towards density peaks on large scales. Here we show that we are concerned rather by an opposite effect, i.e. our volumes might be artificially elongated along the l.o.s.. [^7]: Excluding the possible uncertainty on the bias factor $b$, which does not depend on our reconstruction of the overdensity field. For instance, if we assume $b=2.59$, as derived in [@bielby13] at $z\sim3$, we obtain a total mass $<1\%$ smaller. [^8]: Hyperion, one of the Titans according to Greek mythology, is the father of the sun god Helios, to whom the Colossus of Rhodes was dedicated. [^9]: This assumption is more suited for a virialised object than for a structure in formation. Nevertheless, our approach does not intend to be exhaustive, and we just want to compute a rough correction. [^10]: First we computed $M_{\rm 200}$ as in Eq. \[m200\_munari\], then converted $M_{\rm 200}$ into $M_{\rm vir}$ based on the same assumptions as for the conversion between $R_{\rm 200}$ and $R_{\rm v}$. This gives $M_{\rm vir} =1.06~M_{\rm 200}$. [^11]: According to Greek mythology, Theia is a Titaness, sister and spouse of Hyperion. Eos, Helios, and Selene are their offspring. [^12]: In our case the zCOSMOS-Deep sample, used together with the VUDS sample, is cut at $I=25$. Moreover we do not use the zCOSMOS-Deep quality flag 1.5. [@diener2013_list] used also flag=1.5 and did not apply any magnitude cut. [^13]: [@lee2016_colossus] mention that from their unsmoothed tomographic map this huge overdensity is composed of several lobes (see e.g. their Figs. 4 and 13), but it is more continuous after applying a smoothing with a $4 h^{-1}$Mpc Gaussian filter.

--- abstract: 'Linear index coding can be formulated as an interference alignment problem, in which precoding vectors of the minimum possible length are to be assigned to the messages in such a way that the precoding vector of a demand (at some receiver) is independent of the space of the interference (non side-information) precoding vectors. An index code has rate $\frac{1}{l}$ if the assigned vectors are of length $l$. In this paper, we introduce the notion of strictly rate $\frac{1}{L}$ message subsets which must necessarily be allocated precoding vectors from a strictly $L$-dimensional space ($L=1,2,3$) in any rate $\frac{1}{3}$ code. We develop a general necessary condition for rate $\frac{1}{3}$ feasibility using intersections of strictly rate $\frac{1}{L}$ message subsets. We apply the necessary condition to show that the presence of certain interference configurations makes the index coding problem rate $\frac{1}{3}$ infeasible. We also obtain a class of index coding problems, containing certain interference configurations, which are rate $\frac{1}{3}$ feasible based on the idea of *contractions* of an index coding problem. Our necessary conditions for rate $\frac{1}{3}$ feasibility and the class of rate $\frac{1}{3}$ feasible problems obtained subsume all such known results for rate $\frac{1}{3}$ index coding.' author: - title: 'Rate $\frac{1}{3}$ Index Coding: Forbidden and Feasible Configurations' --- Introduction ============ Index Coding, introduced in [@BiK], considers the problem of efficiently broadcasting a set of messages available at a source, to a collection of receivers each of which already has a subset of the messages (called *side-information*) and demands certain other subset of messages. Based on the configuration of the messages available as side-information and the demand sets, the general index coding problem is classified into various types including unicast [@BBJK] (with disjoint demand sets), uniprior [@OnHL] (with disjoint side-information sets), and the most general *groupcast* index coding problems (arbitrary side-information and demand sets). The goal of index coding is to find optimal index codes, where optimality means that the number of channel uses is minimum. To determine the maximum rate (inversely related to minimum number of channel uses) of an index coding problem is NP-hard. The landmark paper [@BBJK] famously connected the scalar linear index coding problem to finding a quantity called *minrank* associated with the *side-information graph* related to the given single unicast index coding problem. The minrank problem is NP-hard too, but several approaches have been taken to address this problem, most popularly via graph theoretical ideas to bound the maximum rate (see, for example, [@BiK; @BBJK; @SDL; @BKL1; @BKL2; @TOJ]). The techniques used in these works to derive bounds on maximum rate naturally lead to constructions of (scalar and vector) linear index codes. We focus on scalar linear codes in this paper. In [@Jaf; @MCJ2] the index coding problem has been reformulated in the interference alignment framework, where constructing a scalar linear code was shown to be equivalent to assigning precoding vectors to the messages. The precoding vectors are assigned such that the space spanned by vectors assigned to interfering messages is linearly independent of the vector assigned to the demanded message. An index code is said to be a rate $\frac{1}{L}$ code (for some positive integer $L$) if the messages are scalars (from some field $\mathbb{F}$) and precoding vectors are $L$-length. In [@Jaf], a necessary and sufficient condition for index codes of rate $\frac{1}{2}$ was presented based on the structure of the interference (messages not available as side-information), which is modeled using a *conflict graph*. Consequently, a polynomial time algorithm to identify the feasibility of rate $\frac{1}{2}$ (or equivalently, minrank two) for a given index coding problem was given in [@Jaf]. Unlike the rate $\frac{1}{2}$ feasibility problem which has a polynomial time solution, the rate $\frac{1}{3}$ feasibility problem was shown to be NP-hard [@Pee], when the finite field is fixed. A simple necessary condition for rate $\frac{1}{3}$ feasibility was given in [@BBJK]. A class of rate $\frac{1}{3}$ feasible index coding problems was shown in [@Jaf]. In [@PrL], a stricter necessary condition was derived than what was previously known in [@BBJK]. In addition, a larger class of index coding problems which are rate $\frac{1}{3}$ feasible was presented, which include those given in [@Jaf]. Both the necessary and sufficient conditions obtained in [@PrL] are based on the following two ideas (i) *conflict hypergraphs*, which preserve all the required information in the index coding problem (as opposed to conflict graph) (ii) *type-2 alignment sets*, which are special subsets of messages which must necessarily be assigned vectors from a two dimensional space. In this work, we make further progress on the rate $\frac{1}{3}$ index coding problem. Our contributions are as follows. - *Strictly rate $\frac{1}{L}$ feasible subsets:* We introduce the notion of strictly rate $\frac{1}{L}$ feasible subsets of messages of an index coding problem ${\mathbb I}$. A subset of messages is said to be strictly rate $\frac{1}{L}$ feasible if for any rate $\frac{1}{3}$ solution, there are always $L$ linearly independent vectors amongst the precoding vectors assigned to the subset (Definition \[def:strict\_rate\] in Section \[sec:nec\_condition\] makes this precise). - *Necessary condition based on strictly rate $\frac{1}{L}$ feasible subsets:* We show that a certain method of ‘stitching’ strictly rate $\frac{1}{2}$ feasible subsets together generates larger message subsets which should be strictly rate $\frac{1}{2}$ feasible. (Theorem \[thm:converse\], Section \[subsec:nec\_condition\]). Using this result, we recapture the main result of [@PrL], while significantly generalizing it to include new configurations which must be strictly rate $\frac{1}{2}$. This, in turn, results in a general necessary condition for rate $\frac{1}{3}$ feasibility. - *Contraction of an index coding problem:* We develop the notion of a *contraction* of an index coding problem ${\mathbb I}$. A contraction of an index coding problem ${\mathbb I}$ is another index coding problem ${\mathbb I}'$ such that any solution for ${\mathbb I}'$ gives a solution for ${\mathbb I}$ (Section \[sec:suff\_condition\] gives a formal definition). - *Sufficient condition based on contractions:* We give sufficient conditions on rate $\frac{1}{3}$ feasibility of ${\mathbb I}$ based on the structure of type-2 alignments sets of a maximal contraction of ${\mathbb I}$ (Theorem \[thm:suff\_condition\], Section \[sec:suff\_condition\]). These conditions result in a larger class of problems which are rate $\frac{1}{3}$ feasible. *Notations:* Throughout the paper, we use the following notations. Let $[1:m]$ denote $\{1,2,...,m\}$. For a set of vectors $A$, $sp(A)$ denotes their span. For a vector space $V$, $dim(V)$ denotes its dimension. An arbitrary finite field is denoted by $\mathbb F$. A vector from the $m$-dimensional vector space ${\mathbb F}^m$ is said to be picked *at random* if it is selected according to the uniform distribution on ${\mathbb F}^m$. Review of Index Coding {#sec:review} ====================== Formally, the general index coding problem, called a *groupcast* index coding problem, consists of a broadcast channel which can carry symbols from $\mathbb F$, along with (i) A set of $T$ receivers (ii) A source which has messages ${\cal W}=\{W_i, i\in[1:n]\}$, each of which takes values from $\mathbb F$, (iii) For each receiver $j$, a set $D(j)\subseteq {\cal W}$ denoting the set of messages demanded by the receiver $j$, (iv) For each receiver $j$, a set $S(j)\subseteq {\cal W}\backslash D(j)$ denoting the set of side-information messages available at the $j^{th}$ receiver. *A scalar linear index code of symmetric rate* $\frac{1}{L}$ (for some integer $L\geq 1$) for a given index coding problem consists of an encoding function $ \mathbb{E}:\underbrace{\mathbb F\times\mathbb F\times...\times\mathbb F}_{n~\text{times}}\rightarrow {\mathbb F}^L, $ mapping the messages ($W_i\in{\mathbb F}$) to some $L$-length codeword which is broadcast through the channel, as well as decoding functions at the receivers which use the codeword and the available side-information symbols to decode the demands. The encoding function of a scalar linear index code can be expressed as ${\mathbb E}(W_1,W_2,...,W_n)=\sum_{i=1}^nV_iW_i$, where $V_i$ is a $L$-length vector over $\mathbb F$ called the precoding vector assigned to $W_i$. Finding a scalar linear index code of length $L$ (i.e., with a feasible rate $1/L$) is equivalent to finding an assignment of these $V_i$s to the $n$ messages such that the receivers can all decode their demanded messages. For some receiver $j$ and for some message $W_k \in D(j)$, let $Interf_k(j)\triangleq {\cal W}\backslash({W_k\cup S(j)})$ denote the set of messages (except $W_k$) not available at the receiver $j$. The sets $Interf_k(j), \forall k$ are called the *interfering sets at receiver* $j$. If receiver $j$ does not demand message $W_k$, then we define $Interf_k(j)\triangleq\phi$. If a message set ${\cal W}_i$ is not available at a receiver $j$ demanding at least one message $W_k\notin {\cal W}_i$, then ${\cal W}_i$ is said to *interfere at receiver* $j$, and ${\cal W}_i$ and $W_k$ are said to be *in conflict*. For a set of vertices $A\subseteq {\cal W}$, let $V_{\mathbb E}(A)$ denote the vector space spanned by the vectors assigned to the messages in $A$, under the specified encoding function $\mathbb E$. If $A=\phi$, we define $V_{\mathbb E}(A)$ as the zero vector. A message subset $A$ is said to be $L$-dimensional (under the code $\mathbb E$) if $dim(V_{\mathbb E}(A))=L.$ For a given assignment of vectors to the messages (or equivalently, for a given encoding function $\mathbb E$), we say that all the conflicts *are resolved* if $V_k\notin V_{\mathbb E}(Interf_k(j)), \forall W_k\in D(j), \forall j$. It can easy to show that successful decoding at the receivers is possible if and only if all the conflicts are resolved [@PrL]. We alternatively refer to a collection of messages by only their indices (for example, $\{W_1,W_2\}$ is referred to as $\{1,2\}$). Prior results from [@PrL] on Rate $\frac{1}{3}$ Index Codes {#sec:prior} =========================================================== In [@PrL], we developed a new framework for studying the rate $\frac{1}{3}$ feasibility of groupcast index coding problems. We recall the basic definitions and results from [@PrL]. We build on these results in this paper. The conflict hypergraph is an undirected hypergraph with vertex set $\cal W$ (the set of messages), and its hyperedges defined as follows. - For any receiver $j$ demanding any message $W_k$, $W_k$ and $Interf_k(j)$ are connected by a hyperedge (shown dotted in our figures), which is denoted by $\{W_k,Interf_k(j)\}$. It was shown in [@PrL] that the conflict hypergraph sufficiently captures the essence of the index coding problem. The alignment graph is an undirected graph with vertex set $\cal W$ and edges defined as follows. The vertices $W_i$ and $W_j$ are connected by an edge (called an *alignment edge*, shown in our figures by a solid edge) when the messages $W_i$ and $W_j$ are not available at a receiver demanding a message other than $W_i$ and $W_j$. A connected component of the alignment graph is called an *alignment set*. Let $\mathbb I$ denote an index coding problem with message set $\cal W$. For some ${\cal W}'\subseteq {\cal W}$, a ${\cal W}'$*-restricted index coding problem* is defined as the index coding problem ${\mathbb I}_{{\cal W}'}$ consisting of (i) The messages ${\cal W}'$, (ii)The subset ${\cal T}_{{\cal W}'}$ of the receivers of $\mathbb I$ which demand messages in ${\cal W}'$, (iii) For each $j\in {\cal T}_{{\cal W}'}$ the demand sets $D_{{\cal W}'}(j)$ and the side-information sets $S_{{\cal W}'}(j)$ are restricted within ${\cal W}'$, i.e., $D_{{\cal W}'}(j)= D(j)\cap{\cal W}'$ and $S_{{\cal W}'}(j)= S(j)\cap{\cal W}'.$ The alignment graph and the alignment sets of the restricted index coding problem ${\mathbb I}_{{\cal W}'}$ are called the ${\cal W}'$-*restricted alignment graph* and ${\cal W}'$-*restricted alignment sets* respectively. A ${\cal W}'$-*restricted internal conflict* is a conflict between any two messages within a restricted alignment set of ${\cal W}'$. A subset ${\cal W}''\subset {\cal W}$ of size three is said to be a *triangular interfering set* if all the messages in ${\cal W}''$ interfere simultaneously at some receiver, and at least two of the messages in ${\cal W}''$ are in conflict. Two distinct triangular interfering sets ${\cal W}_1$ and ${\cal W}_2$ are said to be *adjacent* if ${\cal W}_1\cap{\cal W}_2=\{W_i,W_j\}$ such that $W_i$ and $W_j$ are in conflict. \[type2align\] Two triangular interfering sets ${\cal W}_1$ and ${\cal W}_2$ are said to be *connected* if there exists a *path* (i.e., a sequence) of adjacent triangular interfering sets starting from ${\cal W}_1$ and ending at ${\cal W}_2$. A *type-2 alignment set* is a maximal set of triangular interfering sets which are connected to each other. The following theorem gives a necessary and sufficient condition for assigning vectors from a two dimensional space to the type-2 alignment sets. \[norestrconflictstype2\] Let ${\cal W}'$ be a type-2 alignment set of the given index coding problem $\mathbb I$. If $\mathbb I$ is rate $\frac{1}{3}$ feasible, then ${\mathbb I}_{{\cal W}'}$ must be rate $\frac{1}{2}$ feasible which holds if and only if there are no ${\cal W}'$-restricted internal conflicts. \[thm:main\] A rate $\frac{1}{2}$ infeasible index coding problem $\mathbb I$ is rate $\frac{1}{3}$ feasible if every alignment set of $\mathbb I$ satisfies *either* of the following conditions. 1. It does not have both forks and cycles (a *fork* is a vertex connected by three or more edges). 2. It is a type-2 alignment set with no restricted internal conflicts. A general necessary condition for rate $\frac{1}{3}$ feasibility {#sec:nec_condition} ================================================================ We begin with the formal definition of strictly rate $\frac{1}{L}$ feasible and infeasible subsets. \[def:strict\_rate\] Let ${\mathbb I}$ be an index coding problem with message set ${\cal W}$. A subset ${\cal W}'\subseteq {\cal W}$ is said to be *strictly rate* $\frac{1}{L}$ *feasible* (or simply, *strictly rate* $\frac{1}{L}$) if for any rate $\frac{1}{3}$ code given by an encoding function $\mathbb E$ (if it exists), we have $dim(V_{\mathbb E}({\cal W}'))=L$. A subset ${\cal W}'$ is called strictly rate $\frac{1}{L}$ infeasible if $dim(V_{\mathbb E}({\cal W}'))\neq L$ for any rate $\frac{1}{3}$ encoding function $\mathbb E$. Examples of strictly rate $\frac{1}{2}$ subsets include triangular interfering sets and type-2 sets. In the forthcoming part of this paper, we also construct other classes of strictly $\frac{1}{2}$ subsets as well as examples of strictly rate $1$ and rate $\frac{1}{3}$ subsets. Some of such sets help us to further characterize index coding problems which are rate $\frac{1}{3}$ feasible based on the existence of some substructures, like the type-2 set conditions given by Theorem \[norestrconflictstype2\]. Two Interference Configurations and their strictly rate $\frac{1}{L}$ feasible (infeasible) subsets {#subsec:inter_config} --------------------------------------------------------------------------------------------------- Here, we will present two interference configurations (i) Successive triangular interference configuration (STIC) and (ii) Square pyramid interference configuration (SPIC). We will identify strictly rate $\frac{1}{L}$ feasible (and infeasible) subsets in these two configurations. We will also present an example based on two STIC subsets which is rate $\frac{1}{3}$ infeasible. A collection of messages ${\cal W}_{STIC}=\{1,2,3,4,5,6\}$ is said to be a *successive triangular interference configuration* (STIC) subset (Fig. \[fig:STICseta\]) if the following conditions hold. - The sets $\{1,2,3\}$, $\{2,4,5\}$, and $\{3,5,6\}$ interfere at some receivers demanding messages other than $\{1,2,3,4,5,6\}$. - The sets $\{1,2\}$ and $\{2,4\}$ interfere at $W_6$, $\{1,3\}$ and $\{3,6\}$ interfere at $W_4$, and $\{4,5\}$ and $\{5,6\}$ interfere at $W_1$ (and there are no other conflicts amongst $\{1,2,3,4,5,6\}$). \[STIClemma\] Let $\{1,2,3,4,5,6\}$ be a STIC set as in Fig. \[fig:STICseta\]. Then the following statements are true. 1. The set $\{2,3,5\}$ is strictly rate $\frac{1}{2}$ infeasible. 2. The sets $\{1,2,3\}$, $\{2,4,5\}$, and $\{3,5,6\}$ are strictly rate $\frac{1}{2}$ feasible. Proof of 1): Suppose $\{2,3,5\}$ is two dimensional for some rate $\frac{1}{3}$ index code given by $\mathbb E$. WLOG, let us assume that $V_2$ and $V_3$ are linearly independent, and thus $V_5\in sp(\{V_2,V_3\})$. Furthermore, $V_1\in sp(\{V_2,V_3\})$ as $\{1,2,3\}$ is interfering at some receiver. Now, if $V_5=\alpha V_2+\beta V_3$ for some $\alpha,\beta\in{\mathbb F}$ both non-zero, then it must be that $V_4=V_5$ (or a scalar multiple). Otherwise $V_4$ and $V_5$ will be independent, and hence we will have $sp(\{V_2,V_3\})=sp(\{V_2,V_5\})=sp(\{V_4,V_5\})$ and thus $V_1\in sp(\{V_4,V_5\})$, which is not allowed. Similarly, we should have $V_6=V_5$ too. This means that $V_4\in sp(\{V_3,V_6\})$, which is not allowed. Hence we cannot have $V_5=\alpha V_2+\beta V_3$ with $\alpha,\beta$ both non-zero. Thus, WLOG, let $V_5=V_2$ (or equivalently, some constant multiple). Suppose $V_6$ (which is in $sp(\{V_3,V_5\})$ as $V_3$ and $V_5$ are independent) and $V_5$ are independent. Then $V_1\in sp(\{V_5,V_6\})$ which is not allowed. Thus $V_6=V_5=V_2$. However this means that $V_6\in sp(\{V_2,V_1\})$ for any choice of $V_1$, which is not allowed again. A similar contradiction arises if $V_5=V_3$. This concludes proof of 1). Proof of 2): Clearly as the sets $\{1,2,3\}$, $\{2,4,5\}$, and $\{3,5,6\}$ are all interfering sets, we must have that each of them is strictly rate $\frac{1}{3}$ infeasible. Suppose $\{1,2,3\}$ is one dimensional. Then $\{2,3,5\}$ must also be one dimensional as $\{2,3,5\}$ cannot be two dimensional by 1). But then, $V_1=V_5$ which is not allowed. Thus $\{1,2,3\}$ must be two dimensional, and a similar argument applies for the other two sets. The following example, based on STIC sets, gives a rate $\frac{1}{3}$ infeasible problem based on an alignment structure constructed using two STIC sets, but does not have any triangular interfering sets (unlike all known previous examples). This also suggest the hardness of deciding the feasibility of rate $\frac{1}{3}$ index coding. Consider an index coding problem ${\mathbb I}$ containing the alignment set structure shown in Fig. \[fig:STICsetDOUBLE\]. There are two STIC sets $\{1,2,3,4,5,6\}$ and $\{7,4,2,8,5,9\}$ which share the common set $\{2,4,5\}$. Consider the index coding problems ${\mathbb I}_1$ (respectively, ${\mathbb I}_2$) restricted to the STIC set $\{1,2,3,4,5,6\}$ (equivalently, $\{7,4,2,8,5,9\}$) and all the messages at which these messages interfere. Clearly, by Lemma \[STIClemma\], $\{2,4,5\}$ is strictly rate $\frac{1}{2}$ feasible in ${\mathbb I}_1$, while the same set is strictly rate $\frac{1}{2}$ infeasible in ${\mathbb I}_2$. Note that any rate $\frac{1}{3}$ feasible code for ${\mathbb I}$ is also feasible for ${\mathbb I}_1$ and ${\mathbb I}_2$. This means $\{2,4,5\}$ is simultaneously two dimensional and also not so. This is meaningless, hence we have that ${\mathbb I}$ is rate $\frac{1}{3}$ infeasible. As a general rule, if there is a subset of messages which is strictly rate $\frac{1}{L}$ infeasible in a restricted index coding problem of the given problem, and strictly rate $\frac{1}{L}$ feasible in another restricted problem, then the given problem must clearly be rate $\frac{1}{3}$ infeasible. We now define the SPIC sets, using which we obtain a new class of strict $\frac{1}{2}$ feasible sets. ![A *square pyramid interference configuration (SPIC)* set: The receivers, at which the sets $\{1,2,3\},\{1,3,4\},\{3,4,5\}$ and $\{2,3,5\}$ are conflicting, are suppressed. Definition \[squareinterferenceconfig\] has the complete details.[]{data-label="fig:squareset"}](SQUARESET_STRICTLYRATEHALF.pdf){width="1.5in"} \[squareinterferenceconfig\] A collection of messages ${\cal W}_{SPIC}=\{1,2,3,4,5\}$ is said to be a *square pyramid interference configuration (SPIC)* set (see Fig. \[fig:squareset\]) if the following conditions hold. - Message sets $\{1,2,3\},\{1,3,4\},\{3,4,5\}$ and $\{2,3,5\}$ interfere at some receivers which do not demand the messages $\{1,2,3,4,5\}$. - Message set $\{1,2\}$ (denoted by $I_5$) interferes at a receiver demanding $W_5$, and $\{4,5\}$(denoted by $I_2$) interferes at a receiver demanding $W_2$. - Messages $W_1$ and $W_3$ are in conflict. - No conflicts other than the above exist between the messages $\{1,2,3,4,5\}$. We now show that any SPIC set has to be strictly rate $\frac{1}{2}$, and also give other properties which it must satisfy, in the following lemma. \[lemmaSPICconditions\] A SPIC set ${\cal W}_{SPIC}$ as shown in Fig. \[fig:squareset\] satisfies the following conditions. 1. The triangular interfering sets $\{1,2,3\},\{1,3,4\},\{2,3,5\}$ and the set $\{1,2,3,4\}$ are strictly rate $\frac{1}{2}$. 2. ${\cal W}_{SPIC}$ is strictly rate $\frac{1}{2}$. 3. The sets $\{1,4\},$ and $\{2,5\}$ are strictly rate $\frac{1}{2}$. 4. The sets $\{4,5\}$ and $\{1,2\}$ must be strictly rate $1$. Proof of 1): This follows by Theorem \[norestrconflictstype2\] as $\{1,2,3,4\}$ must be a subset of a type-2 alignment set, and also because $\{2,3,5\}$ is a triangular interference set. Thus, we must have $V_{\mathbb E}(\{1,2,3,4\})=V_{\mathbb E}(\{1,3\})$ for any rate $\frac{1}{3}$ feasible assignment $\mathbb E$. Proof of 2): Let $\mathbb E$ be the encoding function of some rate $\frac{1}{3}$ index code. Clearly $2\leq dim(V_{\mathbb E}({\cal W}_{SPIC}))\leq 3$. Suppose that $dim(V_{\mathbb E}({\cal W}_{SPIC}))=3$. By 1), we must therefore have that $V_5$ is independent of the space $V_{\mathbb E}(\{1,2,3,4\})$. Clearly, $V_2$ and $V_4$ must be linearly independent, as $W_2$ and $W_4$ are in conflict. By 1), we have $V_3\in sp(\{V_4,V_2\})$. It is easy to see that any such assignment for $V_3$ makes at least one of the sets $\{3,4,5\}$ and $\{2,3,5\}$ as three dimensional, which is not allowed. This proves 2). Proof of 3) and 4): By 2), it must be that $\{1,2,5\}$ must also be assigned vectors from the space spanned by $\{V_1,V_3\}$. Now, if $V_1$ and $V_2$ are linearly independent, then the conflicts will not be resolved at $W_5$. Thus, we must have $V_2=V_1$ (or a constant multiple). By the similar arguments, we must have $V_4=V_5=\alpha V_1+\beta V_3$ (for some $\alpha,\beta\in {\mathbb F}, \beta \neq 0$). This is the only assignment which is possible and it can be checked that 3) and 4) hold. Necessary condition based on strictly rate $\frac{1}{2}$ feasible subsets {#subsec:nec_condition} ------------------------------------------------------------------------- In the following theorem, we show that if the strictly rate $\frac{1}{2}$ subsets are appended together in a certain way, then the resultant subset must be strictly rate $\frac{1}{2}$. This can also be seen as a general necessary condition for rate $\frac{1}{3}$ feasibility based on the properties of strictly rate $\frac{1}{2}$ sets and their intersections. \[thm:converse\] Let ${\cal W}_i:i=1,2,...,r$ ($r\geq 2$) be strictly rate $\frac{1}{2}$ sets of a rate $\frac{1}{3}$ feasible index coding problem ${\mathbb I}$ with message set ${\cal W}$, such that the following holds. - The sets $\left(\cup_{i=1}^{t-1}{\cal W}_i\right)\bigcap {\cal W}_t, 2\leq t\leq r,$ are strictly rate $1$ infeasible. Then the set $\cup_{i=1}^r {\cal W}_i$ must be strictly rate $\frac{1}{2}$. Thus, if the index coding problem ${\mathbb I}$ restricted to $(\cup_{i=1}^r{\cal W}_i)$ has any $(\cup_{i=1}^r {\cal W}_i)$-restricted internal conflicts, then ${\mathbb I}$ is not rate $\frac{1}{3}$ feasible. If we are given that $\cup_{i=1}^r {\cal W}_i$ is strictly rate $\frac{1}{2}$, then the claim about rate $\frac{1}{3}$ infeasibility follows from the necessary condition for rate $\frac{1}{2}$ feasibility in [@Jaf]. Hence, it is enough to show that $\cup_{i=1}^r {\cal W}_i$ is strictly rate $\frac{1}{2}$. Let $\mathbb E$ be the encoding function of some $\frac{1}{3}$ index code. It is clear that $dim(V_{\mathbb E}(\cup_{i=1}^r {\cal W}_i))\geq 2$, as each set ${\cal W}_i$ is strictly rate $\frac{1}{2}$. We prove the claim by induction on $r$. The claim is true for $r=1$ by assumption that each set ${\cal W}_i$ is strictly rate $\frac{1}{2}$. Now suppose that the statement is true upto $r-1$. Then we have $\cup_{i=1}^{r-1} {\cal W}_i$ and ${\cal W}_r$ are strictly rate $\frac{1}{2}$ and $(\cup_{i=1}^{r-1} {\cal W}_i) \cap{\cal W}_r$ is strictly rate $1$ infeasible. This means that $(\cup_{i=1}^{r-1} {\cal W}_i) \cap {\cal W}_r$ must be strictly rate $\frac{1}{2}$ feasible as ${\cal W}_r$ is strictly rate $\frac{1}{2}$. Thus, we have $V_{\mathbb E}(\cup_{i=1}^{r-1} {\cal W}_i)=V_{\mathbb E}({\cal W}_r)=V_{\mathbb E}((\cup_{i=1}^{r-1} {\cal W}_i) \cap {\cal W}_r)$. Thus $V_{\mathbb E}((\cup_{i=1}^{r-1} {\cal W}_i) \cup {\cal W}_r)=V_{\mathbb E}(\cup_{i=1}^{r-1} {\cal W}_i)$ as well, and hence $(\cup_{i=1}^{r-1} {\cal W}_i) \cup {\cal W}_r$ must be strictly rate $\frac{1}{2}$. Now, we will give three examples of sets (Corollary \[cor:type2\],\[cor:xtype2\], and \[cor:spic\]) which satisfy the condition in Theorem \[thm:converse\]. \[cor:type2\] Any type-2 alignment set of an index coding problem ${\mathbb I}$ is strictly rate $\frac{1}{2}$. Let ${\cal W}_i:i=1,...,r$ be the set of all triangular interfering sets in the given type-2 set. Clearly each one of them is strictly rate $\frac{1}{2}$. Furthermore, as the type-2 set is a maximally connected set of triangular interfering sets, we can assume WLOG that the sets ${\cal W}_i$ are in an order such that $(\cup_{i=1}^{t-1}{\cal W}_i)\cap {\cal W}_t, 2\leq t\leq r$ contains a conflict, and thus is strictly rate $1$ infeasible. The corollary follows by applying Theorem \[thm:converse\]. Consider ${\cal W}_i:i=1,...,r$ be a maximal collection of type-2 alignment sets such that $(\cup_{i=1}^{t-1}{\cal W}_i)\cap {\cal W}_t, 2\leq t\leq r$ contains at least one conflict. Then the union $\cup_{i=1}^{r}{\cal W}_i$ is termed as extended type-2 alignment set (Xtype-2 set in short). We then have the following corollary, which we state without proof as it is a direct application of Theorem \[thm:converse\]. \[cor:xtype2\] Any Xtype-2 set of an index coding problem ${\mathbb I}$ is strictly rate $\frac{1}{2}$. For our final corollary, we define SPIC alignment sets as follows, similar to type-2 alignment sets (Definition \[type2align\]). Two SPIC subsets of ${\cal W}_1$ and ${\cal W}_2$ are said to be *adjacent* SPIC subsets if ${\cal W}_1\cap {\cal W}_2$ is strictly rate $1$ infeasible. The SPIC sets ${\cal W}_1$ and ${\cal W}_2$ are called connected if there exists a sequence of adjacent SPIC sets from ${\cal W}_1$ to ${\cal W}_2$. A SPIC alignment set is a maximal set of SPIC subsets which are connected to each other. ![A SPIC Alignment set. The sets $\{14,1,3,15,4\}$, $\{2,1,3,5,4\}$, $\{2,6,7,5,8\}$, $\{9,6,10,11,8\}$ and $\{9,12,10,11,13\}$ are all SPIC sets. The pair-wise intersections in the same sequence are $\{1,3,4\}$,$\{2,5\}$,$\{6,8\}$ and $\{9,10,11\}$ respectively, and all such intersections are strictly rate $1$ infeasible. The messages within the ellipses interfere at the messages as mentioned in the ellipse.[]{data-label="fig:squaresetconnected"}](SQUARESET_STRICTLYRATEHALF_CONNECTED.pdf){width="3.5in"} Fig. \[fig:squaresetconnected\] shows an example shows an example of a SPIC alignment set. We now give the related result. \[cor:spic\] A SPIC alignment set of ${\mathbb I}$ is strictly rate $\frac{1}{2}$. Let ${\cal W}_{a}$ be the SPIC alignment set. Let ${\cal W}_i:i=1,...,r$ be the set of all SPIC sets which comprise ${\cal W}_a$, ordered such that ${\cal W}_t$ is connected to $(\cup_{i=1}^{t-1}{\cal W}_i), 2\leq t\leq r$ (we can assume this WLOG as ${\cal W}_a$ is maximally connected set of SPIC sets). The corollary follows by applying Theorem \[thm:converse\]. The sets described in Corollaries \[cor:type2\], \[cor:xtype2\], \[cor:spic\] are only a few examples of the possible configurations of strictly rate $\frac{1}{2}$ sets in ${\mathbb I}$. We can imagine several other configurations as well, involving a mixture of Xtype-2 sets, SPIC sets, and other strictly rate $\frac{1}{2}$ sets (if they exist), satisfying the conditions in Theorem \[thm:converse\], leading to bigger structures which must be strictly rate $\frac{1}{2}$. An interesting open question is to determine if there a finite number of basic substructures, like the triangular interfering sets, SPIC sets, such that their maximally connected versions should satisfy specific properties for rate $\frac{1}{3}$ feasibility. A Sufficient Condition for Rate $\frac{1}{3}$ Feasibility {#sec:suff_condition} ========================================================= In this section, we introduce the notion of contraction of an alignment edge and contraction of an index coding problem. We prove that an index code for the contraction of an index coding problem can be extended to that of the original index coding problem. We also give a sufficient condition in terms of Xtype-2 sets in the maximal contraction of an index coding problem for the original index coding problem to be rate $\frac{1}{3}$ feasible. \[def:contraction\] An alignment edge connecting two vertices (messages) $W_i$ and $W_j$ which are not in conflict is said to be *contracted* by identifying the vertices as a single vertex thereby making all the edges (alignment edges and conflict hyperedges) that were incident on $W_i$ and $W_j$ now incident on the newly created vertex. When we contract any alignment edge, note that we get a derived index coding problem with one less the number of messages. The derived index coding problem is completely characterised by the contracted alignment graph and the correspondingly altered conflict graph. We can envision a sequence of such derived index coding problems, obtained by a sequence of contractions of the alignment set of the original problem, each with one lesser message than the previous. Abusing the definition, an index coding problem ${\mathbb I}'$ is said to be a *contraction of* an index coding problem ${\mathbb I}$ if it is obtained by a finite sequence of contractions of the alignment edges of ${\mathbb I}$. The following definition captures the maximal cases after which we cannot contract anymore. Let $\mathbb I$ be a given index coding problem. Let ${\mathbb I}_{max}$ be an index coding problem obtained by a sequence of contractions of ${\mathbb I}$, such that any two messages of ${\mathbb I}_{max}$ connected by an alignment edge are also in conflict in ${\mathbb I}_{max}$. Then ${\mathbb I}_{max}$ is said to be a maximal contraction of ${\mathbb I}$. Clearly there could be multiple distinct maximal contractions of a given index coding problem, depending on the sequence of alignment edges which are contracted. \[lem:contract\] Let ${\mathbb I}'$ be a contraction of ${\mathbb I}$. Any scalar linear index coding solution of rate $R$ for ${\mathbb I}'$ can extended to a scalar linear index coding solution of rate $R$ for ${\mathbb I}$. WLOG, we assume that ${\mathbb I}'$ is obtained from ${\mathbb I}$ by a single contraction of an alignment edge. Let ${\boldsymbol W}=(W_1,...,W_n)$ denote the vector of messages in ${\mathbb I}$. WLOG, consider that symbols $W_{n-1}$ and $W_n$ in ${\mathbb I}$ were contracted to get ${\mathbb I}'$. We denote the new symbol (vertex) which is created as the message $W'_{n-1}$. The vector of messages, the demand sets, the side-information sets and the interfering sets of ${\mathbb I}'$ (denoted as $\boldsymbol{W'}, D'(j), S'(j),$ and $Interf_j^{'}(k)$ respectively) can be obtained from those of ${\mathbb I}$ by replacing $W_i:1\leq i \leq n-1$ with $W_i':1\leq i \leq n-1$, and replacing $W_{n}$ with $W'_{n-1}$. We note that the number of receivers remains unchanged in ${\mathbb I}'$. Let ${\mathbb E}'$ be the encoding function of a solution to ${\mathbb I}'$. Thus we must have $$\begin{aligned} \label{eqn100} V'_k\notin V_{\mathbb E}'(Interf_j^{'}(k)),\end{aligned}$$ for all $W'_k\in D(j),~\forall~\text{receivers}~j$, where $V'_k$ is the vector which is assigned to $W_k'$ in ${\mathbb E}'$. Consider the encoding function ${\mathbb E}$ for ${\mathbb I}$ defined as follows. $$\begin{aligned} V_k= \begin{cases} V_k' & \forall 1\leq k\leq n-1.\\ V_{n-1}' & \text{if}~k=n. \end{cases}\end{aligned}$$ We will now show that this assignment resolves all conflicts. Consider a message $W_k$ for some $k\leq n-2$ demanded at some receiver $j$. Thus $V_k=V_k'$. For some message $W_k$ demanded at receiver $j$, suppose $Interf_j(k)$ does not contain $W_{n}$. Then $Interf_j(k)=Interf_j^{'}(k)$ (except for the ‘dashes’). If $Interf_j(k)$ contains $W_{n},$ then $Interf_j^{'}(k)$ contains $W'_{n-1}$ in the place of $W_n$, apart from having other ‘dashed’ messages corresponding to those in $Interf_j(k)$. In any case, we must have $$\begin{aligned} \label{eqn101} V_{\mathbb E}(Interf_j(k))=V_{{\mathbb E}'}(Interf_j^{'}(k)), \forall j,\forall k \leq n-2.\end{aligned}$$ By (\[eqn100\]) and (\[eqn101\]), we have $V_k\notin V_{\mathbb E}(Interf_j(k))$. Now we consider $W_{n-1}$ (equivalently, $W_{n}$) demanded at receiver $j$. The receiver $j$ in ${\mathbb I}'$ demands the symbol $W'_{n-1}$, and it is given that (\[eqn100\]) holds. Clearly, $Interf_j^{'}(n-1)$ does not contain $W'_{n-1}$. This means that $Interf_j(n-1)$ (equivalently, $Interf_j(n)$) does not contain $W_n$ or $W_{n-1}$. Thus, by similar arguments as above, we must have $$V_{\mathbb E}(Interf_j(n-1))=V_{{\mathbb E}'}(Interf_j^{'}(n-1))$$ (equivalently, $V_{\mathbb E}(Interf_j(n))=V_{{\mathbb E}'}(Interf_j^{'}(n-1))$). By assignment ${\mathbb E}$, we have $V_{n-1}\notin V_{\mathbb E}(Interf_j(n-1))$ (equivalently, $V_n\notin V_{\mathbb E}(Interf_j(n))$). Thus all the conflicts in ${\mathbb I}$ are resolved. This concludes the proof. \[thm:suff\_condition\] A rate $\frac{1}{2}$ infeasible index coding problem $\mathbb I$ is rate $\frac{1}{3}$ feasible if there exists a maximal contraction $\mathbb I'$ of the index coding problem $\mathbb I$ such that the following conditions hold: (a) Any Xtype-2 set in $\mathbb I'$ has no restricted internal conflicts. (b) No three Xtype-2 sets have a message vertex in common. (c) For any distinct Xtype-2 sets ${\cal W}_i, {\cal W}_j$, if ${\cal W}_i \cap {\cal W}_j \neq \phi$, then there is no conflict between any two messages in ${\cal W}_i \cap {\cal W}_j$. To prove the theorem, we will give a solution to index coding problem $\mathbb I'$ and then apply Lemma \[lem:contract\] to extend the solution to the actual index coding problem $\mathbb I$. For notational convenience, we will drop the dash associated with the variables corresponding to the index coding problem $\mathbb I'$ (We will however refer to index coding problem as $\mathbb I'$). Let ${\cal W}_i, 1 \leq i \leq r$ denote the Xtype-2 sets of ${\mathbb I}'$. By assumption, we have ${\cal W}_i \cap {\cal W}_j \cap {\cal W}_k = \phi$, for all distinct $i,j,k$. Note that the condition (c) implies that there is no alignment edge between any two messages in ${\cal W}_i \cap {\cal W}_j$, as $\mathbb I'$ is a maximal contraction. Consider a graph which has $r$ vertices, where each vertex represents an Xtype-2 set. There is an edge between two vertices in this graph, if the two Xtype-2 sets intersect in one or more messages. We will refer to this graph as the Extended Type-2 Intersection Graph (ETIG). We will first assign vectors to the edges in ETIG and then use that in turn to come up with an assignment for the messages in the index coding problem $\mathbb I'$. The algorithm to assign vectors to the edges of the ETIG is as follows. We start with the assumption that the edge set is non-empty, else the algorithm terminates straightaway. We repeat the below steps until all edges have been assigned vectors. 1. Pick an unassigned edge $e_{ij}$ between vertices $i$ and $j$. Let ${\cal V}_i$ denote the set of vectors already assigned to edges incident on vertex $i$ and similarly ${\cal V}_j$. 2. Suppose both $sp({\cal V}_i)$ and $sp({\cal V}_j)$ are not $2$-dimensional, then assign a random $3\times 1$ vector to the edge. 3. Suppose exactly one of $sp({\cal V}_i)$ or $sp({\cal V}_j)$ is $2$-dimensional. For example, let $sp({\cal V}_i)$ be $2$-dimensional. Then, a random $3 \times 1$ vector from $sp({\cal V}_i)$ is assigned to the edge. 4. Suppose both $sp({\cal V}_i)$ and $sp({\cal V}_j)$ are $2$-dimensional. Then, a $3 \times 1$ vector from the intersection $sp({\cal V}_i) \cap sp({\cal V}_j)$ is assigned to the edge. Such a vector always exists in the intersection since both $sp({\cal V}_i)$ and $sp({\cal V}_j)$ are $2$-dimensional subspaces of a $3$-dimensional space. We note that at the end of the algorithm, each vertex with degree at least one in ETIG has a set of vectors associated with edges incident on them, the span of which is either $1$-dimensional or $2$-dimensional. An example ETIG and assignment of vectors to the edges in the graph is illustrated in Fig. \[fig:xtype2\_vectors\]. We now describe the procedure to assign vectors to the messages in the index coding problem $\mathbb I'$. 1. Firstly, vectors are assigned to the messages in the intersections of Xtype-2 sets. We assign the same vector to all the messages in the intersection ${\cal W}_i \cap {\cal W}_j$. We pick the vector which was assigned to the edge which joined the vertices corresponding to $i^{\text{th}}$ and $j^{\text{th}}$ Xtype-2 set in ETIG and assign it to all the messages in the intersection ${\cal W}_i \cap {\cal W}_j$. This step is repeated for all intersections of Xtype-2 sets. 2. At the end of the above step, the messages in ${\cal W}_i$s maybe partially assigned. To assign vectors to remaining messages (if such non-assigned messages exist) in the Xtype-2 sets ${\cal W}_i$s, we consider the following three subcases: 1. For any $i$, if there is just one index $j$ such that ${\cal W}_i \cap {\cal W}_j \neq \phi$, then we have assigned only one vector to ${\cal W}_i$ at the end of step 1). Let the vector be denoted by $\underline{v}^{(i)}_1$. Now, we pick another $3 \times 1$ random vector $\underline{v}^{(i)}_2$. Then, we assign a random vector in the span of $\underline{v}^{(i)}_1$ and $\underline{v}^{(i)}_2$ to each message in ${\cal W}_i$. 2. For any $i$, if there is more than one index $j$ such that ${\cal W}_i \cap {\cal W}_j \neq \phi$, then we have assigned a $2$-dimensional space to messages in ${\cal W}_i$ at the end of step 1). To each of the remaining messages in ${\cal W}_i$, we assign a random vector from the same $2$-dimensional space. 3. For any $i$, if there is no index $j$ such that ${\cal W}_i \cap {\cal W}_j \neq \phi$ (corresponds to isolated vertices in ETIG), then we first pick a randomly generated $2$-dimensional space. Then, we assign a random $3\times 1$ vector in the $2$-D space to each message in the $i^{\text{th}}$ Xtype-2 set. 3. For all other messages in ${\mathbb I}'$ (not in Xtype-2 sets), we assign a random $3\times 1$ vector. Let $\mathbb E$ denote the encoding function corresponding to this assignment and $V_k$ denote the vector assigned to the message $W_k$. We now show that this assignment resolves all conflicts in $\mathbb I'$, i.e., with high probability $V_k\notin V_{\mathbb E}(Interf_k(j))$. Consider a receiver $j$ which requests a message $W_k$. We will classify all the conflicts into the following cases and verify that conflicts are resolved in each case. 1. If $|Interf_k(j)| = 1$, then the conflict is resolved, since any two messages which are in conflict are assigned linearly independent vectors according to the scheme. It is only the intersections of Xtype-2 sets which are assigned the same vector and we have already noted earlier that all the messages in any given intersection do not have any conflicts within them. 2. Consider the case when $|Interf_k(j)| = 2$. The two messages within the set $Interf_k(j)$ should have a conflict between them since $\mathbb I'$ is the maximal contraction. Now, we have the following two sub cases: 1. Both $W_k$ and $Interf_k(j)$ belong to a Xtype-2 alignment set. This is not possible since then the Xtype-2 set will have a restricted internal conflict. 2. For any configuration of $W_k$ and $Interf_k(j)$ other than the case considered above, the conflict is resolved. This is true because any three messages which do not all belong to an Xtype-2 alignment set and which have conflicts between them are assigned three linearly independent vectors according to the scheme. Here again, note that $Interf_k(j)$ cannot be contained within the intersection of two Xtype-2 sets (since all the messages in any given intersection do not have any conflicts within them). 3. Consider the case when $|Interf_k(j)| \geq 3$. We note that in the maximal contraction $\mathbb I'$, any triangular interfering set of $\mathbb I'$ has to be contained within a type-2 alignment set and hence within a Xtype-2 set. Thus, $Interf_k(j)$ is contained within a Xtype-2 set. Also since Xtype-2 sets of $\mathbb I'$ are assumed to have no restricted internal conflicts, $W_k$ has to be outside the Xtype-2 set. Since $Interf_k(j)$ lies within a Xtype-2 set and the messages within a Xtype-2 set are assigned vectors from a two dimensional space, $V_{\mathbb E}(Interf_k(j))$ is a two dimensional space. Noting that $V_k$ is a random $3\times 1$ vector, we have that $V_k\notin V_{\mathbb E}(Interf_k(j))$ with high probability. Hence, the conflict is resolved. Thus, all the conflicts in $\mathbb I'$ are resolved. Now, the solution given above can be extended to give a solution of the original index coding problem $\mathbb I$ by ’uncontracting’ one alignment edge at a time and doing an assignment as described in the proof of Lemma \[lem:contract\]. This concludes the proof. We claim that Theorem \[thm:main\] can be seen as a special application of Theorem \[thm:suff\_condition\] (which is the most general result known so far for rate $\frac{1}{3}$ feasible problems), and this can be shown using the proof details of Theorem \[thm:main\] which is available in [@PrL]. We leave the details to the reader. [160]{} Y. Birk and T. Kol, “Coding on Demand by an Informed Source (ISCOD) for Efficient Broadcast of Different Supplemental Data to Caching Clients“, IEEE Transactions on Information Theory, Vol. 52, No. 6, June, 2006, pp. 2825-2830. Z.Bar-Yossef, Y. Birk, T.S. Jayram, T. Kol, “Index Coding with Side Information”, IEEE Transactions on Information Theory, Vol. 57, No. 3, March 2011, pp. 1479-1494. L. Ong, C. K. Ho, and F. Lim, “The single-uniprior index-coding problem: The single-sender case and the multi-sender extension," IEEE Transactions on Information Theory, Vol. 62, No. 6, June 2016, pp. 3165-3182. K. Shanmugam, A. G. Dimakis, and M. Langberg, “Local graph coloring and index coding”, Proceedings of IEEE International Symposium on Information Theory (ISIT) 2013, Istanbul, Turkey, Jul 7-12, pp. 1152-1156. A. Blasiak, R. Kleinberg, and E. Lubetzky, “Index Coding via Linear Programming”, ArXiv-CoRR, Jul. 2011, Available at http://arxiv.org/abs/1004.1379. A. Blasiak, R. Kleinberg, and E. Lubetzky, ”Broadcasting with side information: Bounding and approximating the broadcast rate”, IEEE Transactions on Information Theory, Vol. 59, No. 9, Sep. 2013, pp. 5811-5823. C. Thapa, L. Ong, S. J. Johnson, “Generalized interlinked cycle cover for index coding”, IEEE ITW (Fall) 2015, Jeju Island, South Korea, Oct. 11-15 2015. S. A. Jafar, “Topological Interference Management Through Index Coding”, IEEE Transactions on Information Theory, Vol. 60, No. 1, Jan. 2014, pp. 529-568. H. Maleki, V. R. Cadambe, and S. A. Jafar, “Index Coding-An Interference Alignment Perspective”, IEEE Transactions on Information Theory, Vol. 60, No. 9, Sep. 2014, pp. 5402-5432. R. Peeters, “Orthogonal representations over finite fields and the chromatic number of graphs”, Combinatorica, Vol. 16, No. 3, pp. 417-431, Sept 1996. P. Krishnan and V. Lalitha, “A class of index coding problems with rate $\frac{1}{3}$”, Proceedings of IEEE International Symp. on Info. Theory (ISIT), 2016, July 10-15, Barcelona, Spain, pp. 130-134.

--- abstract: 'The development of machine learning in particular and artificial intelligent in general has been strongly conditioned by the lack of an appropriate interface layer between deduction, abduction and induction [@Domingos09]. In this work we extend traditional algebraic specification methods [@Adamek94] in this direction. Here we assume that such interface for AI emerges from an adequate Neural-Symbolic integration [@Avila08]. This integration is made for universe of discourse described on a Topos[@Goldblatt06] governed by a many-valued [Ł]{}ukasiewicz logic. Sentences are integrated in a symbolic knowledge base describing the problem domain, codified using a graphic-based language, wherein every logic connective is defined by a neuron in an artificial network. This allows the integration of first-order formulas into a network architecture as background knowledge, and simplifies symbolic rule extraction from trained networks. For the train of such neural networks we changed the Levenderg-Marquardt algorithm [@HaganMenhaj99], restricting the knowledge dissemination in the network structure using soft crystallization. This procedure reduces neural network plasticity without drastically damaging the learning performance, allowing the emergence of symbolic patterns. This makes the descriptive power of produced neural networks similar to the descriptive power of [Ł]{}ukasiewicz logic language, reducing the information lost on translation between symbolic and connectionist structures. We tested this method on the extraction of knowledge from specified structures. For it, we present the notion of fuzzy state automata, and we use automata behaviour to infer its structure. We use this type of automata on the generation of models for relations specified as symbolic background knowledge. Using the involved automata behaviour as data sets, we used our learning methodology, to extract new insights about the models, and inject them into the specification. This allows the improvement about the problem domain knowledge.' author: - Carlos Leandro --- ———————————————————————– Category Theory generalized the use of graphic language to specify structures and properties through diagrams. These categorical techniques provide powerful tools for formal specification, structuring, model construction, and formal verification for a wide range of systems, presented on a grate variety of papers. The data specification requires finite, effective and comprehensive presentation of complete structures, this type of methodology was explored on Category Theory for algebraic specification by Ehresmann[@Ehresm68]. He developed sketches as a specification methodology of mathematical structures and presented it as an alternative to the string-based specification employed in mathematical logic. The functional semantic of sketches is sound in the informal sense that it preserves by definition the structure given in the sketch. Sketch specification enjoy a unique combination of rigour, expressiveness and comprehensibility. They can be used for data modelling, process modelling and meta-data modelling as well thus providing a unified specification framework for system modelling. For our goal we extend the syntax of sketch to multi-graphs and define its models on the Topos (see e.g. for definition [@Johnstone02]), defined by relation evaluated in a many-valued logic. We named *specification system* too our version of Ehresmanns sketch, and on its definition we developed a conservative extension to the notions of commutative diagram, limit and colimit for many-valued logic. In this work, we use background knowledge about a problem to specify its domain structures. This type of information is assumed to be vague or uncertain, and described using multi-diagrams. We simplify the exposition and presentation of this notions using a string-based codification, for this type of multi-diagrams, named *relational specification*. We use this description for presenting structures extracted from data and on its integration. There are essentially two representation paradigms to represent the extracted information, usually taken very differently. On one hand, symbolic-based descriptions are specified through a grammar that has fairly clear semantics. On the other hand, the usual way to see information presented using a connectionist description is its codification on a neural network (NN). Artificial NNs, in principle, combine the ability to learn and robustness or insensitivity to perturbations of input data. NNs are usually taken as black boxes, thereby providing little insight into how the information is codified. It is natural to seek a synergy integrating the *white-box* character of symbolic base representation and the learning power of artificial neuronal networks. Such neuro-symbolic models are currently a very active area of research. In the context of classic logic see [@Bornscheuer98] [@Hitzler04] [@Holldobler00], for the extraction of logic programs from trained networks. For the extraction of modal and temporal logic programs see [@Avila07] and [@Avila08]. In [@Komendantskaya07] we can find processes to generate connectionist representation of multi-valued logic programs and for [Ł]{}ukasiewicz logic programs ([Ł]{}L) [@Klawonn92]. Our approach to the generation of neuro-symbolic models uses [Ł]{}ukasiewicz logic. This type of many-valued logic has a very useful property motivated by the ”linearity” of logic connectives. Every logic connective can be defined by a neuron in an artificial network having, by activation function, the identity truncated to zero and one [@Castro98]. This allows the direct codification of formulas into network architecture, and simplifies the extraction of rules. Multilayer feed-forward NN, having this type of activation function, can be trained efficiently using the Levenderg-Marquardt (LM) algorithm [@HaganMenhaj99], and the generated network can be simplified using the “Optimal Brain Surgeon” algorithm proposed by B. Hassibi, D. G. Stork and G.J. Stork [@Hassibi93]. We combine specification system and the injection of information extracted, on the specification, in the context of structures generated using a fuzzy automata. This type of automata are presented as simple process to generate uncertain structures. They are used to describe an example: where the generated data is stored in a specified structure and where we apply the extraction methodology, using different views of the data, to find new insights about the data. This symbolic knowledge is inject in the specification improving the available description about the data. In this sense we see the specification system as a knowledge base about the problem domain . In this section, we present some concepts that will be used throughout the paper. [Ł]{}ukasiewicz logics ---------------------- Classical propositional logic is one of the earliest formal systems of logic. The algebraic semantics of this logic are given by Boolean algebra. Both, the logic and the algebraic semantics have been generalized in many directions [@Jipsen03]. The generalization of Boolean algebra can be based in the relationship between conjunction and implication given by $(x\wedge y)\leq z \Leftrightarrow x\leq (y \rightarrow z). $ These equivalences, called *residuation equivalences*, imply the properties of logic operators in Boolean algebras. In applications of fuzzy logic, the properties of Boolean conjunction are too rigid, hence it is extended a new binary connective, $\otimes$, which is usually called *fusion*, and the residuation equivalence $(x\otimes y)\leq z \Leftrightarrow x\leq (y \Rightarrow z)$ defines *implication*. These two operators induce a structure of *residuated poset* on a partially ordered set of truth values $P$[@Jipsen03]. This structure has been used in the definition of many types of logics. If $P$ has more than two values, the associated logics are called a *many-valued logics*. We focused our attention on many-valued logics having a subset of interval $P=[0,1]$ as set of truth values. In this type of logics the fusion operator $\otimes$ is known as a *t*-norm. In [@Gerla00], it is described as a binary operator defined in $[0,1]$ commutative and associative, non-decreasing in both arguments, $1\otimes x= x$ and $0\otimes x= 0$. An example of a continuous $t$-norms is $x\otimes y=\max(0,x+y-1)$, named *[Ł]{}ukasiewicz* $t$-norm, used on definition of [Ł]{}ukasiewicz logic ([Ł]{}L)[@Hajek95]. Sentences in [Ł]{}L are, as usually, built from a (countable) set of propositional variables, a conjunction $\otimes$ (the fusion operator), an implication $\Rightarrow$, and the truth constant 0. Further connectives are defined as: $ \neg\varphi_1\text{ is }\varphi_1\Rightarrow 0,\;\; 1 \text{ is }0\Rightarrow 0 \text{ and } \varphi_1\oplus\varphi_2\text{ is }\neg\varphi_1\Rightarrow\varphi_2.$ The interpretation for a well-formed formula $\varphi$ in [Ł]{}logic is defined inductively, as usual, assigning a truth value to each propositional variable. The [Ł]{}ukasiewicz fusion operator $x\otimes y=\max(0,x+y-1)$, its residue $x\otimes y=\min(1,1-x+y)$, and the lattice operators $x\vee y=\max\{x,y\}$ and $x\wedge y=\min{x,y}$, defined in $\Omega=[0,1]$ a structure of *resituated lattice* [@Jipsen03] since: 1. $(\Omega,\otimes,1)$ is a commutative monoid 2. $(\Omega,\vee,\wedge,0,1)$ is a bounded lattice, and 3. the residuation property holds, $$\text{for all }x,y,z\in \Omega, x\leq y \Rightarrow z \text{ iff } x\otimes y\leq z.$$ This structure is divisible, $x\wedge y=x \otimes(x\Rightarrow y)$, and $\neg\neg x = x$. Structures with this characteristics are usually called MV-algebras [@Hajek98]. However truth table $f_\varphi$ is a continuous structure, for our computational goal, it must be discretized, ensuring sufficient information to describe the original formula. A truth table $f_\varphi$ for a formula $\varphi$, in [Ł]{}L, is a map $f_\varphi:[0,1]^m\rightarrow [0,1]$, where $m$ is the number of propositional variables used in $\varphi$. For each integer $n>0$, let $S_n$ be the set $\{0,\frac{1}{n},\ldots,\frac{n-1}{n},1\}$. Each $n>0$, defines a sub-table for $f_\varphi$ defined by $f_\varphi^{(n)}:(S_n)^m\rightarrow S_m$, given by $f_\varphi^{(n)}(\bar{v})=f_\varphi(\bar{v})$, and called the $\varphi$ *(n+1)-valued truth sub-table*. Since $S_n$ is closed for the logic connectives defined in [Ł]{}L, we define a *(n+1)-valued [Ł]{}ukasiewicz logic* ($n$-[Ł]{}L), as the fragment of [Ł]{}L having by truth values $\Omega=S_n$. On the following we generic call them “a [Ł]{}L”. Fuzzy logics, like [Ł]{}L, deals with degree of truth and its logic connectives are functional, whereas probability theory (or any probabilistic logic) deals with degrees of degrees of uncertainty and its connectives aren’t functional. If we take two sentence from $\L$ the language of [Ł]{}L, $\varphi$ and $\psi$, for any probability defined in $\L$ we have $P(\phi\oplus\varphi)=P(\phi)\oplus P(\varphi)$ if $\neg(\phi\otimes\varphi)$ is a boolean tautology, however for a valuation $v$ on $\L$ we have $v(\phi\oplus\varphi)=v(\phi)\oplus v(\varphi)$. The divisibility in $\Omega$, is usually taken as a fuzzy modus ponens of [Ł]{}L, $\varphi,\varphi\rightarrow \psi\vdash \psi $, where $v(\psi)=v(\varphi)\otimes v(\psi)$. This inference is known to preserve lower formals of probability, $P(\phi)\geq x$ and $P(\varphi\rightarrow \psi)\geq y$ then $P(\psi)\geq x\otimes y$. Petr Hájek presented in [@Hajek952] extends this principle by embedding probabilistic logic in [Ł]{}L, for this we associated to each boolean formula $\varphi$ a fuzzy proposition “$\varphi$ is provable”. This is a new propositional variable on [Ł]{}L, where $P(\varphi)$ is now taken to be its degree of truth. We assume in our work what the involved entities or concepts on a UoD can be described, or characterize, through fuzzy relations and the information associated to them can be presented or approximated using sentences on [Ł]{}L. In next section we describe this type of relations in the context of allegory theory [@Freyd90]. Relations --------- A vague relation $R$ defined between a family of sets $(A_i)_{i\in Att}$, and evaluated on $\Omega$, is a map $R:\prod_{i\in Att}A_i\rightarrow \Omega$. Here we assume that $\Omega$ is the set of truth values for a [Ł]{}L. In this case we named $Att$ the *set of attributes*, where each index $\alpha\in Att$, is called an *attribute* and the set indexed by $i$, $A_i$, represents the *set of possible values* for that attribute on the relation or its domain. In relation $R$ every instance $\bar{x}\in \prod_{i\in Att}A_i$, have associated a level of uncertainty, given by $R(\bar{x})$, and interpreted as the truth value of proposition $\bar{x}\in R$, in $\Omega$. Every partition $Att_i\cup Att_a\cup Att_o=Att$, where the sets of attributes $Att_i$, $Att_a$ and $Att_o$ are disjoint, define a relation $$G:\prod_{i\in Att_i}A_i\times\prod_{i\in Att_o}A_i\rightarrow \Omega,$$ by $$G(\bar{x},\bar{z})=\bigoplus_{\bar{y}\in\prod_{i\in Att_a}A_i}R(\bar{x},\bar{y},\bar{z}),$$ and denoted by $G:\prod_{i\in Att_i}A_i\rightharpoonup\prod_{i\in Att_o}A_i$, this type of relation we call a *view* for $R$. For each partition $Att_i\cup Att_a\cup Att_o=Att$ define a view $G$ for $R$, where $Att_i$ and $Att_o$ are called, respectively, the set of $G$ inputs and the set of its outputs. This sets are denoted, on the following, by $I(G)$ and $O(G)$. Graphically a view $$_{G:A_0\times A_1\times A_2\rightharpoonup A_3\times A_4\times A_5},$$ can be presented by multi-arrow on figure \[graph1\]. $$\tiny \xymatrix @=7pt { &&&*++[o][F-]{G}\ar `r[rd][rd]\ar `r[rrd][rrd]\ar `r[rrrd][rrrd]&&&\\ A_0\ar `u[urrr][urrr]&A_1\ar `u[urr][urr]& A_2\ar `u[ur][ur]&&A_3&A_4&A_5 }$$ A view $S:A\rightharpoonup A$ is called a *similarity relation* is 1. $S(\bar{x},\bar{x})=1$ (reflexivity), 2. $S(\bar{x},\bar{y})=S(\bar{y},\bar{x})$ (symmetry), and 3. $S(\bar{x},\bar{y})\otimes S(\bar{y},\bar{z})\leq S(\bar{x},\bar{z})$ (transitivity). We use Greek lets for similarity relation relations, and if $\alpha:A\rightharpoonup A$ is a similarity relation we write $\alpha:A$, and call to $A$ the support for similarity $\alpha$. The similarity using $\alpha$ between to elements $\bar{x}$ and $\bar{y}$ is denoted by $[\bar{x}=\bar{y}]_\alpha$ to mean $\alpha(\bar{x},\bar{y})$. We see a similarity relation $\alpha:A$ as a way to encoded fuzzy sets in [Ł]{}L. We do this interpreting, for $\bar{x}\in A$, its diagonal $[\bar{x},\bar{x}]_\alpha$ as the degree of true for proposition $\bar{x}\in \alpha$. Given two elements in the support set $\bar{x},\bar{y}\in A$, we interpret $[\bar{x},\bar{y}]_\alpha$ as the degree of true for proposition $\bar{x}=\bar{y}$ in $\alpha$. This offer us a way to evaluate the equality and de membership relation on the [Ł]{}L. Let $\Omega$-$Set$ be the class of views defined by relations evaluated in $\Omega$. We define a monoidal structure in $\Omega$-$Set$ for every pair of views $$G:\prod_{i\in I(G)}A_i\rightharpoonup\prod_{i\in O(G)}A_i \text{ and } R:\prod_{i\in I(R)}A_i\rightharpoonup\prod_{i\in O(R)}A_i$$ we define, $$R\otimes G:\prod_{i\in I(G)\cup I(R)\setminus O(G)}A_i\rightharpoonup\prod_{i\in O(R)\cup O(G)\setminus I(R)}A_i,$$ given, for every $$(\bar{x},\bar{z})\in \prod_{i\in I(G)\cup I(R)\setminus O(G)}A_i\times\prod_{i\in O(R)\cup O(G)\setminus I(R)}A_i,$$ by $$(R\otimes G)(\bar{x},\bar{z})=\bigoplus_{\bar{y}\in O(R)\cap I(G)}(R(\bar{x},\bar{y})\otimes S(\bar{y},\bar{z})).$$ We call to this tensor product *composition of view*. This operation extends composition of functions: if relation $G$ is a function between sets $A$ and $B$, and if $R$ is a function between sets $B$ and $A$, then for this two views in $\Omega$-$Set$, $G\otimes R$ is the function $R\circ G$. While composition between maps is a partial operator, it is defined only for componible maps, the tensor product $\otimes$ is total, it is defined for every pair of relations. In figure \[graph2\] for two multi-arrow $R$ and $G$ representing views such that $I(R)=\{A_0,A_1\}$, $O(R)=\{A_2,A_3,A_4\}$, $I(G)=\{A_2,A_3\}$, and $O(G)=\{A_5\}$, for the resulting view $R\otimes G$ we have $I(R\otimes G)=\{A_0,A_1,A_2\}$ and $O(R\otimes G)=\{A_4,A_5\}$. $$\tiny \xymatrix @=7pt { &&&&&*++[o][F-]{G}\ar `r[rdd][rdd] &\\ &&*++[o][F-]{R}\ar `r[rrd][rrd]\ar `r[rrrd][rrrd] &&&&\\ A_0\ar `u[urr][urr]\ar `d[drrrr][drrrr]&A_1\ar `u[ur][ur]\ar `d[drrr][drrr]& &A_2\ar `u[uurr][uurr]\ar `d[dr][dr]&A_3\ar `u[uur][uur]&A_4&A_5\\ &&&&*+++[o][F-]{R\otimes G}\ar `r[ru][ru]\ar `r[rru][rru]&&\\ }$$ In $\Omega$-$Set$ we denote by $I:\ast\rightarrow \Omega$ the relation defined on a singleton set by $I(\ast)=1$. This relation is the identity for $\otimes$; $R\otimes I \approx I\otimes R \approx R$. The class $\Omega$-$Set$ have a natural structure of category, having by objects $\Omega$-sets and by morphisms view, such that $R:\alpha\rightarrow \beta$ is a morphism from $\Omega$-set $\alpha:A$ to $\beta:B$ if $R:A\rightharpoonup B$ and 1. $\alpha \otimes R \leq R$, and 2. $R \otimes \beta \leq R$. Note that every object $\alpha:A$ have by identity the relation $\bar{\alpha}:A\rightharpoonup A$ defined by reflexive the close of $\alpha$, define making $\bar{\alpha}(\bar{x},\bar{x})=1$. The category $\Omega$-$Set$ is a symmetric monoidal closed category [@Borceux94], where the tensor product of $\Omega$-sets is given for $\alpha:A$ and $\beta:B$ by $$\alpha\otimes\beta:A\times B$$ defined $$(\alpha\otimes\beta)(a,b)=\alpha(a)\otimes\beta(b).$$ This can be used to describe a functor $$\alpha\otimes\_ : \Omega-Set \rightarrow \Omega-Set,$$ given for morphisms $R:\gamma \rightharpoonup \beta$, with support $f:A \rightharpoonup B$, by $$\alpha\otimes f:\alpha\otimes\gamma\rightharpoonup \alpha\otimes\beta$$ having by support $\alpha\otimes f:A\times C\rightharpoonup A\times B$, described by $$(\alpha\otimes f)(a,c,a',b)=\alpha(a,a')\otimes f(c,b).$$ Functor $\alpha\otimes\_$ have by left adjunct a functor $$\alpha\multimap\_ : \Omega\text{-}Set \rightarrow \Omega\text{-}Set,$$ defined for $\Omega$-sets $\beta:B$ by $$\alpha\multimap \beta:[A,B],$$ construct as the internalization for $\Omega$-set $Hom$ [@Clementino04] $$(\alpha\multimap \beta)(t,h)=\bigvee_{b_0,b_1}\bigoplus_{a}( \alpha(a,a)\otimes t(a,b_0)\otimes h(a,b_1)\otimes \beta(b_0,b_1)),$$ for relations $f:\gamma\rightharpoonup \beta$, with support $f:C\rightharpoonup D$, we have $$(\alpha\multimap f):(\alpha\multimap \gamma)\rightharpoonup (\alpha\multimap \beta),$$ a relation with support $\alpha\multimap f:[A,C]\rightharpoonup [A,B]$, described by $$(\alpha\multimap f)(h,g)(a,c,a',b)=h(a,c)\otimes g(a',b)\otimes \alpha(a,a').$$ This adjunction $\alpha\otimes\_ \;\vdash\; \alpha\multimap\_ $ have by unit [@Borceux94] the natural transformation, $\lambda$ defined for each $\Omega$-set $\gamma:C$, by a multi-morphism $$\lambda_\gamma:(\alpha\multimap \gamma)\otimes \alpha \rightharpoonup \gamma,$$ with support $\lambda_\gamma:[A,C]\times A \rightharpoonup C$, by $$\lambda_\gamma(h,a,b)=h(a,b),$$ the relation $h$ evaluation evaluated in $(a,b)\in A\times B$. The $\alpha\multimap \beta:[A,B]$ reflexive closure defines a similarity relation in $[A,B]$, we use this relation in the following to quantify the similarity between relation form $S$ and $B$, and we call them *power similarity relation*. In the follow we use this relation to compare models or on the quantification of model quality. Two views $R$ and $G$, in $\Omega$-$Set$, are called *independents* if $R\otimes G=G\otimes R$. By this we mean what the $R$ output not depend on $G$ inputs and the $G$ output not depend on $R$ input. Given a view $R:A\rightharpoonup B$, we define projections $R_A:B\rightarrow\Omega$ and $R_B:B\rightarrow\Omega$, respectively, by $R_A(\bar{b})=\bigoplus_{\bar{a}\in A}R(\bar{a},\bar{b})$ and $R_B(\bar{a})=\bigoplus_{\bar{b}\in B}R(\bar{a},\bar{b})$. In the following we used $R(\bar{a},\_)$ to denote the relation defined from $R$ by fixing a input vector $\bar{a}\in A$, $R(\bar{a},\_)(\bar{b})=R(\bar{a},\bar{b})$. Inference --------- Generically inference is a process used to generate now facts based on known facts. On the context of multi-valued logic, the inference allows fining the degree of two for a new proposition based on the known degree of truth for propositions [@hajek97]. This inference can be described using the composition operator defined in $\Omega$-$Set$ [@Zadeh75].The *syllogism* describe by the rule: --------------------------------------------------- $R$: If $a\in \alpha$ then $b\in \beta$ $S$: If $b\in \beta$ then $c\in \gamma$ $R\otimes S$: If $a\in \alpha$ then $c\in \gamma$ --------------------------------------------------- This rule is interpreted saying that: If 1. $R(a,b)\geq([a]_\alpha\Rightarrow [b]_\beta)$, and 2. $S(b,c)\geq([b]_\beta\Rightarrow [c]_\gamma)$ then $(R\otimes S)(a,c) \geq [a]_\alpha\Rightarrow [c]_\gamma$. This gives us a lower bond for degree of truth. However this strategy works better on the version of *Modus Ponens*: --------------------------------------------------- $R$: $a\in \alpha$ $S$: If $a\in \alpha$ then $b\in \beta$ $R\otimes S$: $a\in \alpha\; \wedge\; b\in \beta$ --------------------------------------------------- Since [Ł]{}L is a divisible logic we can write: --------------------- ----- -------------------------------------------------------- $(R\otimes S)(a,b)$ $=$ $R(a,a)\otimes S(a,b)$ $=$ $[a]_\alpha \otimes ([a]_\alpha\Rightarrow [b]_\beta)$ $=$ $[a]_\alpha\wedge [b]_\beta$ --------------------- ----- -------------------------------------------------------- Applying this rule to a simple relation $H:\alpha\rightarrow \beta$, we have --------------------------------------------------- $R$: $a\in \alpha$ $S$: If $a\in \alpha$ then $(a,b)\in H$ $R\otimes S$: $a\in \alpha\; \wedge\; (a,b)\in H$ --------------------------------------------------- since $[a]_\alpha=\bigoplus_b H(a,b)$, the degree of truth of $a\in \alpha\; \wedge\; (a,b)\in H$ is the degree of truth for $(a,b)\in H$, then: $$[a]_\alpha \otimes ([a]_\alpha\Rightarrow H(a,b))= H(a,b).$$ We simplified this excretion defining $$H(\beta | a)(b)=[a]_\alpha\Rightarrow H(a,b),$$ and we write $$[a]_\alpha \otimes H(\beta | a)(b)= H(a,b).$$ In this context $H(\beta | a)(b)$ is interpreted as the degree of truth for the proposition: ”A class associated by relation $H$ is $b$, if its input is $a$”, given by the result for the evaluation of $(a,b)$ by $H$, conditionated to the degree belonging of $a$ on $f$ input domain. In classic logic, when $A$ is finite, this is express by $\forall a\in \alpha: H(a,b)$. Given a faithful view ${R:A\rightharpoonup B,}$ and $\bar{a}\in A$ and $\bar{b}\in B$ from $\Omega$-Set. The equations $$R(\bar{a})_B\otimes x = R(\bar{a},\_)\text{ and }R(\bar{b})_A\otimes x = R(\_,\bar{b}),$$ have by solution, relation $R(\_|\bar{a})$ and $R(\_|\bar{b})$, respectively, defined by $R(\_|\bar{a})=R(\bar{a})_B\Rightarrow R(\bar{a},\_)$ and $R(\_|\bar{b})=R(\bar{b})_A\Rightarrow R(\_,\bar{b})$. $$\tiny \xymatrix @=9pt { &&A \ar[lld]^{R(\_)_B}\ar `l[lld]_{R(\_\mid \bar{b})}[lld]\ar@_{->}[dd]^R\ar[rr]_{i_A}&&A\ar[rd]^{R(\_,\bar{b})}&&\\ \Omega & & &&&\Omega\\ &&B \ar[llu]_{R(\_)_A}\ar[rr]_{i_B}\ar `l[llu]^{R(\_\mid \bar{a})}[llu]&&B\ar[ru]_{R(\bar{a},\_)}&& }$$ We use this rule to solve inference problems in $\Omega$-Set. Given two compatible views $R:A\rightharpoonup B$ and $G:B\rightharpoonup C$, i.e. such that the output attributes for view R are the input attributes for G. For observable descriptions $\bar{a}\in A$ and $\bar{c}\in C$, we have --------------------------------------------------------- --- --------------------------------------------------------------------------------------------- $R(\bar{a})_B\otimes (R\otimes G)(\_|\bar{a})(\bar{c})$ = $(R\otimes G)(\bar{a},\bar{c})$ = $\bigoplus_{\bar{b}} R(\bar{a},\bar{b})\otimes S(\bar{b},\bar{c})$ = $\bigoplus_{\bar{b}} R(\bar{a})_B\otimes R(\_|\bar{a})(\bar{b})\otimes G(\bar{b},\bar{c})$, --------------------------------------------------------- --- --------------------------------------------------------------------------------------------- then $$(R\otimes G)(\_|\bar{a})(\bar{c})=R(\bar{a})_B\Rightarrow (R(\bar{a})_B\otimes\bigoplus_{\bar{b}} R(\_|a)(b)\otimes S(b,c)),$$ i.e. $$(R\otimes G)(\_|\bar{a})=\bigoplus_{\bar{b}} R(\_|\bar{a})(\bar{b})\otimes S(\bar{b},\_).$$ When views $R:A\rightharpoonup C$ and $G:B\rightharpoonup D$ are independent we have $$(R\otimes G)(\_|\bar{a},\bar{b})(\bar{c},\bar{d})= R(\_|\bar{a})(\vec{c}) \otimes S(\_|\bar{b})(\bar{d}).$$ Naturally, if $C=D$ we write $(R\otimes G)(\_|\bar{a},\bar{b})(\bar{d})$ for $(R\otimes G)(\_|\bar{a},\bar{b})(\bar{d},\bar{d})$. Limit sentences and colimit sentences ------------------------------------- A multi-arrow defines a link between a set of input nodes and a set of output nodes, we can see an example of this on figure \[graph1\]. We can use multi-arrows to generalize the notion of arrow in a graph. This allows the definition of a multi-graph as a set of nodes linked together using multi-arrows. Examples of multi-graphs can be seen on figures \[graph2\] and \[multidiagram\]. A multi-diagram in $\Omega$-$Set$, defined having by support a multi-graph ${\mathcal{G}}$, is a multi-graph homomorphism $D:{\mathcal{G}}\rightarrow \Omega\text{-}Set$, where each node in ${\mathcal{G}}$ is mapped to a $\Omega$-set $\alpha:A$, and each multi-arrows in ${\mathcal{G}}$ is mapped to a relation view. In this sense, every set of views in $\Omega$-$Set$ defines a multi-diagram, having by support the multi-graph where the selected views are multi-arrows, and the $\Omega$-set used on this views as nodes. The classically definition of limit for a diagram, in the category of sets, can been as a way to internalize the structure of a diagram in form of a table [@Borceux94]. Given a diagram $D:{\mathcal{G}}\rightarrow Set$ with vertices $V=\{a_i\}_{i\in I}$ and arrows $A=\{f_j\}_{j\in J}$, its limit is a table or a subset of the cartesian product $\prod_{i\in I}D(a_i)$ given by $$Lim\;D=\{(\ldots,x_i,\ldots,x_j,\ldots)\in \prod_{i}D(a_i):\forall_{f:a_i\rightarrow a_j}D(f)(x_i)=x_j\}.$$ were the relation is evaluated on classic logic. We present as limit for a multi-diagram $D:{\mathcal{G}}\rightarrow \Omega\text{-}Set$ a conservative extension from the classical limit definition. Let $D:{\mathcal{G}}\rightarrow \Omega\text{-}Set$ be a multi-diagram with vertices $(v_i)_{i\in L}$. The *limit* of diagram $D$ is a relation denoted by $Lim\;D$, and defined as $$Lim\; D: \prod_{i\in L} D(v_i) \rightarrow \Omega,$$ such that $$(Lim\;D)(\ldots,\overline{x}_{i},\ldots,\overline{x}_{j},\ldots)=\bigotimes_{f:v_i\rightharpoonup v_j\in {\mathcal{G}}} D(f)(\overline{x}_{i},\overline{x}_{j}).$$ The limit for multi-diagram on figure \[multidiagram\] is the relation $ Lim\;D:A_0\times \ldots \times A_5\rightarrow \Omega $ given for every $(a_0,\ldots,a_5)\in A_0\times \ldots\times A_5$ by $$(Lim\;D)(a_0,a_1,a_3,a_4,a_5)= f(a_0,a_1,a_3,a_4,a_5)\otimes g(a_1,a_2,a_4,a_5)\otimes h(a_2,a_3).$$ $$\tiny \xymatrix @=7pt { &&&*+[o][F-]{f}\ar `r[rd][rd]\ar `r[rrd][rrd]\ar `r[rrrd][rrrd]&&&\\ A_0\ar `u[urrr][urrr]&A_1\ar `u[urr][urr]\ar `d[drr][drr]& A_2\ar `d[dr][dr]\ar[r]&*+[o][F-]{h}\ar[r] &A_3&A_4&A_5\\ &&&*+[o][F-]{g}\ar `r[rru][rru]\ar `r[rrru][rrru]&&& }$$ In this sense for parallel views $R,S:X\rightharpoonup Y$, they define a multi-diagram, and its limit is the relation $$Lim(R=S):X\times Y\rightarrow \Omega,$$ given by\ $$Lim(R=S)(x,y)=R(x,y)\otimes S(x,y).$$ This relation is denoted by $[R=S]$ and usually called, on Classic logic, $R$ and $S$ *equalizer*. If $R:X\rightharpoonup U$ and $S:Y\rightharpoonup U$ are views its *pullback*, denoted by $R\otimes_US$ is defined by the limit $$Lim(R\otimes_U S):X\times U\times Y\rightarrow \Omega,$$ given by $$Lim(R\otimes_U S)(x,u,y)= R(x,u)\otimes S(y,u).$$ Given a family of views, having the some output, $(R_i:X_i\rightharpoonup U)_{i\in L},$ its *wide-pullback* is the relation $Lim(\otimes_U R_i):\otimes_{i\in L}R_i$. \[def:lambdaLim\] A relation $R$ described in [Ł]{}L, is the $\lambda$-limit for a multi-diagram $D$ if $R:A\rightarrow \Omega$ is $\lambda$-similar to $Lim\;D:A\rightarrow \Omega,$ i.e if $$(R\multimap Lim\;D)\geq \lambda,$$ when this is the case we write $$R=Lim_\lambda\;D.$$ We used the definition of limit to extend the notion of commutative diagram. The idea was to characterize a commutative diagrams using its internalization on a table. \[Comutatividade Pdiagramas\] If $D:{\mathcal{G}}\rightarrow \Omega\text{-}Set$ is a multi-diagram with vertices in $V$, and associated $\Omega$-sets $(\alpha_i)_{i\in V}$, where we selected a set $s(D)$ of input vertices. Assuming that the sub-graph of ${\mathcal{G}}$ defined by vertices $s(D)$ is acyclic and that $P$ is the Cartesian product defined by each vertices on $D$ with not belong to $s(D)$. The multi-diagram $D$ is commutative with inputs if $s(D)$ if $$\bigvee_{\bar{n} \in P}(Lim\;D)(\bar{s},\bar{n})=\bigvee_{\bar{n}\in V}(\bigotimes_i\;\alpha_i)(\bar{s},\bar{n}),$$ for every $\bar{s}\in \prod_{i\in s(D)}D(i)$. A diagram is $\lambda$-commutative if $$\left(\bigvee_{\bar{n}\in V}(Lim\;D)(\_,\bar{n})\multimap \bigvee_{\bar{n}\in V}(\prod_i\;D(i))(\_,\bar{n})\right)\geq\lambda,$$ Limits, colimits and commutativity can be used on the specification of structures [@Adamek94]. We use the conservative extensions to this notions for the detrition of fuzzy structures. However the notion of colimit is more difficult to present generically. The construction of a colimit reduces to that of two coproducts and a coequalizer, siting [@Borceux94], in the category of sets governed by classic logic the explicit description of a coequalizer is generically very technical since it involves the description of the equivalence relation generated by a family of pairs. This complexity is incased when we extend this notion to relations evaluated on multi-valued logics. We present bellow two examples. The coproduct of $\Omega$-sets $\alpha:A$ and $\beta:B$ is a relation $R$ having by support set $A\coprod B$ given by $$R(a,a')=[a=a']_\alpha\oplus [a=a']_\beta.$$ Where, for simplicity, we assume what relations $\alpha$ and $\beta$ assume the value $0$ when are evaluating pairs outside its support sets. We denote the coproduct for $\alpha:A$ and $\beta:B$ by $\alpha\oplus\beta:A\coprod B$. The diagram defined by a parallel pair of multi-morphisms $f:\alpha:A\rightarrow \beta:B$ and $g:\alpha:A\rightarrow \beta:B$ have by colimite a $\Omega$-set, with support $A\coprod B$, given by $ R(a,a')= \begin{array}{cl} & \bigoplus_{b,b'\in B}f(a,b)\oplus f(a',b')\oplus [b=b']_B\\ \oplus & \bigoplus_{b,b'\in B}g(a,b)\oplus g(a',b')\oplus [b=b']_B\\ \oplus & \bigoplus_{b,b'\in A}f(b,a)\oplus g(b',a')\oplus [b=b']_A \\ \oplus & [a=a']_A \\ \oplus & [a=a']_B \\ \end{array}, $ where $a,a'\in A\coprod B$, for simplicity, we assume what relations $\alpha$ and $\beta$ assume the value $0$ when are evaluated on pairs outside its support sets. Concepts -------- We describe a table or a concept using relation views. A table or a concept description using values in the family $(A_\alpha)_{\alpha\in Att}$, for attributes $Att$, is a view $ {R:O\rightharpoonup\coprod_{\alpha\in Att}A_\alpha}, $ where $O$ is a set of keys identifying concept instances. We use $R(o,\alpha=x)=\lambda$ to denote that, in instance $o\in O$, the uncertainty of an attribute $\alpha$ to be equal to value $x\in A_\alpha$ is $\lambda$. This mean that, in an instance, an attribute may assume different values, associated with different uncertain levels expressed by truth values. When we have $R(o,\alpha=x)\geq\lambda$, for every entity $o\in O$, we write $R(\alpha=x)\geq\lambda$ or just $\alpha\sim_\lambda x$ in $R$. A concept description have different presentations, corresponding to each of the perspectives taken to data. Each partition $Att=V\cup U$, defines a perspective through the view $ _{R_{V,U}:O\times\prod_{\alpha\in V}A_\alpha\rightharpoonup\coprod_{\alpha\in U}A_\alpha}, $ given by $$_{R_{V,U}(o,\overline{\alpha=x},y)=\bigotimes_{\alpha\in V,x\in A_{\alpha}}R(o,\alpha=x)\otimes R(o,y)},$$ where $\overline{\alpha=x}$ abbreviates the tuple defined using family $(\alpha=x)_{\alpha\in V,x\in A_{\alpha}}$. Relation between information on a data set can be defined as a diagram $D$, from a multi-graph ${\mathcal{G}}$ to $Set(\Omega)$, where each multi-arrow is mapped to a view of a concept description. Every multi-graph homomorphism $I:{\mathcal{I}}\rightarrow {\mathcal{G}}$ defines a query in the structure $D$, having by answer the concept description defined by $Lim\;D\circ I$. Where $D\circ I$ denotes the composition between graph homomorphisms. If we assume that $\Omega=[0,1]$, given a pair of concept presentations defined using a finite set of keys $O$, $$_{R_0,R_1:O\times\prod_{\alpha\in V}A_\alpha\rightharpoonup\coprod_{\alpha\in U}A_\alpha},$$ we measure the similarity between this two views using relation $$\Gamma(R_0,R_1)=e^{-\frac{1}{|O|}\sum_{\bar{x}\in\prod_{\alpha\in Att} A_\alpha}\neg(R_0(\bar{x})\Leftrightarrow R_1(\bar{x}))},$$ where $|O|$ is the number of keys. Relation $\Gamma$ is a *similarity relation* between pairs of concept described using a tuple in $\prod_{\alpha\in Att} A_\alpha$ since: 1. $\Gamma(R_0,R_0)=1$ (reflexivity), 2. $\Gamma(R_0,R_1)=\Gamma(R_1,R_0)$ (symmetry), and 3. $\Gamma(R_0,R_1)\otimes\Gamma(R_1,R_2)\leq\Gamma(R_0,R_2)$ (transitivity). The transitivity is a consequence of, in any ML-algebra $\Omega$, for all $\lambda_0,\lambda_1,\lambda_2\in \Omega$,$(\lambda_0\Rightarrow \lambda_1)\otimes(\lambda_1\Rightarrow \lambda_2)\leq \lambda_0\Rightarrow \lambda_2$. When $\Gamma(R_0,R_1)=\lambda$ we write $R_0\sim_\lambda R_1$ and we say that, $R_0$ is $\lambda$-similar to $R_1$, for $R_0\sim_1 R_1$ we write $R_0=R_1$. We named this similarity measure of *exponential similarity*. In the literature we can find other measures for relation similarity measurement. Like the *inf-similarity*, used in Possibilistic logic, $$_{\Gamma(R_0,R_1)=\bigwedge_{\bar{x}\in\prod_{\alpha\in Att} A_\alpha}(R_0(\bar{x})\Leftrightarrow R_1(\bar{x})),}$$ or the *and-similarity*, used in Boolean logic, $$_{\Gamma(R_0,R_1)=\bigotimes_{\bar{x}\in\prod_{\alpha\in Att} A_\alpha}(R_0(\bar{x})\Leftrightarrow R_1(\bar{x})),}$$ however these relations are to crispy for model evaluation. We need to be able to quantify who models are similar to a concept description. This notion is fundamental to make fuzzy some key concepts of relational algebra, useful on data structure specification. Example of this is the description of a “is\_a” relation evaluated on multi-valued logic. For that, let $R:A\rightharpoonup B$ be a concept description, here we assume the existence of a similarity $\Gamma_A$ defined in $A$. The concept description $R$ defines a “is\_a” relation, for similarity $\Gamma_A$, if sentence $$_{R(a_0,b)\otimes R(a_1,b) \Rightarrow \Gamma_A(a_0,a_1),}$$ have by truth-value 1, for every $a_0,a_1\in A$ and every $b\in B$, i.e. $$_{\bigotimes_{a_0,a_1\in A}\bigotimes_{b\in B}(R(a_0,b)\otimes R(a_1,b) \Rightarrow \Gamma_A(a_0,a_1))=1.}$$ In this sense we call to view $R$ a *mono-view* or a *clustering*. Views $R:A\rightharpoonup B$ such that $_{\bigotimes_{b\in B}\bigoplus_{a\in A}R(a,b)}=1$, are called *epi-views*. All the widely used data specification mechanisms (like Entity Relationship Model [@Chen76], the Fundamental Data Model [@Ship81], the Generic Semantic Model [@Abiteboul95]), OOA&D-schemas in a million of versions and UML which itself comprises a host of various notations, have a strong graphical component. They are essentially graphs with special markers in them. Usually the semantics of these markers is defined in an ad-hoc and sometimes non-formal way. An important component of the mathematical structure that will be used to formalized knowledge, are multi-graph homomorphisms into the class $Set(\Omega)$ of relations views. When specifying an information system, it will be necessary to formulate constrains on such graph homomorphisms. A identical notion of a specification whose models are graphs homomorphisms into a category theory are known on the category community under the name of sketch. Sketches where invented by Charles Ehresmann and can be perceived as a graphic based logic, which formalizes in a precise and uniform way the semantic of graph with marks [@piessen00]. For our propose we extended Ehresmann’s sketch to formalize a graphic based fuzzy logic. We named, this mathematical structure, *specification system*. On information specification, specification system will be used to specify finite fuzzy structures. Semantic data specification have been used for may years in the early stages of database design, and they have become key ingredients of object-oriented software. The goal of a semantic data specification is to build a mathematical abstraction of a small part of the real word. This small part of the word is usually called the *universe of discourse* (UoD) in the database literature. The models of the data specification are possible states of the UoD, and will be the structures stored in an information system. The mathematical structure that will be used to describe the UoD are finite models of specification systems together with a labeling of all the elements of these models. Hence a data specification will consist of two parts. The first part will be a specification system, and it describes the fuzzy structure and the interdependencies of the various entities about which we want to store information. The second part indicates what kind of information we want to store about each type of entity: for each type of entity, we give its set of possible attribute values, i.e. the set of all possible labels that an entity of the given type can have. The structure defined, by specification system $S$ and model $M$, we named *the semiotic* $(S,M)$. By a *specification system* $S$ we understood as a structure $S=({\mathcal{G}},C,L,coL)$, where ${\mathcal{G}}$ is a multi-graph, $C$ is a set of pairs $(G,\lambda)$, and $L$ and $coL$ are sets of tuples $(f,G,i(G),o(G),\lambda)$, such that $f$ is a multi-arrow in ${\mathcal{G}}$, $D\subset{\mathcal{G}}$ is a multi-graph, $\lambda\in \Omega$ and $i(G)$ and $o(G)$ are sets of nodes from ${\mathcal{G}}$. Given a specification system $S=({\mathcal{G}},C,L,coL)$ a model for $S$ is a diagram $M:{\mathcal{G}}\rightarrow Set(\Omega)$, mapping multi-arrows to concept description, such that: 1. for every $(G,\lambda)\in C$, $M(G)$ is $\lambda$-commutative, 2. for every $(f,G,i(G),o(G),\lambda)\in L$, $M(f)$ have by input $M(i(G))$ and by output $M(o(G))$ and is $\lambda$-equivalent to relation $Lim\;M(G)$, and 3. for every $(f,G,i(G),o(G),\lambda)\in L$, $M(f)$ have by input $M(i(G))$ and by output $M(o(G))$ and is $\lambda$-equivalent to relation $coLim\;M(G)$. The pair $(S,M)$ defined by a model $M:{\mathcal{G}}\rightarrow Set(\Omega)$ for a specification system $S$ is called a *semiotic*, where the multi-graph ${\mathcal{G}}$ describes a *library of components*. A specification system $S$, if consistent, describes the fuzzy structure for a class of UoD. If the set of all models for specification system $S$ is denoted by $Mod(S)$, every $M\in Mod(S)$ can be seen as a system state. Two states $M_0,M_1\in Mod(S)$ have similar structures and the specification $S$ can be enriched with now knowledge to increase the dissimilarity between the states. The knowledge need to distinguish between $M_0$ and $M_1$ can be extracted querying the state $M_0$, trying to find its particularities. For that, every multi-graph homomorphism $I:{\mathcal{I}}\rightarrow{\mathcal{G}}$ defines a query to a model $M:{\mathcal{G}}\rightarrow Set(\Omega)\in Mod(S)$. And this query $I$ have by answer the relation given by $$Lim\;M\circ I,$$ and each of its views can be used as a data set, usable to feed a data mining processes, to extracted insights about the model. Knowledge integration via specification systems ----------------------------------------------- Specification systems are graphic specifications formalisms (its components are multi-graphs and markers in these graphs) and can be taken as repositories of knowledge about the $UoD$. They are described using a rigorous graphic language with a precise semantic, where we have a methodology to querying its models. Your goal is the enrichment of this structure with knowledge extracted from one of its models. We simplify this process by expressing constraints of first-order logic formulae into a graphical marker. To be able to do that, we consider the multi-graph used on the definition of the specification system component library as a presentation of a first-order many-sorted signature: nodes of the graph are interpreted as data sorts, and multi-arrows are interpreted as relations evaluated in $\Omega$. A structure for this signature is exactly a multi-graph homomorphism from ${\mathcal{G}}$ to $Set(\Omega)$. Every formula for this signature are particularly simple, since there are only relations. A formula $R(x_1,x_2,\ldots,x_n)$ is defined through a multi-graph homomorphism $R:{\mathcal{I}}\rightarrow{\mathcal{G}}$ and its interpretation in model $M$, is the relation $Lim\;M\circ R$. Where the interpretation of each initial and terminal node of the multi-diagram $R$ defines the sorts for variables $x_1,x_2,\ldots,x_n$. Every atomic formula is a relation $R(x_1,x_2,\ldots,x_n)$. The finite conjunction of atomic formulas on variables $x_1,x_2,\ldots,x_n$, using relations $R_1,R_2,\ldots,R_m$ is denoted by: $(R_1\otimes R_2\otimes\ldots\otimes R_m)(x_1,x_2,\ldots,x_n)$ and it is interpreted as the relation $Lim\;M\circ(R_1\otimes R_2\otimes\ldots\otimes R_m)$. In this sense, every disjunctive formula defined using relation can be interpreted as the limit of a finite diagram. The translation in the other direction is also simple. Given a limit mark $(f,G,i(G),o(G),\lambda)$ in $S$, the meaning for sign $f$, is the interpretation of a disjunctive formula defined trough the interpretation of each multi-arrows used on the definition of multi-graph $G$ having its interdependencies (gluing order in $G$) defined using variable repetition. Similarly, every conjunctive formula defined using relation on semiotic $(S,M)$ can be interpreted as the colimit of a finite diagram. For colimit makes\ $(f,G,i(G),o(G),\lambda)$ is $S$, the interpretation for $f$, can be defined as the interpretation for a conjunctive formula defined using each multi-arrow in $G$ . Given a semiotic $(S,M)$ every formula $\varphi$, extracted from model $M$ using a query $I$, can be described by a set of limit marks and colimit marks. This allows the enrichment of specification system $S$, defining a new system $S'$, such that $Mod(S')\subseteq Mod(S)$ and $M\in Mod(S')$. However, it is known from [@Adamek94] what where are structures specifiable using sketches but not in first-order logic. Then first-order logic have less expressive power as specification systems. This result allows to use a mixture of limits, colimits and formulas in the UoD specification and be sure that the resulting specification can always be translated to a specification system [@piessen00]. A specification system description ---------------------------------- A specification system is by nature a graphic specification described, however, some times like in this exposition, it is preferable a description for the specification system using a string-based presentation. For this we used a string-based codification for specification systems named *relational specification*, generalizing the notion of essentially algebraic specification: the original idea goes back to Freyd [@Freyd72] and having the same expressive power as finite limit sketches [@Barr90]. The essentially algebraic fragment, are interesting computer sciences mainly because theories, of many kinds of specification formalisms, are in fact initial algebras for some essentially algebraic specification. A number of proof systems for essentially algebraic specification have been introduced [@Makkai93]. Recall that an ordering on a set $X$ is well-founded iff every strictly decreasing chain of elements of $X$ must be finite. A relational specification consists of: 1. A set of sorts. 2. A set of relational signs, with a well-founded ordering (called a *library of components*). 3. A set of diagrams defined using relational views, with a well-founded ordering (called a *a set of diagrams*). 4. A set of condition build on relational signs and diagrams. Every relation sign or view sign $\omega$ have an arity and a set $Def(\omega)$ of relations with the same arity as $w$, called the *set of domain condition*. For our propose sorts are nominal and finite, and we list its possible values by writing $A:\{a_1,a_2,\ldots,a_t\},$. These are the basic structures for UoD specification, used on the description of relations and views. Every relation have an arity described by a list of data sorts. We denote a relation symbol $R$, with arity the list of sorts $A_1,A_2,\ldots,A_n$, as $R:A_1, A_2, \ldots, A_n$. Interpreted as a concept description $M(R):O\rightharpoonup\coprod_i A_i$. A view of $R$ having by source arity the list $A_1,A_2,\ldots,A_n$ and by target arity $B_1,B_2,\ldots,B_m$ is denoted by: -------- --------------------------------------------------------------- -- $R:\{$ $A_1, A_2, \ldots, A_n\rightharpoonup B_1, B_2, \ldots, B_m$; } -------- --------------------------------------------------------------- -- These are interpreted as concept descriptions $M(R):O\times\prod_i A_i\rightharpoonup \coprod_j B_j$. The relationship between relations and views may be defined using marked diagrams. A diagram is described by a set of views where we marked some of its sorts as input or output sort. We write -------- ---------------------------------------------------------- -- $D:\{$ $A_1,A_2,\ldots,A_n\rightharpoonup B_1,B_2,\ldots,B_m$; D : $D_1\otimes D_2\otimes \ldots\otimes D_k$; } -------- ---------------------------------------------------------- -- where $(D_i)_{i\in I}$ is a list of diagram signs, what are smaller than $D$, describing the proper sequence of gluing to form $D$. Note what, every view can be seen as a diagram with only one multi-arrow. The commutativity or $\lambda$-commutativity for a diagram $D$, for view $G$, in the model, is denote, respectively, by $G:[D]$ or $G:[D]_\lambda$. When the interpretation for a relation or view $G$ must satisfy a limit or a colimit, we writing $_{G:lim\;D,\;\; G:\lambda-lim\;D,\;\; G:colim\;D,}$ or $_{G:\lambda-colim\;D.}$ In specification: [rll]{} $G:\{$ &$A_1, A_2, \ldots, A_n\rightharpoonup B_1, B_2, \ldots, B_m$; &\ & -------- ---------------------------------------------------------- -- $D:\{$ $A_1,A_2,\ldots,A_n\rightharpoonup B_1,B_2,\ldots,B_m$; D : $D_1\otimes D_2\otimes \ldots\otimes D_k$; } -------- ---------------------------------------------------------- -- &\ & $G: \lambda_0$-$lim\;D$;&\ & --------- ---------------------------------------------------------- -- $D':\{$ $A_1,A_2,\ldots,A_n\rightharpoonup B_1,B_2,\ldots,B_m$; $D' : D'_1\otimes D'_2\otimes \ldots\otimes D'_k$; } --------- ---------------------------------------------------------- -- &\ & G:$[D']_{\lambda_1}$;&\ }\ every interpretation for view $G$ must be $\lambda_0$-similar to $Lim\;D$, there $D$ is diagram defined gluing diagrams or multi-arrows $(D_i)_{i\in I}$, and must transform $D'$ is a $\lambda_0$-commutative diagram. However, this type of properties are very generic and some times difficult to understand. When we want to be more specific, we describe some of the relations or views properties using first-order formulas. We will express internal properties on a relation or view $G$ using formulas, with the same arity as $G$, and defined using only signs that are smaller than $G$ in the library of component. For it we write, --------- ------------------------------------------------------------------------------------------------------------ -- $G:=\{$ $A_1,A_2,\ldots,A_n$; $G(x_1,x_2,\ldots,x_n):P(R_1(x_1,x_2,\ldots,x_n),R_1(x_1,x_2,\ldots,x_n),\ldots,R_m(x_1,x_2,\ldots,x_n))$; } --------- ------------------------------------------------------------------------------------------------------------ -- if view or relation $G$ interpretation satisfies formula $P$ having the some arity as $R$ and, dependente from relations $R_1,R_2,\ldots,R_n$ that are smaller than $R$ in the library of componentes. We simplify the specification using some special meta-signs, following the spirit of M. Makkai [@Makkai93] and Z. Diskin [@Diskin99]. If a view $D$ is a *is\_a* relation or a clustering relation using the similarity relation $\Gamma$, we write: [rll]{} $D:\{$ &$A_1,A_2,\ldots,A_n\rightharpoonup B$; &\ & ------------- ------------------------ -- $\Gamma:\{$ $A_1,A_2,\ldots,A_n$; $\Gamma$ : similarity; } ------------- ------------------------ -- \ & $D$ : is\_a$(\Gamma)$;\ }\ In the next section we describe a process useful for knowledge extraction. We are particularly interest in symbolic representation to simplify knowledge integration via relational specifications. There are many methodologists available for this task, however our method uses the same logic assumed to govern the UoD. Given a concept description ${R:O\rightharpoonup\coprod_{\alpha\in Att}A_\alpha}, $ our goal is the extraction of knowledge from one of its views ${R_{V,U}:O\times\prod_{\alpha\in V}A_\alpha\rightharpoonup\coprod_{\alpha\in U}A_\alpha}$. For that the information structure is crystallized in a neural network and codified in string-based notation as a formula. Different concept can be seen as answers to different queries to a UoD models. In this sense different concepts represent different perspective for the available data, allowing the enrichment of a knowledge base or specification systems with new insights. In this section, we present a methodology to extract first-order formulas using neural networks describing available information in a [Ł]{}logic. As mentioned in [@Amato02] there is a lack of a deep investigation of the relationships between logics and NNs. In [@Castro98] it is shown how, by taking as activation function, $\psi$, the identity truncated to zero and one, $$_{\psi(x)=\min(1,\max(x, 0)),}$$ it is possible to represent the corresponding NN as a combination of propositions of [Ł]{}ukasiewicz calculus and *vice-versa* [@Amato02]. For used NNs to learn [Ł]{}ukasiewicz sentences, we define the first-order language as a set of circuits generated from the plugging of atomic components. For this, we used the library of components presented in table \[semioticaLuk\], interpreted as neural units and we gluing them together, to form NNs having only one output, without loops. This task of construct complex structures based on simplest ones can be formalized using generalized programming [@Fiadeiro97]. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Formula: Configuration: Formula: Configuration: Formula: Configuration: Formula: Configuration: ------------------ ---------------------------------------------------------------- -------------------- ---------------------------------------------------------------- ------------------- ---------------------------------------------------------------- ------------------------ ---------------------------------------------------------------- $\neg x\oplus y$ $\xymatrix @R=6pt @C=6pt { x\ar[dr]_{-1} & \ar@{-}[d]^{1} & \\ $ x\otimes \neg y$ $\xymatrix @R=6pt @C=6pt { x\ar[dr]_{1} & \ar@{-}[d]^{0} & \\ $x\oplus y$ $\xymatrix @R=6pt @C=6pt { x\ar[dr]_{1} & \ar@{-}[d]^{0} & \\ $\neg x\otimes \neg y$ $\xymatrix @R=6pt @C=6pt { x\ar[dr]_{-1} & \ar@{-}[d]^{1} & \\ & *+[o][F-]{\varphi} \ar[r]& \\ & *+[o][F-]{\varphi} \ar[r]& \\ & *+[o][F-]{\varphi} \ar[r]& \\ & *+[o][F-]{\varphi} \ar[r]& \\ y\ar[ur]^{1} & & \ y\ar[ur]^{-1} & & \ y\ar[ur]^{1} & & \ y\ar[ur]^{-1} & & \ }$ }$ }$ }$ $x\oplus \neg y$ $\xymatrix @R=6pt @C=6pt { x\ar[dr]_{1} & \ar@{-}[d]^{1} & \\ $x\otimes y$ $\xymatrix @R=6pt @C=6pt { x\ar[dr]_{1} & \ar@{-}[d]^{-1} & \\ $\neg x\otimes y$ $\xymatrix @R=6pt @C=6pt { x\ar[dr]_{-1} & \ar@{-}[d]^{0} & \\ $\neg x \oplus \neg y$ $\xymatrix @R=6pt @C=6pt { x\ar[dr]_{-1} & \ar@{-}[d]^{2} & \\ & *+[o][F-]{\varphi} \ar[r]& \\ & *+[o][F-]{\varphi} \ar[r]& \\ & *+[o][F-]{\varphi} \ar[r]& \\ & *+[o][F-]{\varphi} \ar[r]& \\ y\ar[ur]^{-1} & & \ y\ar[ur]^{1} & & \ y\ar[ur]^{1} & & \ y\ar[ur]^{-1} & & \ }$ }$ }$ }$ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Possible configurations for a neuron in a [Ł]{}NN a its interpretation.[]{data-label="semioticaLuk"} The neurons of these types of networks, which have two inputs and one output, can be interpreted as a function (see figure \[interpretation\]) and are generically denoted, in the following, by $\psi_b(w_1x_1,w_2x_2)$, where $b$ represent the node bias, $w_1$ and $w_3$ are the weights and, $x_1$ and $x_2$ input values. We simplify exposition by calling to $w_1x_1$ and $w_2x_2$ *variables* in $\psi_b$. In this context a network is the *functional interpretation* of a formula in the string-based notation when the relation, defined by network execution, corresponds to the formula truth table. $ \xymatrix @R=5pt @C=8pt { x\ar[dr]_{w_1} & \ar@{-}[d]^{b} & \\ & *+[o][F-]{\psi} \ar[r]&z\;\;\Leftrightarrow\;\; z=\min(1,\max(0,w_1x+w_2y+b)) \\ y\ar[ur]^{w_2} & & \;\;=\psi_b(w_1x,w_2y) \ } $ The use of NNs as interpretation of formulas simplifies the transformation between string-based representations and the network representation, allowing one to write: \[prop1\] Every well-formed formula in the [Ł]{}logic language can be codified using a NN, and the network defines the formula interpretation, when the activation function is the identity truncated to zero and one. For instance, the semantic for sentence $_{\varphi=(x\otimes y\Rightarrow z)\oplus(z \Rightarrow w),}$ can be described using the bellow network or can be codified by the presented set of matrices. From this matrices we must note that the local interpretation of each unit is a simple exercise of pattern checking, where we take by reference the existent relation between formulas and configuration described in table \[semioticaLuk\]. $ \xymatrix @R=6pt @C=8pt { x\ar[dr]_{1} & \ar@{-}[d]^{-1} & \\ & *+[o][F-]{\otimes} \ar[rd]^{-1}& \ar@{-}[d]^{1} \\ y\ar[ur]^{1} & *+[o][F-]{=}\ar[r]_{1} & *+[o][F-]{\Rightarrow} \ar[rd]_{1} &\ar@{-}[d]^{0}\\ z\ar[dr]_{-1} \ar[ur]^{1}& \ar@{-}[d]^{1}\ar@{-}[u]_{0} & \ar@{-}[d]_{0} &*+[o][F-]{\oplus} \ar[r]&\\ & *+[o][F-]{\Rightarrow} \ar[r]^{1}&*+[o][F-]{=} \ar[ru]^{1}\\ w\ar[ur]^{1} & & \\ } $ -------------------- -------------------------------- ------------------------- -------------------------------- $\begin{array}{cccc} $b$’s partial interpretation \;x &\; y &\; z &\; w \\ \end{array}$ $ \begin{array}{c} $\left[ $ \left[ $\begin{array}{l} i_1 \\ \begin{array}{cccc} \begin{array}{c} x \otimes y \\ i_2 \\ 1 & 1 & 0 & 0 \\ -1 \\ z \\ i_3 \\ 0 & 0 & 1 & 0 \\ 0 \\ z\Rightarrow w\\ \end{array}$ 0 & 0 & -1 & 1\\ 1 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\begin{array}{ccc} \;i_1 & \;i_2 & \;i_3 \\ \end{array}$ $ $\left[ $ \left[ $\begin{array}{l} \begin{array}{c} \begin{array}{ccc} \begin{array}{c} i_1\Rightarrow i_2 \\ j_1 \\ -1 & 1 & 0 \\ 1 \\ i_3 \\ j_2 \\ 0 & 0 & 1 \\ 0 \\ \end{array}$ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\begin{array}{cc} \;j_1 &\;j_2 \\ \end{array}$ $\left[ $\left[ $j_1\oplus j_2$ \begin{array}{cc} \begin{array}{c} 1 & 1 \\ 0 \\ \end{array} \end{array} \right] $ \right]$ -------------------- -------------------------------- ------------------------- -------------------------------- ----------------- ------------------------------------------------------------------------------------------------------ -- INTERPRETATION: $j_1\oplus j_2=(i_1\Rightarrow i_2)\oplus (i_3)=((x \otimes y)\Rightarrow z)\oplus (z\Rightarrow w)$ ----------------- ------------------------------------------------------------------------------------------------------ -- In this sense this NN can be seen as an interpretation for sentence $\varphi$; it codifies $f_\varphi$, the proposition truth table. $$_{f_\varphi(x,y,z,w)=\psi_0(\psi_0(\psi_1(-z,w)),\psi_1(\psi_0(z),-\psi_{-1}(x,y)))}$$ However, truth table $f_\varphi$ is a continuous structure, for our goal, it must be discretized and represented using a finite structure, ensuring sufficient information to describe the original formula. A truth table $f_\varphi$ for a formula $\varphi$, in a fuzzy logic, is a map $f_\varphi:[0,1]^m\rightarrow [0,1]$, where $m$ is the number of propositional variables used in $\varphi$. For each integer $n>0$, let $S_n$ be the set $\{0,\frac{1}{n},\ldots,\frac{n-1}{n},1\}$. Each $n>0$, defines a sub-table for $f_\varphi$ defined by $f_\varphi^{(n)}:(S_n)^m\rightarrow [0,1]$, given by $f_\varphi^{(n)}(\bar{v})=f_\varphi(\bar{v})$, and called the $\varphi$ *(n+1)-valued truth sub-table*. Similarity between a configuration and a formula ------------------------------------------------ We call *Castro neural network* (CNN) a type of NN having as activation function $\psi(x)=\min(1,max(0,x))$, where its weights are -1, 0 or 1 and having by bias an integer. A CNN is called *[Ł]{}ukasiewicz neural network* ([Ł]{}NN) if it can be codified as a binary NN: i.e. a CNN where each neuron has one or two inputs. A CNN is *representable* when can be codified using an equivalent [Ł]{}NN. Each neuron, with $n$ inputs, in a CNN can be described using configuration $$_{\alpha=\psi_b(x_1,x_2,\ldots,x_{n-1},x_n)}$$ and it is representable when can be describes by a [Ł]{}NN $$_{\alpha=\psi_{b_1}y_1,\psi_{b_2}(y_2,\psi_{b_3}(\ldots,\psi_{b_{n-1}}(y_{n-1},y_n))).}$$ A CNN is representable if each of its neurons is representable. Note that, a representable CNN can be translated directly into [Ł]{}ukasiewicz first-order language, using the correspondences between configurations and formulas described on table \[semioticaLuk\]. Given the configuration $_{\alpha=\psi_b(x_1,x_2,\ldots,x_n)}$, in a CNN, with $_{0\leq x_1+x_2+\ldots+x_n+b\leq 1}$, we have $_{0\leq x_1+(x_2+\ldots+x_n+b_2)+b_1\leq 1}$, where $_{b=b_1+b_2}$ for integers $b_0$ and $b_1$. And we have $$_{\psi_b(x_1,x_2,\ldots,x_n)=\psi_{b_1}(x_1,\psi_{b_2}(x_2,\ldots,x_n))}.$$ Naturally, a neuron configuration - when representable - can by codified by different [Ł]{}NN. Particularly, we have: If the neuron configuration $_{\alpha=\psi_b(x_1,x_2,\ldots,x_{n-1},x_n)}$ is representable, but not constant, it can be codified in a [Ł]{}NN with the following structure:\ $$_{\alpha=\psi_{b_1}(x_1,\psi_{b_2}(x_2,\ldots,\psi_{b_{n-1}}(x_{n-1},x_n)\ldots)),}$$ where $_{b_1,b_2,...,b_{n-1}}$ are integers, and $_{b=b_1+b_2+...+b_{n-1}}$. And, since the $n$-nary operator $\psi_b$ is commutative, variables $_{x_1,x_2,\ldots,x_{n-1},x_n)}$ could interchange its position in function $_{\alpha=\psi_b(x_1,x_2,\ldots,x_{n-1},x_n)}$ without changing the operator output. By this we mean that, for a three input configuration, when we permutate variables, we generate equivalent configurations: $$_{\psi_b(x_1,x_2,x_3)=\psi_b(x_2,x_3,x_1)=\psi_b(x_3,x_2,x_1)=\ldots}$$ When these are representable, they can be codified in string-based notation using logic connectives. But these different configuration only generate equivalent formulas if these formulas are disjunctive or conjunctive. A disjunctive formulas is formula written using the disjunction of propositional variables or negation of propositional variable. Similarly, a conjunctive formulas are formulas written using only the conjunction of propositional variables or its negation. If $_{\alpha=\psi_b(x_1,x_2,\ldots,x_{n-1},x_n)}$ is representable, it is the interpretation of a disjunctive formula or a conjunctive formula. This leave us with the task of classifying a neuron configuration according to its representation. For that, we must note what, if $$_{\alpha=\psi_b(-x_1,-x_2,\ldots,-x_n, x_{n+1},\ldots,x_m)}$$ is representable: 1. When $b=n$ is the number of negative inputs, in $\alpha$, we have $$_{\alpha=\psi_{1}(-x_1,\psi_{1}(-x_2,\ldots\psi_{1}(-x_{n}, \psi_{0}(x_{n+1},\ldots\psi_{0}(x_{m-1},x_m))\ldots)\ldots)),}$$ using Table \[semioticaLuk\], the configuration $\alpha$ is the interpretation for $$_{\neg x_1\oplus\ldots\oplus\neg x_n\oplus x_{n+1}\oplus\ldots\oplus x_m.}$$ 2. When $b=-p+1$ is the number of negative inputs, in $\alpha$, we have $$_{\alpha=\psi_{1}(-x_1,\psi_{1}(-x_2,\ldots\psi_{0}(-x_{n}, \psi_{-1}(x_{n+1},\ldots\psi_{-1}(x_{m-1},x_m))\ldots)\ldots)),}$$ an interpretation for formula $$_{\neg x_1\otimes\ldots\otimes\neg x_n\otimes x_{n+1}\otimes\ldots\otimes x_m.}$$ This establishes a relationship between the formula structure and the configuration bias, the number of negative and positive weights. \[conf classification\] Given the neuron configuration $ _{\alpha=\psi_b(-x_1,-x_2,\ldots,-x_n, x_{n+1},\ldots,x_m)} $ with $m=n+p$ inputs and where $n$ and $p$ are, respectively, the number of negative and the number of positive weights, on the neuron configuration: 1. If $b=-p+1$ the neuron is called a *conjunction* and it is an interpretation for $ _{\neg x_1\otimes\ldots\otimes\neg x_n\otimes x_{n+1}\otimes\ldots\otimes x_m.} $ 2. When $b=n$ the neuron is called a *disjunction* and it is an interpretation of $ _{\neg x_1\oplus\ldots\oplus\neg x_n\oplus x_{n+1}\oplus\ldots\oplus x_m.} $ Imposing some structural order on the neural network transformation: Every conjunctive or disjunctive configuration\ $$_{\alpha=\psi_b(x_1,x_2,\ldots,x_{n-1},x_n),}$$ can be codified by a [Ł]{}NN\ $$_{\beta=\psi_{b_1}(x_1,\psi_{b_2}(x_2,\ldots,\psi_{b_{n-1}}(x_{n-1},x_n)\ldots)),}$$ where $$_{b_1,b_2,...,b_{n-1}\text{ are integers, }b=b_1+b_2+\cdots+b_{n-1}\text{ and }b_1\leq b_2\leq \cdots\leq b_{n-1}.}$$ This property can be translated in the following rewriting rule, $$\tiny \xymatrix @R=6pt @C=10pt { \ar[rd]_{w_1} & \ar@{-}[d]^{b} & & \\ \vdots & *+[o][F-]{\psi} \ar[r] & \ar[r]^{R} &\\ \ar[ru]^{w_n} & &\\ } \xymatrix @R=7pt @C=8pt { \ar[rd]_{w_1} & \ar@{-}[d]^{b_0} & & \\ \vdots & *+[o][F-]{\psi} \ar[rd]^{1} & \ar@{-}[d]^{b_1}\\ \ar[ru]^{w_{n-1}} & &*+[o][F-]{\psi}\ar[r]&\\ \ar[rru]^{w_{n}} & &\\ }$$ linking equivalent networks, when the integers $b_0$ and $b_1$ satisfy $b=b_0+b_1$ and $b_1\leq b_0$, and are such that neither of the involved neurons have constant output. Note that, a representable CNN can be transformed by the application of rule R in a set of equivalent [Ł]{}NN with simplest neuron configuration: Un-representable neuron configurations are those transformed by rule R in, at least, two non-equivalent NNs. For instance, the un-representable configuration $\psi_0(-x_1,x_2,x_3)$, is transformed by rule R in three non-equivalent configurations: ------------------------------------------------------------------- ----------------------------------------------------------------------- $\psi_0(x_3,\psi_0(-x_1,x_2))=f_{x_3\oplus(\neg x_1\otimes x_2)}$ $\psi_{-1}(x_3,\psi_{1}(-x,x_2))=f_{x_3\otimes(\neg x_1\otimes x_2)}$ $\psi_0(-x_1,\psi_0(x_2,x_3))=f_{\neg x_1\otimes(x_2\oplus x_3)}$ ------------------------------------------------------------------- ----------------------------------------------------------------------- The representable configuration $\psi_2(-x_1,-x_2,x_3)$ is transformed by rule R on only two distinct but equivalent configurations: --------------------------------------------------------------------- ------------------------------------------------------------------------- $\psi_0(x_3,\psi_2(-x_1,-x_2))=f_{x_3\oplus \neg (x_1\otimes x_2)}$ $\psi_1(-x_2,\psi_1(-x_1,x_3))=f_{\neg x_2\oplus (\neg x_1\oplus x_3)}$ --------------------------------------------------------------------- ------------------------------------------------------------------------- For the extraction of knowledge from trained NNs, we translate neuron configuration in propositional connectives to form formulas. However, not all neuron configurations can be translated in formulas, but they can be approximate by one. To quantify the approximation quality we used the exponential-similarity. Two neuron configurations $\alpha=\psi_{b}(x_1,x_2,\ldots,x_n)$ and $\beta=\psi_{b'}(y_1,y_2,\ldots,y_n)$, are called $\lambda$-similar, in a $(m+1)$-valued [Ł]{}logic, if ${\lambda=e^{-\frac{1}{|O|}\sum_{\bar{x}\in T}\neg(\alpha(\bar{x})\Leftrightarrow\beta(\bar{x}))}}$, we write $ \alpha\sim_\lambda\beta$. If $\alpha$ is un-representable and $\beta$ is representable, the second configuration is called *a representable approximation* to the first. On the $2$-valued [Ł]{}logic (the Boolean logic case), we have for the un-representable configuration $\alpha=\psi_0(-x_1,x_2,x_3)$: ---------------------------------------------------------------- --------------------------------------------------------------------- $\psi_0(-x_1,x_2,x_3)\sim_{0.883}\psi_0(x_3,\psi_0(-x_1,x_2))$ $\psi_0(-x_1,x_2,x_3)\sim_{0.883}\psi_{-1}(x_3,\psi_{1}(-x_1,x_2))$ $\psi_0(-x_1,x_2,x_3)\sim_{0.883}\psi_0(-x_1,\psi_0(x_2,x_3))$ ---------------------------------------------------------------- --------------------------------------------------------------------- In this case, the truth sub-tables of, formulas $\alpha_1=x_3\oplus(\neg x_1\otimes x_2)$, $\alpha_1=x_3\otimes(\neg x_1\otimes x_2)$ and $\alpha_1=\neg x_1\otimes(x_2\oplus x_3)$ are both $\lambda$-similar to $\psi_0(-x_1,x_2,x_3)$, where $\lambda=0.883$, since they differ in one position on 8 possible positions. This means that both formulas are 92% accurate. For an un-representable configuration, $\alpha$, we can generate the finite set $S(\alpha)$, of representable networks similar to $\alpha$, using rule R. Given a $(n+1)$-valued logic, from that set of formulas we select to approximate $\alpha$ the formula having the interpretation more similar to $\alpha$. This identification of un-representable configuration, using representable approximations, is used to transform networks with un-representable neurons into representable structures. The stress associated with this transformation characterizes the translation quality. Bellow we present an example of a un-representable CNN: $$\tiny \begin{tabular}{lll} $\left[ \begin{array}{ccccccc} -1 & 1 & -1 & 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & -1 \\ 1 & 1 & 0 & 0 & 0 & 0 & -1 \\ \end{array} \right] $ &$ \left[ \begin{array}{c} 0 \\ 1 \\ 0 \\ \end{array} \right] $& $\begin{array}{l} i_1\text{ un-representable} \\ A4\oplus A5\oplus \neg A7 \\ i_3\text{ un-representable} \\ \end{array}$\\ $\left[ \begin{array}{ccc} 1 & -1 & 1 \\ \end{array} \right] $&$ \left[ \begin{array}{c} 0 \\ \end{array} \right] $& $\begin{array}{l} j_1 \text{un-representable} \\ \end{array}$\\ \end{tabular}$$ For each local un-representable configuration $\alpha$, we selected the most similar representable configuration on $S(\alpha)$, after applying rule R, we have in this case: 1. $i_1\sim_{0.9387}((\neg A1\otimes A4) \oplus A2)\otimes \neg A3 \otimes \neg A6$, 2. $i_3\sim_{0.8781}(A1\oplus\neg A7)\otimes A2$, and 3. $j_1\sim_{0.8781}(i_1\otimes\neg i_2)\oplus i_3$. Using this substitutions we reconstructed the formula: $\alpha=(((((\neg A1\otimes A4) \oplus A2)\otimes \neg A3 \otimes \neg A6)\otimes\neg (A4\oplus A5\oplus \neg A7))\oplus ((A1\oplus\neg A7)\otimes A2)$, $\lambda$-similar to the original CNN, with $\lambda=0.7323$, in a $5$-valued [Ł]{}logic. Crystallizing trained neural networks ------------------------------------- Standard error back-propagation algorithm (EBP) is a gradient descent algorithm, in which the network weights are moved along the negative of the gradient of the performance function. EBP algorithm has been a significant improvement in NN research, but it has a weak convergence rate. Many efforts have been made to speed up the EBP algorithm. The Levenberg-Marquardt (LM) algorithm [@HaganMenhaj99] [@Andersen95] ensued from the development of EBP algorithm-dependent methods. It gives a good exchange between the speed of the Newton algorithm and the stability of the steepest descent method [@Battiti92]. The basic EBP algorithm adjusts the weights in the steepest descent direction. When training with the EBP method, an iteration of the algorithm defines the change of weights and has the form $ _{w_{k+1}=w_k-\alpha G_k,} $ where $G_k$ is the gradient of performance index $F$ on $w_k$, and $\alpha$ is the learning rate. Note that, the basic step of Newton’s method can be derived from Taylor formula and is $ _{w_{k+1}=w_k-H_k^{-1}G_k,} $ where $H_k$ is the Hessian matrix of the performance index at the current values of the weights. Since Newton’s method implicitly uses quadratic assumptions, the Hessian matrix dos not need be evaluated exactly. Rather, an approximation can be used, such as $ _{H_k\approx J_k^TJ_k,} $ where $J_k$ is the Jacobian matrix that contains first derivatives of the network errors with respect to the weights $w_k$. The simple gradient descent and Newtonian iteration are complementary in the advantages they provide. Levenberg proposed an algorithm based on this observation, whose update rule blends aforementioned algorithms and is given as $$_{w_{k+1}=w_k-[J_k^TJ_k+\mu I]^{-1}J_k^Te_k},$$ where $e_k$ is a vector of current network errors and $\mu$ is the learning rate. This update rule is used as follows. If the error goes down following an update, it implies that our quadratic assumption on the function is working and we reduce $\mu$ (usually by a factor of 10) to reduce the influence of gradient descent. In this way, the performance function is always reduced at each iteration of the algorithm [@Megan96]. On the other hand, if the error goes up, we would like to follow the gradient more and so $\mu$ is increased by the same factor. We can obtain some advantage out of the second derivative, by scaling each component of the gradient according to the curvature. This should result in larger movements along the direction where the gradient is smaller so the classic “error valley” problem does not occur any more. This crucial insight was provided by Marquardt. He replaced the identity matrix in the Levenberg update rule with the diagonal of Hessian matrix approximation resulting in the LM update rule. In each LM iteration we restricted the NN representation bias, making its structure similar to a CNN. For that, we used a *smooth crystallization* procedure resulting from function, $$_{\Upsilon_n(w)=sign(w).((\cos(1-abs(w)-\lfloor abs(w)\rfloor).\frac{\pi}{2})^n+\lfloor abs(w)\rfloor),}$$ iteration, where $sign(w)$ is the sign of $w$ and $abs(w)$ its absolute value. Denoting by $\Upsilon_n(N)$ the function having by input and output a NN, where the weights on the output network results of applying $\Upsilon$ to all the input network weights and neurons biases. Each interactive application of $\Upsilon$ produce a networks progressively more similar to a CNNs. For show that, we define by *representation error*, for a network $N$, with weights $w_1,\ldots,w_n$, $$\Delta(N)=\sum^n_{i=1}(w_i-\lfloor w_i\rfloor).$$ When $N$ is a CNNs we have $\Delta(N)=0$. Since, for every network $N$ and $n>0$, $\Delta(N)\geq \Delta(\Upsilon_n(N))$, we have Given a neural networks $N$ with weights in the interval $[0,1]$. For every $n>0$ the function $\Upsilon_n(N)$ have by fixed points [Ł]{}ukasiewicz neural networks $N'$. We changed the LM algorithm by applying a soft crystallization step after the LM update rule: $$_{w_{k+1}=\Upsilon_2(w_k-[J_k^TJ_k+\mu .diag(J_k^TJ_k)]^{-1}J_k^Te_k})$$ This can be seen as a process to regularize the network and improves drastically the convergence to a CNN preserving its ability to learn. On our method network regularization is made using three different strategies: 1. using the described soft crystallization process, where we restricted the knowledge dissemination on the network structure, information is concentrated on some weights; 2. after the training we use crisp crystallization, where links between neurons with weights near 0 are removed and weights near -1 or 1 are consolidated; 3. the resulting crisp network is pruned using ”*Optimal Brain Surgeon*” method. The first regularization technic avoids knowledge dissemination on the NN. The last regularization technic avoid redundancies, in the sense that the same or redundant information can be codified at different locations. We minimized this by selecting weights to eliminate. The ”*Optimal Brain Surgeon*” method uses the criterion of minimal increase in training error. It uses information from all second-order derivatives of the error function to perform network pruning. The *Optimal Brain Surgeon* method is derived from Taylor series of the error with respect to weights, $$\Delta E=J_w^T.\Delta w + \frac{1}{2}\Delta w^TH_w\Delta w+O(\|\Delta w\|^3).$$ For a network trained to a local minimum in error, $\Delta w\approx0$, the first linear term vanishes, third and all higher order terms are ignored. The method finds a weight to be set to zero (which we call $w_q$) to minimize $\Delta E$ the increase in error. Making $w_q$ zero correspond to change its value using $\Delta w_q$ making $\Delta w_q+w_q=0$ or more generally: $$e^T_q\Delta w+w_q=0,$$ where $e_q$ is the unit vector in weight space corresponding to (scalar) weight $w_q$. This reduces our goal to solve: $$\min_q\{\min_{\Delta w}\frac{1}{2}\Delta w^TH_w\Delta w\;|\;e^T_q\Delta w_q+w_q=0\}$$ The optimization problem is constrained, following [@Hassibi93], we form the Lagrangian operator $$L=\frac{1}{2}\Delta w^TH_w\Delta w+\lambda(e^T_q\Delta w_q+w_q),$$ where $\lambda$ is a Lagrange undetermined multiplier. After take functional derivatives, employ the constraint $e^T_q\Delta w+w_q=0$, and use matrix inverse to find that the optimal weight change and resulting change in error are respectively $$\Delta w= - \frac{w_q}{[H_w^{-1}]_{qq}}H_w^{-1}e_q\text{ and } L_q=\frac{1}{2}\frac{w^2_q}{[H_w^{-1}]_{qq}}.$$ The method recalculates the magnitude of all the weights in the NN. Hassibi, Stork and Stork called $L_q$ the “saliency” of weight $q$, the increase in error that results when the weight is eliminated. Algorithm \[RevEng\] describes our methodology for training CNN and extraction of symbolic pattern descriptions. Given a concept description evaluated on a (n+1)-valued [Ł]{}ukasiewicz logic Define an initial network complexity Generate an initial neural network Apply the LM algorithm with soft crystallization If need increase network complexity Try a new network. Go to 3 Crisp crystallization on the trained NN. Try a new network. Go to 3 Refine the crystallized NN using ”*Optimal Brain Surgeon*” algorithm Identify un-representable configurations Replace each un-representable configurations, using a similar representable configuration, selected from the set of configurations generated using rule $R$. Evaluated the procured NN performance on the original concept description. Translated the procured NN on string-based notation. Given a view for a concept description, we try to find a CNN describing information. For that our implementation generates neural networks with a fixed number of hidden layers (in our implementation we used three). When a network have bad learning performance, training is stopped, and train is initiate for a new network, with random heights. After a fixed number of tries the network topology is changed. This number iterations depends on the number of network inputs. After trying configure a set of networks with a given complexity and bad learning performance, the system tries to apply LM algorithm, with soft crystallization, for a more complex set of networks. The process stops when a adequate description for data is find. And after the network be pruned, un-representable configurations are approximated using representable ones. This defines a description for the information using a [Ł]{}NN. We tested our methodology in real data sets [@Leandro09] and on artificially generated data structures. On this section we present our approach using artificially generated data sets. For that, we developed a simple way to generate complex fuzzy structures. Our method is based on a non-deterministic state machine. In this machine every states have associated a level of uncertainty quantified by a truth value, from a many-valued logic $\Omega$. These type of machines have its stats changed based on the reading a word and this change is described by a relation defining stat transition, from the actual state and the signs that are being read. This relation, for stat change, is a multi-morphism evaluated in $\Omega$ and the actual signs being read are described by a $\Omega$-set. We called to these type of state machines a $\Omega$-*automata*. In this sections we describe an automata as a concept to be learned, and show how reverse engineering its structure using the data generated from its execution. We do this translating a CNN structure in a formula, representable on a specification system. Beginning with a structural description of a problem, in the from of a specification system, and a model, we enrich the specification structure using knowledge extracted from data generated querying the model and crystallized on neuronal networks. [Ł]{}ukasiewicz automatas ------------------------- A $\Omega$-automata is an non-deterministic state machine, where state transition relation. This change is made after reading a sign from a word. Each symbol used to define automata input string is a sign having the form $\alpha=_{\lambda}x$, where $\alpha\in Att$ is an attribute, $x$ a possible value for $\alpha$, $x\in A_\alpha$, and $\lambda$ quantifies the truth value of $\alpha=x$ evaluated in $\Omega$. The input sting at position $i$ defines the truth of $\alpha=_{\lambda}x$ for each $\alpha\in Att$. The automata is defined by a set of states $E$, and at each moment this states have associated a level of uncertainty in $\Omega$ of being the automata state. In this sense an $\Omega$-automata actual state is describes by a relation evaluated in $\Omega$, $e_i:E\rightarrow \Omega$, having by support the set of automata possible states. The states of an automata are classified in three classes: input states, auxiliary states and output states. The input states have its uncertainty directly assigned by the uncertainty on the input signs. Each of this state is associated directly with one sign $\alpha= x$, used on the input string, and its uncertainty is the truth value of equality $\alpha= x$ on the actual reading position. After all input string have been read, the level of uncertainty on output states defines the $\Omega$-automata output. An $\Omega$-automata begins its activity in a initial state $e_0:E\rightarrow \Omega$, reads a string $s_1s_2\ldots s_n$, in each iteration a position $s_i$ is read and on the $n$-iteration reach its final state $e_n:E\rightarrow \Omega$. The final state is used to describe the automata output $o:E_O\rightarrow \Omega$. Formally An $\Omega$-automata $A$ is a structure described by $$(\text{\L},(A_i)_{i\in Att},E,E_O,\{M_\lambda,M_{\neg\lambda}\}_{\lambda\in\Sigma},e_0)$$ where: 1. [Ł]{} is a finite multi-valued logic having truth values in $\Omega$; 2. $(A_\alpha)_{\alpha\in Att}$ a family $\Omega$-sets used as domain for attributes in construction of signs $'\alpha=x'$; 3. $E$ is the set of states, where each input sign $'\alpha=x'$ have a state associated. The states, with a sign associated, are called *input states*; 4. $E_O$ is a subset of $E$, called set *output states*; 5. two boolean matrices $M_0$ and $M_1$, describing state transition. $M_0$ describes negative uncertainty propagation and $M_1$ positive uncertainty propagation; 6. $e_0:E\rightarrow \Omega$ is a relation describing the initial automata state; 7. if the automata state on iteration $k$ is $e_k:E\rightarrow \Omega$ and let $X_k:\coprod_{\alpha \in Att} A_\alpha \rightarrow \Omega$ describe the input sign on position $k$, the new automata state is given by $$e_{k+1}=M_1(e_k)\oplus M_0(\neg e_k),$$ where in $e_k:E$ we update, the input states, with the sign uncertainty on the reading position, described by vector $X_k$. A $\Omega$-automata is called a *[Ł]{}ukasiewicz automata* when the system is governed by a finite [Ł]{}ukasiewicz logic. We named *relational automata* to an extension to the $\Omega$-automata defined using transition matrices $M_0$ and $M_1$ with values in $\Omega$. This type of fuzzy automata have its behaviour described using formulas of Relational [Ł]{}ukasiewicz logic, introduced in [@Gerla01], outfitting the scope of this work. In this sense we interpret a word as a string of begs, where the possibility of a sign is in the beg is described by a relation $X_k:\coprod_{\alpha \in Att} A_\alpha \rightarrow \Omega$. In each iteration the $\Omega$-automata reads a position on the string. If in the iteration $k$ the position $k$ is read, on the position $k+1$ it reads the string position $k+1$. Each position in the string can have more than a sign or can be empty. On iteration $k$ the automata reads the position $k$, updating the uncertainty on each input states, using the sign uncertainties on the reading positions. This uncertainty is propagating to other states, applying the state transformation matrices. The update and the propagation of state uncertainty is done for each iteration. The input string length determines the number of automata iteration. In this context a word $w$ can be define as a sequence of relations$$\tiny s_1, s_2,\ldots s_i\ldots,s_n$$ having the type $s_i:\coprod_{\alpha\in Att}A_\alpha\rightarrow \Omega$. Using $w$ as input to the automata, it generates a sequence of states described using a sequence of relation $$\tiny e_1,e_2,\ldots e_i\ldots,e_{n},$$ having the type $e_i:E\rightarrow \Omega$, describing the change on automata state between iterations. The $\Omega$-set $e_1$ is defined using the automata initial state $e_0$, where the input states were update with the reading position $s_0$. The state $e_i$, when $i>1$, depends on state $e_{i-1}$ updated with the uncertainty on input sign, described by the reading position $s_i$. The automata output is defined by the uncertainty in each output state $E_O$ in the automata state $e_{n}$. The following example illustrates an automata execution. \[exemautbin\] The string of a binary [Ł]{}ukasiewicz automata is defined using only an attribute, having two possible values. Let this attribute be $a$ and its possible values $0$ or $1$. Words interpreted using this automata are described using a sequence of $\Omega$-sets having by support the set of signs $\{'a=0','a=1'\}$. An example, of this type of words is presented on table \[word\], where each column codifies the existence of each sign in that position. --------- --- --- ----- --- ----- ----- --- --- ----- ----- --- --- ’$a$=1’ 1 1 1/2 0 1/4 1/2 1 1 1/2 1/4 0 0 ’$a$=0’ 0 0 1/2 1 3/4 1/2 0 0 1/2 3/4 0 1 --------- --- --- ----- --- ----- ----- --- --- ----- ----- --- --- : Word defined using signs ’$a$=1’ and ’$a$=0’.[]{data-label="word"} For this example, we fixed the finite [Ł]{}ukasiewicz logic having by truth values $_{\Omega=\{0,1/4,1/2,3/4,1\}}$. This table can be interpreted by saying what: first and second marks in the word are a ’1’, position 4 and 12 have a ’0’, and in position 11 we not know the symbol used, on position 6 we can not distinguished between a ’0’ or a ’1’. For simplify the graphic presentation, we labelled each input state by its associates sign and to each not input state we labelled it with a number. Lets $$_{\tiny\langle I(a=1), I(a=0), 1, 2, 3, 4, 5, 6\rangle}$$ be the list of states, where $I(a=1)$ and $I(a=0)$ represent the input states, its uncertainty is indexed to the uncertainty on the reading of sign $'a=1'$ and $'a=0'$, and let $4,5,$ and $6$ be the automata output states. Let the state change boolean relations described using the graph presented on figure \[transition\], where the arrows labelled with 0 represents the propagation of negative uncertainty and the arrows labelled with 1 represents the propagation of positive uncertainty. $$\tiny \xymatrix@=12pt{ I_1(a=1)\ar[r]_1 & 2 \ar[rr]_1& & 4 \ar[dl]_0 \ar[dr]_1& \\ & & 3 \ar[dr]_0 \ar[ul]_1& & 6 \ar[dl]_0 \\ I_2(a=0)\ar[r]_1 & 1 \ar[ur]_0& & 5 & }$$ This graph can be codified using two graph adjacency matrices: $M_0$, for sub-graph having arrows labelled with 0, describing the state change when a state isn’t active, and $M_1$ the adjacency for sub-graphs having arrows labelled with 1, describing behaviour when a state is active. $$\tiny M_0= \left[ \begin{array}{cccccccc} 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &0& 1 & 0 & 0 & 1 & 0 & 0 \\ 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &0& 0 & 0 & 1 & 0 & 0 & 1 \\ 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right] \;\;\; M_1= \left[ \begin{array}{cccccccc} 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &1& 0 & 0 & 0 & 0 & 0 & 0 \\ 1 &0& 0 & 0 & 1 & 0 & 0 & 0 \\ 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &0& 0 & 1 & 0 & 0 & 0 & 0 \\ 0 &0& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &0& 0 & 0 & 0 & 1 & 0 & 0 \\ \end{array} \right]$$ Executing this automata, for word described on table \[word\], and taking the automata initial state uncertain, i.e. making the initial state $_{ e_1=[1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2]'}$. After reading the first position, we update $e_1$ using the uncertain for each signs on first position, $_{ e_1=[1,0,1/2,1/2,1/2,1/2,1/2,1/2]'}$, from it we compute $e_2$ by: $$_{e_2= M_0\otimes(\neg e_1)\oplus M_1\otimes e_1= \left[0, 0, 0, 1, 1, 1/2, 1, 1/2 \right]'.}$$ Updating the resulting state, since the second mark is ’a=1’, we have $$_{e_2=[1,0,0,1,1,1/2,1,1/2]'}.$$ Using the same procedure $_{e_3=[1/2,1/2,0,0,1,1,1,1/2,1/2]'}$, since we not know if in position 3 we have ’a=1’ or ’a=0’. Repeating the process for other word positions we have: $$\tiny \begin{array}{ll} e_4=[0,1,1/2,1,1,1,1/2,1]' & e_8=[1,0,0,1,3/4,1/2,3/4,3/4]' \\ e_5=[1/4,3/4,1,1,1/2,1,0,1]' & e_9=[1/2,1/2,0,1,1,1,1/2,1/2]' \\ e_6=[1/2,1/2,3/4,3/4,0,1,1/2,1]' & e_{10}=[1/4,3/4,1/2,1,1,1,1/2,1]' \\ e_7=[1,0,1/2,1/2,1/4,3/4,1,1]' & e_{11}=[0,0,2/3,1,1/2,1,0,1]' \\ & e_{12}=[0,1,1,1/2,1/4,1,1/2,1]' \end{array}$$ The automata final state is defined using, $_{e_{13}=[0,0,1,1/4,0,1/2,3/4,1]'}$, and it is $_{A(w)=[1/2,3/4,1]'}$ since these are the uncertainties in $e_{13}$ for output states $\{4,5,6\}$. We will interpret $_{A(w)=[1/2,3/4,1]'}$ as the output for automata $A$ when it is executed over the fuzzy string $w$. Since in this type of automata transitions are defined using only boolean relations, they can be described using CNNs. Each node depends on a positive or on a negative way from its neighbours, and the relation can be expressed through a sentence in disjunctive form. We can see this translation from a local configuration to an admissible configuration on figure \[actconf\]. Note what, neuron bias is defined by the number of arrows labelled with zero. In string-base notation it could be write as $ _{c_i=c_1\oplus c_2\oplus c_3\oplus \neg c_4\oplus \neg c_5,} $ which can also be seen as a colimit sentence. $$\tiny \begin{array}{ccc} \xymatrix @=15pt { c_1\ar[rrd]^1 & c_2\ar[dr]^1 &c_3\ar[d]^1& c_4\ar[dl]_0 & c_5\ar[lld]^0 \\ &&*+[o][F-]{c_i}& \\ } & \xymatrix @=10pt { & & \\ & & \\ \ar[rr] & & \\ & & \\ & & } & \xymatrix @=15pt { c_1\ar[rrd]^1 & c_2\ar[rd]^1 & c_3\ar[d]^1 & c_4\ar[ld]_{-1} & c_5\ar[lld]^{-1} \\ \ar@{-}[rr]^2 & &*+[o][F-]{c_i}& \\ } \end{array}$$ Reverse engineering [Ł]{}ukasiewicz automatas --------------------------------------------- The extraction of knowledge from data generated by automata execution offers a rich framework to present a simple knowledge integration methodology on [Ł]{}logic. For this we used automata to generate complex artificial data sets. $$\tiny \begin{array}{cc} \xymatrix @=10pt {I(a=1)\ar[r]_1 & A_3 \ar[rr]_0& & A_5 & \\ & & A_4 \ar[ur]_1 \ar[ul]_0& & A_7 \ar[ul]_0\\ I(a=0)\ar[r]_1 & A_2 \ar[ur]_1\ar[rr]_0& & A_6\ar[ur]_1\ar[ul]_0 & } & \xymatrix @=10pt { I(a=1)\ar[r]_1 & A_3 \ar[rr]_0 & & A_5\ar[dd]_1 & \\ & & A_4 \ar[ur]_1\ar[ul]_0 & & A_7 \ar[ul]_0\\ I(a=0)\ar[r]_1 & A_2 \ar[ur]_1\ar[rr]& & A_6\ar[ur]_1 \ar[ll]_0 \ar[lu]_0& } \end{array}$$ We selected two binary automata described graphically on figure \[exmp2\], and constructed data sets using information generated through automata execution. This was made selecting possible input configuration with length 6, in a 5-valued [Ł]{}logic. Where we impose a consistence principle in the automata reading sensor: the uncertainty for sign ’$s=0$’, is the negation of sign ’$s=1$’ uncertainty. For both automata we generated a data set describing the dependence between states, relating state uncertainty on iteration $t$ with the state in iteration $t+1$. With this data set we reverse engineering the automata structure, using Algorithm \[RevEng\] presented in last section. Trying predict the uncertainty in each state on iteration $t+1$ using the information, about the automata state uncertainty, in iteration $t$ and the input uncertainty. For each prediction task we selected a rule describing the relationships between states. Figure \[neuroConf\] describes the reverse engineering output for each state, generated using the data produced by the acyclic automata from figure \[exmp2\]. [ccc]{} ----------------------------------------- ------------------------- --------------------------- $\left[ $ \left[ $\begin{array}{l} \begin{array}{cccccccc} \begin{array}{c} \neg I(a_t=1) \\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $ \left[ $\begin{array}{l} \begin{array}{c} \begin{array}{c} i_1 \\ 1 \\ 0 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $\left[ $j_1$ \begin{array}{cc} \begin{array}{c} 1 \\ 0 \\ \end{array} \end{array} \right] $ \right]$ ----------------------------------------- ------------------------- --------------------------- & ----------------------------------------- ------------------------- -------------------------------------- $\left[ $ \left[ $\begin{array}{l} \begin{array}{cccccccc} \begin{array}{c} \neg I(a_t=1) \otimes A_4\\ -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $ \left[ $\begin{array}{l} \begin{array}{c} \begin{array}{c} \neg i_1 \\ -1 \\ 1 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $\left[ $j_1$ \begin{array}{cc} \begin{array}{c} 1 \\ 0 \\ \end{array} \end{array} \right] $ \right]$ ----------------------------------------- ------------------------- -------------------------------------- \ $_{A_2(t+1)= I(a_t=1)}$&$_{A_3(t+1)=\neg(\neg I(a_t=1)\oplus A_4(t))}$\ \ ------------------------------------------ ------------------------- -------------------------------- $\left[ $ \left[ $\begin{array}{l} \begin{array}{cccccccc} \begin{array}{c} \neg A_2 \oplus A_6\\ 0 & 0 & -1 & 0 & 0 & 0 & 1 & 0\\ 0 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $ \left[ $\begin{array}{l} \begin{array}{c} \begin{array}{c} \neg i_1 \\ -1\\ 1 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $\left[ $j_1$ \begin{array}{cc} \begin{array}{c} 1 \\ 0 \\ \end{array} \end{array} \right] $ \right]$ ------------------------------------------ ------------------------- -------------------------------- & ------------------------------------------ ------------------------- ----------------------------------------------------------- $\left[ $ \left[ $\begin{array}{l} \begin{array}{cccccccc} \begin{array}{c} \neg A_3\oplus A_4\oplus A_6 \oplus \neg A_7 \\ 0 & 0 & 0 & -1 & 1 & 0 & 1 & -1\\ 2 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $ \left[ $\begin{array}{l} \begin{array}{c} \begin{array}{c} i_1 \\ 1 \\ 0 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $\left[ $j_1$ \begin{array}{cc} \begin{array}{c} 1 \\ 0 \\ \end{array} \end{array} \right] $ \right]$ ------------------------------------------ ------------------------- ----------------------------------------------------------- \ $_{A_4(t+1)=\neg (\neg A_2(t) \oplus A_6(t))}$&$_{A_5(t+1)\sim_{0.97}\neg A_3(t)\oplus A_4(t)\oplus A_6(t) \oplus \neg A7}$\ \ ---------------------------------------- ------------------------- ---------------------- $\left[ $ \left[ $\begin{array}{l} \begin{array}{cccccccc} \begin{array}{c} \neg A_2 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 1 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $ \left[ $\begin{array}{l} \begin{array}{c} \begin{array}{c} i_1 \\ 1 \\ 0 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $\left[ $j_1$ \begin{array}{cc} \begin{array}{c} 1 \\ 0 \\ \end{array} \end{array} \right] $ \right]$ ---------------------------------------- ------------------------- ---------------------- & ----------------------------------------- ------------------------- ---------------------- $\left[ $ \left[ $\begin{array}{l} \begin{array}{cccccccc} \begin{array}{c} \neg A_6 \\ 0 & 0 & 0 & 0 & 0 & 0& -1 & 0 \\ 1 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $ \left[ $\begin{array}{l} \begin{array}{c} \begin{array}{c} \neg i_1 \\ -1 \\ 1 \\ \end{array}$ \end{array} \end{array} \right] \right] $ $ $\left[ $\left[ $j_1$ \begin{array}{cc} \begin{array}{c} 1 \\ 0 \\ \end{array} \end{array} \right] $ \right]$ ----------------------------------------- ------------------------- ---------------------- \ $_{A_6(t+1)=\neg A_2(t)}$&$_{A_7(t+1)=A_6(t)}$\ Each configuration can be expressed or approximate using a string-based presentation. Each of this formulas is interpreted as knowledge extracted from different views of the data set. The set of all extracted formulas (equalities) we call a *theory*, and can be codified as an specification system. For acyclic automata, on figure \[exmp2\], and translating configurations presented in figure \[neuroConf\], we have the theory: $$_{T_0=\{A_2(t+1)=I(a_t=1),\;A_3(t+1)=\neg(\neg I(a_t=1)\oplus A_4(t)),\;A_4(t+1)=\neg (\neg A_2(t) \oplus A_6(t)), }$$ $$_{A_5(t+1)\sim_{0.97}\neg A_3(t)\oplus A_4(t)\oplus A_6(t) \oplus \neg A_7(t),\; A_6(t+1)=\neg A_2(t),\;A_7(t+1)=A_6(t)\}}$$ This symbolic description allows forecast the automata behaviour. In this case theory $T_0$ isn’t a prefect automata description, since we only have a approximation to automata beaver on state $A_5$. The consistence level of a equational theory $T$, with a model $D$, is describes by the disjunction of each similarity level associated to each equality in $T$. In the case of equational theory $T_0$, we have a consistence level of $0.97$. Given a generic word $w=s_0s_1s_2s_3s_4$ in the interaction $t=5$, the uncertainty on state $A_4$ depends on signs from positions $s_2$ and $s_3$. $ \begin{array}{rcl} A_4(5) & = & A_2(4)\otimes \neg A_6(4) \\ & = & I(s_3=1)\otimes \neg A_2(3) \\ & = & I(s_3=1)\otimes I(s_2=1) \\ \end{array} $ In interaction $t=6$, the uncertainty on state $A_5$, is given by: $ \begin{array}{rcl} A_5(6) & \sim_{0.97} & \neg A_3(5)\otimes A_4(5)\otimes A_6(5) \otimes \neg A_7(5)\\ & \sim & \neg I(s_2=1)\oplus ( I(s_3=1)\otimes I(s_2=1))\oplus \neg A_3(4) \oplus \neg A_6(4) \\ & \sim & \neg I(s_2=1)\oplus I(s_3=1)\oplus \neg I(s_3=1) \oplus A_4(3) \oplus A_3(3)\\ & \sim & 1\\ \end{array} $ From a functional point of view, knowledge integrated in theory $T_0$, are interpreted using the first [Ł]{}ukasiewicz neural network on figure \[knowledgeInteg\]. This NN configuration results from integrating NNs presented on figure \[neuroConf\]. The resulting NN have 6 outputs, one for each non input state. $$\tiny \begin{tabular}{ccc} \begin{tabular}{lll} $\left[ \begin{array}{ccccccc} -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 1 & 0 \\ 0 & 0 & -1 & 1 & 0 & 1 & -1 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ \end{array} \right] $ &$ \left[ \begin{array}{c} 1 \\ 0 \\ 0 \\ 2 \\ 1 \\ 1 \\ \end{array} \right] $& $\begin{array}{l} I(a=1) \\ \neg I(a=1) \\ \neg A_2 \oplus A_6 \\ \neg A_3 \oplus A_4\oplus A_6\oplus \neg A_7 \\ \neg A_2 \\ \neg A_7 \\ \end{array}$\\ $\left[ \begin{array}{ccccccc} 1 & 0& 0& 0& 0& 0 \\ 0& -1& 0& 0& 0& 0 \\ 0& 0& -1& 0& 0& 0 \\ 0& 0& 0& 1& 0& 0 \\ 0& 0& 0& 0& 1& 0 \\ 0& 0& 0& 0& 0& -1 \\ \end{array} \right] $&$ \left[ \begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ \end{array} \right] $& $\begin{array}{l} i_1 \\ \neg i_2 \\ \neg i_3 \\ i_4 \\ i_5 \\ \neg i_6 \\ \end{array}$\\ \end{tabular} & \;\;\; & \begin{tabular}{lll} $\left[ \begin{array}{cccccccc} 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0\\ 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1 & 1 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \end{array} \right] $ &$ \left[ \begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \\ 2 \\ 0 \\ \end{array} \right] $& $\begin{array}{l} I(a=0)\oplus \neg A_6 \\ I(a=1)\oplus \neg A_4 \\ \neg I(a=1) \oplus A_4 \\ \neg A_2\oplus A_6 \\ \neg A_3 \oplus A_4 \oplus \neg A_7\\ A_6 \\ \end{array}$\\ $\left[ \begin{array}{ccccccc} 1 & 0& 0& 0& 0& 0 \\ 0& 1& 0& 0& 0& 0 \\ 0& 0& 1& 0& 0& 0 \\ 0& 0& 0& -1& 0& 0 \\ 0& 0& 0& 0& 1& 0 \\ 0& 0& 0& 0& 0& 1 \\ \end{array} \right] $&$ \left[ \begin{array}{c} 0 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ \end{array} \right] $& $\begin{array}{l} i_1 \\ i_2 \\ \neg i_3 \\ i_4 \\ i_5 \\ \end{array}$ \end{tabular} \end{tabular}$$ Integrating configuration representing knowledge in an NN usually introduce redundancies. In the sense that the same or similar information is codified by different configurations in different locations. This can be minimized by regularizing the produced NN using “Optimal Brain Surgeon” method [@Hassibi93]. In the other direction, every formula in [Ł]{}ukasiewicz logic can be used to describe a state uncertainty. By this we mean that, for every formula we can define an automata with a state having by uncertainty level the result of formula evaluation, after some interactions. For this the automata must have by input a string, having by first position the uncertainty on formula arguments, followed by marks with uncertainty zero for every possible sign. The number of positions in input sting must be equal to the formula parsing tree height. Injecting a formula in an automata structure is a simple process. For this we must note what every formula can be expressed in the disjunctive normal form, i.e. using only the connectives disjunction and negation. For instance the formula $((a\otimes b) \oplus c)\rightarrow d$, is equivalent to $\neg(\neg(\neg a\oplus \neg b) \oplus c)\oplus d$, evaluated as the uncertainty on state $E$, after 4 interaction, using automata describe in figure \[forAut\]. Note that, this defines a concept view $_{R:O\times \{a,b,c,d\}\rightharpoonup \{E\}}$. $$\tiny \xymatrix @=10pt{ d\ar[r]_1 & \bullet\ar[r]_1 &\bullet\ar[r]_1 &\bullet\ar[rd]_1 & & \\ a\ar[rd]_0 & & & &E & \\ & \bullet\ar[r]_0 &\bullet\ar[rd]_1 & & & \\ b\ar[ru]_0 & & &\bullet\ar[ruu]_0 & & \\ c\ar[r]_1 & \bullet\ar[r]_1 &\bullet\ar[ru]_1 & & & \\ }$$ The integration of knowledge using connective models is very restrictive, and difficult to be used directory by domain experts. However, it can be useful to deploy analytic models, or on procedures for redundancy minimization in a knowledge base. We want to present specification systems as the suitable framework for symbolic knowledge integration. Given a semiotic $(S,M)$ our goal is to complete the semiotic by enriching specification $S$, such that the new specification $S'$ is a better description for $M$. This is made by enriching structure $S$ with knowledge extracted from different views on $M$. $$\tiny \xymatrix @=30pt{A' \ar[rr]^{I_a}\ar[rrd]^{G_{a'}}& &A \ar[rrd]^{R_a} \ar[d]^{G_a} & & \\ & &E\ar@(dr,ur)[]_{T_a}\ar@(dl,ul)[]_{T_b} \ar[rr]^{P}& & F \\ B' \ar[rr]^{I_b}\ar[rru]^{G_{b'}}& &B \ar[rru]_{R_b} \ar[u]_{G_b} & & \\ }$$ For that, in this section, we extract knowledge from a structure construct using fuzzy relations. This relations are known to be defined using fuzzy automata, but this fact is hidden to the completion methodology. Figure \[GraphP\] describe the UoD structure, where concept views $G_a$ and $G_b$ are known to be described, respectively, by the acyclic and cyclic automata on figure \[exmp2\], where $A$ and $B$ are sets of input strings and $E$ is the set of automata possible states. This relations are described by two data sets with 15625 cases, using 14 attributes. Relations $T_a$ and $T_b$ are defined by the state transformation relations, codified on two data sets with 15625 cases, using 16 attributes, each. In the specification, views $R_a$ and $B_b$ denote the set of automata output states, given through the selection by $F$ of automata final states in $E$, satisfying: $R_a=G_a\otimes F$ and $R_a=G_b\otimes F$. Generically, we can formalize this structure using the following string-based relational specification: [cc]{} [rcl]{} %Nodes:\ $I$&:&$\{0,1\};$\ $A,B,A',B'$&:&I,I,I,I,I,I,I,I;\ $F$&:&I;\ $E$&:&$\{A_0,A_1,A_2,A_3,A_4,A_5,A_6,A_7\};$\ \ %Arrows:\ $R_a$&:&$\{A\rightharpoonup F\};$\ $G_a$&:&$\{A\rightharpoonup E\};$\ $R_b$&:&$\{B\rightharpoonup F\}; $\ $G_b$&:&$\{B\rightharpoonup F\}; $\ $P$&:&$\{E\rightharpoonup F\}; $\ $T_a,T_b$&:&$\{E\rightharpoonup E\}; $\ $I_a$&:&$\{A'\rightharpoonup A;$\ & & --------------- -------------------------- -- $\Gamma_a:\{$ $A'$; $\Gamma_a$ : similarity; } --------------- -------------------------- -- \ && $I_a$ : is\_a$(\Gamma_a)$;\ && $\};$\ $I_b$&:&$\{B'\rightharpoonup B;$\ & & --------------- -------------------------- -- $\Gamma_b:\{$ $B'$; $\Gamma_b$ : similarity; } --------------- -------------------------- -- \ & &$I_b$ : is\_a$(\Gamma_b)$;\ && $\};$\ \ & [rcl]{} %Commutative diagrams:\ $D_1$&:&$\{A\rightharpoonup F;$\ & &$D_1:R_a\otimes G_a\otimes P$;\ & &}\ $[D_1]$\ \ $D_2$&:&$\{B\rightharpoonup F;$\ & &$D_2:R_b\otimes G_a\otimes P$;\ & &}\ $[D_2]$\ \ $D_3$&:&$\{A'\rightharpoonup E;$\ & &$D_3:I_a\otimes G_a\otimes G_{a'}$;\ & &}\ $[D_3]$\ \ $D_4$&:&$\{B'\rightharpoonup E;$\ & &$D_4:I_b\otimes G_b\otimes G_{b'}$;\ & &}\ $[D_4]$\ \ \ This specification have the model, described by a set of data sets generated using fuzzy automata. This data sets are interpretations for each of the arrows used on the specification. Using the extraction process, described in the last section, we can extract rules with insights about the UoD structure. The knowledge extracted from crystallized rules from CNN, when trained over views $G_a$ and $G_b$, describe the relation between sign uncertainty on input strings, with length 6, and the uncertainty for each state after the string have been read. Since extraction is made in a supervising learning context, the process is oriented for the perdition of each automata state, using input string. Every obtain configuration, in this cases was representable, simplifying the translation for the string-based rules presented bellow for the knowledge base enrichment. ------- --- ------------------------------------------------------------------------------------------ $G_a$ : $\{A\rightharpoonup E;$ $\begin{array}{ccl} G_a(s_1,s_2,s_3,s_4,s_5,s6) & : & \\ & : & A_0 = s_6; \\ & : & A_1 = \neg s_6; \\ & : & A_2 = \neg s_5; \\ & : & A_3 = \neg(\neg s_6)\oplus (s_3 \otimes s_4);\\ & : & A_4 = \neg(s_4\oplus s_5);\\ & : & A_5 = \neg (s_3 \oplus s_4) \oplus (\neg s_2\otimes s_3\otimes \neg s_5);\\ & : & A_6 = s5;\\ & : & A_7 = s4;\\ \end{array}$ $\};$ $G_b$ : $\{B\rightharpoonup E; $ $\begin{array}{ccl} G_b(s_1,s_2,s_3,s_4,s_5,s_6) & : & \\ & : & A_0= s_6; \\ & : & A_1= \neg s_6; \\ & : & A_2\sim_{0.9391} (s_2\otimes \neg s_4)\oplus \neg s_6; \\ & : & A_3\sim_{0.9961} \neg(\neg s_4 \otimes \neg s_6);\\ & : & A_4\sim_{0.9785} \neg (s_2\otimes s_3 \otimes \neg s_4);\\ & : & A_5\sim_{0.9947} \neg s_4 \oplus ( \neg s_3\otimes \neg s_5);\\ & : & A_6\sim_{0.9429} \neg (s_3\otimes \neg s_5);\\ & : & A_7\sim_{0.9767} \neg s_2\oplus \neg s_3 \oplus s_4;\\ \end{array}$ $\};$ ------- --- ------------------------------------------------------------------------------------------ The knowledge presented on figure \[knowledgeInteg\] describes the relation between automata states. It was generated predicting the uncertainty in each state in interaction $i+1$ using automata state uncertainty in interaction $i$ and input signs uncertainty. We present bellow the best rules generated by the extraction process for each task. ------- --- -------------------------------------------------------------------------------- $T_a$ : $\{E\rightharpoonup E$; $\begin{array}{ccl} T_a(A_0,A_1,A_2,A_3,A_4,A_5,A_6,A_7) & : & \\ & : & A_2 = \neg A_0; \\ & : & A_3 = \neg(\neg A_0 \oplus A_4); \\ & : & A_4 = \neg(\neg A_2 \oplus A_6); \\ & : & A_5\sim_{0.9747} \neg A_3\oplus A_4\oplus A_6 \oplus \neg A_7);\\ & : & A_6 = \neg A_3;\\ & : & A_7 = A_6;\\ \end{array}$ } $T_b$ : $\{E\rightharpoonup E$; $\begin{array}{ccl} T_b(A_0,A_1,A_2,A_3,A_4,A_5,A_6,A_7) & : & \\ & : & A_2 = A_1 \oplus \neg A_6; \\ & : & A_3 = \neg A_0 \otimes A_4; \\ & : & A_4 = ;\\ & : & A_5 = \neg(\neg A_2 \otimes A_6); \\ & : & A_6\sim_{0.9993} \neg A_3\oplus A_4 \oplus \neg A_7);\\ & : & A_7 = A_6;\\ \end{array}$ } ------- --- -------------------------------------------------------------------------------- We can improve the description of our UoD, presenting constrains valid on queries to the specification model. For example, from query defined using equalizer $G_{a'\otimes b'}:A'\times B'\rightharpoonup E$ for view $G_{a'}$ and $G_{b'}$, we can generate new insights in the model structure, which can be used in knowledge base enrichment. ------------------- --- ---------------------------------------------------------------------------------------------------------------------------------- %Limit sentences: $D_3$ : $\{A',B'\rightharpoonup E;$ $D_3:=G_{a'}\otimes G_{b'}$; } $G_{a\otimes b}$ : $\{ A,B,E;$ $G_{a\otimes b}:Lim\;D_3$; $\begin{array}{ccl} G_{a\otimes b}(s_1,\ldots,s_6,s'_1,\ldots,s'_6) & : & \\ & : & A_0= s_6 \otimes s'_6; \\ & : & A_1= \neg(s_6 \oplus s'_6); \\ & : & A_2\sim_{0.9878} (s_6 \oplus \neg s'_6)\oplus (\neg s_6 \otimes s'_2\otimes s'_3\otimes \neg s'_4\otimes s'_6); \\ & : & A_3\sim_{0.9873}\neg(\neg s_6\oplus \neg s'_4)\oplus(s_6\otimes s'_6);\\ & : & A_4\sim_{0.9869}(\neg s_5\oplus \neg s'_2\oplus s_4)\oplus(s'_1\oplus s'_3\oplus \neg s'_5)\oplus(s_5\oplus s'_5);\\ & : & A_5\sim_{0.9609}((\neg s'_4\oplus \neg s'_5)\oplus(s'_3\otimes s'_4)\oplus\neg(s_4\oplus\neg s_5))\otimes\\ & & \otimes\neg((s'_3\otimes s'_4)\otimes\neg(s_4\oplus s_5));\\ & : & A_6=(s_4\oplus s_5)\otimes\neg(s'_3\otimes s'_4\otimes \neg s'_5)\otimes(s'_2\oplus s'_3\oplus s'_5);\\ & : & A_7\sim_{0.9526}\neg s'_2\oplus \neg s'_3\oplus s_4;\\ \end{array}$ } ------------------- --- ---------------------------------------------------------------------------------------------------------------------------------- This methodology to codify and extract symbolic knowledge from a NN is very simple and efficient for the extraction of comprehensible rules from medium-sized data sets. It is, moreover, very sensible to attribute relevance. In the theoretical point of view it is particularly interesting that restricting the values assumed by neurons weights restrict the information propagation in the network, thus allowing the emergence of patterns in the neuronal network structure. For the case of linear neuronal networks, having by activation function the identity truncate to 0 and 1, these structures are characterized by the occurrence of patterns in neuron configuration directly presentable as formulas in [Ł]{}logic. The application of procedures like the one above, on information systems, generates grates amount of information. We organize this information in a specification systems using a relational language. And we propose this language as an Interface Layer for AI. Here, classic graphical models like Bayesian and Markov networks have to some extent played the part of an interface layer, but one with a limited range having insufficiently expressive for general AI [@Domingos06].

--- abstract: 'A kind of transformation media, which we shall call the “anti-cloak”, is proposed to partially defeat the cloaking effect of the invisibility cloak. An object with an outer shell of “anti-cloak” is visible to the outside if it is coated with the invisible cloak. Fourier-Bessel analysis confirms this finding by showing that external electromagnetic wave can penetrate into the interior of the invisibility cloak with the help of the anti-cloak.' author: - 'Huanyang Chen$^*$, Xudong Luo, and Hongru Ma' - 'C.T. Chan' title: 'The Anti-Cloak' --- The ideas of transformation optics and cloaking \[1-4\] have attracted keen interest both in theory \[1-18\] and in experiment \[19\]. The cloaking effect has been proved using different methods, such as ray tracing \[8\], full wave simulation employing finite element methods \[9\] and the finite difference time domain methods \[10\], and the Mie scattering models \[11,12\]. In particular, Ruan et al. \[11\] employed Mie scattering models to confirm that a cylindrical cloak with the ideal material parameters is indeed a perfect invisibility cloak using Fourier-Bessel analysis. Similar approaches \[12\] were used to confirm the perfect cloaking effect of the spherical cloak. However, essentially all the aforementioned examples that demonstrated the perfect cloaking effect did not consider embedded objects with material anisotropy inside the cloak. In this paper, we show that the perfect cloaking effect can be defeated by adding another kind of transformation media inside the cloak (i.e., the anisotropy inside the cloak is considered, see in Fig. 1 schematically). We shall construct an example which demonstrates that an object with an outer shell of a specific form of negative index anisotropic material cannot be made entirely invisible by the transformation media cloak. Starting from the mapping \[6, 14\] (see in Fig. 2(a)), $r = \frac{b - r_0 }{b - a}(r' - b) + b,$ the required parameters for a partial cylindrical cloak (transverse electric (TE) mode is considered here) are obtained as follow, $$\label{eq1} \mu _r = \frac{r' - a_1 }{r'},\;\mu _\theta = \frac{r'}{r' - a_1 },\;\varepsilon _z = (\frac{b - r_0 }{b - a})^2\frac{r' - a_1 }{r'},$$ where $a_1 = \frac{a - r_0 }{b - r_0 }b.$ This partial cloak can reduce the total scattering cross section of a perfect electrical conductor (PEC) cylinder from its radius $r' = a$ to an equivalent PEC cylinder whose radius is $r = r_0 $. In the limit as $r_0 $ goes to zero, the partial cloak becomes perfect \[1\]. Now let us add another coordinate transformation inside the cloak ($c < r' < a)$ as depicted in Fig. 2(a), $r = \frac{d - r_0 }{c - a}(r' - c) + d.$ The corresponding material parameters are then, $$\label{eq2} \mu _r = \frac{r' - a_2 }{r'},\;\mu _\theta = \frac{r'}{r' - a_2 },\;\varepsilon _z = (\frac{d - r_0 }{c - a})^2\frac{r' - a_2 }{r'},$$ where $a_2 = \frac{ad - cr_0 }{d - r_0 }.$ We note that these values are negative. We call this kind of transformation media the “anti-cloak” as we shall see that they cancel partially the effect of an invisibility cloak. In the same spirit of the partial cloak, when a PEC cylinder with a radius $r' = c$ is coated with the anti-cloak in direct contact with the partial cloak, the total scattering cross section will be changed into that of an equivalent PEC cylinder whose radius is $r = d$. We note that there are no PEC boundary between the cloak and the anti-cloak (at $r' = a$), they are in direct contact. Doing the same Fourier-Bessel analysis in \[11\], we can obtain the electric fields in each region, $$\label{eq3} \begin{array}{l} (b \le r):E_z = \sum\limits_l {\alpha _l^{in} J_l (k_0 r)\exp (il\theta )} \\ \mbox{ } + \alpha _l^{sc} H_l (k_0 r)\exp (il\theta ), \\ (a \le r < b):E_z = \sum\limits_l {\alpha _l^1 J_l (k_1 (r - a_1 ))\exp (il\theta )} \\ \mbox{ } + \alpha _l^2 H_l (k_1 (r - a_1 ))\exp (il\theta ), \\ (c \le r < a):E_z = \sum\limits_l {\alpha _l^3 J_l (k_2 (r - a_2 ))\exp (il\theta )} \\ \mbox{ } + \alpha _l^4 H_l (k_2 (r - a_2 ))\exp (il\theta ). \\ \end{array}$$ where $J_l \backslash H_l $ are the $l$-order Bessel$\backslash $Hankel function of the 1st kind, $k_0 $ is the wave vector of the light in vacuum, $k_1 = \frac{b - r_0 }{b - a}k_0 ,\;k_2 = \frac{d - r_0 }{c - a}k_0 ,$ $\alpha _l^{in} $ and $\alpha _l^{sc} $ are the incident and scattering coefficients outside the cloak, $\alpha _l^i (i = 1,2,3,4)$ are the expansion coefficients for the field in the cloak and anti-cloak. The primes are dropped for aesthetic reasons from here. From the continuous boundary conditions (at $r = b$ and $r = a$) and the PEC boundary ($E_z = 0$ at $r = c$), we can obtain that, $$\label{eq4} \begin{array}{l} \alpha _l^1 = \alpha _l^3 = \alpha _l^{in} , \\ \alpha _l^2 = \alpha _l^4 = \alpha _l^{sc} , \\ \alpha _l^{sc} = - \frac{J_l (k_0 d)}{H_l (k_0 d)}\alpha _l^{in} . \\ \end{array}$$ This result confirms that the PEC cylinder with its radius $r = c$ coated with the anti-cloak and cloak is equivalent to a PEC cylinder with its radius $r = d$ in the view of outside world. \ To demonstrate the properties of the anti-cloak, we set $a = 0.1m$, $b = 0.2m$, $c = 0.05m$, $d = 0.02m$, $r_0 = 0.001m$. We plot the parameters of the cloak and anti-cloak at different radial positions in Fig. 2(b)-(d). All the parameters of anti-cloak are negative because of the negative slope of the coordinate transformation. A plane wave is incident from left to right with the frequency $2GHz$. In Fig. 3(a), we plot the scattering pattern of a PEC cylinder with a radius $r_0 $. The tiny PEC cylinder causes little scattering for the incoming plane wave which can be treated as almost invisible. In Fig. 3(b), we plot the scattering pattern of a PEC cylinder with a radius $a$ coated by a partial cloak. The outer radius of the cloak is $b$. We see that the partial cloak reduces substantially the scattering of the PEC cylinder with its radius $a$ when we compare Fig. 3(a) and Fig. 3(b). When $r_0 $ is made as small as we like, the scattering becomes vanishing small. In Fig. 3(c), we plot the scattering pattern of a PEC cylinder with a radius $c$ coated by an anti-cloak and a partial cloak \[20\]. The anti-cloak is located in $c < r < a$, the cloak locates in $a < r < b$. Without the anti-cloak, the wave basically goes around the shielded region, but if the anti-cloak is in contact with the cloak, EM wave from outside can go into the anti-cloak to interact with the object inside. The scattering of the cloak is enlarged again to that of an equivalent PEC cylinder whose radius is $d$. We plot the scattering pattern of the equivalent PEC cylinder in Fig. 3(d). When $c = d$, the anti-cloak together with the partial cloak becomes invisible, that means one can directly see the PEC cylinders with radius $r = c$, and the anti-cloak cancels out the effect of the partial cloak completely. For aesthetic reasons, if the electric field is larger than the maximum value in color bar in Fig. 3(c), we have replaced this overvalued field with the maximum value when plotting Fig. 3(c). If the electric field is smaller than the minimum value, we have replaced this overvalued field with the minimum value when plotting Fig. 3(c). Due to the continuous coordinate transformation at $r = a$, the impedances are matched at this touching boundary of the cloak and anti-cloak. The electric field is very large at this touching boundary. To show this property, we plot the electric field for different angles at fix radii near $r = a$ in Fig. 4. Three fixed radii are chosen, $r = a - 0.1r_0 $, $r = a$ and $r = a + 0.1r_0 $. We find that the electric field near $r = a$ is very large. Analytically, one can obtain the electric field at $r = a$ as follow, $$\label{eq5} \begin{array}{l} E_z = \sum\limits_l {\alpha _l^{in} J_l (k_0 r_0 )\exp (il\theta ) + \alpha _l^{sc} H_l (k_0 r_0 )\exp (il\theta )} \\ = \sum\limits_l {\alpha _l^{in} \exp (il\theta )[J_l (k_0 r_0 ) - \frac{J_l (k_0 d)}{H_l (k_0 d)}H_l (k_0 r_0 )]} . \\ \end{array}$$ The term $H_l (k_0 r_0 )$ becomes very large when $r_0 $ is small, that is why we obtain large electric field above. Since we can make $r_0 $ as small as we like, we reach the conclusion that an almost perfect cloak can be defeated by an anti-cloak. In other words, the transformation media cloak is not a panacea as there exists some objects that it cannot hide. In the limit that $r_0 $ is exactly zero, the situation requires further mathematical analysis due to the singularity properties of the anti-cloak and cloak ($H_l (k_0 r_0 )$ diverges when $r_0 $ goes to zero). From a physical standpoint, we may argue as follows. Near the inner boundary of the invisibility cloak, $\mu _r $ goes to zero and $\mu _\theta $ goes to infinity and they are positive, while near the outer boundary of the anti cloak, $\mu _r $ goes to zero and $\mu _\theta $ goes to infinity from the negative side. The positive singular values have to come from an in-phase resonance while the negative infinity comes from out-of-phase resonance. If we put them in contact, the system response is canceled out, and the cloaking effect is weaken or even destroyed. The surface mode resonance at $r = a$ is excited and contributes to the large electric field. In addition, if the losses are considered, the electric field will become finite for $r_0 $ is exactly zero. The cylindrical anti-cloak concept could be extended to three dimensions. In conclusion, we find that the invisible cloak cannot hide the enclosed domain if the inside domain has a shell of anti-cloak. The properties are demonstrated by using the Fourier-Bessel analysis and finite-element full wave simulations. The anti-cloak region is an anisotropic negative refractive shell that is impedance matched to the cloak outside, which has a positive refractive index. It is known that \[21\] a negative refractive index medium “cancels” the space of a positive index medium that has the same impedance. So, a heuristic way of understanding the operation of an anti-cloak is that it annihilates the functionality of the interior part of the invisibility cloak, and effectively shifts the enclosed PEC region outwards to make contact with the outer part of the cloaking shell that is not “canceled”. This leads to a finite cross section. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the National Natural Science Foundation of China under grand No.10334020 and in part by the National Minister of Education Program for Changjiang Scholars and Innovative Research Team in University, and Hong Kong Central Allocation Fund HKUST3/06C.\ $^*$Correspondence should be addressed to: kenyon@ust.hk [99]{} J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science [**312,**]{} 1780-1782 (2006). U. Leonhardt, “Optical conformal mapping,” Science [**312,**]{} 1777-1780 (2006). A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. [**10,**]{} 685-693 (2003). A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas [**24,**]{} 413-419 (2003). W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics [**1,**]{} 224-227 (2007). Huanyang Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, and C.T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B [**76,**]{} 241104 (2007). Huanyang Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. [**90,**]{} 241105 (2007). D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express [**14,**]{} 9794-9804 (2006). S.A. Cummer, B.-I. Popa, D. Schurig, D.R. Smith, and J.B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E [**74,**]{} 036621 (2006). Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and the essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. [**92,**]{} 131118 (2008). Z. Ruan, M. Yan, C.W. Neff, and M. Qiu, “Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations,” Phys. Rev. Lett. [**99,**]{} 113903 (2007). H.S. Chen, B.-I. Wu, B. Zhang, and J.A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. [**99,**]{} 063903 (2007). W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. [**91,**]{} 111105 (2007). R.V. Kohn, H. Shen, M.S. Vogelius, and M.I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. [**24,**]{} 015016 (2008). G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. A [**462,**]{} 3027-3059 (2006). G.W. Milton,M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. [**8,**]{} 248-267 (2006). N. A. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express [**15,**]{} 6314-6323 (2007). A. Al$\grave{u}$ and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E [**72,**]{} 016623 (2005). D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science [**314,**]{} 977-980 (2006). We have run simulations with a finite-element code, and obtained the same results as in Fig. 3(c). J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys.: Condens. Matter [**15,**]{} 6345-6364 (2003).

--- abstract: 'In this note, we establish a novel maximal inequality of the 2D Young integral $\int_a^b\int_c^d FdG$ in terms of the $(p,q)$-bivariation norms of the section functions $x\mapsto F(x,y)$ and $y\mapsto F(x,y)$ where $G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ is a controlled path satisfying finite $(p,q)$-variation conditions. The proof is reminiscent from the Young’s original ideas [@young1] in defining two-parameter integrals in terms of $(p,q)$-finite bivariations. Our result complements the standard maximal inequality established by Towghi [@towghi1] in terms of joint variations. We apply the maximal inequality to get novel strong approximations for 2D Young integrals w.r.t the Brownian local time in terms of number of upcrossings of a given approximating random walk.' address: - 'Departamento de Matemática, Universidade Federal da Paraíba, 13560-970, João Pessoa - Paraíba, Brazil' - 'Departamento de Matemática, Universidade Federal da Paraíba, 13560-970, João Pessoa - Paraíba, Brazil' author: - Alberto Ohashi - 'Alexandre B. Simas' title: A Maximal Inequality of the 2D Young Integral based on Bivariations --- Preliminaries and Main Result ============================= One remarkable result in the seminal L.C Young’s article [@young] is the development of a (1-parameter) 1D Riemman-Stieltjes-type integral $\int fdg$ where $f,g:[a,b]\rightarrow \mathbb{R}$ are two functions with suitable finite variations (see e.g [@friz]) $$\|f\|^p_{[a,b],p}:= \sup_{\Pi}\sum_{x_i\in \Pi}|f(x_i)-f(x_{i-1})|^p <\infty, \quad\hbox{and}\quad \|g\|^q_{[a,b],q}:= \sup_{\Pi}\sum_{x_i\in \Pi}|g(x_i)-g(x_{i-1})|^q <\infty,$$ where $\frac{1}{p} + \frac{1}{q}> 1$ and $\sup$ is taken over all partitions of the compact set $[a,b]\subset \mathbb{R}$. Let $\mathcal{W}^p([a,b];\mathbb{R})$ be the linear space of real-valued functions $h$ equipped with the seminorm $\|h\|_{[a,b],p} < \infty$. In his seminal article in 1936, Young proved that if $f\in \mathcal{W}^p([a,b];\mathbb{R})$ and $g\in \mathcal{W}^q([a,b];\mathbb{R})$ are two continuous functions, then there exists an absolute constant $C>0$ such that $$\label{in1} \Big|\int_a^bfdg \Big|\le C \Big[|f(a)| + \|f\|_{[a,b],p}\Big]\|g\|_{[a,b],q},$$ provided $\frac{1}{p} + \frac{1}{q}> 1$. The 1D Young’s integration theory has great importance in many areas in Analysis and Probability. In particular, it was the starting point for T. Lyons (see e.g [@lyons]) to introduce his original ideas on the so-called Rough Path theory where higher-order increments of the functions play a key role in determining integrals beyond the constraint $\frac{1}{p} + \frac{1}{q} > 1$. The 2D Young integral was introduced by L.C Young [@young1] and recently it has been an important tool in the Gaussian rough path theory [@friz2; @hairer; @gubinelli], extensions of Itô formula [@feng] and functional stochastic calculus [@LOS]. See also [@towghi1] for a particular multi-dimensional extension of the 2D Young integral. In the sequel, let us recall some basic definitions from the original article [@young1]. Throughout this article, we are going to fix $-\infty < a < b < +\infty$ and $-\infty < c < d < +\infty$. Let $\Pi(\xi)=\{x_i; 0\le i \le N\}$ be a partition of $[a,b]$ equipped with a set of points $\xi=\{\xi_i; i=1,\ldots,N\}$, where $x_{i-1}\le \xi_i\le x_i$ and $x_0=a,~x_N=b$. We call $\Pi(\xi)$ a tagged partition of $[a,b]$. Similarly, let $\Pi'(\eta)=\{y_j; 0\le j\le N'\}$ be a partition of $[c,d]$ equipped with a set of points $\eta=\{\eta_j; j=1,\ldots,N'\}$ such that $y_{j-1}\le \eta_j\le y_j$ and $y_0=c$ and $y_{N'}=d$. We call $\Pi'(\eta)$ a tagged partition of $[c,d]$. Let $\Pi(\xi)$ be a tagged partition of $[a,b]$ and let $f:[a,b]\to\mathbb{R}$ be a given function. We say $f_{\Pi(\xi)}:[a,b]\to\mathbb{R}$ is a **step function** of $f$ based on $\Pi(\xi)$ if $f_{\Pi(\xi)}(x_i) = f(x_i)$, for $i=0,\ldots, N$, and $f_{\Pi(\xi)}(x) = f(\xi_i)$, if $x_{i-1}<x<x_i$, $i=1,\ldots,N$. It is immediate from the definition that $\|f_{\Pi(\xi)}\|_{[a,b],p}\le \|f\|_{[a,b],p}$ for every tagged partition $\Pi(\xi)$. Throughout this article, we make use of the following terminology. $\Pi(\xi)$ and $\Pi'(\eta)$ will denote tagged partitions, whereas the notations $\Pi_j$ and $\Pi_j'$ stand for “untagged” partitions of $[a,b]$ and $[c,d]$, respectively. Finally, we say that an “untagged” partition $\Pi$ refines a tagged partition $\Pi(\xi)$, if $\Pi$ refines the partition $\Pi(\xi)$ without taking into account the set $\xi$. Let $\Pi(\xi)$ and $\Pi'(\eta)$ be tagged partitions of $[a,b]$ and $[c,d]$, respectively, and let $F:[a,b]\times [c,d]\to\mathbb{R}$ be a given function. We say that $F_{\Pi(\xi),\Pi'(\eta)}:[a,b]\times [c,d]\to\mathbb{R}$ is a **step function of** $F$ **on** $(\Pi(\xi),\Pi'(\eta))$, if $F_{\Pi(\xi),\Pi'(\eta)}(x_i,y_j) = F(x_i,y_j),$ for $i=0,\ldots,N$, $j=0,\ldots,N'$; $F_{\Pi(\xi),\Pi'(\eta)}(x_i,y) = F(x_i,\eta_j)$, if $y_{j-1}<y<y_j$, $i=0,\ldots,N$ and $j=1,\ldots,N'$; $F_{\Pi(\xi),\Pi'(\eta)}(x,y_j) = F(\xi_i,y_j),$ if $x_{i-1}<x<x_i$, $i=1,\ldots,N$ and $j=0,\ldots,N'$; $F_{\Pi(\xi),\Pi'(\eta)}(x,y) = F(\xi_i,\eta_j)$ if $x_{i-1} < x < x_i$ and $y_{j-1}< y < y_j$ for $i=1,\ldots,N$ and $j=1,\ldots, N'$. In the sequel, $\Delta_iH(x_i,y_j): = H(x_i,y_j)-H(x_{i-1},y_j)$ denotes the first difference operator acting on the variable $x$ of a given function $H:[a,b]\times[c,d]\rightarrow \mathbb{R}$, whereas $\Delta_jH(x_i,y_j) = H(x_i,y_j)-H(x_i,y_{j-1})$ denotes the first difference operator acting on the variable $y$ of $H$. Let $F_{\Pi(\xi),\Pi'(\eta)}$ be a step function for a given $F$ and let $G$ be a function such that $(x,y)\mapsto G(x,y)-G(\alpha,y) -G(x,\beta)+G(\alpha,\beta) $ admits only points of discontinuity of first kind for any $\alpha\in [a,b],~\beta\in [c,d]$. Then, we define $$\int_a^b\int_c^d F_{\Pi(\xi),\Pi'(\eta)}(x,y)d_{x,y}G(x,y):=\sum_{j=1}^N\sum_{i=1}^{N'}F(\xi_{i},\eta_{j})\Delta_i\Delta_j G(x_i,y_j).$$ We are now in position to recall the classical definition of the 2D Young integral, see also Section 4 in [@young1]. Let $F,G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ be two functions (which we would like to emphasize that these functions do not need to be continuous). We say that the Young integral $$\int_a^b\int_c^d F(x,y)d_{x,y}G(x,y)$$ exists (in the generalized Moore-Pollard sense) and it is equal to a real number $I$ if for every $\epsilon>0$, there exist finite subsets $E$ and $E'$ of $[a,b]$ and $[c,d]$, respectively, such that for every tagged partitions $\Pi(\xi)$ and $\Pi'(\eta)$, with the partition $\Pi(\xi)$ containing the points in $E$, and $\Pi'(\eta)$ containing the points in $E'$, the Riemann-Stieltjes integral of the step function $F_{\Pi(\xi),\Pi'(\eta)}$ with respect to $G$ satisfies $$\left| \int_a^b\int_c^d F_{\Pi(\xi),\Pi'(\eta)}(x,y)d_{x,y}G(x,y) - I\right| < \epsilon.$$ The following notion is originally due to Young [@young1] and it will play a key role in this work: \[pq\] We say that $F:[a,b]\times [c,d]\rightarrow \mathbb{R}$ has $(p,q)$-bivariation for $p,q >0$ if $$\|F\|_{1;p}:=\sup_{y_1,y_2\in [c,d]^2}\| F(\cdot, y_1) - F(\cdot, y_2)\|_{[a,b],p}< \infty,$$ and $$\|F\|_{2;q}:=\sup_{x_1,x_2\in [a,b]^2}\| F(x_1, \cdot) - F(x_2,\cdot)\|_{[c,d],q}< \infty.$$ There is a very related notion of variation which takes into account joint variation in both variables rather than bivariations  (see e.g [@friz; @friz1; @towghi; @towghi1]): \[jointdef\] Let $p\in [ 1, \infty)$. A function $F:[a,b]\times [c,d]\rightarrow \mathbb{R}$ has finite $p$-variation if $$V_p(F): = \Bigg(\sup_{\Pi,\Pi'}\sum_{\substack{x_i\in \Pi\\y_j\in\Pi'}} |\Delta_i\Delta_j F|^p\Bigg)^{\frac{1}{p}}< \infty,$$ where the supremum varies over all partitions $\Pi$ of $[a,b]$ and $\Pi'$ of $[c,d]$. The linear space of real-valued functions defined on $[a,b]\times [c,d]$ having finite $p$-variation equipped with the seminorm $V_p(F)$ will be denoted by $\mathcal{W}^p([a,b]\times [c,d];\mathbb{R})$. The following remarks show that the the joint variation notion is actually stronger than bivariations. Let $F:[a,b]\times [c,d]\to \mathbb{R}$, then $$\|F\|_{1;p}\leq V_p(F)\quad \text{and}\quad \|F\|_{2;q} \leq V_q(F).$$ We will only prove the first inequality, since the other one is entirely analogous. In the case where the supremum is attained at $y_1=c$ and $y_2=d$, the inequality is obvious. Let us assume $c\le y_1< y_2\le d$, and consider the partition $\Pi'= \{s_j\}$, with $s_0=c; s_1=y_1; s_2 =y_2; s_3 = d$. Then, we have $$\begin{aligned} \|F(\cdot,y_1) - F(\cdot,y_2)\|_{[a,b],p}^p &=& \sup_{\Pi} \sum_{x_i\in\Pi} |F(x_i,y_2)-F(x_i,y_1)-F(x_{i-1},y_2) + F(x_{i-1},y_1)|^p\\ &=& \sup_{\Pi}\sum_{x_i\in\Pi} |F(x_i,s_2)-F(x_i,s_1)-F(x_{i-1},s_2) + F(x_{i-1},s_1)|^p\\ &\leq& \sup_{\Pi}\sum_{s_j\in \Pi'} \sum_{x_i\in\Pi} |\Delta_i\Delta_j F(x_i,s_j)|^p\\ &\leq&V^p_p(F).\end{aligned}$$ Now, one can take the supremum on the left-hand side of the inequality. This concludes the proof. One should notice that $F\in \mathcal{W}^p([a,b]\times [c,d];\mathbb{R})$ if, and only if, both section functions $x\mapsto F(x,\cdot)$ and $y\mapsto F(\cdot,y)$ are $\mathcal{W}^p([\gamma,\eta];\mathbb{R})$-valued $p$-variation functions where $\gamma=c,\eta=d$ and $\gamma=a,\eta=b$, respectively. Moreover, if $F(a,\cdot)=F(\cdot,c)=0$ vanish, then $F$ has $(p,q)$-bivariation if, and only if, $x\mapsto F(x,\cdot)$ and $y\mapsto F(\cdot,y)$ are $\mathcal{W}^q$-valued bounded resp. $\mathcal{W}^p$-valued bounded functions. In this case, both $\big(\sup_{x\in [a,b]}\|F(x,\cdot)\|_{[c,d],q},\|F\|_{2;q}\big)$ and $\big(\sup_{y\in [c,d]}\|F(\cdot,y)\|_{[a,b],p},\|F\|_{1;p}\big)$ are equivalent. The importance of $(p,q)$-bivariation lies in the following result, which is a particular case of a theorem due to L. C. Young. \[tyoung\] Let $p,q>0$, and let $\rho$, $\sigma$, $\mu$ and $\lambda$ be monotone increasing functions such that, $\rho$ and $\sigma$ are subject to $\rho(u)\sigma(u)=u$. Assume that $$\label{i} \sum_{n=1}^\infty \rho\left(\frac{1}{n^{\frac{1}{p}}}\right)\lambda\left(\frac{1}{n}\right)<\infty~~\hbox{and}~~\sum_{n=1}^\infty \sigma\left(\frac{1}{n^{\frac{1}{q}}}\right)\mu\left(\frac{1}{n}\right)<\infty.$$ Let $F:[a,b]\times [c,d]\rightarrow \mathbb{R}$ be a function which vanishes on the lines $x=a$ and $y=c$ and which has bounded $(p,q)$-bivariation. Let $G :[a,b]\times [c,d]\rightarrow \mathbb{R}$ be a function satisfying $$\label{s} |\Delta_i\Delta_j G(x_i,y_j)|\le \lambda(x_i-x_{i-1}) \mu(y_j-y_{j-1}).$$ Then the 2D Young integral $\int_a^b\int_c^d FdG$ exists. That is, for each $\epsilon>0$, there exist finite subsets $E\subset [a,b]$ and $E^{'}\subset [c,d]$ such that $$\left|\int_a^b\int_c^d F(x,y)d_{x,y}G(x,y) - \int_a^b\int_c^d F_{\Pi(\xi),\Pi'(\eta)}(x,y)d_{x,y}G(x,y)\right| < \epsilon$$ for every tagged partitions $\Pi(\xi)$ and $\Pi'(\eta)$ which contain points of $E$ and $E'$, respectively. \[tychoices\] Typical candidates for the monotone increasing functions above are $\lambda(u) = u^{\frac{1}{\tilde{p}} },~\mu(u) = u^{\frac{1}{\tilde{q}}}$, $\rho(u)=u^{\alpha}$, $\sigma(u)=u^{1-\alpha}$ in such way that (\[i\]) and (\[s\]) hold. In the modern language of rough path theory, assumption (\[s\]) precisely says that if $\tilde{p}=\tilde{q}$ then $G$ admits a 2D-control $\omega([x_1,x_2]\times [y_1,y_2]) = |x_1-x_2|^{\frac{1}{\tilde{p}}}|y_1-y_2|^{\frac{1}{\tilde{q}}}$ so that (\[s\]) trivially implies that $G\in \mathcal{W}^{\tilde{p}}([a,b]\times [c,d];\mathbb{R})$. See Section 5.5 in [@friz]. The following result due to Towghi [@towghi] yields the existence of the Young integral under joint variation assumptions for both integrand and integrator as follows. Next, for the convenience of the reader we present his result as stated in [@friz1]. \[tow\] Let $p,q \ge 1$, assume that $\theta = \frac{1}{p}+\frac{1}{q}> 1$, and consider $F,G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ functions of $p$-variation resp. $q$-variation which do not have common jump points and $F(a,\cdot) = 0,~F(\cdot, c)=0$. Then the 2D Young integral $\int_a^b\int_c^d FdG$ exists (in the Riemann-Stieltjes sense) and for every $\alpha\in (1,\theta)$, $$\label{boundint} \Bigg| \int_a^b\int_c^d FdG\Bigg|\le \Bigg[\Bigg(1+\zeta \left(\frac{\theta}{\alpha}\right)\Bigg)^\alpha \zeta(\alpha) +(1+\zeta(\theta))\Bigg] V_p(F) V_q(G),$$ where $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.$ By comparing Theorems \[tow\] and \[tyoung\], we notice that the price we pay when dealing with $(p,q)$-bivariation is the stronger assumption (\[s\]) which provides the necessary smoothness on $G$ in order to get the existence of 2D Young integral. In one hand, one should notice that the mere finiteness of $V_q(G)<\infty$ does not imply (\[s\]) with $\lambda(u)= \mu(u)= u^{\frac{1}{p}}$. On the other hand, when $G$ satisfies (\[s\]) then we shall relax the joint variation property in $F$ by requiring only finite bivariation of a suitable order. Therefore, it is natural to ask a maximal inequality for the 2D Young integral under assumptions in Theorem \[tyoung\]. This issue is particularly important in the theory of local-times of Brownian motion. See Section \[applt\] for further details. Main Result ----------- In this note, our goal is to establish a maximal inequality for the 2D Young integral in terms of $(p,q)$-bivariations rather than the joint variation notion of Def \[jointdef\] and Theorem \[tow\]. We explore the $(p,q)$-bivariation notion pioneered by L.C Young instead of the joint variation in order to obtain the following maximal inequality for the 2D Young integral. \[mainTh\] Under the assumptions of Theorem \[tyoung\], the following estimate holds $$\begin{aligned} \left|\int_{a}^{b}\int_{c}^{d} F(x,y)d_{x,y}G(x,y)-F(b,d)(G(b,d)-G(b,c)-G(a,d)+G(a,c))\right|\\\nonumber \leq K \left(\sum_{m=1}^\infty \rho\left(\frac{\|F\|_{1;p}}{m^{1/p}}\right)\lambda\left(\frac{4}{m}\right)\right)\left(\sum_{m'=1}^\infty\sigma\left(\frac{\|F\|_{2;q}}{{m'}^{1/q}}\right)\mu\left(\frac{4}{m'}\right)\right)\\\nonumber +K_1\mu(d-c) \sum_{m=1}^\infty\frac{\|F\|_{1;p}}{m^{1/p}}\lambda\left(\frac{4}{m}\right)+ K_2\lambda(b-a) \sum_{m'=1}^\infty\frac{\|F\|_{2;q}}{{m'}^{1/q}}\mu\left(\frac{4}{m'}\right).\label{mainbound}\end{aligned}$$ where $K,K_1$ and $K_2$ are absolute constants. The first term on the right-hand side of the above inequality can be seen as a mixture of the $(p,q)$-bivariations, whereas the other terms are purely marginal terms. One observes that the joint $p$-variation is replaced by an equilibrium of the marginal $(p,q)$-bivariations, with the equilibrium being given by the functions $\rho$ and $\sigma$. This relaxation is compensated by controlling the paths of $G$ by means of assumption (\[s\]). In most applications of Theorem \[mainTh\], the statement can be simplified. In fact, as indicated in Remark \[tychoices\], the typical candidates for the functions $\rho(u)$ and $\sigma(u)$ are given by $\rho(u) = u^\alpha$, $\sigma(u) = u^{1-\alpha}$. Furthermore, the functions $\lambda(u)$ and $\mu(u)$ are usually given by $\lambda(u) = u^{1/\tilde{p}}$, and $\mu(u) = u^{1/\tilde{q}}$, with $\tilde{p},\tilde{q}>1$. In this case, we have the following corollary. \[mainbound\] Let $F,G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ be two functions, where $F$ vanishes on the lines $x=a$ and $y=c$ and has bounded $(p,q)$-bivariation, and $G$ satisfies $|\Delta_i\Delta_jG(x_i,y_j)|\leq C|x_i-x_{i-1}|^{1/\tilde{p}}|y_j-y_{j-1}|^{1/\tilde{q}}$, for some constant $C>0$, and $\tilde{p},\tilde{q}>1$. If there exists $\alpha\in (0,1)$ such that $$\alpha/p + 1/\tilde{p}>1\quad \hbox{and}\quad(1-\alpha)/q + 1/\tilde{q}>1,$$ then, the 2D Young integral $\int_a^b\int_c^d F(x,y)d_{x,y}G(x,y)$ exists and the following estimate holds $$\begin{aligned} &&\left|\int_{a}^{b}\int_{c}^{d} F(x,y)d_{x,y}G(x,y)-F(b,d)(G(b,d)-G(b,c)-G(a,d)+G(a,c))\right|\\ &\leq& K(\alpha,p,\tilde{p},q,\tilde{q}) \|F\|_{1;p}^{\alpha}\|F\|_{2;q}^{1-\alpha}+ K_1(p,\tilde{p},\tilde{q})\|F\|_{1;p}+ K_2(q,\tilde{q},\tilde{p})\|F\|_{2;q}, \end{aligned}$$ where $$K(\alpha,p,\tilde{p},q,\tilde{q}) = K 4^{1/\tilde{p}}\zeta\left(\frac{\alpha}{p}+\frac{1}{\tilde{p}}\right)\zeta\left(\frac{1-\alpha}{q} +\frac{1}{\tilde{q}}\right),$$ $$K_1(p,\tilde{p},\tilde{q}) = K_14^{1/\tilde{p}}(d-c)^{1/\tilde{q}}\zeta\left(\frac{1}{p}+\frac{1}{\tilde{p}}\right),$$ $$K_2(q,\tilde{q},\tilde{p}) = K_24^{1/\tilde{q}}(b-a)^{1/\tilde{p}} \zeta\left(\frac{1}{q}+\frac{1}{\tilde{q}}\right),$$ and $\zeta(s) = \sum_{i=1}^\infty \frac{1}{n^s}.$ The importance of Theorem \[mainTh\] and Corollary \[mainbound\] lies in cases when $F$ lacks or it is hard to check joint variation but $G$ satisfies condition (\[s\]). This type of regularity naturally arises in the context of functional Itô formulas (see e.g [@LOS]). See Section \[applt\] for some examples related to space-time local-time integral in the Brownian motion setting. Proof of Theorem \[mainTh\] =========================== Throughout this section, we are going to fix a function $G:[a,b]\times[c,d]\rightarrow \mathbb{R}$ such that $(x,y)\mapsto G(x,y)-G(\alpha,y) -G(x,\beta)+G(\alpha,\beta)$ admits only points of discontinuity of first kind for any $\alpha\in [a,b],~\beta\in [c,d]$. Let $[a,b]\subset\mathbb{R}$, and let $\Pi(\xi)$ be a fixed tagged partition of $[a,b]$. Denote the points of the partition $\Pi(\xi)$ by $x_0,\ldots,x_N$. We will now obtain a new sequence of partitions $\Pi_0,\ldots,\Pi_M$, where $M$ is chosen in such a way that $\Pi_M$ refines $\{x_0,\ldots,x_N\}$. Each partition $\Pi_j$ is chosen such that $\#\Pi_j = 2^j +1; j\ge 1$ and they are constructed inductively. Let $\Pi_0 = \{a,b\}$. Suppose $\Pi_j = \{t_0^{(j)},\ldots,t_{2^j}^{(j)}\}$, and $|t_{k+1}^{(j)} - t_k^{(j)}|<4\cdot2^{-j}$, for $k=0,\ldots, 2^j-1$. Then, let for each $k$, $t^{(j+1)}_{2k} = t_k^{(j)}$, and $t_{2k+1}^{(j+1)}$ be any number in $\{x_0,\ldots,x_N\}\cap (t_{k}^{(j)},t_{k+1}^{(j)})$ such that $$\label{cond1} |t_{2k+1}^{(j+1)} - t_k^{(j)}|<4\cdot2^{-(j+1)}\qquad\hbox{and}\qquad |t_{k+1}^{(j)}-t_{2k+1}^{(j+1)}|<4\cdot 2^{-(j+1)}.$$ If there is no such element we take $t_{2k+1}^{(j+1)}:= \frac{t_{k+1}^{(j)} + t_k^{(j)}}{2}$, which obviously satisfies . It is clear that, since $\#\Pi_M = 2^M+1$, the mesh $\|\Pi_M\|<2^{-(M-2)}$, and $\Pi(\xi)$ is finite, there exists some $M<\infty$ such that $\Pi_M$ refines $\{x_0,\ldots,x_N\}$. Then, we clearly have $$\int_a^b f_{\Pi(\xi)} dg = \sum_{i=0}^{2^M} f_{\Pi(\xi)}(t_i^{(M)}) (g(t_{i}^{(M)})-g(t_{i-1}^{(M)})).$$ \[onedim\] Let $\{t_i^{(n)}\}$ be the points of the partition $\Pi_n$, then, for functions $f,g:[a,b]\to\mathbb{R}$, let $$S_n = \sum_{i=1}^{2^n} f(t_i^{(n)}) (g(t_{i}^{(n)})-g(t_{i-1}^{(n)})).$$ Thus, $$S_n - S_{n-1} = \sum_{j=1}^{2^{n-1}} (f(t_{2j-1}^{(n)})-f(t_{2j}^{(n)}))(g(t_{2j-1}^{(n)})- g(t_{2j-2}^{(n)})).$$ Note that $$\begin{aligned} S_{n-1} &=& \sum_{i=1}^{2^{n-1}} f(t_i^{(n-1)}) (g(t_{i}^{(n-1)})-g(t_{i-1}^{(n-1)}))\\ &=& \sum_{i=1}^{2^{n-1}} f(t_{2i}^{(n)}) (g(t_{2i}^{(n)})-g(t_{2i-2}^{(n)}))\\ &=& \sum_{i=1}^{2^{n-1}} f(t_{2i}^{(n)}) (g(t_{2i}^{(n)})-g(t_{2i-1}^{(n)}))+ \sum_{i=1}^{2^{n-1}} f(t_{2i}^{(n)}) (g(t_{2i-1}^{(n)})-g(t_{2i-2}^{(n)})).\end{aligned}$$ Since $$\begin{aligned} S_n &=& \sum_{i=1}^{2^n} f(t_i^{(n)}) (g(t_{i}^{(n)})-g(t_{i-1}^{(n)}))\\ &=& \sum_{i=1}^{2^{n-1}} f(t_{2i}^{(n)}) (g(t_{2i}^{(n)})-g(t_{2i-1}^{(n)}))+ \sum_{i=1}^{2^{n-1}} f(t_{2i-1}^{(n)}) (g(t_{2i-1}^{(n)})-g(t_{2i-2}^{(n)})),\end{aligned}$$ we have, $$\begin{aligned} S_n - S_{n-1} &=& \sum_{i=1}^{2^{n-1}} (f(t_{2i-1}^{(n)})-f(t_{2i}^{(n)}))(g(t_{2i-1}^{(n)})-g(t_{2i-2}^{(n)})).\end{aligned}$$ We will now prove a two-parameter version of this lemma. We begin with some definitions. Let $F,G:[a,b]\times [c,d]\to\mathbb{R}$ be two functions, $\Pi(\xi), \Pi'(\eta)$ tagged partitions, together with sequences of partitions $\Pi_0,\ldots,\Pi_M$, $\Pi_0',\ldots,\Pi_{M'}'$, where we denote $\Pi_k = \{t_i^{(k)}\}$ and $\Pi_l'= \{s_j^{(l)}\}$. As before, it is easy to see that $$\int_a^b \int_c^d F_{\Pi(\xi),\Pi'(\eta)}(x,y)d_{x,y} G(x,y) = \sum_{i=1}^{2^M}\sum_{j=1}^{2^{M'}} F_{\Pi(\xi),\Pi'(\eta)}(t_i^{(M)},s_j^{(M')}) \Delta_i\Delta_j G(t_i^{(M)},s_j^{(M')}),$$ where $M$ and $M'$ are such that $\Pi_M$ and $\Pi_{M'}$ refine $\Pi(\xi)$ and $\Pi'(\eta)$, respectively. For a two-indexed sequence $S_{n,n'}$, we denote $\Delta_1 S_{n,n'} := S_{n,n'}- S_{n-1,n'}$ and $\Delta_2 S_{n,n'} := S_{n,n'} - S_{n,n'-1}$. Then, we have $S_{n,n'} - S_{n-1,n'}-S_{n,n'-1}+S_{n-1,n'-1} = \Delta_1\Delta_2 S_{n,n'}$. \[d1d2S\] Let $F,G:[a,b]\times [c,d]\to\mathbb{R}$ be two functions, and $\Pi_0,\ldots,\Pi_M$, $\Pi_0',\ldots,\Pi_{M'}'$ sequences of partitions of $[a,b]$ and $[c,d]$, respectively. Then, if we denote $$S_{n,n'} = \sum_{i=1}^{2^n}\sum_{j=1}^{2^{n'}} F(t_i^{(n)},s_j^{(n')}) \Delta_i\Delta_j G(t_i^{(n)},s_j^{(n')}),$$ we have that $$\Delta_1\Delta_2 S_{n,n'} = -\sum_{i=1}^{2^n-1}\sum_{j=1}^{2^{n'}-1} \Delta_i\Delta_jF(t_{2i}^{(n)},s_{2j}^{(n')})\Delta_i\Delta_j G(t_{2i-1}^{(n)},s_{2j-1}^{(n')}).$$ By setting $f_i(y) = F(t_i^{(n)},y)$ and $g_i(y) = \Delta_i G(t_i^{(n)},y)$ in Lemma \[onedim\], it is easy to see that for each $i$, $$\begin{aligned} \sum_{j=1}^{2^{n'}} F(t_i^{(n)},s_j^{(n')}) \Delta_i\Delta_j G(t_i^{(n)},s_j^{(n')}) - \sum_{j=1}^{2^{n'-1}} F(t_i^{(n)},s_j^{(n'-1)}) \Delta_i\Delta_j G(t_i^{(n)},s_j^{(n'-1)})\\ = \sum_{j=1}^{2^{n'-1}} (F(t_i^{(n)},s_{2j-1}^{(n')})-F(t_i^{(n)},s_{2j}^{(n')}))(\Delta_i G(t_i^{(n)},s_{2j-1}^{(n')}) - \Delta_i G(t_i^{(n)},s_{2j-2}^{(n')})).\end{aligned}$$ Thus, we have that $$\begin{aligned} \Delta_2 S_{n,n'} &=& S_{n,n'}-S_{n,n'-1}\\ &=& \sum_{i=1}^{2^n}\sum_{j=1}^{2^{n'-1}} (F(t_i^{(n)},s_{2j-1}^{(n')})-F(t_i^{(n)},s_{2j}^{(n')}))(\Delta_i G(t_i^{(n)},s_{2j-1}^{(n')}) - \Delta_i G(t_i^{(n)},s_{2j-2}^{(n')})).\end{aligned}$$ Applying Lemma \[onedim\] again, we can proceed in a similar manner to obtain the desired result: $$\Delta_1\Delta_2 S_{n,n'} =$$ $$\sum_{i=1}^{2^n-1}\sum_{j=1}^{2^{n'}-1} (F(t_{2i-1}^{(n)},s_{2j-1}^{(n')})-F(t_{2i-1}^{(n)},s_{2j}^{(n')})-F(t_{2i}^{(n)},s_{2j-1}^{(n')})+F(t_{2i}^{(n)},s_{2j}^{(n')}))\Delta_i\Delta_j G(t_{2i-1}^{(n)},s_{2j-1}^{(n')}).$$ We recall the following elementary remark for reader’s convenience. \[boundsigma\] Let $A,B,C\geq 0$, and let $\alpha>0$. Let $\rho,\sigma:[0,\infty)\to [0,\infty)$ be two non-decreasing functions such that $\rho(u)\sigma(u) = u$. Then, $$A\leq \alpha B\quad\hbox{and}\quad A\leq \alpha C \Rightarrow A \leq \alpha \rho(B)\sigma(C).$$ Let us now present a suitable bound for the double difference $\Delta_i\Delta_j F$ in terms of bivariations and a pair of monotone functions. \[boundd1d2f\] Let $\Pi = \{t_0,\ldots,t_{2^m}\}$ and $\Pi'=\{s_0,\ldots,s_{2^{m'}}\}$ be any partitions of the intervals $[a,b]$ and $[c,d]$, respectively. Then, for any function $F:[a,b]\times[c,d]\to\mathbb{R}$ with finite $(p,q)$-bivariation, the following inequality holds $$\sum_{i=1}^{2^m}\sum_{j=1}^{2^{m'}} |\Delta_i\Delta_j F(t_i,s_j)| \leq 4\cdot 2^{m+m'}\rho\left( \frac{\|F\|_{1;p}}{2^{m/p}}\right)\sigma\left(\frac{\|F\|_{2;q}}{2^{m'/q}}\right),$$ where $\rho$ and $\sigma$ are non-decreasing functions such that $\rho(u)\sigma(u)=u$. $$\begin{aligned} \sum_{i=1}^m\sum_{j=1}^{m'} |\Delta_i\Delta_j F(t_i,s_j)| &\leq&\sum_{i=1}^m\sum_{j=1}^{m'} \Big[ |\Delta_j F(t_i,s_j)|+|\Delta_jF(t_{i-1},s_j)|\Big]\\ &=& 2^{m'}\sum_{i=1}^m\left[\left(\sum_{j=1}^{2^{m'}}\frac{1}{2^{m'}}\Big[ |\Delta_j F(t_i,s_j)|+|\Delta_jF(t_{i-1},s_j)|\Big]\right)^{q}\right]^{1/q}\\ &\leq& 2^{m'}\sum_{i=1}^m\left[\sum_{j=1}^{2^{m'}}\frac{1}{2^{m'}}\Big[ |\Delta_j F(t_i,s_j)|+|\Delta_jF(t_{i-1},s_j)|\Big]^{q}\right]^{1/q}\\ &\leq& 2\cdot 2^{m'} \sum_{i=1}^{2^m}\left[\frac{1}{2^{m'}}\sum_{j=1}^{2^{m'}} \Big[ |\Delta_j F(t_i,s_j)|^q+|\Delta_jF(t_{i-1},s_j)|^q\Big]\right]^{1/q}\\ &\leq& 4\cdot 2^{m+m'} \frac{\|F\|_{2;q}}{2^{m'/q}}. \end{aligned}$$ A similar reasoning yields $$\sum_{i=1}^m\sum_{j=1}^{m'} |\Delta_i\Delta_j F(t_i,s_j)|\leq 4\cdot 2^{m+m'} \frac{\|F\|_{1;p}}{2^{m/p}}.$$ From lemma \[boundsigma\], the result follows. \[propdesig\] Let $F,G:[a,b]\times [c,d]\to\mathbb{R}$ be two functions, together with sequences of partitions $\Pi_0,\ldots,\Pi_M$, $\Pi_0',\ldots,\Pi_{M'}'$ of the intervals $[a,b]$ and $[c,d]$, respectively. Assume that assumptions of Theorem \[tyoung\] hold. Then, if we denote $$S_{n,n'} = \sum_{i=1}^{2^n}\sum_{j=1}^{2^{n'}} F(t_i^{(n)},s_j^{(n')}) \Delta_i\Delta_j G(t_i^{(n)},s_j^{(n')}),$$ we have that $$|S_{M,M'} - S_{0,M'}-S_{M,0}+S_{0,0}| \leq K \left(\sum_{m=1}^\infty \rho\left(\frac{\|F\|_{1;p}}{m^{1/p}}\right)\lambda\left(\frac{4}{m}\right)\right)\left(\sum_{m'=1}^\infty\sigma\left(\frac{\|F\|_{2;q}}{{m'}^{1/q}}\right)\mu\left(\frac{4}{m'}\right)\right),$$ where $K$ is an absolute constant. We begin by noting from Lemma \[d1d2S\] that $$\Delta_1\Delta_2 S_{k,k'} = -\sum_{i=1}^{2^k-1}\sum_{j=1}^{2^{k'}-1} \Delta_i\Delta_j F(t_{2i}^{(k)},s_{2j}^{(k')})\Delta_i\Delta_j G(t_{2i-1}^{(k)},s_{2j-1}^{(k')}).$$ Therefore, $$|\Delta_1\Delta_2 S_{k,k'}|\leq \sum_{i=1}^{2^k-1}\sum_{j=1}^{2^{k'}-1} |\Delta_i\Delta_j F(t_{2i}^{(k)},s_{2j}^{(k')})| \lambda(2^{-k+2})\mu(2^{-k'+2}),$$ and from Lemma \[boundd1d2f\], we have $$|\Delta_1\Delta_2 S_{k,k'}|\leq 4\cdot 2^{k+k'}\rho\left(\frac{\|F\|_{1;p}}{2^{k/p}}\right)\sigma\left(\frac{\|F\|_{2;q}}{2^{k'/q}}\right) \lambda(2^{-k+2})\mu(2^{-k'+2}).$$ Note that, $$S_{M,M'} - S_{0,M'}-S_{M,0}+S_{0,0} = \sum_{k=1}^M\sum_{k'=1}^{M'} \Delta_1\Delta_2 S_{k,k'}.$$ Now, there is an elementary inequality (see, for instance, [@feng p. 181-182]) that says that if $f$ is non-decreasing and non-negative, the following bound holds true $$\sum_{k=1}^\infty 2^{k-1}f\left(\frac{1}{2^k}\right) \leq \sum_{m=1}^\infty f\left(\frac{1}{m}\right).$$ Applying this inequality twice, we obtain $$\begin{aligned} |S_{M,M'} - S_{0,M'}-S_{M,0}+S_{0,0}|&\leq& \sum_{k=1}^n\sum_{k'=1}^{n'} |\Delta_1\Delta_2 S_{k,k'}|\\ &\leq&\sum_{k=1}^n\sum_{k'=1}^{n'} 4\cdot 2^{k+k'}\rho\left(\frac{\|F\|_{1;p}}{2^k/p}\right)\sigma\left(\frac{\|F\|_{2;q}}{2^{k'/q}}\right) \lambda(2^{-k+2})\mu(2^{-k'+2})\\ &\leq& 16 \sum_{m=1}^\infty \rho\left(\frac{\|F\|_{1;p}}{m^{1/p}}\right)\lambda\left(\frac{4}{m}\right)\sum_{m'=1}^\infty \sigma\left(\frac{\|F\|_{2;q}}{{m'}^{1/q}}\right)\mu\left(\frac{4}{m'}\right).\end{aligned}$$ This concludes the proof, and shows that $K\leq 16$. In a similar manner, but much more easily, one can prove the following lemma: \[desigmarginal\] Let $F,G:[a,b]\times [c,d]\to\mathbb{R}$ be two functions, together with sequences of partitions $\Pi_0,\ldots,\Pi_M$, $\Pi_0',\ldots,\Pi_{M'}'$ of the intervals $[a,b]$ and $[c,d]$, respectively. Assume that assumptions of Theorem \[tyoung\] hold. Then, if we denote $$S_{M,M'} = \sum_{i=1}^{2^M}\sum_{j=1}^{2^{M'}} F(t_i^{(n)},s_j^{(n')}) \Delta_i\Delta_j G(t_i^{(n)},s_j^{(n')}),$$ we have that $$|S_{M,0} - S_{0,0}| \leq K_1\mu(d-c) \sum_{m=1}^\infty\frac{\|F\|_{1;p}}{m^{1/p}}\lambda\left(\frac{4}{m}\right),$$ and $$|S_{0,M'} - S_{0,0}| \leq K_2\lambda(b-a) \sum_{m'=1}^\infty\frac{\|F\|_{2;q}}{{m'}^{1/q}}\mu\left(\frac{4}{m'}\right),$$ where $K_1$ and $K_2$ are absolute constants. One corollary of Proposition \[propdesig\] is Theorem 4.1 in Young’s original article [@young1]. Theorem 4.1 in [@young1] cannot be used directly to prove Theorem \[mainTh\], because it only works for integrands defined in terms of very specific double differences. Combining all the above results, we arrive at the following result. \[ineqstep\] Assume that $F,G:[a,b]\times[c,d]\rightarrow \mathbb{R}$ satisfy assumptions in Theorem \[tyoung\]. Then, for any step function $F_{\Pi(\xi),\Pi'(\zeta)}$, the following inequality holds $$\left|\int_a^b\int_c^d F_{\Pi(\xi),\Pi'(\zeta)}(x,y)d_{x,y}G(x,y)-F(b,d)(G(b,d)-G(a,d)-G(b,c)+G(a,c))\right|$$ $$\leq K \left(\sum_{m=1}^\infty \rho\left(\frac{\|F\|_{1;p}}{m^{1/p}}\right)\lambda\left(\frac{4}{m}\right)\right)\left(\sum_{m'=1}^\infty\sigma\left(\frac{\|F\|_{2;q}}{{m'}^{1/q}}\right)\mu\left(\frac{4}{m'}\right)\right)$$ $$\label{prebound} +K_1\mu(d-c) \sum_{m=1}^\infty\frac{\|F\|_{1;p}}{m^{1/p}}\lambda\left(\frac{4}{m}\right)+ K_2\lambda(b-a) \sum_{m'=1}^\infty\frac{\|F\|_{2;q}}{{m'}^{1/q}}\mu\left(\frac{4}{m'}\right).$$ Let $F,G:[a,b]\times [c,d]\to\mathbb{R}$ be two functions, $\Pi(\xi), \Pi'(\eta)$ tagged partitions, together with sequences of partitions $\Pi_0,\ldots,\Pi_M$, $\Pi_0',\ldots,\Pi_{M'}'$ of the intervals $[a,b]$ and $[c,d]$, respectively. Denoting $$S_{n,n'} = \sum_{i=1}^{2^n}\sum_{j=1}^{2^{n'}} F_{\Pi(\xi),\Pi'(\eta)}(t_i^{(n)},s_j^{(n')}) \Delta_i\Delta_j G(t_i^{(n)},s_j^{(n')}),$$ we have that, for $M$ and $M'$, $$\int_a^b \int_c^d F_{\Pi(\xi),\Pi'(\eta)}(x,y)d_{x,y} G(x,y) = S_{M,M'}.$$ Observe, also, that $S_{0,0} = F(b,d)(G(b,d)-G(a,d)-G(b,c)+G(a,c))$. The result is thus a simple consequence of Proposition \[propdesig\] and Lemma \[desigmarginal\]. From Proposition \[ineqstep\], the bound (\[prebound\]) holds uniformly for step functions of $F$. The 2D Young integral of $F$ w.r.t $G$ is defined as a Moore-Pollard-type limit of integrals of step functions and hence, we shall conclude the proof. An Application to the Brownian Motion Local-Time {#applt} ------------------------------------------------ In this section, we illustrate the importance of Theorem \[mainTh\] and Corollary \[mainbound\] with an application to local-times. In the sequel, $B = \{B(s); s\ge 0\}$ is a standard Brownian motion defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. The goal of this section is o provide strong approximations for the two-parameter random integral process $$\label{st} \int_{0}^t\int_{-2^m}^{2^m} g(s,x)d_{(s,x)}\ell^x(s); ~0\le t \le T,$$ where $$\ell^x(t): = \lim_{\varepsilon\rightarrow 0}\frac{1}{2\varepsilon}\int_0^t1\!\!1_{\{|B(s) - x| < \varepsilon\}} ds\quad \text{almost surely};~(t,x)\in [0,T]\times [-2^m,2^m]$$ is the so-called local-time of the Brownian motion on a bounded rectangle $[0,T]\times[-2^m,2^m]\subset \mathbb{R}^2$ with $m\in \mathbb{N}$, and $g:\Omega\times [0,T]\times [-2^m,2^m]\rightarrow \mathbb{R}$ is a two-parameter stochastic process with jointly continuous and controlled sample paths in the sense of (\[s\]). We are interested in strong approximations (in $L^1(\mathbb{P})$-sense) for (\[st\]) in terms of the number of upcrossings of an embedded random walk based on $B$. Similar to identity (4.5) in [@feng], we shall write $$\begin{aligned} \sum_{i=0}^{l-1}\sum_{j=0}^{p-1} g(s_j,x_i) \Delta_i\Delta_j \ell^{x_{i+1}}(s_j+1)&=&\sum_{i=1}^{l}\sum_{j=1}^{p}\ell^{x_{i}}(s_j) \Delta_i\Delta_j g(s_j,x_i)\\ & &\\ &-& \sum_{i=1}^l\ell^{x_i}(t)\Delta_i g(t,x_i).\end{aligned}$$ From Lemmas 2.1, 2.2 in [@feng], we know that $\{\ell^x(s); 0\le s\le T; -2^m\le x \le 2^m\}$ has $(1,2+\delta)$-bivariations a.s for every $\delta > 0$. Then under conditions of Theorem \[tyoung\] and the classical 1D-Young integral (see [@young]), we have $$\label{int} \int_0^t\int_{-2^m}^{2^m}g(s,x)d_{(s,x)}\ell^x(s) =\int_0^t\int_{-2^m}^{2^m}\ell(s,x)d_{(s,x)}g(s,x) - \int_{-2^m}^{2^m}\ell^x(t)d_xg(t,x); ~0\le t \le T.$$ The importance of Theorem \[mainTh\] (in particular, Corollary \[mainbound\]) lies on the fact that the paths of the Brownian local time is only known to be of finite $(1,2+\delta)$-bivariation (See Lemma 2.1 in [@feng]) for any $\delta>0$. In this case, the usual Towghi inequality (see Theorem \[tow\]) does not hold so the maximal inequality in Corollary \[mainbound\] plays a key role for the study of processes of the form (\[int\]). Since the 1D-Young integral in the right-hand side of (\[int\]) can be treated by means of standard Young estimates (see [@young]), we concentrate our example on the 2D-Young integral $$\label{st22} \int_0^t\int_{-2^m}^{2^m}\ell(s,x)d_{(s,x)}g(s,x);~0\le t\le T.$$ In the sequel, in order to approximate (\[st22\]), let us introduce $T^k_0:= 0$ and $$T^k_n: = \inf\{t> T^k_{n-1}; |B(t) - B(T^k_{n-1})|=2^{-k}\}; n\ge 1.$$ We set $A^k(t):= \sum_{n=1}^\infty B(T^k_n)1\!\!1_{\{T^k_n < t \le T^k_{n+1}\}}; 0\le t \le T$. In the sequel, for a given $x\in \mathbb{R}$, let $j_k(x)$ be the unique integer such that $(j_k(x)-1)2^{-k} < x \le j_k(x)2^{-k}$. Let us define $$u(j_k(x)2^{-k},k,t):= \#\ \Big\{n \in \{0, \ldots, N^k(t)-1\}; A^k(T^k_{n}) =(j_k(x)-1)2^{-k}, A^k(T^k_{n+1}) =j_k(x)2^{-k}\Big\};$$ for $x\in \mathbb{R}, k\ge 1, 0\le t \le T.$ Here, $N^k(t): = \max \{n; T^k_n \le t\}$ is the length of the embedded random walk until time $t$. By the very definition, $u(j_k(x)2^{-k},k,t) :=$ number of upcrossings of $A^k$ from $(j_k(x)-1)2^{-k}$ to $j_k(x)2^{-k}$ before time $t$. To shorten notation, we denote $$U^k(t,x):= 22^{-k}u(j_k(x)2^{-k},k,t);x\in I_m, 0\le t\le T,$$ where $I_m:= [-2^m,2^m]$ for a given positive integer $m\ge 1$. Assumption **(H1)**: Let $g^k:\Omega\times [0,T]\times I_m\rightarrow\mathbb{R}$ be a sequence of stochastic processes such that $$g^k(t,y)\rightarrow g(t,y)~a.s~\text{uniformly in}~(t,y)\in [0,T]\times I_m$$ and $g$ has jointly continuous paths a.s. Assumption **(H2.1)**: Assume for every $L>0$, there exists a positive constant $M$ such that $$\label{l2.1} |\Delta_i\Delta_jg(t_i,x_j)|\le M|t_i-t_{i-1}|^{\frac{1}{q_1}}|x_j-x_{j-1}|^{\frac{1}{q_2}}~a.s$$ for every partition $\Pi= \{t_i\}_{i=0}^N\times \{x_j\}_{j=0}^{N^{'}}$ of $[0,T]\times [-L,L]$, where $q_1,q_2>1$. In addition, there exists $\alpha\in (0,1)$ and $\delta >0$ such that $\min\{\alpha + \frac{1}{q_1}, \frac{1-\alpha}{2+\delta} + \frac{1}{q_2}\} > 1$. Assumption (**H2.2**): In addition to assumption **(H2.1)**, let us assume $\forall L >0$, there exists $M>0$ such that $$\label{l2.3} \sup_{k\ge 1} |\Delta_i\Delta_jg^k(t_i,x_j)|\le M |t_i-t_{i-1}|^{\frac{1}{q_1}}|x_j-x_{j-1}|^{\frac{1}{q_2}} ~a.s.$$ for every partition $\Pi= \{t_i\}_{i=0}^N\times \{x_j\}_{j=0}^{N^{'}}$ of $[0,T]\times [-L,L]$. Concrete examples for $(g^k,g)$ in terms of suitable functional derivatives of a given non-anticipative functional $F_t:C([0,t];\mathbb{R})\rightarrow \mathbb{R}$ of Brownian paths are illustrated by [@LOS] in the framework of functional Itô formulas. In particular, the authors show that suitable 2D-Young integral w.r.t local-times represents the unbounded variation components for functionals $F_t:C([0,t];\mathbb{R})\rightarrow \mathbb{R}$ of the Brownian paths under controlled sample paths assumptions. We refer the reader to Section 8.1-8.2 in [@LOS] for further details. Now let us recall a technical lemma describing some necessary bounds for the number of upcrossings. In the sequel, we always consider the stopped Brownian motion at $S_m := \inf\{t\ge 1 ; |B(t)|> 2^m\}\wedge T$. \[lemmaLk\] For each $m\ge 1$, the following properties hold: \(i) $U^{k}(t,x)\rightarrow \ell^x(t)\quad\text{a.s uniformly in}~(x,t)\in I_m\times [0,T]$ as $k\rightarrow \infty.$ \(ii) $\sup_{k\ge 1}\mathbb{E}\sup_{x\in I_m}\|U^{k}(\cdot,x)\|^p_{[0,T]; 1}< \infty$ and $\sup_{k\ge 1}\sup_{x\in I_m}\|U^{k}(\cdot,x)\|_{[0,T]; 1}< \infty~a.s$ for every $p\ge 1$. \(iii) $\sup_{k\ge 1}\mathbb{E}\sup_{t\in [0,T]}\| U^k(t)\|^{2+\delta}_{I_m;2+\delta}< \infty$ and $\sup_{k\ge 1}\sup_{t\in [0,T]}\| U^k(t)\|^{2+\delta}_{I_m;2+\delta}< \infty~a.s$ for every $\delta>0$. The proof can be founded in Corollary 2.1 in [@OS] and Lemma 8.1 in [@LOS]. Under assumption **(H1-H2)**, the following approximation holds $$\label{convergence} \int_0^t\int_{-2^m}^{2^m} U^{k}(s,x)dg^k(s,x)\rightarrow \int_0^t\int_{-2^m}^{2^m}\ell(s,x)dg(s,x)\quad \text{in}~L^1(\mathbb{P})$$ as $k\rightarrow \infty$, for every $t\in [0,T]$. By **(H2.2)**, (ii,iii) in Lemma \[lemmaLk\] and Theorem \[tyoung\], we know that the 2D Young integral $\int_0^t\int_{-2^m}^{2^m} U^{k}(s,x)dg^k(s,x)$ exists for every $k\ge 1$. We apply Lemma \[lemmaLk\], Th. 6.3 and 6.4 in [@young1] to get $$\lim_{k\rightarrow \infty}\int_0^t\int_{-2^m}^{2^m} U^{k}(s,x)dg^k(s,x) = \int_0^t\int_{-2^m}^{2^m}\ell(s,x)dg(s,x)$$ almost surely up to some vanishing conditions on $t=0$ and $x=-2^m$. They clearly vanish for $t=0$. For $x=-2^m$ we have to work a little. In fact, we will enlarge our domain in $x$, from $[-2^m,2^m]$ to $[-2^m-1,2^m]$, and define for all functions with $-2^m-1\le x < -2^m$, the value $0$, that is, for the functions $U^{k}(s,x), \ell^x(s), g(s,x)$ and $g^k(s,x)$ we put the value $0$, whenever $x\in [-2^m-1,-2^m)$. Then, it is easy to see, that all the conclusions of Lemma \[lemmaLk\] still hold true, and in this case, $U^{k}(s,-2^m-1)=0$ and $\ell^{-2^m-1}(s)=0$ for all $s$. Thus, we can apply Theorems 6.3 ad 6.4 in [@young1] on the interval $[0,t]\times [-2^m-1,2^m]$. It remains to show uniform integrability. By Corollary \[mainbound\], we have $$\begin{aligned} \nonumber \Bigg|\int_0^t\int_{-2^m}^{2^m}U^k(s,x)d_{(s,x)}g^k(s,x)\Bigg|&\le& K_0U^k(2^m,T) + K\|U^k\|^{\alpha}_{1;1} \|U^k\|^{1-\alpha}_{2;2+\delta}\\ \nonumber & &\\ \label{better}&+& K_1\|U^k\|_{1;1} + K_2\| U^k \|_{2;2+\delta}.\end{aligned}$$ Here $K_0$ is a constant which comes from assumption (\[l2.3\]) and $K,K_1,K_2$ are positive constants which only depend on the constants of assumption $\textbf{(H2.1, H2.2)}$ namely $\alpha,q_1,q_2,\delta, T, m$. From Lemma \[lemmaLk\], we have $\sup_{k\ge 1}\mathbb{E}\|U^{k}\|^{2+\delta}_{2;2+\delta} < \infty$, $\sup_{k\ge 1}\mathbb{E}\|U^{k}\|^{r}_{1;1} < \infty$ for every $r\ge 1$ and $\delta>0$. From Th.1 in [@Barlow], $\{U^{k}(2^m,T);k\ge 1\}$ is uniformly integrable, so we only need to check uniform integrability of $\{\|U^{k}\|^\alpha_{1;1}\|U^{k}\|^{1-\alpha}_{2;2+\delta}; k\ge 1 \}$. For $\beta > 1$, we apply Hölder inequality to get $$\mathbb{E}\|U^{k}\|^{\beta\alpha}_{1;1}\|U^{k}\|^{(1-\alpha)\beta}_{2;2+\delta}\le \big(\mathbb{E}\|U^k\|_{2;2+\delta}\big)^{1/b}\big(\mathbb{E}\|U^k\|^{\alpha\beta d}_{1;1}\big)^{1/d};~k\ge 1$$ where $b=\frac{1}{(1-\alpha)\beta} >1,~d = \frac{b}{b-1} = \frac{1}{1-(1-\alpha)\beta}$ with $\alpha \in (0,1)$. Lemma \[lemmaLk\] allows us to conclude the proof. [15]{} Barlow, M. (1984). A maximal inequality for upcrossings of a continuous martingale. *Z. Wahrch.Verw. Gebiete*, **67**, 2, 169-173. Borodin, A.N (1986). On the character of convergence to Browanian local time.*Z. Wahrch.Verw. Gebiete*. **72**, 231-250. Cass, T., Friz, P. and Victoir, N. (2009). Non-degeneracy of Wiener functionals arising from rough differential equations. *Trans. Amer.Math. Soc.* **361**, 6, 3359-3371. Cass, T., Hairer, M., Litterer, C. and and Tindel, S. Smoothness of the density for solutions to Gaussian rough differential equations. Forthcoming Annals of Probab. Chouk, K. and Gubinelli, M. Rough sheets. arXiv:1406.7748v1. Feng, C. and Zhao, H. (2006). Two-parameter $p,q$-variation Paths and Integrations of Local-Times. *Potential Anal*, **25**, 165-204 Friz, P. Victoir, N. *Multidimensional stochastic processes as rough paths. Theory and Applications*. Cambridge University Press. 2011. Friz, P. and Victoir, N. (2011). A note on higher dimensional $p$-variation. *Eletron. J. Probab*. **16**,1880-1899. Khoshnevisan, D. (1994). Exact Rates o Convergence to Brownian Local Time. *Ann. of Probab*.**22**, 3, 1295-1330. Leão, D. Ohashi, A. and Simas, A. B. On the Weak Functional Itô Calculus and Applications. arXiv: 1408.1423. Ohashi, A. and Simas, A. B. A note on the sharp $L^p$-convergence rate of upcrossings to the Brownian local time. arXiv:1408.1426. Lyons, T. J. (1998). Differential equations driven by rough signals. *Rev. Mat. Iberoam.* **14**, 215–310. Towghi, N. (2002). Littlewood’s inequalities for $p$-bimeasures. *Journal of Inequalities in Pure and Applied Maths*, **3**, 2, 19. Towgui, N. (2002). Multidimensional extension of L.C Young’s inequality. *Journal of Inequalities in Pure and Applied Maths*, **3**, 2, 22 19. Young, L.C. (1936). An inequality of Holder type, connected with Stieltjes integration. *Acta Math*, **67**, 251-282. Young, L.C. (1937). General inequalities for Stieltjes integrals and the convergence of Fourier series. *Mathematische Annalen*, 581-612.

--- abstract: 'Computers are widely utilized in today’s weather forecasting as a powerful tool to leverage an enormous amount of data. Yet, despite the availability of such data, current techniques often fall short of producing reliable detailed storm forecasts. Each year severe thunderstorms cause significant damage and loss of life, some of which could be avoided if better forecasts were available. We propose a computer algorithm that analyzes satellite images from historical archives to locate visual signatures of severe thunderstorms for short-term predictions. While computers are involved in weather forecasts to solve numerical models based on sensory data, they are less competent in forecasting based on visual patterns from satellite images. In our system, we extract and summarize important visual storm evidence from satellite image sequences in the way that meteorologists interpret the images. In particular, the algorithm extracts and fits local cloud motion from image sequences to model the storm-related cloud patches. Image data from the year 2008 have been adopted to train the model, and historical thunderstorm reports in continental US from 2000 through 2013 have been used as the ground-truth and priors in the modeling process. Experiments demonstrate the usefulness and potential of the algorithm for producing more accurate thunderstorm forecasts.' author: - 'Yu Zhang, Stephen Wistar, Jia Li, Michael Steinberg and James Z. Wang [^1]' bibliography: - 'ref.bib' title: Storm Detection by Visual Learning Using Satellite Images --- Acknowledgments {#acknowledgments .unnumbered} =============== Acknowledgment {#acknowledgment .unnumbered} ============== This material is based upon work supported by the National Science Foundation under Grant No. 1027854. Shared computational infrastructure was provided by the Foundation under Grant No. 0821527. Part of the work was done when J. Z. Wang and J. Li were with the Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Foundation. We also thank the US National Oceanic and Atmospheric Administration (NOAA) for providing the data used in this research. Siqiong He assisted with data collection. The discussions with Jose A. Piedra-Fernandez of the University of Almeria, Spain, has been very helpful. [^1]: Manuscript received –; revised –.

--- abstract: 'Using 0.37 megaton$\cdot$years of exposure from the Super-Kamiokande detector, we search for 10 dinucleon and nucleon decay modes that have a two-body final state with no hadrons. These baryon and lepton number violating modes have the potential to probe theories of unification and baryogenesis. For five of these modes the searches are novel, and for the other five modes we improve the limits by more than one order of magnitude. No significant evidence for dinucleon or nucleon decay is observed and we set lower limits on the partial lifetime of oxygen nuclei and on the nucleon partial lifetime that are above $4\times 10^{33}$ years for oxygen via the dinucleon decay modes and up to about $4 \times 10^{34}$ years for nucleons via the single nucleon decay modes.' title: 'Dinucleon and Nucleon Decay to Two-Body Final States with no Hadrons in Super-Kamiokande' --- One of the biggest unanswered questions about our universe is the origin of the matter/antimatter asymmetry that we observe. Non-conservation of baryon number, $\mathcal{B}$, is one of the three necessary conditions to create a baryon asymmetry where none previously existed [@Sakharov]. Since $\mathcal{B}$ is an accidental symmetry in the Standard Model (SM) of particle physics, observation of $\mathcal{B}$ violation would imply new physics beyond the Standard Model. Many theoretical extensions of the SM allow violation of $\mathcal{B}$ and/or lepton number, $\mathcal{L}$, and predict experimentally observable processes (see reviews in [@2013snowmass] and [@2006Nath]). The searches for ten such $\mathcal{B}$-violating processes via nucleon or dinucleon decay in Super-Kamiokande are detailed in this Letter. Four of the eight dinucleon decay modes studied here have $\Delta(\mathcal{B-L})=-2$, with two nucleons decaying to a lepton and an antilepton. A scenario in which baryon asymmetry would remain after $\Delta(\mathcal{B-L})=-2$ decays in the early universe is discussed in Ref. [@babu1]. Three of the eight dinucleon decay modes, with two nucleons decaying to two antileptons, violate each of $\mathcal{B}$ and $\mathcal{L}$ by two units, but conserve the quantity $(\mathcal{B-L})$. These modes are interesting in the context of models such as [@1985Alberico; @1982Arnellos; @1982Mohapatra; @2009Ajaib], and are shown in Ref. [@2015Learned] to be competitive with LHC measurements in probing the mass scale of new physics. The final dinucleon decay mode and the two single-nucleon decay modes studied here are radiative; these decay modes can arise in various models of grand unification, but are often predicted to have suppressed decay rates [@1982Lucha; @1983Lucha]. The radiative modes have similar experimental signatures as the other modes studied; they also have similar signatures to the previously searched $p \rightarrow e^{+}\pi^{0}$ and $p \rightarrow \mu^{+}\pi^{0}$ modes, but have the benefit of higher detection efficiency due to the lack of hadronic interactions. The ten decay modes we search for in Super-Kamiokande data are characterized by two back-to-back Cherenkov rings and no hadrons. The dinucleon decay modes in these three categories are: (i) $pp \rightarrow e^{+}e^{+}$, $nn \rightarrow e^{+}e^{-}$, $nn \rightarrow \gamma\gamma$, (ii) $pp \rightarrow e^{+}\mu^{+}$, $nn \rightarrow e^{+}\mu^{-}$, $nn \rightarrow e^{-}\mu^{+}$, and (iii) $pp \rightarrow \mu^{+}\mu^{+}$, $nn \rightarrow \mu^{+}\mu^{-}$. We classify the modes as follows: (i) both rings are showering ($NN\rightarrow ee$), (ii) one ring is showering and the other is non-showering ($NN\rightarrow e\mu$), and (iii) both rings are non-showering ($NN\rightarrow \mu \mu$). Figure \[fig:evtdisp\] illustrates how distinct these final states are seen in Super-Kamiokande, due to their well-separated, bright rings. The nucleon decay modes with identical experimental signatures, but lower invariant mass, are (i) $p \rightarrow e^{+}\gamma$ and (ii) $p \rightarrow \mu^{+}\gamma$. We do not include the search for dinucleon decays into tau leptons because there would be missing momentum and some subsequent tau decay modes are hadronic. The Super-Kamiokande (SK) water Cherenkov detector, with a fiducial volume of 22.5 kilotons, contains $1.2\times10^{34}$ nucleons. SK lies one kilometer under Mt. Ikenoyama in Japan’s Kamioka Observatory. The detector is cylindrical with a diameter of 39.3 meters and a height of 41.4 meters, optically separated into an inner and an outer region. Eight-inch photomultiplier tubes (PMTs) line the outer detector facing outwards and serve primarily as a veto for cosmic ray muons, and 20-inch PMTs face inwards to measure Cherenkov light in the inner detector [@SKNIM]. SK has collected data for four different detector periods, accumulating 91.5, 49.1, 31.8 and 199.3 kiloton$\cdot$years of exposure during SK-I, SK-II, SK-III, and SK-IV, respectively. During SK-I, the inner detector photocathode coverage was 40$\%$, but the SK-II period had a reduced photo-coverage of 19$\%$ after recovery from an accident. For SK-II, the remaining PMTs were evenly distributed to maintain isotropic detector uniformity. SK-II efficiency is only $\sim$2$\%$ lower than the other detector periods for these dinucleon and nucleon decay searches because the rings still have many hits. In the subsequent periods, SK-III and SK-IV, we restored the original photo-coverage of 40$\%$. The SK-IV period benefited from an electronics upgrade described in Ref. [@2010elec]: a “deadtime free” data acquisition system enables SK-IV to detect the 2.2 MeV gamma ray emission from neutron capture on hydrogen, which occurs about 200 $\mu$sec after the primary event. For each dinucleon or nucleon decay mode studied, we simulated 100,000 signal Monte Carlo (MC) events with vertices uniformly distributed throughout the detector and final state particle momenta uniformly distributed in phase space. Fermi motion, nuclear binding energy, and correlated decay are simulated in the dinucleon and nucleon decay signal MC [@2017mine; @2015jeff]. Unlike the atmospheric $\nu$ MC, where the Fermi momentum distribution of the nucleons follows the Fermi gas model, the signal MC Fermi momentum distribution follows a spectral function fit to electron-$^{12}$C scattering data [@197612c]. We address this difference between signal and atmospheric $\nu$ event samples by computing the systematic uncertainty in signal efficiency based on our choice of nuclear model. Correlated decay is a hypothesized effect where the total mass and momentum distributions are smeared out in a “tail" due to the correlated motion of a nearby nucleon. For both nucleons and pairs of nucleons, we assume that 10$\%$ of such decays are affected by the correlated motion of an additional nucleon [@1999corr]. Lepton rescattering within the nucleus is negligible. The atmospheric $\nu$ MC sample corresponds to an exposure of 500 years for each of the four SK periods, 2000 years in total. Events in this sample are weighted assuming two-flavor mixing as is done in recent dinucleon and nucleon analyses [@2017mine; @2016miura; @2015jeff]. Details of the cross-sections and flux modeling used in this sample are discussed in recent SK nucleon decay analyses [@2016miura; @2017mine]. Event rates obtained from this sample are normalized to the relevant SK detector livetime. Details of the neutron simulation and neutron tagging algorithm used for both the signal and atmospheric $\nu$ MC samples can be found in Ref. [@2016miura]. Neutron tagging can only be done for the SK-IV period; it reduces the expected number of background events by about 50% for our searches, and impacts signal efficiency by only a few percent.  ![(color online) Total mass ($M_{tot}$) and total momentum ($P_{tot}$) projections for $p \rightarrow \mu^+ \gamma$ after cut (A4). The red histogram shows atmospheric $\nu$ MC corresponding to 2000 years of SK exposure normalized to SK-I through SK-IV data. The selection criteria are indicated by the vertical blue lines.[]{data-label="fig:projection"}](Fig3.png) Although the selection criteria for all ten modes are similar, the two single-nucleon decay modes have more background due to their lower total mass. We adapt our strategy, as is done in Ref. [@2016miura], to perform a two-box analysis which allows us to study free and bound protons separately. The following selection criteria are applied to signal MC, atmospheric $\nu$ MC, and data: 1. [Events must be fully contained in the inner detector with the event vertex within the fiducial volume (two meters inward from the detector walls),]{} 2. [There must be two Cherenkov rings,]{} 3. [Both rings must be showering for the $pp \rightarrow e^{+}e^{+}$, $nn \rightarrow e^{+}e^{-}$, $nn \rightarrow \gamma\gamma$ and $p \rightarrow e^{+}\gamma$ modes; one ring must be showering and one ring must be non-showering for the $pp \rightarrow e^{+}\mu^{+}$, $nn \rightarrow e^{+}\mu^{-}$, $nn \rightarrow e^{-}\mu^{+}$ and $p \rightarrow \mu^{+}\gamma$ modes; both rings must be non-showering for the $pp \rightarrow \mu^{+}\mu^{+}$, $nn \rightarrow \mu^{+}\mu^{-}$ modes (see note in [^1]),]{} 4. [There must be zero Michel electrons for the $pp \rightarrow e^{+}e^{+}$, $nn \rightarrow e^{+}e^{-}$, $nn \rightarrow \gamma\gamma$ and $p \rightarrow e^{+}\gamma$ modes; there must be less than or equal to one Michel electron for the $pp \rightarrow e^{+}\mu^{+}$, $nn \rightarrow e^{+}\mu^{-}$, $nn \rightarrow e^{-}\mu^{+}$ and $p \rightarrow \mu^{+}\gamma$ modes; there is no Michel electron cut for the $pp \rightarrow \mu^{+}\mu^{+}$, $nn \rightarrow \mu^{+}\mu^{-}$ modes (see note in [^2]),]{} 5. [The reconstructed total mass, $M_{tot}$, should be $1600\leq M_{tot}\leq 2050$  MeV/c$^2$ for the dinucleon decay modes; the reconstructed total mass should be $800\leq M_{tot}\leq 1050$ MeV/c$^2$ for the nucleon decay modes,]{} 6. [The reconstructed total momentum, $P_{tot}$, should be $0 \leq P_{tot} \leq 550$ MeV/c for the dinucleon decay modes; for the nucleon decay modes, it should be $100 \leq P_{tot} \leq 250$ MeV/c for the event to be in the “High $P_{\text{tot}}$" signal box and $0 \leq P_{tot} \leq 100$ MeV/c for the event to be in the “Low $P_{\text{tot}}$" signal box,]{} 7. [\[SK-IV nucleon decay searches only\] There must be zero tagged neutrons.]{} Figure \[fig:mass-momentum\] shows the distributions of signal MC events (left panels), atmospheric neutrino background (middle), and data (right) as a function of $P_{tot}$ versus $M_{tot}$ after cut (A4). The signal selection efficiencies and background rates are summarized in Table \[tab:effic-bg\] for each of the decay modes and each of the SK running periods. The signal efficiency for the two nucleon decay modes is $\sim 50\%$ for the “High $P_{\text{tot}}$"signal box and $\sim 28\%$ for the “Low $P_{\text{tot}}$" signal box for each SK period. It is worth noting that these signal efficiencies are significantly higher than those of the similar event signature in the $p \rightarrow \ell^+ \pi^0$ decay mode searches. These differences are due to the fact that the $\pi^0$ undergoes nuclear effects before exiting the nucleus while the $\gamma$ does not. For the eight dinucleon decay modes, the signal efficiency is $\sim$$80\%$ for each SK period. Due to the high total mass required in (A5), the modes are virtually background-free (as shown in the middle panels of Fig. \[fig:mass-momentum\]). Background estimates are done in one of the two following ways, depending on the number of background events that fall in the signal box: (1) for signal regions that contain more than 10 events from 2000 years of atmospheric $\nu$ MC, the background is estimated by the traditional method of counting the number of events that fall inside the signal region; or, (2) for signal regions that are nearly background-free, an extrapolation method is used to estimate the expected background using the distribution of events nearby (but outside) the signal region. The background extrapolation is done by measuring the distance from the center of the signal box to the location of each nearby event in mass-momentum parameter space, and then fitting an exponential to the distribution of distances. Integration of the exponential function up to the radius which approximates the signal box (250 units in mass-momentum parameter space) gives the estimated background inside the signal region. A similar estimation method was done in Ref. [@shiozawathesis]. Background rate for $p \rightarrow e^+ \gamma$ “Low $P_{\text{tot}}$" is estimated by extrapolation to be 0.089 events/Mton$\cdot$yr; we take double this value (0.18 $\pm$ 0.18 events/Mton$\cdot$yr) as a conservative estimate of the background rate for this decay mode. Similarly, we extrapolate for all of the dinucleon decay modes, finding background rates of 0.008 ($NN \rightarrow ee$), 0.033 ($NN \rightarrow e \mu$), and 0.006 ($NN \rightarrow \mu \mu$) events/Mton$\cdot$yr. We conservatively take the largest of these and double it as our estimate of expected background for all of the dinucleon decay modes: $0.07\pm 0.07$ events/Mton$\cdot$yr. [l l @ cccc @cccc]{} & &\ & & SK-I & SK-II & SK-III & SK-IV & SK-I & SK-II & SK-III & SK-IV\ & High $P_{\text{tot}}$ & $51.0\pm0.2$ & $49.5 \pm 0.2$ & $50.8 \pm 0.2$ & $50.6 \pm 0.2$ & $0.01 \pm 0.01$ & $0.02 \pm 0.02$ & $< 0.01$ & $0.07 \pm 0.07$\ & Low $P_{\text{tot}}$ & $27.6 \pm 0.1$ & $26.1 \pm 0.1$ & $27.6 \pm 0.1$ & $27.5 \pm 0.1$ & $0.02 \pm 0.02 $ & $0.01 \pm 0.01$ & $0.01 \pm 0.01$ & $0.04 \pm 0.04$\ & High $P_{\text{tot}}$ & $50.2\pm 0.2$ & $49.7\pm0.2$ & $51.0 \pm 0.2$ & $48.1 \pm 0.2$ & $0.22 \pm 0.14$ & $0.14 \pm 0.11$ & $0.07 \pm 0.07$ & $0.23 \pm 0.14$\ & Low $P_{\text{tot}}$ & $29.1 \pm 0.1$ & $28.3 \pm 0.1$ & $29.0 \pm 0.1$ & $29.4 \pm 0.1$ & $0.02 \pm 0.02$ & $0.01 \pm 0.01$ & $<0.01$ & $0.02 \pm 0.02$\ $NN \rightarrow ee$ & & $80.9 \pm 0.1$ & $77.2 \pm 0.1$ & $79.5 \pm 0.1$ & $78.6 \pm 0.1$ & $0.01 \pm 0.01$ & $<0.01$ & $<0.01$ & $0.01 \pm 0.01$\ $NN \rightarrow e\mu$ & & $84.1 \pm 0.1$ & $83.7 \pm 0.1$ & $83.4 \pm 0.1$ & $81.7 \pm 0.1$ & $0.01 \pm 0.01$ & $<0.01$ & $<0.01$ & $0.01 \pm 0.01$\ $NN \rightarrow \mu\mu$ & & $86.3 \pm 0.1$ & $85.9 \pm 0.1$ & $86.0 \pm 0.1$ & $82.8 \pm 0.1$ & $0.01 \pm 0.01$ & $<0.01$ & $<0.01$ & $0.01 \pm 0.01$\ We find zero candidate events for the eight dinucleon decay modes. For the nucleon decay mode $p \rightarrow e^+ \gamma$, we also find zero candidate events. We observe two candidate events during the SK-IV period for the $p \rightarrow \mu^+ \gamma$ decay mode in the “High $P_{tot}$" signal box when $0.23\pm0.14_{stat}\pm0.07_{sys}$ events were expected. The Poisson probability to see two or more events in the SK-IV livetime given an expected rate of 0.23 events is 2.3%. One of the two candidates was previously found in Ref. [@2016miura]. The other candidate is more ambiguous since it lacks a Michel electron. This may be an indication that the event is due to a $\nu_{e}n \rightarrow e^{-}p$ charged-current quasielastic interaction, where the non-showering ring is due to a proton rather than a muon. Requiring a Michel electron would have eliminated this event, however such a requirement was not applied for the $p \rightarrow \mu^{+}\gamma$ mode in order to be consistent with cut (A4) for the dinucleon decay mode. Fig. \[fig:projection\] shows the agreement of data and atmospheric $\nu$ MC for $p \rightarrow \mu^+ \gamma$. [l l @ ccc @ c ]{} & & Background rate uncertainty(%)\ & & & Correlated & Nuclear &\ & & & Decay & Model &\ & High $P_{\text{tot}}$ & 10.5 & 3.5 & 2.4 & 40.4\ & Low $P_{\text{tot}}$ & 8.1 & 2.9 & 5.3 & 100\ & High $P_{\text{tot}}$ & 10.3 & 3.5 & 3.7 & 31.0\ & Low $P_{\text{tot}}$ & 8.0 & 3.1 & 5.8 & 44.0\ $NN \rightarrow ee$ & & 5.7 & 8.0 & — & 100\ $NN \rightarrow e\mu$ & & 2.6 & 8.4 & — & 100\ $NN \rightarrow \mu\mu$ & & 4.4 & 8.7 & — & 100\ $$\label{eq:probability} P(\Gamma | n_i)\! = \! \int^{\lambda}\! \int^{\epsilon}\! \int^{b} \! \frac{e^{-(\Gamma\lambda_i(\lambda)\epsilon_i(\epsilon)+b_i(b))}(\Gamma \lambda_i(\lambda)\epsilon_i(\epsilon)+b_i(b))^{n_i}}{n_i !}\! P(\Gamma)\! P(\lambda_i(\lambda) | \lambda_{i,0}, \sigma_{\lambda_{i,0}})\!P(\epsilon_i(\epsilon) | \epsilon_{i,0}, \sigma_{\epsilon_{i,0}})\!P(b_i(b) | b_{i,0}, \sigma_{b_{i,0}})d\lambda\,d\epsilon\,db$$ Table \[tab:systematics\] summarizes the systematic uncertainties in the signal efficiency and in the background rate for each of the nucleon and dinucleon decay modes. The dominant contributions to uncertainty in the signal efficiency arise from uncertainties in the areas of reconstruction, correlated decay, and nuclear model. To assess the impact of differences in the reconstruction of data and MC, for every variable used in the selection, we compute the percent shift of the atmospheric $\nu$ MC distribution necessary to minimize its chi-square against the corresponding data distribution. The cut value in the event selection is then shifted by that percentage and applied to the signal MC to recalculate the efficiency. The total systematic uncertainty due to reconstruction is calculated by summing in quadrature the independent percent changes in signal efficiency due to each percent-shifted cut. For nucleon decay in the SK-IV period only, we also add in quadrature with the other reconstruction uncertainties an additional 10% uncertainty due to neutron tagging, as was done in Ref. [@2016miura]. This is the reason that the reconstruction uncertainties for nucleon decay are $\sim$6% larger than the corresponding uncertainties for dinucleon decay. To estimate the uncertainty in the signal efficiency arising from uncertainties in correlated decay, we assume 100% uncertainty on the correlated decay effect, reweight the correlated decay events accordingly, and recalculate the signal efficiency, taking the overall change in signal efficiency as the systematic uncertainty. The nuclear model uncertainty is estimated as the percent change in signal efficiency when the Fermi gas model is used to compute the true momentum of the protons within the signal MC events instead of the spectral function fit to data described earlier. The systematic uncertainty on the rate of background events is conservatively taken to be 100% for decay modes where the background events are scarce (all dinucleon decay modes, and the $p \rightarrow e^+ \gamma$ “Low $P_{\text{tot}}$" nucleon decay). For the other nucleon decay signal regions, we use an event-by-event database with uncertainty weights from 73 sources of background systematic uncertainty including uncertainties in flux, cross section and energy calibration, as described in the 2018 SK oscillation analysis [@2018osc]. Lifetime limits are computed using a Bayesian method, assuming that the SK-I through SK-IV datasets have correlated systematic uncertainties [@2018pdg]. For the nucleon decay modes, the systematic uncertainties of the “High $P_{\text{tot}}$" and “Low $P_{\text{tot}}$" search boxes are treated as independent datasets with fully correlated systematic uncertainties. The conditional probability distribution for the decay rate is given by Eq. \[eq:probability\], where $\Gamma$ is the decay rate and for dataset $i$, $\lambda_i$ is the exposure (given in proton-years for nucleon decay and in oxygen-years for dinucleon decay), $\epsilon_i$ is the efficiency, $b_i$ is the number of background events, and $n_i$ is the number of candidate events. Since the systematic errors are correlated for all datasets, integrating the prior probability distribution up to $b$ in some dataset implies that we integrate the prior distribution in dataset $i$ up to $b_i(b)$. We assume a Gaussian prior distribution $P(\lambda_i(\lambda) | \lambda_{i,0}, \sigma_{\lambda_{i,0}})$ for $\lambda_i$ with a mean value of $\lambda_{i,0}$ and $\sigma_{\lambda_{i,0}}$ given by the $1\%$ percent systematic uncertainty in exposure. We also assume Gaussian priors $P(\epsilon_i(\epsilon) | \epsilon_{i,0}, \sigma_{\epsilon_{i,0}})$ and $P(b_i(b) | b_{i,0}, \sigma_{b_{i,0}})$ for $\epsilon_i$ and $b_i$ with standard deviations set to the total percent systematic uncertainties in efficiency and background, respectively. To require a positive lifetime, $P(\Gamma)$ is 1 for $\Gamma \geq 0$ and otherwise 0. We calculate the upper bound of the decay rate $\Gamma_{\text{limit}}$ as in Eq. \[eq:conflevel\], with a 90% confidence level (CL): $$\label{eq:conflevel} \text{CL} = \frac{\int_{\Gamma=0}^{\Gamma_{\text{limit}}}\prod_{i=1}^{N} P(\Gamma|n_i)d\Gamma}{\int_{\Gamma=0}^{\infty}\prod_{i=1}^{N} P(\Gamma|n_i)d\Gamma}.$$ Therefore we obtain the lower bound on the partial lifetime limit of a decay mode: $\tau/\text{B} = 1/\Gamma_{\text{limit}}$. Table \[tab:lifetimes\] summarizes the partial lifetime limits obtained for the ten decay modes studied, and these are also shown in relation to previous measurements in Fig. \[fig:moneyplot\]. [l @c @c]{} &\ & per oxygen nucleus & per nucleon\ & ($\times 10^{33}$ years) & ($\times 10^{34}$ years)\ $pp \rightarrow e^+ e^+$ & 4.2 & —\ $nn \rightarrow e^+ e^-$ & 4.2 & —\ $nn \rightarrow \gamma \gamma$ & 4.1 & —\ $pp \rightarrow e^+ \mu^+$ & 4.4 & —\ $nn \rightarrow e^+ \mu^-$ & 4.4 & —\ $nn \rightarrow e^- \mu^+$ & 4.4 & —\ $pp \rightarrow \mu^+ \mu^+$ & 4.4 & —\ $nn \rightarrow \mu^+ \mu^-$ & 4.4 & —\ $p \rightarrow e^+ \gamma$ & — & 4.1\ $p \rightarrow \mu^+ \gamma$ & — & 2.1\ We searched for the 10 dinucleon and nucleon decay modes characterized by a two-body final state with no hadrons in the Super-Kamiokande data with an accumulated exposure of 0.37 megaton$\cdot$years. No significant evidence for dinucleon or nucleon decay was observed, and we set lower limits on the partial lifetimes that are above $4 \times 10^{33}$ years for the dinucleon decay modes, $4.1 \times 10^{34}$ years for $p \rightarrow e^+ \gamma$, and $2.1 \times 10^{34}$ years for $p \rightarrow \mu^+ \gamma$. For five of the modes, the limits are novel, and the limits for all 10 modes are the most stringent by over one order of magnitude. ![(color online) The partial lifetime limits set by Super-Kamiokande for these ten modes, compared with previous limits set by the Fréjus and IMB detectors [@frejus; @IMB]. Note that Fréjus set dinucleon decay lifetime limits per iron nucleus rather than per oxygen nucleus.[]{data-label="fig:moneyplot"}](Fig4.png) We gratefully acknowledge the cooperation of the Kamioka Mining and Smelting Company. The Super-Kamiokande experiment has been built and operated from funding by the Japanese Ministry of Education, Culture, Sports, Science and Technology, the U.S. Department of Energy, and the U.S. National Science Foundation. [25]{}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 [****,  ()](\doibase 10.1070/PU1991v034n05ABEH002497) ()  [****,  ()](\doibase 10.1016/j.physrep.2007.02.010),  [****,  ()](\doibase 10.1103/PhysRevLett.109.091803) [****,  ()](\doibase 10.1103/PhysRevC.32.1722) [****,  ()](\doibase 10.1103/PhysRevLett.48.1708) [****,  ()](\doibase 10.1103/PhysRevLett.49.7) [****,  ()](\doibase 10.1103/PhysRevD.80.125026) [****,  ()](\doibase 10.1103/PhysRevD.91.035012) [****,  ()](\doibase 10.1016/0550-3213(83)90580-1) [****,  ()](\doibase 10.1007/BF01578149) [****,  ()](\doibase 10.1016/S0168-9002(03)00425-X) [****, ()](\doibase 10.1109/TNS.2009.2034854) [****,  ()](\doibase 10.1103/PhysRevD.96.012003) [****,  ()](\doibase 10.1103/PhysRevD.91.072009) [****,  ()](\doibase 10.1016/0375-9474(76)90539-X) [****,  ()](\doibase https://doi.org/10.1016/S0370-2693(99)00163-X) [****,  ()](\doibase 10.1103/PhysRevD.95.012004) **, @noop [Ph.D. thesis]{},  () [****,  ()](\doibase 10.1103/PhysRevD.97.072001) [****, ()](\doibase 10.1103/PhysRevD.98.030001) [****,  ()](\doibase 10.1016/0370-2693(91)91479-F) [****,  ()](\doibase 10.1103/PhysRevD.59.052004) [^1]: A note on cut (A3): Due to the two photons in the final state of $nn \rightarrow \gamma\gamma$, there is no visible energy in the first radiation-length of the two showers: this slightly impacts particle identification. We are about $1\%$ less efficient at particle identification for $nn \rightarrow \gamma\gamma$ than we are for similar dinucleon decay modes $pp \rightarrow e^{+}e^{+}$ and $nn \rightarrow e^{+}e^{-}$. [^2]: The decay electron cut is loose for the non-showering dinucleon decay modes because there is almost zero background to eliminate.

--- abstract: 'This study evaluates the performances of an LSTM network for detecting and extracting the intent and content of commands for a financial chatbot. It presents two techniques, sequence to sequence learning and Multi-Task Learning, which might improve on the previous task.' author: - Marc Velay - Fabrice Daniel date: July 2018 title: '**Seq2Seq and Multi-Task Learning for joint intent and content extraction for domain specific interpreters**' --- [**Keywords**]{}: Deep Learning, LSTM, MTL, Natural Language Processing, Seq2Seq

--- abstract: 'If there is explicit violation of baryon plus lepton number at some energy scale, then the electroweak theory depends upon a $\theta$-angle. Due to a singular integration over small scale size instantons, this $\theta$-dependence is sensitive to very high momentum scales. Assuming that there is no new physics between the electroweak and Planck scales, for an electroweak axion the energy difference between the vacuum at $\theta \neq 0$, and that at $\theta = 0$, is of the correct order of magnitude to be the dark energy observed in the present epoch.' author: - 'Larry McLerran$^{(1,2)}$, Robert D. Pisarski$^{(1,2)}$, and Vladimir Skokov$^{(1)}$' title: '**Electroweak Instantons, Axions, and the Cosmological Constant** ' --- 1. Nuclear Theory, Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA 2. RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA Introduction ============ In a gauge theory, a $\theta$-angle appears by adding a term $$\theta \; {\alpha \over {8\pi}} \int~d^4x~ {\rm tr} \left(F_{\mu \nu} \widetilde{F}^{\mu \nu} \right) \label{theta}$$ to the action, where $F_{\mu \nu}$ is the field strength, and $\widetilde{F}_{\mu \nu} = {1 \over 2} \epsilon_{\mu \nu \lambda \sigma} F^{\lambda \sigma}$ its dual. Here $N$, $$N = {\alpha \over {8\pi}} \int~d^4x~ {\rm tr} \left(F_{\mu \nu} \tilde{F}^{\mu \nu} \right)$$ is the winding number of an Euclidean field configuration [@'tHooft:1976fv; @Callan:1976je; @Coleman:1978ae; @Krasnikov:1978dg; @'tHooft:1986nc]. Instantons are solutions of the field equation with finite action and nonzero topological charge, $N \neq 0$. For the $SU(3)$ color gauge field, the physics of such configurations are well known. One can add a term such as Eq. (\[theta\]) to the action, which violates CP symmetry. A small value of $\theta$ can be attained by using the Peccei-Quinn symmetry [@Peccei:1977hh], and coupling to an axion field [@Wilczek:1977pj]. In the electroweak theory, for the $SU(2)_L$ gauge field, the associated $\theta$ angle has no physical significance. Since the electroweak action conserves baryon number, it does not change under a rotation of $(B+L)$ number, where $B$ and $L$ denote baryon and lepton number, respectively. The only place where electroweak instantons contribute are in amplitudes connecting states with different numbers of baryons [@'tHooft:1976fv; @Krasnikov:1978dg]. For such processes, if $(B+L)$ changes by by $\Delta(B+L) = 3N$, then $\theta$ appears in the path integral as ${\rm e}^{i N \theta}$. The factor of three arises here because each generation of quark is produced. The basic instanton process therefore involves 9 colored quarks and three leptons. In amplitudes squared, the phase disappears and the $\theta$-angle is no consequence. Beyond the Standard Model ========================= In generalizations of the standard model, one can have processes that violate $(B+L)$ explicitly. Following Anselm and Johansen [@Anselm:1993uj], let us assume there is an explicit $(B+L)$ violating interaction of the form $$S_{(B+L)} = {1 \over M^2}~ \int~ d^4x \left\{ \lambda \; l_L \, q_L \, q_L \, q_L + c.c. \right\} . \label{BLviolation}$$ Here $l_L$ is a left handed lepton field and $q_L$ is a left handed quark field. The scale $M$ is the energy scale at which lepton and baryon number changing interactions are important, and is presumably a scale of a Grand Unified Theory (GUT) or higher. The matrix $\lambda$ is of order $1$, and contracts various spinor, color and flavor indices. This interaction violates both $(B+L)$ and chirality. The basic process that can generate vacuum to vacuum overlap is shown in Fig. (\[instanton\]). A $SU(2)_L$ instanton emits three baryons and one lepton per generation; this is compensated by vertices for the $(B+L)$ process of Eq. (\[BLviolation\]), so that in all, the total amplitude does not change $(B+L)$. One might expect that since the scale of explicit $(B+L)$ violation in Eq. (\[BLviolation\]) is much larger than the electroweak scale, that surely integrating over the instanton scale size cuts off such efforts. This is wrong. After integrating over the external lines of quarks and leptons, the instanton amplitude is $$I = \int d^4x \int {{ d\rho} \over {\rho^5}} \; {1 \over {\rho^6 M^6}} \; {\rm e}^{-2\pi /\alpha(\rho)}.$$ For the electroweak theory, the coupling constant decreases somewhat at high momentum, or small $\rho$, $$\frac{1}{\alpha(\rho)} = \frac{1}{\alpha(\rho_0)} +{{(22/3-N_f/3 - N_h/6)} \over {2\pi}} \; \ln(\rho_0/\rho) ,$$ where $N_h$ is the number of Higgs particles and $N_f$ is the number of electroweak doublets. Using the values for the standard model, $N_f = 12$ and $N_h = 1$, we obtain $22/3-N_f/3-N_h/6 = 19/6$. This means that for small instantons, $\rho \gg 1/M$, the integration over instanton scale size behaves as $$\sim \frac{1}{M^6} \int {d \rho \over {\rho^{\, 47/6}}} \; .$$ This integral does not converge, even for very small scale instantons above the scale $M$. Presumably above this scale, new physics enters which makes the $\rho$ integral convergent. This lack of sensitivity to the low energy integration also occurs in supersymmetric GUT’s [@Nomura:2000yk; @Takahashi:2005kp]. In such theories, the electroweak coupling actually increases until the unification scale, with the $\rho$ integral convergent only above the GUT scale. Thus in supersymmetric GUT’s, the integration over instanton scale size is less convergent than without supersymmetry. We conclude that whatever physics is associated with a $SU(2)_L$ theta angle and vacuum to vacuum transitions in the electroweak theory is sensitive to physics at super high energy scales. Let us assume that the scale of explicit chiral $(B+L)$ violation is the scale at which new physics makes the integral over $\rho$ converge. This scale is $M$. It could be a scale somewhat less than that of the Planck scale, where Grand Unification might occur, or it might be the Planck scale itself. Including the effects of zero modes [@Anselm:1993uj], the rate for instanton processes is $$S_{\rm I} = \kappa \; \left(\frac{2 \pi}{\alpha_{ \rm W}}\right)^4 \; {\rm e}^{-2\pi /\alpha(M)} \; M^4 \; ,$$ where $\kappa$ is a constant of order 1. If we use electroweak theory with no Grand Unification, this formula becomes $$S_{\rm I} = \kappa \; \left(\frac{2 \pi}{\alpha_{ \rm W}}\right)^4 \; \left( {M_{\rm EW} \over M} \right)^{19/6} \; e^{-2\pi/\alpha_{\rm W}(M_{\rm EW})} \; M^4 \; .$$ If we take the energy scale $M$ to be the Planck mass, and $1/\alpha_{\rm W} \sim 1/29$, we find that $$S_{\rm I} \sim 10^{-122} \cdot M_{\rm pl}^4. \label{darkenergy}$$ This is remarkably close to the value of dark energy presently observed cosmologically, $\epsilon_{\rm DE} \sim 10^{-123} \cdot M_{\rm pl}^4 $ [@Peebles:2002gy]. In Eq. (\[darkenergy\]), $M_{\rm pl} \sim 10^{19}$ GeV is the Planck mass, and not the reduced mass, $M_{\rm pl}/\sqrt{8 \pi}$. Note that in Grand Unified theories, the coupling decreases less rapidly, or increases at higher energies, making the exponential suppression less important. At the Planck scale, one expects that this result is larger than the vacuum energy, although one might argue that in a GUT, one would generically go to a scale lower in energy. An estimate similar to ours has been performed in a supersymmetric theory by Nomura, Watari, and Yanagida [@Nomura:2000yk], see also Ref. [@Takahashi:2005kp]. In supersymmetric GUT theories, the electroweak coupling increases more rapidly than without supersymmetry. Consequently, to obtain an energy density as above, it is necessary to have a low value of the GUT scale [@Nomura:2000yk; @Takahashi:2005kp]. The purpose of this note is to show that the numbers in a model [*without*]{} supersymmetry are extremely interesting in their own right. Tegmark, Aguirre, Rees, and Wilczek, and later Hertzberg, Tegmark, and Wilczek [@Tegmark:2005dy], consider axion models with an axion scale on order of the Planck mass. They also find a reasonable value for the dark energy from the axion potential. Their model does not involve electroweak axions, though, and so differ in some important details from ours. There are numerical factors that need to be computed, in order to get a more precise estimate of instanton induced processes within the electroweak theory. There is a coefficient associated with the precise normalization of zero modes in the one loop computation [@Anselm:1993uj]. We have checked that using the running of $\alpha_W$ to two loop order changes the exponent in Eq. (\[darkenergy\]) by $\sim 1\%$. Given all the uncertainty, the estimate we have obtained is remarkably close to the observed value of the dark energy. It is interesting to consider how our estimate changes by altering the matter of the theory. Adding a single Higgs field has relatively little effect, $\sim 10^{- 3}$. In contrast, a fourth generation contributes three quarks and one lepton, and so suppresses the estimate in Eq. (\[darkenergy\]) by a signficant factor, $\sim 10^{-21}$. One might worry that even though the instanton amplitude contributes to the energy of the $\theta$-vacuum ground state, that the rate for any such process, which involves square of the amplitude, is so small that it never happens in the lifetime of the universe. In the evolution of the universe from its initial conditions, though, it should be suffice that there be an overlap between the initial set of states and a $\theta$-vacua. Since the lifetime of the universe, times the splitting of energy between the states include instantons and those which do not include instantons, is large, presumably one projects onto the proper ground state. If so, electroweak instantons to contribute to the ground state energy as above. Electroweak Axions ================== If we promote the electroweak $\theta$-angle to an axion [@Peccei:1977hh; @Wilczek:1977pj] we generate a vacuum energy of order $S_I$. Let us take the axion field modulus, $F_A \sim M$, as is natural if there is one high energy scale. If $M$ is the Planck or a GUT scale, a very small axion mass is generated. Such a light axion does not affect the long range gravitational force. The electroweak axion is a pseudo-scalar, so that due to derivative couplings, at large distances the exchange of the electroweak axion is suppressed by two powers of $r$, relative to the $1/r$ potential of gravity. In the absence of Grand Unification, it is natural to assume that $M \sim M_{\rm pl}$ [@Shaposhnikov:2009pv; @Shaposhnikov:2010zz], and the electroweak coupling runs up to the Planck scale without substantial modification. In this case, the Compton wavelength for the electroweak axion is larger than the size of the universe. This is automatic since $S_{\rm I} \sim \epsilon_{\rm vac}$, and $M_A \sim 1/R_{\rm universe}$ by Einstein’s equations. For such a large value of $F_A$, the electroweak axion is very weakly coupled, and the lower bound on $F_A$, from the cooling of large stars by axion emission [@Raffelt:1990yz], is not a problem. There may be additional constraints on $F_A$ where axion emission is competitive with gravitational radiation. Because black holes have no hair, global symmetries can be broken by quantum effects at the Planck scale [@Kamionkowski:1992mf]. If we take a unification scale at the Planck scale, our model is sensitive to this effect. Having such a light axion mass may allow us to evade this criticism; in any case, we ignore it. The argument that the energy density trapped in electroweak axions is the dark energy has a naturalness problem. There are other vacuum energy effects that are much larger, and must be artificially set to zero. We apply a naturalness condition: - After the axion field has had time to relax to zero, the cosmological constant should vanish. Therefore at late times, the universe has vanishing cosmological constant. If this condition is true, then the electroweak axion energy is non zero only because the modulus of the axion field has not yet relaxed to zero. Recall that the vacuum energy is $$\epsilon_{A} \sim \; M_A^2 \; a^2 \sim \; M_A^2 \; F_A^2 \; \theta^2 \; \sim S_{\rm I} \; \theta^2.$$ Because the inverse axion mass is of the order of the present size of the universe, there has not been time for the axion field to relax to zero. This happens only at some very late time, where the axion energy becomes dynamical, and is ultimately a type of matter energy. Assuming that far in the future there is no cosmological constant, in the present epoch the vacuum energy in the electroweak axion field has physical significance. That is, the cosmological constant is a temporary phenomenon that awaits its ultimate decay. Summary ======= Of course the picture we paint of the electroweak axion as a source of the cosmological constant is very speculative. Electroweak theory could have other intermediate energy scales that are important. There could be compensations and tuning of various energy scales so that within such a picture, one would get the correct dark energy. It may also be that there is no electroweak axion, or that even if there is, the source of dark energy is not in the axion field. Perhaps most interesting in these considerations is that instantons can be sensitive to physics in the far ultraviolet, as is also seen in the supersymmetric case [@Nomura:2000yk; @Takahashi:2005kp]. There is no decoupling of high energy and low energy degrees of freedom. This, and the sensitivity of instantons to scale breaking effects suggests their importance in other contexts. Perhaps in technicolor theories where the running of the coupling constant is assumed to be quite slow, effects of broken symmetries on higher energy scale might be important [@Dimopoulos:1979es; @Appelquist:1986an]. It is also quite amusing that there is another indication that there may be no intermediate energy scale physics between the electroweak scale and that of the Planck scale. This comes from requiring that electroweak theory is sensible at intermediate scales, and that there are fixed points of coupling constant evolution at the Planck scale. These considerations lead to a prediction of the Higgs boson mass at 126 GeV [@Shaposhnikov:2009pv]. Such a value is suggested by recent results from the LHC [@Aad:2012vn; @Chatrchyan:2012tx]. Schaposhnikov has argued that including massive right handed singlet neutrinos gives a reasonable cosmology for the standard model, with appropriate values for dark matter and the baryon asymmetry [@Shaposhnikov:2009zz]. Lastly, we note that Zhitnitsky has argued that there may be other processes, related to intantons in strongly interactions, that might be responsible for the cosmological constant [@Urban:2009yg]. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Eric Zhitnitksy for discussions and seminars where he argued, with other mechanisms, that instantons in QCD might be responsible for the cosmological constant. We also thank Fodor Bezrukov, Hooman Davoudiasl, and Bill Marciano for discussions. The research of L. McLerran, R. D. Pisarski, and V. Skokov is supported under DOE Contract No. DE-AC02-98CH10886. [00]{} G. ’t Hooft, Phys. Rev.  D [**14**]{}, 3432 (1976) \[Erratum-ibid.  D [**18**]{}, 2199 (1978)\]. C. G. Callan, Jr., R. F. Dashen and D. J. Gross, Phys. Lett. B [**63**]{}, 334 (1976); R. Jackiw and C. Rebbi, Phys. Rev. Lett.  [**37**]{}, 172 (1976). S. R. Coleman, Subnucl. Ser.  [**15**]{}, 805 (1979). N. V. Krasnikov, V. A. Rubakov and V. F. Tokarev, Phys. Lett.  B [**79**]{}, 423 (1978). G. ’t Hooft, Phys. Rept.  [**142**]{}, 357 (1986). R. D. Peccei and H. R. Quinn, Phys. Rev. Lett.  [**38**]{}, 1440 (1977); Phys. Rev.  D [**16**]{}, 1791 (1977). S. Weinberg, Phys. Rev. Lett.  [**40**]{}, 223 (1978); F. Wilczek, [*ibid.*]{} [**40**]{}, 279 (1978); A. A. Anselm and A. A. Johansen, Nucl. Phys.  B [**412**]{}, 553 (1994) \[arXiv:hep-ph/9305271\]. Y. Nomura, T. Watari and T. Yanagida, Phys. Lett. B [**484**]{}, 103 (2000) \[hep-ph/0004182\]. F. Takahashi and T. T. Yanagida, Phys. Lett. B [**635**]{}, 57 (2006) \[hep-ph/0512296\]. P. J. E. Peebles and B. Ratra, Rev. Mod. Phys.  [**75**]{}, 559 (2003) \[astro-ph/0207347\]. M. Tegmark, A. Aguirre, M. Rees and F. Wilczek, Phys. Rev. D [**73**]{}, 023505 (2006) \[astro-ph/0511774\]; M. P. Hertzberg, M. Tegmark and F. Wilczek, [*ibid.*]{} [**78**]{}, 083507 (2008) \[arXiv:0807.1726 \[astro-ph\]\]. M. Shaposhnikov and C. Wetterich, Phys. Lett. B [**683**]{}, 196 (2010) \[arXiv:0912.0208 \[hep-th\]\]. M. Shaposhnikov, “New physics without new energy scale”, unpublished. G. G. Raffelt, Phys. Rept.  [**198**]{}, 1 (1990). M. Kamionkowski and J. March-Russell, Phys. Lett. B [**282**]{}, 137 (1992) \[arXiv:hep-th/9202003\]. S. Dimopoulos and L. Susskind, Nucl. Phys.  B [**155**]{}, 237 (1979). T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett.  [**57**]{}, 957 (1986). G. Aad [*et al.*]{} \[ATLAS Collaboration\], arXiv:1203.1303 \[hep-ex\]. S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], arXiv:1202.1488 \[hep-ex\]. M. Shaposhnikov, Prog. Theor. Phys.  [**122**]{}, 185 (2009). F. R. Urban and A. R. Zhitnitsky, Nucl. Phys. B [**835**]{}, 135 (2010) \[arXiv:0909.2684 \[astro-ph.CO\]\]; Phys. Rev. D [**82**]{}, 043524 (2010) \[arXiv:0912.3248 \[astro-ph.CO\]\]; Phys. Rev. D [**82**]{}, 123532 (2011) \[arXiv:1011.2425 \[astro-ph.CO\]\]; A. R. Zhitnitsky, arXiv:1112.3365 \[hep-ph\]; E. Thomas and A. R. Zhitnitsky, arXiv:1203.6073 \[hep-ph\].

--- abstract: 'Spin splitting in *p*-type semiconductor nanowires is strongly affected by the interplay between quantum confinement and spin-orbit coupling in the valence band. The latter’s particular importance is revealed in our systematic theoretical study presented here, which has mapped the range of spin-orbit coupling strengths realized in typical semiconductors. Large controllable variations of the $g$-factor with associated characteristic spin polarization are shown to exist for nanowire subband edges, which therefore turn out to be a versatile laboratory for investigating the complex spin properties exhibited by quantum-confined holes.' author: - 'D. Csontos' - 'U. Z[ü]{}licke' title: Tailoring hole spin splitting and polarization in nanowires --- Engineering spin splitting of charge carriers in semiconductor nanostructures may open up intriguing possibilities for realizing spin-based electronics [@sciencerev] and quantum information processing [@lossbook]. Due to the generally strong dependence of $g$-factors on band structure [@roth:pr:59], it is expected that spatial confinement will have an important effect on Zeeman splitting when bound-state quantization energies are no longer negligible compared with the separation of bulk-material energy bands. The degeneracy of heavy-hole (HH) and light-hole (LH) bulk dispersions at the zone center makes the spin properties of valence-band states especially susceptible to such confinement engineering [@roland:prl:00; @uz:prl:06; @pryor:prl:06; @haug:prl:06]. Recent advances in fabrication technology [@nanoWireRev2; @nanoWireRev1; @defran:nlett:06; @janik:apl:06; @johan:natmat:06; @dick:jcg:07; @piccio:apl:05; @romain:apl:06; @oleh:apl:06] have created opportunities to investigate hole spin physics in semiconductor nanowires made from a range of different materials. ------------ ------------ ------------ ------------ ------------ ------------ ZnTe/ZnS AlAs/AlP AlSb CdTe GaN/AlN GaAs/InP 0.28[^1] 0.31[^2] 0.32$^{b}$ 0.34$^{a}$ 0.36$^{b}$ 0.37$^{b}$ Ge InN GaSb InAs InSb GaP 0.38$^{a}$ 0.40$^{b}$ 0.41$^{b}$ 0.45$^{b}$ 0.46$^{b}$ 0.48$^{b}$ ------------ ------------ ------------ ------------ ------------ ------------ : \[gammamatdep\] Relative spin-orbit coupling strength $\gamma= \gamma_{s}/\gamma_{1}$ in the valence band of common semiconductors. Here $\gamma_{s}=(2\gamma_{2}+3\gamma_{3})/5$, and $\gamma_{1,2,3}$ denote the Luttinger parameters [@luttham2]. In contrast to previous theoretical work [@kossut:prb:00; @kita:prb:06; @xia:jpd:07; @uz:prb:07b] on hole spin splitting in quantum wires, we focus here on the influence of the spin-orbit coupling strength on Zeeman splitting of wire-subband edges. A suitable parameter $\gamma$ quantifying spin-orbit coupling in the valence band can be defined in terms of the effective masses $m_{\text{HH}}$ and $m_{\text{LH}}$ associated with the HH and LH bands [@warpNote], respectively: $2 \gamma = (m_{\text{HH}} - m_{\text{LH}})/\left(m_{\text{HH}} + m_{\text{LH}} \right)$. Table \[gammamatdep\] lists values for $\gamma$ in common semiconductors and states its relation to basic band-structure parameters [@luttham2]. A large part of the theoretically possible range $0\le \gamma \le 1/2$ is covered by available materials [@realNote], enabling a detailed study of the interplay between spin-orbit coupling in the valence band and nanowire confinement. Our theoretical investigation reveals surprising qualitative differences in the hole spin properties of nanowires depending on the value of $\gamma$, showing that spin splitting (and polarization) of zone-center valence-band edges in nanowires is highly tunable and has a complex materials dependence. A detailed understanding of these properties is vital for proper interpretation of optical and transport measurements as well as for the design of spintronic applications involving *p*-doped semiconductor nanowires. We use the Luttinger model [@luttham2] in the spherical approximation [@lip:prl:70] for the top-most bulk valence bands. Including the bulk Zeeman term $H_{\text{Z}} =-2 \kappa \mu_{\text{B}} B \hat{J}_{z}$, the Hamiltonian is given by $$H=-\frac{\gamma_{1}}{2m_{0}}p^{2}+\frac{\gamma_{s}}{m_{0}}\left [ ({\mathbf p}\cdot \hat {\mathbf J})^{2}-\frac{5}{4}p^{2}{\mathbf 1}_{4\times 4} \right ]+H_{\text{Z}} \quad . \label{Hamiltonian}$$ Here ${\mathbf p}$ is the linear orbital momentum, $\hat{\mathbf J}$ the vector of spin-3/2 matrices, $m_{0}$ the electron mass in vacuum, $\gamma_{s}=(2\gamma_{2}+3 \gamma_{3}) /5$ in terms of the Luttinger parameters [@luttham2], $\mu_{\text{B}}$ is the Bohr magneton and $\kappa$ the bulk hole $g$-factor. We neglect the small anisotropic part of the bulk-hole Zeeman splitting. A hard-wall confinement in the $xy$ plane defines the quantum wire with either cylindrical or square cross-section. Our method for finding the zone-center subband edges and calculating their $g$-factor $g^\ast$ in a magnetic field parallel to the wire axis has been described elsewhere [@uz:prb:07b; @uz:physe:07b]. An intriguing universal behavior of wire-subband spin splittings emerges when the bulk-Zeeman term dominates the orbital effects which, in principle, also contribute to the effective $g$-factor. This universal regime, which is characterized by $g^\ast$ scaling with $\kappa$ and being independent of wire diameter, is accessible in real nanowire systems [@defran:nlett:06] where $\kappa$ is enhanced by the *p-d* exchange interaction with magnetic acceptor ions [@dietl:prb:01]. Figure \[orbVSkappa\] illustrates that, for the highest (i.e., closest to the top of the valence band) GaAs hole-wire levels, only a moderate enhancement of $\kappa$ is needed to quench orbital contributions to the $g$-factor. Similar results are obtained for other materials. In the following, we focus entirely on the properties of hole-wire subband-edge $g$-factors in the universal regime where orbital contributions can be neglected. ![\[orbVSkappa\] (Color online) Effective $g$-factors for the six highest zone-center subband edges in a GaAs wire with square cross-section, plotted as a function of the bulk-hole $g$-factor $\kappa$. An order of magnitude enhancement in $\kappa$ leads to saturation, in effect quenching orbital contributions to the Zeeman splitting.](fig1){width="3in"} ![\[gFactResults\] (Color online) Effective $g$-factors for the ten highest zone-center subband edges in cylindrical hole nanowires, calculated for various spin-orbit coupling strengths.](fig2){width="3in"} Our results are summarized in Figure \[gFactResults\] where we show $g$-factors for the ten highest zone-center subband edges in cylindrical hole nanowires, calculated for various spin-orbit coupling strengths $\gamma$. A naïve assumption that the hole spin projection parallel to the wire axis should be quantized would lead us to expect to find only two possible values for the $g$-factor; namely $6\kappa$ and $2\kappa$ for the HH and LH states, respectively. Evidently, our results are quite different. Firstly, for any given material, the $g$-factor values vary strongly between the different wire-subband edges, some levels even displaying vanishing $g$-factors. Such seemingly random fluctuations can be explained [@uz:prb:07b; @uz:physe:07b] by nontrivial microscopic hole spin-polarization profiles of wire-subband bound states. Large $g$-factors are found for subband edges with predominantly HH or LH character, whereas subbands with mixed HH-LH character or with vanishing hole-spin polarization have strongly suppressed $g$-factors. We will see below that the intrinsic connection between hole spin splittings and polarizations holds for all materials considered. Secondly, focusing on individual wire levels, it is found that their $g$-factor can vary substantially between different materials. For some subbands, e.g., the third and seventh, the $g$-factors span almost the entire range of values between $0$ and $6 \kappa$. For other subbands, $g$-factors cluster around certain values, as is the case of the first, sixth, and tenth levels. Yet other subbands display a seemingly random sequence of alternatingly increasing and decreasing values of $g^{\ast}$ as the relative spin-orbit coupling strength $\gamma$ is varied. ![\[SpinPolProfile\] (Color online) Squared normalized spin-3/2 dipole (spin-polarization) density, $\rho_{1}^{2}(r)/ \rho_{0}^{2}(r)$, for (a) the highest subband with $F_z=1/2$, and (b) the second-highest subband with $F_z=3/2$. The values of spin-orbit coupling parameter $\gamma$ and corresponding $g$-factor $g\equiv g^\ast/\kappa$ are indicated.](fig3){width="3in"} The anomalous spin splittings in hole nanowires can be attributed to strong HH-LH mixing that is present even at the wire-subband edges. The relative spin-orbit coupling strength $\gamma$ determines this mixing. To be able to characterize the spin properties of individual subband-edge bound states independent of any particular spin-projection basis, we utilize scalar invariants of the spin-3/2 density matrix. See Refs.   for details of the formalism. In particular, we consider the radial variation of the normalized hole-spin dipole density, denoted by $\rho_{1}^{2}/\rho_{0}^{2}$, which provides a measure of the local hole spin polarization. A uniform value of $\rho_{1}^{2}/\rho_{0}^{2}=9/5$ ($1/5$) indicates a HH (LH) state characterized by a ${\hat J}_z$-projection quantum number $\pm 3/2$ ($\pm 1/2$). As previously discussed, Zeeman splitting for such a state in a magnetic field parallel to the $z$ axis arises with effective $g$-factor $6\kappa$ ($2\kappa$) [@luttham2]. Figure \[SpinPolProfile\] shows the radial spin-polarization profiles $\rho_{1}^{2}(r)/\rho_{0}^{2} (r)$, for the highest hole-wire subband edges with (a) $F_z=1/2$, and (b) the second-highest subbands with $F_z=3/2$, for different representative values of $0.28\leq \gamma \leq 0.48$. Here, $F_z$ is the eigenvalue of $\hat J_z + \hat L_z$, i.e., the sum of the $z$ components of spin and orbital angular momentum, which is the good quantum number labelling wire-subband bound states [@sercel:apl:90; @uz:prb:07b]. Deviations of the hole-spin polarization from the values $9/5$ and $1/5$ is an indication of the, in principle, ever-present HH-LH mixing in hole wires. Interestingly, states with $F_z=1/2$ that form the highest subband edge in systems with $\gamma\le 0.37$ are quite close to a pure LH character, having $\rho_{1}^{2}(r)/\rho_{0}^{2}(r) \approx 0.2$ across most of the wire radius. However, a continuously increasing trend to develop a HH-LH texture is exhibited for larger $\gamma$. As can be seen, this feature is concomitant with a drastic reduction of the $g$-factor from its value close to $2\kappa$ that is expected for pure LH states. A related trend is exhibited by the highest subband edges with $F_z=3/2$ (not shown here) where, for small values of $\gamma$, the normalized dipole moment is close to the value $9/5$ corresponding to a pure HH state. With increasing $\gamma$, however, the dipole moment is increasingly suppressed. The $g$-factors show a corresponding monotonous suppression, from values close to $6\kappa$ to values close to 0. ![\[avGfacts\] (Color online) Mean g-factors $g_{\text{av}}^{\ast}=\frac{1}{N}\sum_{i=1}^{N} g^\ast_{i}$, obtained by averaging over the $N$ highest wire levels, plotted as a function of relative spin-orbit coupling strength $\gamma$. Inset: Wire geometry and orientation of the magnetic field.](fig4){width="3in"} In contrast to the previous two examples, a very nonmonotonous behavior as a function of $\gamma$ is observed for the second-highest subband edge with $F_z=3/2$. See Fig. \[SpinPolProfile\](b) where, for small $\gamma$-values, suppressed polarization profiles correlate with very small effective $g$-factors. As $\gamma$ is increased, the spin dipole moment of the state increases dramatically, approaching values associated with HH character. \[See the dashed-dotted and dashed curves corresponding to $\gamma=0.37,0.41$ in Fig. \[SpinPolProfile\](b).\] The corresponding $g^\ast$ values come close to $6\kappa$. For yet higher values of $\gamma$, the polarization is again suppressed, with concomitantly vanishing $g$-factors. A general comparison of polarization profiles for various subband edges with their $g$-factors shows that, as the hole-spin dipole moment vanishes and/or HH-LH mixing in the radial profile increases, $g^\ast$ is increasingly suppressed. Thus, a direct correlation emerges between the relative spin-orbit coupling strength $\gamma$, the hole-spin polarization, and the Zeeman spin splitting. However, on average, the hole-spin polarization and effective $g$-factors decrease as the relative spin-orbit coupling strength $\gamma$ is increased. This is illustrated by the calculated mean $g$-factors shown in Fig. \[avGfacts\]. Such mean values will describe Zeeman splitting in experimental situations where single wire subbands are not resolved. Extrapolating to $\gamma=0.38$, which corresponds to Ge, the value found is consistent with the hole $g$-factor measured recently [@stefanoSUB] in rod-shaped quantum dots fabricated from Ge/Si core-shell nanowires. DC acknowledges support from the Massey University Research Fund. The authors benefited from useful discussions with P. Brusheim, A. Führer, S. Roddaro, and H.Q. Xu. [25]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , , , , , ****, (). , , , eds., ** (, , ). , , , ****, (). , , , , ****, (). , , , , , , , , , , ****, (). , ****, (). , , , , , ****, (). , ****, (). , ****, (). , , , , , , , , , , , , , , , ****, (). , , , , , , , , , , , , ****, (). , , , , , , , , ****, (). , , , , ****, (). , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , ****, (). , ****, (). , , , , ****, (). , ****, (); ****, (). , ****, (). In general, the HH and LH effective masses have a slight dependence on the [*direction*]{} of wave vector $\mathbf k$. We neglect this, typically very small, band warping in the following. , ****, (). In the following, we present results for $0.28 \le \gamma \le 0.48$ because nanowires have recently been grown from ZnTe [@janik:apl:06] ($\gamma=0.28$), GaAs [@defran:nlett:06] ($\gamma=0.37$), Ge [@stefanoSUB] ($\gamma=0.38$), InAs [@dick:jcg:07] ($\gamma=0.45$), and GaP [@johan:natmat:06] ($\gamma=0.48$). , , , ****, (). , , , ****, (). , ****, (). , . , ****, (). , ****, (). S. Roddaro, A. Fuhrer, C. Fasth, L. Samuelson, J. Xiang, and C. M. Lieber, arXiv:0706.2883 (unpublished). [^1]: From Ref.   [^2]: From Ref.